Recovering Parametric Geons from Multiview Range Data - CiteSeerX

Recovering Parametric Geons from Multiview Range Data Kenong Wu and Martin D. Levine Center for Intelligent Machines & Dept. of Electrical Engineering McGill University, Montreal, Quebec, Canada, H3A 2A7 [email protected] [email protected]

Abstract

new set of volumetric models, parametric geons, as a qualitative shape description [19]. Unlike generalized cones which parameterize attributes of volumes [6], parametric geons are a set of distinctive shapes de ned by implicit equations. The major dierence between parametric and conventional geons is that the former are characterized by global shape constraints which explicitly restrict the resulting descriptions to a set of speci c shape types. This facilitates shape approximation. In this paper, we focus on the problem of recovering parametric geons from multiview range data of a single object part. The related problem of decomposing a more complicated object into parts is currently under study. The strategy for recovering parametric geons is similar to that for other parametric primitives. That is, a tting scheme is used to minimize an objective function which measures some property dierence between an object and a model [14, 16, 10]. However, there is an additional requirement for parametric geon recovery. The process must also produce discriminative information such that the resulting metric data can be converted to a qualitative description. We note that the only previously reported attempt to obtain qualitative shapes from parametric models is due to Raja and Jain [15]. They explored the recovery of geons from single-view range images by classifying the actual parameters of globally-deformed superellipsoids. It was found that the estimated parameters were extremely sensitive to viewpoint, noise, and objects with coarse surfaces. In contrast to Raja and Jain's work, we use a new model and a new strategy to recover qualitative shape models. They used superellipsoids which are nonunique and cause uncertainties in the estimated model parameters, especially when representing noisy and partially-viewed data [17]. Instead, we propose the parametric geons as volumetric primitives. In addition, multiview range data are employed to reduce ambiguous information in the object data. Another issue is that their error measure was the weighted

This paper focuses on approximating object part shapes by distinctive types of volumetric primitives. Shape approximation is accomplished by tting volumetric models called parametric geons to multiview range data of single-part objects and classifying the tting residuals. Parametric geons are seven qualitative shape types de ned by parameterized equations which control the size and degree of tapering and bending. Model tting is performed by minimizing an objective function which measures the similarity in both size and shape between models and objects. Multiple view data, global shape constraints and global optimization are employed to obtain unique models and to compensate for noise and minor variations in object shape. This approach has been studied in experiments with both synthetic 3D data and actual range nder data of perfect and imperfect geon-like objects.

1 Introduction The interest in the derivation of part-based descriptions of 3D objects arises in part because such descriptions re ect natural structures of the real world and support ecient object recognition [5, 14]. Biederman has proposed geons - a nite set of distinct volumetric shapes - as qualitative shape models for parts [5]. The key issue in geon-based representation is how to approximate various shapes of object parts by a set of restricted shape types. However, the original geon de nition oered by Biederman was based on the attributes of generalized cones [6] manifested in 2D line drawings. These are frequently dicult to detect, and thus, the rst approaches in the literature were limited to input data obtained from perfect geon-like objects [3, 8], that is, the shape of their components was assumed to be exactly the same as the shape of the perfect geons. The problem of shape approximation was not addressed. We have proposed a 1

inside-outside function for superellipsoids [16], which is not proportional to sensor error. As well their optimization method - an iterative gradient decent method perturbed by Poisson noise - may not always converge to the global minimum. In our approach, model tting and selection are combined into one process. An objective function is de ned which measures (i) the spatial distance between data points and the model surface and (ii) the dierence between the normal vectors of the model and object. Model tting is performed by minimizing this function using a stochastic global optimization approach, which statistically guarantees nding the global minimum. The selection of the model which best ts the data is based on the tting residuals, rather than on the model parameters. Thus, using parametric geons and the proposed model recovery scheme, we can robustly obtain qualitative shape descriptions from object data even though object shapes do not exactly conform to the shape of parametric geons.

ellipsoid: ε 1 = 1 ε 2 = 1

ε 1 = 0.1

BENDING

TAPERING

cylinder: ε 1 = 0.1 ε 2 = 1

tapered cylinder

curved cylinder

ε2 = 0.1

BENDING

TAPERING

tapered cuboid

Figure 1:

cuboid: ε 1 = 0.1 ε 2 = 0.1

curved cuboid

The seven parametric geons.

 1 = 0:1, 2 = 0:1: Cuboid.

Here, 1 is the \squareness" parameter in the northsouth direction; 2 is the \squareness" parameter in the east-west direction. a1; a2; a3 are scale parameters along the x; y; z axes, respectively. Three of the parametric geons can be derived from Equation (1) by specifying 1 and 2 as follows:  1 = 1, 2 = 1: Ellipsoid.  1 = 0:12 2 = 1: Cylinder.

