Stochastic 3D Geometric Models for Classification ... - CiteSeerX

Stochastic 3D Geometric Models for Classification, Deformation, and Estimation by

Andrew R. Willis B. S. Computer Science, Worcester Polytechnic Institute, U.S.A., 1994 B. S. Electrical Engineering, Worcester Polytechnic Institute, U.S.A., 1995 M. S. Engineering Sciences, Brown University, U.S.A., 2002 M. S. Applied Math, Brown University, U.S.A., 2003

A dissertation submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in the Division of Engineering at Brown University

PROVIDENCE, RHODE ISLAND, U.S.A.

May 2004

c Copyright 2004 by Andrew R. Willis

This dissertation by Andrew R. Willis is accepted in its present form by the Division of Engineering as satisfying the dissertation requirements for the degree of Doctor of Philosophy

Date Professor David B. Cooper, Director Division of Engineering Recommended to the Graduate Council

Date Professor Benjamin B. Kimia, Reader Division of Engineering Date Professor David B. Mumford, Reader Division of Applied Math Date Professor Gabriel Taubin, Reader Division of Engineering Approved by the Graduate Council Date Karen Newman, Ph.D. Dean of the Graduate School

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Vitae Andrew R. Willis was born in Wilmington, Delaware on June 9, 1972. He lived in Holliston, Massachusetts, where he graduated from Holliston High School in 1990. Mr. Willis graduated from Worcester Polytechnic Institute cum laude in 1994 and in 1995, with degrees in Computer Science and Electrical Engineering respectively. In 2002 and 2003 he finished Masters degrees in Electrical Sciences and Applied Math respectively at Brown University. In 2002 he was admitted to Scientific Research Society Sigma Xi.

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Acknowledgments My Ph.D. studies were supported in part by a fellowship offered by Analog Devices for the academic years 2000 and 2001 and from the NSF KDI grant #BCS-9980091 and ITR grant #0205477, as research-assistantships for the calendar years of 2002 and 2003. The support of my family, friends, co-workers, and especially my fiancé Julianna, has made the completion of this work a reality. The pottery assembly system reflects discussions with a diverse group of people from several disciplines including Applied Math : David Mumford, Archaeology : Martha Sharp Joukowsky, and Engineering : David Cooper, and Benjamin Kimia. Some programming and significant creative input was provided by Jasper Speicher with regard to the interface design of the presented sculpting system. Without the helpful discussions and direction provided by my advisor, David Cooper, this thesis would not have been possible.

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Contents 1

Introduction

1

1.1

Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

3D Puzzle Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3.1

Estimating individual sherd Geometries . . . . . . . . . . . . . . . . . . .

3

1.3.2

Estimating sherd Configuration Geometries . . . . . . . . . . . . . . . . .

6

1.3.3

Bayesian geometric model estimation . . . . . . . . . . . . . . . . . . . .

8

1.3.4

Search algorithms for puzzle solving . . . . . . . . . . . . . . . . . . . . .

13

Deformable 3D Surface Models . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

1.4.1

Shape Representation

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

1.4.2

Stochastic 3D Surface Models : Virtual Sculpting . . . . . . . . . . . . . .

18

1.4

2

Pottery Assembly

21

2.1

Estimating Sherd Surface Geometries . . . . . . . . . . . . . . . . . . . . . . . .

21

2.1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.1.2

Surfaces of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.1.3

Axis / Profile Curve Model . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.1.4

Axis / Profile Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.1.5

Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.1.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

Assembling Configurations of Sherds . . . . . . . . . . . . . . . . . . . . . . . .

38

2.2.1

38

2.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

vii

CONTENTS

2.3

2.4

3

2.2.2

Parameters to Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

2.2.3

Data Generation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.2.4

Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

2.2.5

Estimating the Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

2.2.6

Estimating the Transformation . . . . . . . . . . . . . . . . . . . . . . . .

44

2.2.7

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

2.2.8

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

The Bayesian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

2.3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

2.3.2

Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

2.3.3

Probability and Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . .

47

2.3.4

Physical Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

2.3.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

The Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

2.4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

2.4.2

Treatment of the assembly primitives : Pairwise Hypotheses . . . . . . . .

58

2.4.3

A word on multiscale boundary matching . . . . . . . . . . . . . . . . . .

61

2.4.4

Detecting Duplicate Configurations . . . . . . . . . . . . . . . . . . . . .

63

2.4.5

Controlling the complexity . . . . . . . . . . . . . . . . . . . . . . . . . .

64

2.4.6

Pot Assembly Search Algorithm . . . . . . . . . . . . . . . . . . . . . . .

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2.4.7

Analysis of the Solution . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

2.4.8

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

Stochastic 3D Surface Models

75

3.1

Markov Random Fields for 3D Surfaces . . . . . . . . . . . . . . . . . . . . . . .

75

3.1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

3.1.2

Surface Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

3.1.3

Data Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

3.1.4

Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

3.1.5

Time Varying Surface Behaviors . . . . . . . . . . . . . . . . . . . . . . .

81

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CONTENTS

3.1.6 3.2

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Example : 3D Sculpting

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

3.2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

3.2.2

Surface Behaviors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

3.2.3

Sculpting Potentials

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.2.4

The Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

3.2.5

The Physical Input Device . . . . . . . . . . . . . . . . . . . . . . . . . .

87

3.2.6

The sculpting process . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

3.2.7

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

Appendix A

93

Bibliography

96

Chapter 1

Introduction 1.1 Main Contributions Current trends in computational shape theory are expanding from the traditional models of 3D surface differential geometry originally specified by early geometers such as Gauss and Frenet more than 100 years ago. New technologies such as 3D scanners and 3D reconstruction from images have created an abundant source of measured 3D surface data for a wide variety of applications. This dissertation proposes two new 3D stochastic shape models : (1) a model for the estimation and classification of 3D axially symmetric shapes, and (2) a model for the deformation of 3D free-form shapes. In (1), a complete system is described which automatically estimates complete mathematical models for 3D ceramic pots given 3D measurements of their fragments commonly called sherds. The unique approach integrates solutions to 4 different problems : (i) an algorithm for accurately estimating the surface geometry of an individual sherd from dense-data 3D laser scans, (ii) an algorithm for accurately aligning assemblies of sherds, called configurations, (iii) a Bayesian performance measure for sherd configurations, and (iv) a performance-driven search algorithm. For individual sherds, estimation of the outer surface geometry is implemented as maximum likelihood estimation (MLE) of the axially-symmetric surface parameters given the measured sherd data. Sherd configurations are aligned along break-point segments which lie on the boundary of the sherd’s outer surface. An algorithm is proposed for accurately aligning configurations of N sherds given a hypothesized set of correspondences between the sherd break-point segments. This is also imple1

1.1. MAIN CONTRIBUTIONS

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mented as MLE where the estimated parameters are the N − 1 sherd alignment transformations, the matched break-point segment parameters, and the global configuration surface parameters. A common Bayesian framework provides a performance measure for sherd configurations which is the log of the probability of the measured sherd data given the computed configuration MLEs, referred to as the configuration cost. The search mechanism is of the nature of a uniform cost search (in AI) where cost is the log of probability . However, a considerable number of modifications are made to eliminate redundant comparisons and reduce both time and space complexity. The assembly process starts with a fast clustering scheme which approximates the MLE solution for all sherd pairs, i.e., configurations of size 2, using a subspace of the geometric parameters, specifically, the sherd break curves. More accurate MLE values based on all parameters are computed when sherd pairs are merged with other sherd configurations. Merging takes place in order of constant probability starting at the most probable configuration. In (2), a new stochastic surface model for deformable 3D surfaces is proposed and its application to unconstrained 3D free-form surface deformation is presented. A 3D surface is a sample of a Markov Random Field (MRF) defined on the vertices of a 3D mesh where MRF sites coincide with mesh vertices and the MRF cliques consist of subsets of sites. Each site has 3D coordinates (x, y, z) as random variables and is a member of one or more clique potentials which are functions defined on sites within a clique and describe stochastic dependencies among sites. Data, used to deform the surface can consist of, but is not limited to, an unorganized set of 3D points and is modeled by a conditional probability distribution given the 3D surface. A deformed surface is a MAP (Maximum A posteriori Probability) estimate and is the surface for which the joint distribution of the MRF surface model and the data is maximum. The idea of modeling surface deformation as MAP estimation using mesh vertices as MRF site variables provides a new and powerful tool for generic shape modeling. Further, the MRF model is extended to accommodate cases where the surface deforms according to a series of time-dependent data sets thus having the flavor of a tracking problem. The generality and simplicity of the MRF model provides the ability to incorporate completely arbitrary local and global deformation properties. A specific instance of the generic MRF shape model is defined for the purpose of virtual surface sculpting. This is the problem of simple-to-use, intuitively interactive 3D free-form model building. Development of the virtual sculpting system demonstrates the simplicity and power of the proposed MRF surface model and includes a new data model, new Stochastic 3D Geometric Models for Classification, Deformation, and Estimation by Andrew R. Willis Ph.D., Brown University, May 2004

1.2. STRUCTURE OF THE THESIS

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anisotropic clique potentials, and an example of global surface deformations using cliques involving sites that are spatially far apart.

1.2 Structure of the thesis The remainder of Chapter 1 details the problems addressed by the thesis. This is divided into two sections : (1) stochastic models for axially symmetric geometries, and (2) stochastic models for free-form surfaces and their deformation. Subsections detail the different topics in which the thesis makes a contribution. For each topic a general overview of the addressed problem is presented, followed by a review of related previous work on the problem, and concluded with an overview of how the thesis makes a contribution to each of these fields. Chapter 2 discusses the system for pottery assembly and is divided into four parts : (1) the estimation of individual sherd geometries, (2) the estimation of geometries involving 2 or more sherds, (3) the stochastic geometry model for sherds and (4) the search problem. Chapter 3 discusses stochastic deformable surface models and the implemented 3D virtual sculpting system.

1.3 3D Puzzle Solving 1.3.1 Estimating individual sherd Geometries Problem Summary Automatic assembly of pots from their sherds is facilitated by solving the difficult problem of extracting an accurate geometric model of the a priori unknown local pot geometry from measurement data of a sherd. It is assumed the ceramic pot was made on a potter’s wheel and is consequently an axially symmetric surface. Such a surface may be represented in terms of its axis of symmetry and a profile or generating curve with respect to the axis of symmetry (see Fig. 1.1). The task is to obtain a highly accurate estimate of the unknown axis of symmetry and profile curve given 3D measurements of the sherd outside surface. Measurement data for a sherd provides information about the portion of the pot axis/profile curve combination that is subtended by the sherd. Hence, as seen in Fig. 2.10, measurements from more than one sherd are necessary for estimating the axis/profile-curve for a pot. Stochastic 3D Geometric Models for Classification, Deformation, and Estimation by Andrew R. Willis Ph.D., Brown University, May 2004

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1.3. 3D PUZZLE SOLVING

Figure 1.1: An axially-symmetric surface Previous Work Over the past decades there has been increasing interest in understanding axially symmetric shapes from 3D surface data. Early work was done on the subject by Nevatia et. al. in [57, 58] and continued in their paper [90]. Most published work concentrates on extracting the axis and profile of pieces using shape from silhouette [59]. However, methods which use one image may only capture the vessel profile curve and can do this only if the axis of the vessel is orthogonal to the camera view. Note that the occluding contour of a fragment does not provide the information necessary for estimation of the axis location. This requires the occluding contour from both sides of the vessel or 3D data from possibly a 3D scanner or multiple images to estimate. Due to these problems, the methods proposed for estimation of axially symmetric shapes have not shown to be effective for small surface patches. But these cases are important in practice and challenging in concept. Some initial work on the subject was proposed in [65] where the authors apply concepts from algebraic geometry to develop a linear algorithm for estimating the axis of symmetry. This method has the benefit of providing a quick and reasonable estimate. Yet, in the interest of preserving linear computational complexity, the authors do not enforce the Plücker relation which guarantees the solution is a valid Euclidean line. Additionally, by estimating the axis as a line intersected by locally Stochastic 3D Geometric Models for Classification, Deformation, and Estimation by Andrew R. Willis Ph.D., Brown University, May 2004


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estimated surface normals, the method cannot make use of the fact that the measured data lies on a continuous axially symmetric surface patch. Hence, the estimation accuracy is not sufficient for our applications. In [55], a method of axis estimation is presented based on the fact that, for a surface of revolution, maximal spheres tangent to the surface will have centers on the axis of symmetry. This method differs from the method proposed in this thesis since the authors are estimating osculating spheres for each data point/normal pair to obtain an estimate of the axis of symmetry. The centers of these spheres depend upon the principal curvature of the surface parallel which passes through each of the point/normal pairs (see Fig. 2.1 and associated text of §2.1 for a definition of parallel). The authors add robustness to their estimator by detecting outliers in a weighted iterative least-squares framework. Having computed their axis estimate, the authors then use this estimate to compute the profile curve using a cubic spline model fit to the sherd data. Another technique developed at the same time seeks the axis that generates the best set of cross-sectional circular curves [61]. Here, the data is binned according to height intervals along the axis where each bin represents a parallel of the unknown axially-symmetric surface. An energy function is proposed which seeks the axis which, when the data is binned, provides the best set of circular cross-sections. The proposed estimation method, developed at the same time as [55, 61], fits algebraic axially symmetric surfaces of specified degree to all surface data/normal pairs simultaneously using a weighted iterative leastsquares fitting method. In doing so, the model incorporates information from both the meridians and the parallels of the surface of revolution and does not require use of any local operators such as differentiation which amplifies the measurement noise. §2.1.3 illustrates the distinction between the methods which lead to different axis/profile estimates. Researchers at the Technical University of Vienna have developed a system for sherd classification based on qualitative sherd features, e.g., global shape of profiles, with human-driven pre-processing [81, 39, 38]. Contribution The thesis describes an automatic method for accurately estimating an axis/profile curve pair for each sherd (even when they are small) based on axially symmetric algebraic, i.e., implicit polynomial, surface models. The method estimates the axis/profile curve for a sherd by finding the axially symmetric algebraic surface which best fits the measured set of dense 3D points and associated normals. Note that in contrast to other methods, the proposed method can be directly applied to Stochastic 3D Geometric Models for Classification, Deformation, and Estimation by Andrew R. Willis Ph.D., Brown University, May 2004


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3D point data alone and does not require any local surface computations such as curvature and normal estimation. Axis/profile curve estimates are accompanied by a detailed statistical error analysis. Estimation and error analysis are illustrated with application to a number of sherds. These fragments, excavated from Petra, Jordan, are chosen as exemplars of the families of geometrically diverse sherds commonly found on an archaeological excavation site.

1.3.2 Estimating sherd Configuration Geometries Problem Summary Consider the situation where an axially symmetric surface is broken into a set of pieces or sherds. A subset of the sherds are available and for each of them noisy measurements of its surface and boundary are obtained. Denote the portions of the sherd boundary where the original surface locally broke into two pieces as the sherd break-curves. Using the sherd surface and break-curve measurements and knowledge of which sherds share a common break-curve, the problem is to automatically estimate the unknown axially-symmetric global surface. The estimation is an alignment problem where the unknown axially-symmetric surface and break-curves must be estimated while simultaneously estimating the Euclidean transformation that positions each measured sherd with respect to the a-priori unknown global surface. During the assembly process, the system constructs puzzle solutions by combining individual sherds with larger sherd configurations. Sherds are combined by adding a new sherd to a configuration along one of its un-matched break-curves. The resulting arrangement of sherds must be adjusted, i.e., its parameters must be re-estimated, due to the newly added sherd. Hence, estimation of sherd configurations is crucial to the puzzle assembly process. The proposed estimation algorithm is invoked each time the system constructs new configurations and allows the system to convert to the collection of measured sherd data into a single scalar value which determines how well the data represents an axially-symmetric surface. Previous Work The alignment of multiple 3D point sets is a common problem in computer vision which has been incrementally improving in accuracy and performance for the past decade [12, 15]. Alignment

Stochastic 3D Geometric Models for Classification, Deformation, and Estimation by Andrew R. Willis Ph.D., Brown University, May 2004


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problems typically involve a set of 3D surface-point measurements taken from different viewpoints of an object. The data sets measured from different viewpoints are then aligned with one another to generate an improved model of the measured object. The alignment process starts by identifying viewpoint pairs which contain measurements from the same portion of the object surface. These viewpoint pairs are said to be overlapping and the points shared between the two point sets are control points. The Iterative Closest Point (ICP) algorithm proposed by [12, 15] aligns pairs of views using a two-step iterative process. In step 1, a correspondence is established between the control points in each view. In step 2, an alignment error is minimized which reduces the distance between corresponding points in each view according to some defined metric, e.g., the Euclidean distance. Since its introduction, numerous variations of the algorithm have been proposed (see [44, 68] for a survey of these variants for view pair alignment). Other researchers have proposed model-based methods for point alignment [73]. In this case, a surface or curve model is fit to one data set and the points of another data set are aligned with the estimated model. Since both methods depend upon aligning control points which come from corresponding portions of the object surface, alignment is difficult if there are few control points. In multiple viewpoint alignment, such as the pot sherd alignment problem, there exist at least 2 viewpoint pairs and we must align the three or more viewpoints simultaneously. These methods typically align point sets by first performing a pairwise viewpoint alignment and subsequently minimizing the global alignment error, i.e., the alignment error between the control points of all corresponding viewpoints. Examples of such work include [26, 15, 22, 11]. Contribution In the context of automatic axially symmetric surface estimation, each sherd data set is a partial viewpoint of the unknown axially symmetric surface. Sherd pairs are related by their corresponding break-curve segments, i.e, the curve segments along which the two sherds were separated, and their shared axis/profile curve. The thesis treats the difficult problem of aligning axially symmetric surfaces where the surface overlap is limited to only the break-curve segments, and incorporates the constraint of axial symmetry into the global fitting error. The alignment problem is treated as a two-fold problem of (1) estimating the unknown surface geometric parameters consisting of an axially-symmetric surface and a set of break-curve segments and (2) estimating the alignment Stochastic 3D Geometric Models for Classification, Deformation, and Estimation by Andrew R. Willis Ph.D., Brown University, May 2004


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parameters of the pieces which consists of N − 1 Euclidean 3D transformations for a group of N pieces. A stochastic approach is taken which computes the Maximum Likelihood Estimate (MLE) of all the unknown parameters given the measured data where the unknown parameter vector, θ, consists of : (1) the parameters of the axially-symmetric surface, which are the coefficients of an axiallysymmetric implicit polynomial surface; (2) the true break-curve, modeled as a sequence of 3D points; and (3) the Euclidean transformation parameters for each of the pieces. The formulation of geometric alignment of point and surfaces as MLE is new and provides robust, fast, and accurate estimates of the configuration geometry. The formulation uses two different data types in the alignment process, i.e., break-curve points and axis/profile-curve pairs, which is also new. The alignment algorithm handles chipped, eroded free-form fragments and noisy data. Experimental results are presented which solves an application of interest, specifically the reconstruction of archaeological pots from subsets of their surface fragments.

1.3.3 Bayesian geometric model estimation Problem Puzzle assembly solutions are constructed by combining puzzle pieces into larger configurations. Given our assumption that the measured sherds come from an axially symmetric object, a set of parameters θ1 , θ2 , ...θN are extracted which model the measured datasets D 1 , D2 , ..., DN from each of the N sherds. The parameters provide compact representations each sherd’s data as an axis/profile-curve and an associated set of boundary curves. Typical puzzle assembly solutions seek to define a similarity or affinity metric, between puzzle pieces. Metrics are formed by defining a function, f (θ1 , θ2 ), which operates on pairs of of points in parameter space Ω θ and satisfies a set of defined constraints [67]; one of which is f (θ 1 , θ2 ) ∈ R > 0. For puzzle solutions in 2D, affinity measures reflect the similarity between matched 2D boundaries of objects [69, 49, 30]. In the case of assembling axially symmetric objects, this function must take into consideration the matched break-curves between sherd pairs and the global axis/profile-curve of the unknown pot. Hence, the problem is to define a metric which acts locally to estimate vessel break-curves from sherd boundary curve data and globally to estimate the vessel axis/profile curve from sherd surface data.


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(a)

(b)

Figure 1.2: An example match between two pieces. The illustrated configuration pair represents one of many possible matches between the sherd pair (A, B). Each sherd match assumes the following : “Sherd A (top –in pink) and sherd B (bottom –in brown) broke apart along the boundary regions J and K”. The boundary regions J and K from sherds A and B respectively are indicated by a series of spheres and cylinders representing the sherds break segment points and their corresponding surface normals respectively. Fig. (a) shows sherds A and B in close proximity but unaligned. Fig. (b) illustrates shows the best alignment between sherds A and B assuming that they broke apart along the shown boundary regions. Solving the problem from §1.3.2 determines the optimal alignment of the sherd pair given the stated assumption. The system constructs new configurations using the proposed solution to the search problem (§1.3.4). The proposed solutions to all stated problems relies on an underlying Bayesian framework from §1.3.3 which allows the system to construct puzzle solutions by adding pieces to configurations. Fig. 1.3 shows one such configuration.


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In solving a puzzle problem, the system combines sherds with previously constructed sherd configurations. Each time a sherd is added to a configuration, the similarity function must be evaluated and the result stored for future use. Hence, one also seeks a metric which minimizes the computational cost of evaluating the similarity function as well as the storage space required to hold the computed solution. This is referred to as the computational complexity in time and space of the similarity metric. Hence, for puzzle assembly, the problem is to define a function on the chosen parameters which satisfies the following set of desirable properties : (1) the function must define a valid metric on the space of parameters [67], e.g,. the triangle inequality must be satisfied; (2) the function must provide significant delineation between good matches and bad matches, e.g., f (θ 1 , θ2 ) f (θ1 , θ3 ) if the pair (θ1 , θ2 ) is a “bad” piece match and (θ 1 , θ3 ) is a “good” piece match; (3) the function must incorporate local break-curve matching and global surface matching information simultaneously; (4) the function must be applicable to configurations which may have an arbitrary number of pieces; and (4) the metric should allow for incorporation of additional information by either (a) extending 0

the set of parameters, e.g., color attributes θ = (θ, Y, Cr, Cb)t where we have added the 3D color parameters from the color space (Y, Cr, Cb) t or (b) giving prior information, i.e., all the fragments come from one of ten different types of axially symmetric shapes. Previous Work There are many examples of affinity measures for 2D fragments. One example is [49], where fragment boundaries are extracted from images of the 2D pieces. Here, the metric is based on the extracted 2D boundary shape and intensity gradient of the image in the vicinity of the boundary points. Another notable example is [69], where the authors define a metric based on the distance between matched boundary points alone. Work on defining similarity metrics for puzzle assembly in 3D are restricted to defining similarities between 3D curves [41, 82]. In this case the metric is an energy expressed in terms of difference between the curvature and torsion matched 3D curves. There are currently no existing similarity metrics for axially symmetric puzzle assembly.


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(a)

(b)

(c)

Figure 1.3: A correct seven piece configuration. Fig. (b) shows a view orthogonal to that shown in (a), i.e., the camera is looking at the side of the vessel. Local costs correspond to error in the matched break-curves of the configuration. Figs. (a) and (b) show these matched curves as rendered spheres on the sherd boundary and the surface normal at each point is rendered as a cylinder passing through the sphere center. There is also a single global cost, i.e., all 7 sherds lie on a common axially symmetric surface. Fig. (b) shows the estimated axes for each of the 7 sherds as red cylinders. These are noisy estimates of the true vessel axis, l, shown as a black line. In Fig. (c), a cylindrical coordinate system with axis z is defined and the measured sherd data points from (b) are projected into a (r, z) plane where they are shown as dark green points. In Fig. (c), the profile curve, α(r, z) = 0, is plotted in light green and interpolates the projected (r, z) data points. As shown in §2.1 the axis/profile-curve model is equivalent to fitting a 3D axially symmetric surface to the data shown in (b). Stochastic 3D Geometric Models for Classification, Deformation, and Estimation by Andrew R. Willis Ph.D., Brown University, May 2004


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Contribution A probability metric is defined using an assumed Gaussian probability distribution on the sherd surface and break curve data. In this framework, each configuration of N sherds has parameters θ = (β, l, α, T) where β denotes the locally matched break-curves, l, α denotes a single vessel axis/profile-curve pair, and T denotes the parameters of the N − 1 transformations. The similarity function is the negative log of the probability of the configuration’s sherd data given the MLE of the parameters θ. For a configuration containing a set of sherds S, the similarity function is then b = − ln p(∪i∈S Di |θ) b . Evaluation of the similarity function is a two step process, the first f (θ)

b and the second step is to evaluate f ( θ) b whose value is referred to as step is to compute the MLE θ,

the cost of the configuration.

The MLE of the geometric parameters for each configuration integrates the local cost of estimating the vessel break-curves and global cost of estimating the vessel axis/profile-curve into a single framework. Vessel break curve costs are considered local since only 2 sherds may share a common break-segment (see Figs. 1.2(a) and 1.3(a)). The vessel axis/profile-curve costs are considered global since many sherds may describe the same portion of the pot profile (see Fig. 1.3(b,c)). Since the sherds are measured in different coordinate systems and our chosen parameterization is not Euclidean invariant, the parameters of the 3D Euclidean transformation T B which aligns sherd B to sherd A must be known to evaluate the hypothesis. We now see that the problem of puzzle assembly has been posed as a recognition problem. In this framework, the system seeks to find the value of the parameters which best explains the sherd data. The value of the MLE reflects the best axially symmetric model possible given the sherd data and the assumed correspondence between the break-curve segments. The probabilistic approach to puzzle assembly implemented as MLE of the unknown vessel geometry is a heretofore unexplored viewpoint. This viewpoint has the benefit of being able to improve the parameter estimates of all the geometric parameters as new data is supplied. Previous proposed assembly schemes work strictly from the estimated piece parameters which makes global re-estimation of the solution difficult. The probabilistic framework for puzzle assembly is new and its demonstrated utility for addressing this computationally difficult task marks an important theoretical contribution to the computer vision community and an important practical contribution to the archaeological community. In §1.3.4, a



13

Figure 1.4: A set of sherd hypothesis for the sherds A and B where the boundary correspondences J and K are allowed to vary. search method is chosen which complements the chosen metric.

1.3.4 Search algorithms for puzzle solving Problem As detailed in §1.3.3, configurations of sherds are generated by making the following assumption : “sherd A and sherd B broke apart along break curve segments J and K.” Since A, B, J, and K can vary, a group of sherds can generate a large number of configurations. Fig. 1.4 shows configurations generated from a subset of the possible hypotheses between a sherd pair. The goal of the search problem is to construct the best configuration of sherds with as little computational cost in space and time as possible. Here best denotes the configuration of sherds which has least cost or, equivalently, highest joint probability. For even a modest number of sherds, the size of the solution space, i.e., all possible configurations, is huge. For illustration, consider a small puzzle having 10 sherds where each sherd has approximately 10 break segments. The number of possible triplets for this modest puzzle is roughly 3E6 and the number of quadruplets exceeds 2E9. Hence, the total number of possible alignments between sherds is huge for even a modest number of sherds.



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Previous Work Early research on the more general problem of puzzle solving in general subject includes works such as [45, 89, 27] and more recently [30]. Here, puzzle pieces are commercially produced jigsaw puzzle pieces which simplifies the problem since jigsaw pieces have similar sizes, easily identifiable areas which need to be matched, and a mostly regular outline, i.e., the piece contour is differentiable with a few exceptions at corner pieces which require special treatment. Solving more difficult puzzles based on real-world data remains an active area of research today with recent results published in [69, 30, 41], which attempt to improve the algorithmic speed and remove these restrictions. Yet, examples of automatic puzzle solving in 3D are limited to [62, 41, 82]. In [41, 82], the authors deal with fragments of arbitrary shape and propose to assemble them elegantly by matching the fragments break curves, i.e., the curves on the pot surface along which the sherds break apart (Fig. 2.10). Both methods match curves based on their curvature and torsion signatures which, unfortunately, is prone to instability when the data is noisy. [82] proposes a greedy search which, in the context of large puzzle problems, is likely to fail due to the number of false positives which are known to commonly occur in solving real life puzzles [69, 49]. [41] proposes a best-first search using observed triplet matching errors. This search method is known to have exponentially increasing complexity in both time and space and no clear criterion for terminating the assembly is given. Hence this method will work well when there is little chance of classifying an incorrect match as a correct curve match, i.e., false positives are rare. However, in practice, false positives are quite common, especially with break curves that have little curvature and torsion which is very common on real-world fragments and especially on archaeological fragments (see Fig. (2.17) for 2 examples). In [62], the authors match broken 3D shapes by matching of their fractured surfaces, i.e., the surface through the pot wall at which the pot breaks, using simulated annealing. Their approach requires both of the fractured surfaces to be parameterizable with respect to a common plane, i.e., for aligning 2 fractured surfaces about the xy-plane the two surfaces may be represented as f 1 (x, y) = z1 and f2 (x, y) = z2 . For all three methods, no results are provided for 3D fragments with more than 2 pieces. Specifically, no results for assembly of more than two 3D sherds are given in [41], a single pairwise match is shown in [82], and several pairwise matches are shown in [62].


1.4. DEFORMABLE 3D SURFACE MODELS

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Contribution The thesis proposes a modified uniform cost search where the metric or cost is the negative loglikelihood of the sherd(s) data given the computed MLE of the geometric parameters. This corresponds to forming new configurations by appending the most probable pairwise configuration to the most probable existing configuration. Note that the most probable existing configuration at any time in the search is the most likely, i.e., lowest cost, collection of aligned sherds which has not been previously considered. There are five contributions proposed by the thesis which expedite the search of the solution space by (1) quickly computing candidate configuration pairs, (2) limiting the breadth of the search to make the search computationally tractable, (3) quickly computing new configurations by using solutions to previously computed pairwise configurations, (4) incorporating a mechanism for detecting the creation of duplicate configurations, and (5) incorporating a mechanism which avoids considering configurations which, when merged, produce solutions of high cost.

1.4 Deformable 3D Surface Models 1.4.1 Shape Representation Problem A mathematical formalism of shape is fundamental to a wide variety of important problems. In fields such as computer-aided design, shape models are used to design and manufacture the tools and machines that power our civilization. Other fields such as medicine and biology use shape for a wide variety of purposes such as diagnosis, analysis and surgery planning. In computer vision and machine intelligence, shape representations provide a computational way to recognize objects and patterns in images, video, or any other device which obtains measurements from real world scenes. It is common to find situations where the evolution of the object shape over time is important, e.g., tracking the growth of an object over time or the deformation of an object under external forces. In situations such as this, the shape model must be dynamic. Deformable models seek to explain how a specific object changes or evolves it shape.



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Previous Work Deformable curve and surface models are commonly used in computer vision for a variety of traditional problems such as shape estimation [53, 78], recognition [18], tracking [77, 8, 52], and have recently become popular for segmentation, particularly in the field of medical imaging [53, 78, 31, 51]. There are three distinct approaches to deformable surface representation: 1) parameterized models; 2) implicit models; and 3) discrete models. Parameterized models are the most popular and may be divided between physics-based Partial Differential Equation (PDE) models such as 3D mass-spring systems [79] and variational models such as 3D snakes [84, 31, 17]. [48] and [10], are representative examples of implicit deformation models which are computationally efficient but these deformation models have been found more difficult to control than parameterized and discrete models. The literature on discrete models represents the family of deformable models which are most similar to the proposed shape model [78, 54, 52, 29, 36]. Discrete models describe the geometry of an object using a set of 3D point samples taken from the object surface and point-to-point connections which provide a polyhedral interpolations of the surface between points, e.g., a surface triangulation. Each point and its associated connections provide a local representation of the surface shape and the totality of these local relationships provides a global shape description. Deformation for discrete models is implemented by defining deterministic functions which operate on the 3D point samples causing them to move and consequently deform the surface. There exists a vast body of MRF models defined for images (see [51, 46] for an overview). Some have also defined MRF deformable models on range maps with a regular grid such as [71]. However, few examples exist of 3D deformable MRF models. An early example of a 3D MRF shape model is provided in [85]. Here, the authors define an MRF surface model based on Terzopoulos’s deformable superquadric models [76] and concentrate on transitioning between local and global models using a wavelet basis as a vehicle for implementing changes in scale. [47] represents a recent example of a 3D MRF shape model. In [47], MRFs are defined on a symmetry-based representation of shape referred to as M-reps (see [64] for details on the shape model). Here, the M-rep surface representation is quite different from the proposed surface-based MRF and their emphasis is placed on understanding shape deformation at different scales. Recent work by Cipolla et. al. has also made use of MRF models for the purpose of 3D surface reconstruction from images [16]. Hence,



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the popularity of stochastic deformable models seems to be increasing rapidly. Contribution A generic stochastic shape and deformation model is specified which can serve as a common framework for defining the discrete models described in [78, 54] as well as some newly introduced statistical shape surface models such as [16]. As in other discrete models, the model is defined on an initial 3D mesh which consists of vertices, points which lie on the original shape, and edges, connections between vertices. An MRF site is assigned to each vertex and a set of cliques, which consist of subsets of sites, are defined. Clique members and the associated clique potentials which operate on cliques must be either learned or specified by the system designer. They represent applicationspecific information regarding the desired deformation behavior. These cliques and clique energies can be anisotropic, i.e., can be different in different directions, and can be non-homogeneous, i.e., can vary over the surface. The complete deformation model is expressed in two parts: 1) a model for the probability density of the surface which is based on cliques involving only MRF sites, i.e., surface vertices; and 2) a conditional probability density for the newly observed data given the surface. The probability density for the surface specifies a functional relationship among MRF sites, or equivalently, surface points. The conditional density for the data specifies a functional relationship between MRF sites and the measurements. Deformed surfaces are sample functions from the joint probability density and correspond to Maximum A-posteriori Probability (MAP) estimates of site random variables given the joint probability of the MRF sites and the data. The notion of surface deformation as MAP estimation using the vertices as site variables is new and its utility and power for specifying surface deformation models is demonstrated. Simple clique potentials coupled with a new data model are combined to generate a simple deformation model useful for the interpolation of data. While it has been known in the literature that deformation can be expressed as MAP estimation (in fact [85, 36, 51] explicitly mention this viewpoint ), there are no instances of stochastic MRF models defined on a generic 3D surface mesh in which mesh vertices are MRF sites. Other important distinctions which identify the proposed MRF as unique are that the mesh vertices may be distributed arbitrarily, i.e., non-uniformly, on the mesh surface and may have arbitrary connectivity. In addition, scale changes in the mesh are handled via standard contemporary graphics remeshing techniques (see §3.2 for Stochastic 3D Geometric Models for Classification, Deformation, and Estimation by Andrew R. Willis Ph.D., Brown University, May 2004


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remeshing details). The thesis extends the stochastic deformation model to deal with sequences of data sets provided at different time instances. In this case, the model becomes a time-varying stochastic process where the system state can in theory consist of the surface vertices at a succession of two or more instants (i.e., cycles). It is this fully 3D stochastic model for the representation and modification of surfaces, and the ability to incorporate unlimited local and global deformation properties (either physically realizable or physically unrealizable) that is new, tremendously powerful and computationally fast (real time on a PC).

1.4.2 Stochastic 3D Surface Models : Virtual Sculpting Problem One application of the shape model from §1.4.1 is the interactive specification of free form surfaces to a computer, which is referred to as virtual 3D surface sculpting. This is a twofold problem of (1) providing a computational model to represent the surface and how it deforms and (2) developing a intuitive interaction by which the user can specify the desired object or deformation. Previous Work Development of an intuitive virtual sculpting paradigm has been pursued by many researchers over the past decade. Early work on deformable models with user interaction was done by Terzopoulos [79]. Since then, numerous examples of systems for the purpose of virtual sculping have emerged. In [80], Terzopoulos and Qin adopted their deformable NURB surface models of [79] to the specific task of virtual sculpting. Several years later, Qin et. al. developed a system for virtual sculpting using a dynamic spring-mass model where mesh vertices are masses and mesh edges are springs which connect 2 masses. Hence, a 3D mesh defines a network of springs and masses and deformation results from introducing a deformation in the spring mass network or by modifying the rest lengths of the springs. Further work regarding physically-based models such as that of Qin is being done for volumetric or voxel representations where the object is modeled as a collection of cubical volume elements [66, 25]. In [91], objects are represented as a dense point cloud of surface measurements. Each point and its associated surface normal defines a small patch of the surface and


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(a)

(b)

Figure 1.5: A sculpture created using the proposed MRF-based virtual sculpting system. The initial capsule shape shown in (a) was deformed by the artist to generate a model of a human head. The head on the right has been flat shaded to enhance the faceting effect of the triangulation and provide a more accurate representation of the underlying geometric mesh structure.



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local re-parameterizations of the surface allow for coherent deformations of adjacent points. In [32, 5, 21, 50] sculpting is done using a haptic device, i.e., the device seeks to simulate how the tool would actually feel if in contact with the virtual surface. This is accomplished by tracking the user’s position and orientation using an articulated mechanical arm. Motors integrated into the articulated joints are controlled by the computer generating force-feedback resistance to the artist’s tool. Interaction of this sort makes a deformation analogy based on surface contact, i.e., the user makes contact with the surface using the tool and introduces a force which serves to deform the surface. Specific instances of the many implemented sculpting systems include all of the mainstream surface models mentioned in 1.4.1. Contribution This thesis details how the MRF surface model from §1.4.1 may be implemented for generating 3D free-form shapes interactively, i.e., virtual sculpting. In contrast to the large number of systems based on haptic devices, the proposed sculpting systems allows the user to sketch 3D curves as a series of points in arbitrary relation to the surface, and the surface deforms to interpolate the sketched 3D curves using a configurable deformation model. In addition to the interactive differences, the proposed sculpting system introduces a new set of specific clique potential functions for virtual sculpting. The potential functions proposed include functions which serve to perform local deformations similar to that in previous physically-based models and a new class of potential functions which serve to enforce global structure such as symmetry about a pre-defined plane. Arbitrary clique configurations and clique potential functions can be designed and used to extend the base system. The sculpting system developed uses a custom-built 3D point-generation pen with an integrated 3D tracker and an immersive 3D virtual environment to allow the user to modify a free-form 3D surface given an initial surface mesh. The system performs in real-time and is being used to create original works of art by professors and students from the Brown University School of Art. In addition, the system has been used for sculptural restoration of damaged archaeological artifacts.


Chapter 2

Pottery Assembly This chapter describes the system developed for the purpose of automatic estimation of axially symmetric geometries from their fragments. The chapter is divided into four sections each of which describe distinct areas to which the thesis contributes. The first section describes a method for estimating the axially symmetric surface model for each of the measured sherds. Section 2 details how configurations of N sherds are aligned given an assumed correspondence between portions of their boundary. Section 3 discusses the stochastic model for sherd assemblies and how this model contributes to the assembly process. Section 4 details how the system assembles individual sherds into axially-symmetric vessels.

2.1 Estimating Sherd Surface Geometries 2.1.1 Introduction This section discusses a highly accurate solution to the difficult problem of extracting a geometric model of the unknown pot structure in the region associated with 3D measurements obtained from a sherd. The model extracted may be used in a variety of applications. Some examples are: shapebased searching of 3D sherd databases; sherd classification; and pot reconstruction.

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2.1. ESTIMATING SHERD SURFACE GEOMETRIES

Figure 2.1: Geometry of a surface of revolution

2.1.2 Surfaces of Revolution A surface of revolution S ∈ R3 is obtained by revolving a planar curve α ∈ R 2 about a line l ∈ R3 . α is called the profile (or generating) curve and l the axis of S. When the z-axis is taken as the axis of revolution with profile curve α(z), the surface S may be represented parametrically as (2.1). 1

S ( θ, z) = (α(z) cos θ, α(z) sin θ, z)

(2.1)

With this parameterization, the curves z = constant are parallels of S and the curves θ = constant are meridians of S. The profile curve characterizes how the radius and height of the surface change for a fixed meridian (see Fig. (2.1)). In equation (2.1), the radius function, r = α(z), is a singlevalued function of z. For archaeological sherds, this presents a problem since profile curves which are multi-valued with respect to the z-axis commonly occur. Examples include sherds which come from pot bases and rims. Figs. 2.5 and 2.7 represent typical examples of sherds which have multivalued profile curves. For this reason, the profile curve model from (2.1) has been generalized to include these situations by using a planar implicit polynomial, i.e., the zero set of an algebraic planar curve : α(r, z) = 0. The problem of interest is: given an unorganized set of 3D measured points of a small patch of a larger axially symmetric surface, estimate a surface geometry model for the patch. This geometry is completely specified by an axis in 3D and a 2D profile curve with respect to that axis. Consequently, estimation of axially symmetric surface models is the estimation of axis/profile curve pairs (l, α(r, z)). The algorithm seeks the axis/profile-curve pair which best fits the measured 1

The adopted notation and terminology for surfaces is that used in classic texts such as [70, 43].


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sherd data.

2.1.3 Axis / Profile Curve Model The pose of a rotationally symmetric object is defined by the position and direction of its axis of symmetry. The axis of symmetry is parameterized using a standard parametric equation of a 3D line:

x = mx z + bx

(2.2)

y = my z + by The equations from (2.2) contain four unknown parameters which describe the 3D axis of symmetry, l = (mx , my , bx , by ). Two of these parameters, mx and my , describe the slope of the line when it is projected into the xz-plane and the yz-plane respectively. The remaining two parameters, b x and by , specify the x and y coordinates where the axis line intercepts the xy-plane at z=0. The profile curve is represented using an implicit polynomial curve model, α(r, z) = 0. Implicit polynomial curves provide a compact and low computational-cost method of representing shape [73, 13]. In addition, closed-form linear least-squares fitting methods have been developed which provide stable and robust curve and surface fits in the presence of noise [72, 13]. The specification of a 2D implicit polynomial curve of degree d has [(d + 1)(d + 2)/2] unknown coefficients and is the set of points satisfying (2.3).

αd (r, z) =

X

ajk r j z k = 0 .

(2.3)

0≤j+k≤d; j,k≥0

Here, d is a parameter which is related to the geometric complexity of the pottery sherd to be estimated. Typically one assigns the smallest value to d which is large enough to represent all objects of interest. In this way, objects which may have little geometric complexity are described as degenerate cases of the more complex model. For the artifacts in this paper, all experiments are performed with d = 6. Axis estimation is posed as a problem where we seek to find the 3D axially symmetric algebraic surface which best approximates the measured sherd data. However, fitting a 3D axially symmetric Stochastic 3D Geometric Models for Classification, Deformation, and Estimation by Andrew R. Willis Ph.D., Brown University, May 2004


0

0

0

24

Figure 2.2: Several (x , y , z ) points (in blue) are projected into the (r, z) plane (in red) defined by the estimated axis l and a vector orthogonal to the axis. The highlighted portion is further discussed in Figure (2.3).


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surface directly to the data is difficult since the coordinate system of the basis monomials is a priori unknown. Instead, an equivalent method is proposed which makes use of the axis of symmetry to simplify the fitting problem. To this end, consider a plane which contains the unknown surface axis, l, in a general 3D position. A local cylindrical coordinate system is defined in this plane with origin o, a point on the axis, and two vectors in the plane, z and r, where z is a unit vector in the direction of l, and r is a unit vector orthogonal to z which passes through the point p i . All of the 3D data points and normals are transformed to the local cylindrical coordinate system using the transformation equations from (2.4). The exact choice of the origin is arbitrary. The system defines 0

0

0

the origin to be the point on the axis closest to the mean of the measured (x , y , z ) surface data. Then the ith measured data point, pi , in 3D becomes the 2D data point (ri , zi ) by rotating pi about the axis l and into the plane via (2.4). The measured surface normal n i associated with the point pi will not lie in the rz-plane. Its projection into the plane is denoted n pi and is computed via (2.5). A graphical example of the projection is shown in Fig. (2.2). ri =

p (pi − o)t (I − zzt )(pi − o)

(2.4)

zi = (pi − o)t · z

npi = (ni · r, ni · z)

(2.5) 0

0

0

This projection (2.4) takes each measured data point, p i , from x y z -space to rz-space and preserves the distance between the axis and the data point. The projection (2.5) takes only those components of the measured surface normal, n i , in the directions of r and z. Here, r is the vector through pi and the axis which is perpendicular to the axis, i.e., r =

(pi −o)−(pi −o)t zz . krk

Hence,

for any 3D normal, ni , the corresponding projected normal, n pi , may not be of unit length, i.e., knpi k ≤ 1. The system then solves for the unknown profile-curve coefficients, i.e., the a jk from (2.3), using the objective function (2.6) defined on the projected data. (2.6) is a least-squares fit of an algebraic implicit polynomial to the projected 2D data and is a modified version of the energy function in [72], which is referred to as the Gradient-1 fitting algorithm.

egrad1 =

I X i=1

(α2 (ri , zi ) +

1 2 knpi − ∇α(ri , zi )k ) , λ


(2.6)


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The Gradient-1 2D curve and 3D surface fitting algorithm appends a penalty term to the usual least-squares objective function (algebraic distance in this case) which makes use of the gradient of the polynomial in order to get a more stable estimate of the curve. This fitting method makes use of all the available information: the hypothesized axis, the measured spatial data and the observed normals. This information is used to compute the global surface model which best fits the data and is t ∂α ∂α denotes the gradient symmetric about the hypothesized axis. Note that ∇α(r i , zi ) = ∂r ∂z and α2 (ri , zi ) is the surface point fitting error in (2.6) and the latter term fits the polynomial normals to the measured surface normals. The value chosen for λ will depend upon the noise present in the measured surface normals. For the experimental results in §2.1.5, λ = 0.01. This corresponds to weighting errors due to the data normals less than errors due to data points. In §2.2.3, a noise model for the measured data is assumed which places more emphasis on normal information. An alternative objective function may make use of the full 3D normals n which

measured  2i 

 ∇α(ri , zi )  2 changes the second error term of (2.6) from kn pi − ∇α(ri , zi )k to ni −   , where

0

a zero has been appended to the profile curve gradient vector. Denote this alternative version of the fitting objective function as equation (2.6)’. For this version, components of the vector n i perpendicular to the (r, z) plane correspond to observed asymmetries in the measured normal data given the assumed axis. The proposed projection of the normal n i → npi disposes of these observed errors and hence causes the curve fit to respect the measured spatial data more when the normals are inaccurate. Note that (2.6)’ is an error function for 2D curve fitting that gives exactly the same result as does the error function in [72] for 3D surface fitting. The scalar residual, egrad1 , is a measure of asymmetry resulting from the axis/profile-curve fit. Fig. 2.3 illustrates how each measured surface point and normal impacts the observed fit error egrad1 . Each of the measured data points will not lie exactly on the profile curve model due to measurement noise and possible asymmetries in the sherd fragment. Fig. 2.3 decomposes this error between the profile curve and the measured data into its r and z components : (∆r i , ∆zi ). Errors in the direction of r, indicated as ∆r i , are associated with asymmetries due to local fluctuations of a surface parallel which passes through the measured surface point. Errors in the direction of z, indicated as ∆zi , are associated with asymmetries due to local fluctuation in the surface meridian which passes through the measured surface point. Fig. 2.3 illustrates these error terms graphically Stochastic 3D Geometric Models for Classification, Deformation, and Estimation by Andrew R. Willis Ph.D., Brown University, May 2004


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Figure 2.3: Decomposition of fit error: An example profile curve for a hypothetical pot is shown as α(r, z) = 0 (in red). Two points are illustrated: (1) a measured sherd sample (r i , zi ), shown as an outlined circle; and (2) a point from the zero set (i.e, a point from the curve α(r, z) = 0 which is closest to the sherd sample, shown as a solid circle (in black). The algebraic distance α(r i , zi )2 approximates the Euclidean distance between the two points (see [74] for details). Two vectors emanating from the measured surface point are labeled n pi , representing the measured surface normal, and 5α(ri , zi ), representing the normal of the implicit polynomial profile curve at the point (ri , zi ). The difference between the vectors is shown as n pi − 5α(ri , zi ). The size of this error term, knp − ∇α(r, z)k, approximates the difference between the axially symmetric algebraic surface normal and the measured sherd surface normal. Minimizing (2.6)’, which is almost the same, is completely equivalent to performing a 3D axially symmetric surface fit using the method from [72].



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: spatial asymmetry for the point (r i , zi ) in the directions of the surface parallel and meridian are indicated with (∆ri , ∆zi ) respectively. The important thing to notice is that the objective function (2.6) seeks to minimize the approximate orthogonal distance of the measured data to the unknown surface. Use of the orthogonal distance as an error measure makes this method distinct from [61]. In [61], a method is proposed which fits a set of circular cross-sections to binned data. The data is binned according to a set of divisions along the z-axis and the circular fit within each bin is a measurement of error in the surface parallels alone, i.e., this method minimizes observed errors in ∆ri . In [55], a method is proposed which approximates the osculating sphere to the unknown surface for each measured data point. This method is more accurate than that of [61] because the errors being minimized here depend upon a second order expansion of the surface at the measured surface point (see [56] for details on the theory). However, the surface continuity at orders higher than two are not taken into consideration in such a model and estimation accuracy decreases with decreasing angle between the surface normal and the object axis. In contrast, the algebraic surface α(r, z) models the measured sherd surface as a C ∞ continuous surface, i.e., the model is infinitely differentiable everywhere. This approximation will generate improved models where the surface is smooth, which is often the case for archaeological sherds (see Figs. 2.4 and 2.6 for examples of such sherds). It is also interesting to note that the approximation can be done without computing any local information from the measured data which makes the estimation procedure robust to noisy sherd data and estimates are still possible when there are no measured surface normals (here the term 2

knpi − ∇α(ri , zi )k from (2.6) is zero).

2.1.4 Axis / Profile Estimation The method utilitizes a two-step iterative algorithm to estimate the axis and associated profile curve which best describes the observed 3D data. 1. Based on the value of the objective function after the preceding iteration, choose a new value for the axis parameters, l. 2. Based on the new l, compute egrad1 , the value of the objective function by solving the weighted linear least-squares problem (2.6). Note that (2.6) has an explicit solution which may be computed quickly from the projected data. Stochastic 3D Geometric Models for Classification, Deformation, and Estimation by Andrew R. Willis Ph.D., Brown University, May 2004


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Since the surface model depends upon the axis, the resulting objective function is highly non-linear. Consequently, convergence to a local minimum may occur if minimization is started far from the true parameter value. For this reason, we use the Nelder-Mead simplex method to minimize this function. While standard gradient-based optimization techniques are possible for solving this problem, the chosen method was found to be more robust with respect to the numerous local minima that can occur in the objective function for a given data set. The estimation algorithm needs only a hypothesized axis of symmetry in order to begin. 2

2.1.5 Experimental Results A series of experiments were run on the sherds obtained from an excavation of the Great Temple located in Petra, Jordan [37]. The sherds were scanned by a ShapeGrabber 3D laser scanner which provides surface point and normal measurements for the sherd’s outer surface [4]. Results are presented for five sherds which are examples of the families of geometrically diverse sherds commonly found on an archaeological excavation site. For each of these sherds there is a detailed error analysis which is discussed to emphasize various subtle aspects of the problem which are more apparent for specific types of sherd geometries. Error analysis proceeds by applying the bootstrap method with a sample size of B = 500. Each bootstrap sample uses the same number of points and normals as the original sherd data set and is generated by a random re-sampling of the sherd point/normal data where a single data point may be selected more than once [19]. For each of the 500 bootstrap samples, an axis/profile curve estimate is generated and the value of the estimated axis is stored. The b th bootstrap axis estimate is referred to as lb . It is also assumed that the resulting 500 axis/profile curve estimates represent independent samples taken from a multivariate normal distribution with mean µ and covariance Σ. Hence, one may estimate the distribution of the axis parameters by computing the estimated P 1 PB b bt b mean µ b = bl = B1 B b=1 li and covariance Σ = B−1 b=1 (lb − l)(lb − l) . These are sufficient to construct an approximation of the true distribution of the axis parameters which are assumed 2

Typically, the initial axis estimate is obtained using the linear algorithm described in [65]. However, an additional validation step is applied to detect degenerate surfaces of revolution by determining if the sherd data is well approximated by a sphere, hyper-ellipsoid, or saddle surface. If the sherd data is determined to be spherical in nature, then no axis direction is estimated and the axis location is taken as the sphere center. If the sherd data lies on a saddle surface or on a nearly spherical ellipsoid such that the principle curvatures of the surface satisfies |κ1 | ≈ |κ2 |, then there are two possible axes of symmetry. In these cases, additional information is necessary to determine which of the two axes corresponds to the unknown vessel axis of symmetry.


30


b Note, the error function minimized could result in slightly to be normally distributed∼ N (b µ, Σ).

biased estimates. Hence, the most important use of the bootstrap results is estimating the posterior

probability density function for the axis given the 3D measurement data set. This information seems to be very important in deciding on the confidence the system can have in hypothesized configurations of sherds. For the purpose of comparing results between sherds of different shapes and sizes, a global coordinate system is defined in which analysis of the axis/profile curve parameters for all sherds takes place. This coordinate system transforms the sherd data such that the mean estimated axis, bl,

is the world coordinate z-axis and the sherd data centroid lies on the x-axis.

Four sets of information are presented for each sherd : 1) an image of each measured sherd; 2) the covariance matrix of the bootstrapped axis parameters (m x , bx , my , by ); 3) a set of three profile curve fits corresponding to the profiles generated from three different axis estimates: the mean bootstrapped axis and two additional axes representing what we refer to as 95% confidence interval for the largest mode of variation around the mean (see discussion for details); and 4) A plot of the standard deviation of the laser scan 3D data points in the direction perpendicular to the axis as a function of height, z, along the axis. The standard deviation is measured with respect to the profile curve estimate obtained from the mean bootstrapped axis in (3). In order to compute the 95% confidence interval for the largest mode of variation, the slope parameters of the axis, (mx , my ), from (2.2) were normalized by the height extent of the sherd, z 0 . The new axis parameterization is given by (2.7). x=

mx z0 z

+ bx

y=

my z0 z

+ by

(2.7)

Here z0 denotes the height range spanned by the profile of a sherd with respect to its mean estimated axis. This normalization changes the axis slope parameters (m x , my ) into parameters which represent slope of the axis line as a fraction of the sherd height when the line is projected into the xz-plane and yz-plane respectively. This provides a meaningful measure which has units,

%sherd height , mm.

for

those components of the eigenvector which characterize the directions of the axis. For the parameter mx , the normalized parameter

mx z0

represents the change in axis direction in the xz-plane as a

fraction of the total sherd height per incremental step, ∆x, in the direction of the x-axis: Stochastic 3D Geometric Models for Classification, Deformation, and Estimation by Andrew R. Willis Ph.D., Brown University, May 2004

∆z(%) ∆x(mm.) .

31


mx bx my by

(a) Sherd image

mx

bx

my

by

0.0111

-0.0661

0.0002

-0.0005

-0.0661

0.4179

-0.0004

-0.0022

0.0002

-0.0004

0.0002

-0.0015

-0.0005

-0.0022

-0.0015

0.0100

(b) Axis parameters covariance matrix (mm.)

(c) profile curve estimates for the largest mode of variation (mm.)

(d) standard deviation (mm.)

Figure 2.4: Experimental Results for sherd 654 Fig. 2.4 : The sherd profile curve has a simple shape which is commonly associated with body sherds found at a typical archaeological excavation. A body sherd is any sherd which comes from neither the rim nor the base of the pot. Since curvatures are small in direction of both sherd meridians and sherd parallels, axis estimates here are difficult. The accuracy in estimating the axis of this piece demonstrates robustness to data noise.


32


mx bx my by

(a) Sherd image

mx

bx

my

by

0.0290

-0.1748

-0.0034

0.0264

-0.1748

1.6484

0.0389

-0.3441

-0.0034

0.0389

0.0013

-0.0109

0.0264

-0.3441

-0.0109

0.0979




Figure 2.5: Experimental Results for sherd 967 Fig 2.5 : The sherd has parallels with small curvatures, yet the meridians of this sherd have significant curvature information. Axis estimation for geometries such as this is especially difficult for methods based on surface normal lines. This is due to several reasons : 1) the piece is very small, 2) it has a multiple-valued profile curve, and 3) it has normals which are almost parallel to the axis of symmetry. Since the proposed method incorporates normal information differently than those based on normal lines [55, 65], these apparent obstacles do not cause the algorithm problems. Uncertainty in the axis location is represented by parallel profile curves in (c) and again in (b) by examining elements (2,2) and (4,4) of the axis parameter covariance matrix.


33


mx bx my by

(a) Sherd image

mx

bx

my

by

0.0979

-0.5373

-0.0255

0.1404

-0.5373

2.9906

0.1406

-0.7820

-0.0255

0.1406

0.0069

-0.0380

0.1404

-0.7820

-0.0380

0.2111




Figure 2.6: Experimental Results for sherd 997 Fig 2.6 : The sherd here has small curvatures along sherd parallels and a small region of high curvature along sherd meridians. As in Fig. 2.5, it is difficult to accurately estimate the axis location when there is little data (i.e. a small sherd) combined with small curvatures in the direction of the sherd parallels. This is evident in the parallel translations of the profile curves in (c) as well as in (b) by examining the covariance matrix. This leads to an important inference: Sherds which have little data with respect to their subtended angle about the axis of revolution and small curvatures in the direction of the surface parallels have more variation in their axis location parameters (b x , by ).


34


mx bx my by

(a) Sherd image

mx

bx

my

by

0.0042

-0.0026

0.0001

0.0004

-0.0026

0.0031

0.0003

-0.0004

0.0001

0.0003

0.0002

-0.0001

0.0004

-0.0004

-0.0001

0.0002


(c) profile curve estimates for largest mode of variation (mm.)


Figure 2.7: Experimental Results for sherd 1135 Fig 2.7 : The sherd here has highly curved parallels and meridians. Yet the sherd ridges are chipped and worn (see part (a)) which generates a large amount of asymmetric data measurements. The axis/profile curve estimates here are very accurate which reflects good robustness to asymmetry in regions of high curvature and to local surface noise.


35


mx bx my by

(a) Sherd image

mx

bx

my

by

0.2317

-0.0771

0.0193

-0.0076

-0.0771

0.0715

-0.0078

0.0058

0.0193

-0.0078

0.0088

-0.0027

-0.0076

0.0058

-0.0027

0.0022




Figure 2.8: Experimental Results for sherd 1313 Fig 2.8 : Particularly interesting aspects of the curve are the points of high curvature along the profile found at the sherd rim and again just slightly down from the sherd rim. These points appear to be nearly discontinuous in their second derivative, i.e., the curvature at these locations jumps. In cases such as this, the actual surface may not be C ∞ continuous and the resulting algebraic surface fit regularizes the locations which violate this assumption. Hence, pieces such as this are more difficult to estimate using our axis approximation method.


36


(a) p967 data projection (mm.)

(b) p1313 data projection (mm.)

Figure 2.9: Sherd data projected into (r, z) plane using the mean axis estimate, bl. Figure 2.9 provides perspective on the axis estimates which are obtained via the method. For Figs. 2.4-2.8, plotted profile curves are zoomed to accentuate the source of the small observed errors in the estimates. Since variations typically range between 0.3-0.001 mm., without zooming in closely to examine small fluctuations, it is difficult to discern any variation in the three 95% confidence interval profile curves for a sherd. In Fig. 2.9, a non-scaled version of the original 3D data points is shown which have been rotated into the (r, z) plane. The y-axis is height along the estimated axis, and the x-axis is distance from the estimated sherd axis for each 3D data point. The best axis estimate will be that for which the scatter is smallest locally in the direction perpendicular to the true surface. The data scatter for sherds 967 and 1313 is shown in Fig. 2.9. What is amazing is the small data scatter, especially in Fig. 2.9(a) where the sherd is almost horizontal with respect to it’s estimated axis. Qualitative confidence in the axis direction estimation is obtained by observing rings on the inside of the sherd resulting from pot construction on a spinning wheel.



37

Having normalized the axis directional parameters, we proceed by taking the eigenvector, v, associated with the maximum eigenvalue of the axis parameter covariance matrix, i.e., the largest mode of variation, and choose the parameter values at the two ends of the 95% confidence interval b This genin the direction of the eigenvector assuming the parameters have the distribution N (b µ, Σ).

erates three axes : the mean axis parameters given by µ b, and the parameters defining the confidence

interval [b µ + 2σ, u b − 2σ]v. Note that this confidence interval does not capture all major modes of

variation for the parameters (there are 3 additional modes of variation). Yet, it provides some insight regarding the dominant direction of variation for the bootstrap estimates. This dominant mode

indicates those parameters which have the most variation or equivalently are less reliable. Three profile curves are shown which are the different profile curve fits obtained by solving (2.6) using the three different axes. The profile curve associated with the mean axis estimate is shown in red and the profile curves associated with the axes from the confidence interval are shown in blue and green.

2.1.6 Conclusion This section of the thesis has detailed a highly accurate method for automatically estimating axially symmetric surface models given dense 3D laser scanner data from pot sherds. The end result is an axis/profile curve pair for a sherd, and this completely specifies the sherd surface geometry. If desired, the accuracy of an estimate can be further improved by not using data that occurs at highcurvature points on the sherd surface in the estimation procedure. This data can be automatically identified by detecting data outliers with respect to the presently estimated profile curve models. The estimated axis/profile curve models are close to being maximum likelihood estimates, and should therefore be close to maximum accuracy because MLEs are known to be asymptotically of minimum variance, and the number of data points used in estimating sherd geometry is of the order of 10,000. Multivariate Gaussian distributions were estimated for the sherd axis for a variety of sherd geometries. The range of geometries included in the experimental data set allows for insightful comments which accompanies the presented statistical data. Finally, these distributions could be used for pot reconstruction because they tell the reconstruction algorithm how much confidence it can have in various estimated parameters.


2.2. ASSEMBLING CONFIGURATIONS OF SHERDS

38

Figure 2.10: A broken vessel and its parametric description.

2.2 Assembling Configurations of Sherds 2.2.1 Introduction This section discusses a solution to the problem of estimating an unknown free-form axially symmetric surface from an incomplete set of its pieces. The pieces individually describe small portions of the unknown surface and are connected to each other by break-segments, i.e., locations where the pieces were broken apart. During the assembly process, the surface reconstruction program must determine which break-segments are shared between individual pieces in order to discover the unknown global structure of the pot. This is a difficult search problem whose solution is given in §2.4. This section, §2.2, treats the problem of aligning N sherds where it is assumed that the correspondence between the sherd break-segments is already known. The problem here is to simultaneously estimate all of the geometric parameters of the unknown axially symmetric surface while simultaneously aligning the N sherds.

2.2.2 Parameters to Estimate The break-curve parameters, β, are points on the surface where the surface has broken into two pieces. These locations include vertices—locations of Y and T junctions (see Fig. (2.10)). Note that junctions denote points which are high-curvature points as one travels along the piece boundaries. T-junctions are points which are high-curvature points on two of the three piece boundaries and



39

Figure 2.11: Break-point segments : a sherd outer surface, in grey, and one of the sherd vertices is shown as a large opaque red sphere. Two sets of break-point data which are called break-point segments are generated shown as light grey and black points on the break-curve. Each break-point segment has four ordered elements, starting with the vertex and then listed in order of increasing distance from the vertex. Note that break-point segment points lie at locations where a sphere (in transparent blue) of radius kR intersects the break-curve where k = [1, 2, 3]. Y-junctions denote points which are high-curvature points for all three piece boundaries. These high-curvature points are referred to as vertices. Each break curve is represented as a sequence of K points starting on a vertex. The points in each sequence occur at successive intervals of fixed length from the vertex (see Fig. (2.11)) and each sequence is referred to as a break-point segment. These points, along with a surface normal at each point, constitute β—the parameterization of the surface break-curves for the axially-symmetric surface. Hence, break-point segment v is written βv = ((p1 , p2 , . . . pK ) , (n1 , n2 , . . . , nK ))t where pk denotes the k th 3D point and nk denotes the k th 3D normal for βv . The group of all break-curve parameters is β = ∪ v=1 βv . The k th point in a break-point segment is the location where the break-curve intersects a sphere of radius kR centered at a vertex point where k = [0, 1, 2, . . . , K − 1] and R is the radius of a sphere centered on the vertex. Figure (2.11) illustrates two break-point data segments for a vertex where K = 4. Note that the break-point segment data used is that which is illustrated in Fig. 2.11.


40


This choice is not rigorous because it is the point on the true break-point segment that should be a distance kR from the vertex. The surface axis of symmetry is parameterized using the standard parametric equation of a 3D line as shown in (2.2). Hence l = (bx , by , mx , my )t and consists of the pair (mx , my ) as specified in §2.1.3. A profile curve α(r, z) with respect to the piece axis l is our model for the unknown 3D axially-symmetric algebraic surface with axis l where r denotes radius, i.e., the shortest distance between a 3D point and the axis and z denotes height along the axis (see Figs. 2.10,2.2,2.3). The surface parameters are the coefficients of the algebraic profile curve in (2.3) and the axis of symmetry in (2.2). Hence α = (∪j,k ajk )t is the vector of coefficients for the implicit polynomial curve of degree d. Note that for all experiments contained in the thesis, d = 6. It is assumed that each piece undergoes an arbitrary rigid Euclidean transformation which moves the piece to its measurement position. Hence, for the i th piece the transformation, Ti , which moves the piece from its measurement position to its aligned position must be estimated. The 3D transformations are parametrized with 6 parameters consisting of 2 parts : (1) a 3D translation vector t, and (2) a 3D rotation R. The 3x3 rigid rotation matrix R is represented using the so called axis-angle parameters which describe rotation in terms of a rotation angle ψ about a 3D unit vector n R . Hence pure rotation is the 3D vector (ψnR ) and the equivalent 3x3 rotation matrix is referred to as R. The 3D transformation parameters are T = (t, ψn R )t (see Appendix A of [63] for additional details on this parameterization). Further, transformations of piece datasets are denoted as T(D) for surface point/normal data and T(B) for break segment point/normal data. In this notation it is assumed that T(D) indicates that a single transformation, T, operates on each of the points and normals in D according to the rules shown in (2.8). Tp = Rp + t Tn =

(2.8)

Rn

2.2.3 Data Generation Model The surface measurement data is provided by a Shapegrabber laser/camera scanner [4]. It produces 15,000 3D points/sec. at a resolution and accuracy of the order of 0.25mm. The data is divided into two sets : (1) 3D surface points and normal measurements, denoted D i for the ith surface piece, and



41

(2) 3D break-segment points and their surface normals, denoted B v for the v th break-segment. Data for each break-segment is extracted according to a simple parameterization of the unknown surface break-curves adopted in §2.2.2. Hence, break-point segment data is simply a special sequence of K measured 3D points and outer surface normals (see the discussion on break-curve parameters in §2.2.2 and Fig. 2.11 for clarification). Assumptions 2 I) Surface measurement points are i.i.d. N(0, σ D

These are independent, identically distributed, spherically symmetric Gaussian perturbations in 32 . See [14] for a justification space about each point on the true surface with mean 0 and variance σ D

of this model. 2 I) Surface measurement normals are i.i.d. N(0, λσ D

These are independent, identically distributed symmetric Gaussian perturbations on the unit sphere, i.e., in SO(3), about the true surface normal for each point on the sherd outer surface with mean 0 2 . These are independent and for the j th piece are distributed over a spherical cap and variance λσD

about a mean that is normal to the surface as represented by an axis/profile-curve for the j th piece. Break-segment measurement points are i.i.d. N(0, σ B2 I) These are independent, identically distributed spherically symmetric Gaussian perturbations in 3space about each point on the true break-curve, with mean 0 and variance σ B2 . Note that more appropriate but more complicated models can be used.

2.2.4 Algorithm Alignment is done by adding one piece at a time, each addition producing a new configuration. For each new configuration, all parameters are re-estimated to minimize the joint probability density for all the break-point segment and surface data associated with the pieces in the configuration. However, this is not done in one simultaneous nonlinear minimization, since that would be both much too time consuming and would likely converge to a local maximum. Note, for an N piece configuration, the number of parameters to be estimated is : 6(N − 1) Euclidean transformation parameters; 4 + d(d+1) global surface parameters for a dth degree surface (if d = 6 then there are 29 2


42


parameters); and a minimum (and likely many more) of 3K(N −1) break-point segment parameters where K is the number of break-points in a segment. Estimating an axis/profile curve for a surface data set of a few thousand points may require up to 1 minute if using an arbitrary starting point. Using a small-error starting point can result in orders of magnitude computation reduction. Such is the case when the system constructs configurations during the search (see §2.2.5 for details on how a suitable initial point is found). Estimating a Euclidean transformation for aligning a pair of matched break-point segments is a linear least squares computation on the order of a millisecond. 1. Given a N piece configuration denoted Conf igN , the algorithm begins by treating the configuration as though a single piece. The global axis/profile-curve parameters for Conf igN are estimated by solving the non-linear equation (2.9) (see §2.2.5 for details). 

el, α e = arg max ln  l,α

Y

Di ∈Conf igN



e T ei  P Di |l, α, β,

(2.9)

Upon convergence, the parameters of the estimated axially symmetric surface are stored : e do not play a role in this optimization e ). Note that the break-point segment parameters β (el, α

yet are included for completeness in (2.9).

2. A new candidate piece N + 1 is taken and the Euclidean transformation is computed which moves this piece so that its break-point data segments align with the matched break-point data segments of Conf igN and its surface data fits the 3D surface model for Conf igN by maximizing (2.10) (see §2.2.6). “ “ ”” e = arg max ln P DN +1 , BN +1 , BcN |el, α e N +1 , β e , β, TN +1 T TN +1 ,β

(2.10)

Where TN +1 denotes the Euclidean transformation for piece N + 1 , D N +1 denotes transformed surface data of piece N + 1. BcN and BN +1 denote the matched break-point data segments of Conf igN and the transformed piece N + 1 respectively. β denotes the true values of the unknown parameters for the matched break-point segments. 3. Add piece N + 1 to Conf igN and go to step (1) if the number of pieces in Conf igN is smaller than the total number of available pieces.


43


It is possible to adjust all N pieces slightly by re-estimating the transformation for piece 1 to make its data better fit the configuration defined by piece 2 through N , then repeat with piece 2, etc., but the improvement is small, and doesn’t appear to be necessary.

2.2.5 Estimating the Surface Surface shape parameters are obtained by finding the axially symmetric algebraic surface which best fits the aligned data sets using the method outlined in §2.1. This is a non-linear minimization consisting of two steps: 1. Based on the value of the objective function after the preceding iteration, choose a new value for the axis parameters, el.

2. Fixing the axis l = el, the system computes the axially symmetric surface α which maximizes

the probability of the data by fitting an axially symmetric surface to the 3D data with axis el. This is equivalent to solving the weighted linear least-squares problem (2.11) which has an

explicit solution and incurs little computational cost. Note that (2.11) is an abbreviated form of (2.6) where the points included in the summation and the specific value of λ may differ. e(α|l = el) =

X

p,n∈Conf igN

1 α (p) + kn − ∇α(p)k2 λ 2

(2.11)

The objective function (2.11) is very similar to (2.6) from §2.1 and is minimized in the same way, i.e., via simulated annealing. However, a very important distinction is the choice of the initial point for the non-linear minimization. Consider a merge between an assembled configuration pair and a new sherd. In this case, the system starts minimizing (2.11) using the MLE value of the axis parameters, el, obtained as a solution for the pairwise configuration. For correctly matched sherds,

this initial point provides a good starting point for the global parameters (l, α) and greatly expedites

the minimization process. Note that the error metric α 2 (p) is the so-called algebraic distance which can be used as an approximation of the Euclidean distance (for details see [74]), and λ1 kn−∇α(p)k2 is for regularization [87].


44


(a) Example of algorithm Step 1.

(b) Example of algorithm Step 2.

Figure 2.12: Surface Alignment : Conf igN consists of two pieces as shown in (a). The axially e , el) obtained from step (1) of the algorithm is shown in transparent cyan and symmetric surface (α e is shown as a black curve. Piece N + 1, in green, is added to the configuration using the profile α the rough alignment given by solving (2.13), the resulting 3-piece configuration is shown in (a). The algorithm then solves (2.15) to obtain an accurate alignment for piece N + 1 using its matched eN +1 . D eN +1 denotes those break point data segments, BN +1 , and a subset of its surface points, D surface points of piece N + 1 whose height along the axis el is within the height interval defined by the Conf igN data set, i.e., the bracketed region in (b).

2.2.6 Estimating the Transformation A coarse estimate for the piece N + 1 transformation is obtained by aligning the subset of its corresponding break point segments to those of Conf igN , i.e., solving the optimization problem (2.12).

TNˇ+1 = min kTN +1 (BN +1 ) − βk2 + kBcN − βk2 TN +1

(2.12)

From the assumptions stated in §2.2.3, this corresponds to solving (2.13). ˇ N +1 = min kBcN − TN +1 (BN +1 )k2 T TN +1

Where the difference between two corresponding break-point segments is defined in (2.14).


(2.13)

45


2

kBcN − TN +1 (BN +1 )k =

K X k=1

kpcN,k − TN +1 pN +1,k k2 +

1 kncN,k − TN +1 nN +1,k k2 (2.14) λ

The optimization (2.13) is commonly referred to as the absolute orientation problem to which there exist several explicit linear solutions [83, 35, 33]. The system uses the solution proposed by [83] (see Appendix A for details). Note that (2.12,2.13,2.14) include a term involving the surface normals at each break segment data point which aligns the measured surface tangent-planes of corresponding points. This adjustment improves the alignment especially in cases where the break-point segment data has little curvature. e ) from step (1) of the algorithm (see §2.2.4), the system solves the Using surface estimate (el, α

ˇ= ˇ N +1 , and β non-linear alignment problem (2.15) using the rough estimate T

ˇ N +1 (BN +1 )+BcN T 2

as

an initial point. e(TN +1 , β) =

1 1 1 2 2 2 e 2 α (TN +1 (DN +1 )) + 2 kTN +1 (BN +1 ) − βk + 2 kBcN − βk σD σB σB

(2.15)

Note that the assumed distribution stated in §2.2.3 makes solving for β trivial. However, the parameters of the unknown break-point segments in (2.15) are present to show that the system seeks to estimate the unknown geometry of the unknown break-point segment β. This solution for β may be non-trivial in a general situation where the measurement noise distributions differ from that in §2.2.3 eN +1 denotes the subset of the piece N +1 surface data as shown in Fig. 2.12(b). This is the subset .D

eN +1 )) that lies within the extent of the estimated profile curve for Conf igN . The term α 2 (TN +1 (D

eN +1 to the estimated axially is the approximate Euclidean distance of the transformed data set D symmetric surface for Conf igN . The surface alignment error being minimized here is the square

of the algebraic distance (see Section XI of [74] for details where a similar alignment problem is solved). The break-segment alignment error is kT N +1 (BN +1 ) − βk2 + kBcN − βk2 which denotes the Euclidean distances between the matched breakpoints on piece N + 1 and Conf igN and the true locations of the break points β. The solution to (2.15) is computed by direct non-linear minimization using the well-known Levenberg-Marquardt algorithm.


2.3. THE BAYESIAN FORMULATION

46

2.2.7 Results Given the large number of unknown parameters being estimated, the algorithm succeeds in finding the global MLE solution quickly. The 3 piece configuration in Fig. 2.12(b) required approximately 10 seconds to compute. Note that this minimization involves 221 parameters (12 transformation parameters, 29 surface parameters, and 180 break segment parameters). Fig. 2.20 shows the solution for a 13 piece axially symmetric vessel where only 10 pieces are available which required approximately 45 seconds to compute, the number of estimated parameters here is 893 (54 transformation parameters, 29 surface parameters, and 810 break segment parameters).

2.2.8 Conclusion A robust and fast method was proposed which estimates an unknown axially symmetric surface from measurements of a subset of size N of its pieces. As others have in the past, the problem is decomposed into a set of N − 1 pairwise piece alignments. The proposed method is unique in that it solves the problem using a 2 step recursive algorithm. In step 1, the geometric constraint of axial symmetry is used to obtain an estimate of the unknown surface for a configuration of aligned pieces. Step 2 uses this surface estimate to obtain accurate estimates of the break segment and transformation parameters for a new piece added to the configuration. There is a rationale for this approach. The algorithm is shown to work and performs well even though dealing with a vast number of unknown parameters.

2.3 The Bayesian Formulation 2.3.1 Introduction Estimation of the true global shape and structure of the assembled pots is implemented as Maximum Likelihood Estimation (MLE) of a set of geometric parameters which characterize the unknown pot surface and the unknown curves along which the pot originally broke. Since these parameters are not Euclidean invariant, we must also estimate the 3D transformation parameters which align the sherds into a global coordinate system. Individual sherds and configurations of more than one sherd are cast into a common Bayesian framework. This section details the structure of the Bayesian



Symbol l α β Ti

47

Significance the vessel axis the axially symmetric surface of the entire vessel (i.e., profile-curve, α(z), with respect to z-axis which is the vessel axis) break-curves for entire vessel a 3D transformation which takes sherd i into its aligned position in the vessel Table 2.1: Basic geometric parameters.

model and how this model is used to represent the geometric knowledge the system has obtained from the measured data.

2.3.2 Model Parameters Axially symmetric vessels are modeled using a subset of the complete geometric information that can be used. It consists of the outer-surface break curves, break curve vertices at sherd junctions (see Fig. (2.10)), an axis/profile curve for the entire pot and portions for individual sherds, and Euclidean transformations that take each sherd from its data-measurement position to its aligned position in a configuration. The goal is to estimate the global pot parameters in Table (2.1) by hypothesizing matches, i.e., transformations, between sherd datasets. The most probable matched set of sherd data given the sherd transformation values and the geometric parameters is considered the most likely pot. §2.1 and §2.2 detail methods for estimating these parameters for individual sherds and sherd configurations.

2.3.3 Probability and Knowledge The Bayesian framework provides a completely general probabilistic framework from which one may construct computational models of knowledge. The knowledge being modeled is application dependent and is represented in terms of a set of parameters, θ, whose true value are assumed to be unknown. The typical Bayesian parameter estimation problem assumes the following [24]: • The form of the distribution p(x|θ) is known but not the value of the parameters θ. • All previous knowledge about the parameters is contained in a known prior distribution p(θ).


48


• All additional knowledge about θ is contained in the data set D which consists of M samples drawn independently according to the unknown distribution p(x). Bayes rule (2.16) provides a method for converting the posterior probability distribution p(D|θ) into a posterior distribution on the parameters p(θ|D).

p(θ|D) = R

p(D|θ)p(θ) p(D|θ)p(θ)dθ

(2.16)

The unknown quantity in (2.16) is p(D|θ), by the assumption of independence this is the product (2.17).

p(D|θ) =

ND Y

p(xi |θ)

(2.17)

i=1

In the context of estimating the geometry of an axially symmetric vessel from its sherds, the data consists of N data sets D1 , D2 , ..., DN , each of which consists of two distinct subsets : (1) the collection of sherd surface measurements, G i , for the ith sherd, and (2) the collection of sherd boundary measurements, Bi , for the ith sherd. The data independence assumption and the form of the distribution p(x|θ) are detailed in §2.2.3. It is also assumed that no prior information is available to the system, i.e., p(θ) ∼ U nif orm(θ). Suppose the system is assembling a single vessel and that knowledge of the original shape of the vessel surface and the shapes of all break curves are known probabilistically. In Bayesian estimation, this knowledge specifies a distribution on the unknown parameters p(D|θ)p(θ) which could be used to improve estimates of the parameters. However, it is assumed that this distribution is not known and further, that the configurations being constructed could come from more than one axially symmetric vessel. Since there is no prior information available about p(θ), the system considers all possible axially symmetric surfaces and all their surface curves as equally likely events in nature. In this case, p(D|θ)p(θ) = kp(D|θ) where k is a constant. The geometric interpretation of this is that the system assumes only that given a sherd pair (i, j) matched along their break segments (m, n), the sherds surface data lies on an axially symmetric surface which contains the surface curve described by their matched space curve pair (m, n). This is actually a combination of 2 assumptions : (1) the sherd break-point segments describe the same 3D space curve, and (2)


49


(a)

(b)

(c)

Figure 2.13: A sherd triplet decomposed into its global and local parameters. There are two local break-curve segments being estimated, i.e., the curves for sherd pair {6,10} and {5,6}. The matched break-curve data is shown in Fig. (a) where points are rendered as spheres on the sherd boundary and the surface normal at each point is rendered as a cylinder passing through the sphere center. There is also a single global cost, i.e., all 3 sherds describe the same axially symmetric surface. Fig. (b) shows the estimated axes for each of the 3 sherds as red cylinders, the global geometric parameters l models these as noisy estimates of the true vessel axis, shown as a black line labeled z. Fig. (c) shows the (r, z) projection of the measured sherd data points for all 3 sherds in dark green, the global geometric parameters for the axis/profile-curve l, α models the measured sherd data lies on a continuous axially symmetric surface, represented as a profile-curve with respect to the global vessel axis. The profile curve, α(r, z) = 0, is plotted in light green and interpolates the projected (r, z) surface data points.



50

when the sherds are aligned along this space curve, their surface data lies on an axially symmetric surface. The assumption p(θ) ∼ U nif orm(θ) is clearly not true for archaeological pottery. For example, ancient pottery objects commonly have an open end for purposes of placing material inside the pot. Knowledge such as this is not reflected in how the system treats configurations. In theory, the parameters of the distribution, p(θ), could be estimated from a set of learning data. In this case, the system can exploit this prior information using (2.16) to define a more accurate probability metric on the parameter space p(θ|D) by computing what is now a Maximum A-Posteriori (MAP) estimate of the unknown parameters θ given the measured data D. It is also theoretically possible to remove the assumption that the sherds from a single vessel have matching boundaries. In this case, the system must uncover the global shape of the vessel by aligning the sherds using only their axis/profile curve pairs. However, the implemented system constructs hypotheses which depend upon an assumed correspondence between sherd boundaries, i.e., break-point segments. Hence, solutions of this type cannot be constructed by the proposed system. In cases where many sherds are missing, this may make it impossible for the proposed solution to construct the global pot solution. Fig. 2.14 provides some information regarding the susceptibility of the system to missing sherd data. The proposed method of break-curve alignment is tractable for all break-curve pairs only due to the fact that the pot vertices provide a method for computing break-curves which have the same parameterization based on the vertex location. Two possible avenues for increasing robustness to missing sherd data is to introduce more break correspondences between sherds by attempting to match triplets as done in [49] or by assembling sherd configurations using their axis and profile curve estimates. While both avenues improve the system robustness this is at the expense of increased computational cost. Specifically, additional correspondences increase both the number of potential matches between sherds and the computational cost of each match. This is because evaluating matches reduces to the problem of matching two curves where each curve has a different parameterization. For additional break-curve matching, this is a 3D curve-matching problem and for axis/profile-curve matches this is a 2D curve matching problem. Cost is used as the metric for determining whether a constructed configuration is correct. It takes all of the evidence available, i.e., all of the sherd data, and maps it to a single positive real number, i.e., for a configuration involving a set of sherds S, cost(∪ i∈S Di ) → R > 0, where S is Stochastic 3D Geometric Models for Classification, Deformation, and Estimation by Andrew R. Willis Ph.D., Brown University, May 2004

51


(a)

(b)

(c) Figure 2.14: Shown are three pots assembled by humans (photograph by Jill L. Baker, courtesy of the Leon Levy Expedition to Ashkelon). Each sherd in the assembled pots is labeled with a number and break-curve segment correspondences between sherds are indicated as edges between the labels. For each of the rendered images, a red circle has been chosen to show the location of a T-junction of interest. In particular, T-junctions on large sherds such as sherd 1 in Figs. (a) and (c) and sherd 10 in Fig. (b) can make computation of global solutions difficult if there are many missing sherds. For example, assume that only the sherds shown in Figs. (a-c) are available for computation, in this case, half of the entire object is missing for both (a) and (c). In this situation, consider how the global solution will depend upon the omission sherds of the shown set of sherds. In Fig. (a) if sherds 11 and 10 are missing from the puzzle, the proposed system will not be able to connect the cap of the amphora to the remaining body of the amphora. In Fig. (b) the omission of sherds 4 and 15 will not allow the system to connect the lower portion of the pot to the remaining pieces. In Fig. (c) omission of sherd 2 will not allow the system to add sherd 1.


52


a subset of sherd indices, e.g., S = {3, 8, 2}defines a triplet configuration involving sherds with indices 3, 8, and 2 for a puzzle which may consist of 10 sherds. The system computes the cost of b of the configuration parameters and then computing the a configuration by estimating the MLE, θ,

logarithm of the probability of the data given the MLE value (2.18).

b = − ln(p(∪i∈S Di |θ)) b cost(∪i∈S Di ) = cost(∪i∈S Di |θ)

(2.18)

As mentioned in §1.3.3, each configuration of N sherds has parameters θ = (β, l, α, ∪ i∈S Ti ) where β denotes the locally matched break-curves, (l, α) denotes a single vessel axis/profile-curve pair, and ∪i∈S Ti denotes the parameters of the N − 1 transformations which align the N − 1 sherds to a global coordinate system defined by the N th sherd. Hence the local match between breaks curves and the global requirement that the sherd surface data lie on an axially symmetric surface are integrated into a single expression. Fig. 2.15 uses a triplet configuration of sherds to illustrate how all of the measured sherd data is taken into account when evaluating the cost of a sample configuration. There is a rationale for using cost as a metric. The MLE estimate represents the most probable value of the unknown vessel parameters given the assumed correspondence between the measured sherd data (2.19).

b = arg max − ln(p(∪i∈S Di |θ)) θ θ

(2.19)

Configurations which result in sherd configurations whose break point segments match well and surface data lie close to a perfect axially symmetric surface will have low cost and vice-versa (see Fig. 2.13). Hence, the MLE provides a natural ordering of configurations in terms of the probability that the configuration is an actual surface of axial symmetry. As discussed previously, the true distribution of the vessel parameters is unknown which makes it difficult to accept or reject any single configuration as correct or incorrect. Hence, in practice, the system retains a sequence of configurations which are ordered by their cost value. Since the number of pairwise configurations for even modest puzzles ( 1 and det(UV t ) = 1 or if the rank of S > 1 The solution for R and the minimum singular value is a simple root.

3. The best translation vector b t is then computed using (12). b N +1 b t = µcN − Rµ The specific form of the solution shown here is based upon a solution from [6].


(12)

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