By applying linear tapering in the x; y coordinates along the z axis to the regular cylinder and cuboid, we obtain a tapered cylinder and cuboid. By applying a circular bending deformation along the z axis in the positive x direction, we obtain a curved cylinder and cuboid. The general implicit equation for the parametric geons is given by gi (~x;~ai) = 0 i = 1; 7 (2) Here ~x = fx; y; z gT de nes the surface points, i is the index of parametric geons and ~ai is a parameter vector in from 9 to 11 dimensions. Seven typical shapes and their relationship are shown in Figure 1. A normal vector at a particular point on the surface of the parametric geons with the three regular shapes can be computed by dierentiating the implicit equations (2) as follows:   @g ( ~ x ;~ a ) @g ( ~ x ;~ a ) @g ( ~ x ;~ a ) i i i i i i ~nm = (3) @x ; @y ; @z The approach to computing normals for the deformed primitives is to apply a transformation to the normal vectors of the three regular shapes given in (3) as follows [2]: ~nXm~ = B~n~xm (4) where ~nXm~ and ~n~xm are the normal vectors of deformed and regular primitives, respectively. B is the inverse

1 This can be a cylindrical shape with an elliptical cross section. 2 Superellipsoid shape changes smoothly with  and  . We 1 2

choose 1 = 0:1 for a cylinder, based on computational robustness and the perceptual acceptance of its shape. The same reasoning applies to the cuboid.

2 Parametric geons Similar to Biederman's geons, parametric geons consist of a nite set of distinct shapes. In order to obtain unique results with parametric models, we use a subset of Biederman's geons. Motivated by the art of sculpture, a more traditional framework for 3D object representation [20], we have selected seven volumetric shape types: the ellipsoid, the cylinder1 , the cuboid, the tapered cylinder, the tapered cuboid, the curved cylinder and the curved cuboid. Most of these shapes were regarded as basic shape forms in sculptures. The parametric forms of these seven shapes are derived from the implicit superellipsoid equations [1]:

x 2=2 + y 2=2 a a1 2





 =1

! 2

2=1 + az = 1 (1) 3



2

transpose of the Jacobian matrix of the deformation function. The details of the parametric geon equations and those for the normals are described in [19].

difference between normals

θi

nm

distance between two surfaces

A xs

3 Model recovery 3.1 The Objective function

O

Image data

Figure 2: De ning the objective function. nm and nd are the model and data surface normals, respectively. O is the origin of the model. A is the distance between a particular data point and the center of the model. xs is a point on the model surface. i is the angle between a model and object surface normals.

In this paper, the procedure for parametric geon recovery is: (i) t models to 3D object data by minimizing an objective function and (ii) select the best model according to the minimum residual. The objective functions for tting parametric models used previously by other researchers were neither intended nor used for step (ii) [9]. However, to identify individual qualitative shapes based on tting residuals, we require objective functions which correctly re ect the dierence in both size and shape between the object data and the parametric models. Our objective function consists of two terms expressed as follows:

E = d1 + d2

O

p=



O

O

O computed distance

e(x i , a)

(b)

Figure 3:

The right cylinders in (a) and (b) are obtained by applying inverse tapering and bending transformations to the left tapered cylinder and curved cylinder, respectively. e(xi ;~a) is the Euclidean distance along a line Oxi in the inverse transformed case. 0

0

where f (~x;~a) = g(~x;~a) + 1 is an implicit function for a parametric geon. Gross and Boult have shown the signi cant advantages of this measure over others [9]. Since tapering and bending signi cantly complicate the implicit primitive equations of tapered and curved primitives, we cannot obtain a closed-form solution for e(~xi ;~a), as in (7). Thus an iterative method might be indicated. Since the objective function evaluation is the largest computational component of the model recovery procedure, for the sake of simplicity, we compute an approximate distance measure for the tapered and curved models. No iteration is required. As shown in Figures 3, rst we apply an inverse tapering or bending transformation to both the data and the model in order to obtain the transformed data xi; this gives either a regular cuboid or cylinder. Second, we use (7) to compute the distance from the transformed data point xi to the transformed model surface along a line passing through xi and the model origin O. We interpret e(xi ;~a) as the approximation of the distance along a line from ~xi to the model surface. Although this approximation creates a small error in the distance measure, it tremendously speeds up computation.

(6)

Here N is the number of data points, f~xi 2 R3; i = 1; :::; N g is the set of data points in a model frame, and ~a is the vector of model parameters. For the three regular primitives (ellipsoid, cylinder and cuboid), e(~xi ;~a) is de ned as the Euclidean distance from a data point to the model surface along a line passing through the origin O of the model and the data point [9, 17] (see Figure 2) given as follows: 1 e(~xi ;~a) = A 1 ? [f (~x ;~ i a)]1=p

xs actual distance

(a)

The rst term of the objective function is given by



computed distance

e(x i, a)

3.1.1 The distance measure

i=1

xi xi

xs

(5)

N X d1 = N1 je(~xi ;~a)j

xi

xi actual distance

Both terms are de ned and discussed in detail in the remainder of this section. When the model and object pose are the same, the intuitive interpretation of these two terms corresponds to size and shape similarity, respectively.



nd

Model

0

0

0

(7)

0

2 for the ellipsoid; 20 for the cylinder and cuboid: 3

3.1.2 The normal measure

We de ne the second term (d2 ) of the objective function by measuring a square dierence between the surface normal vectors ~nd of objects and the surface normal vectors ~nm of models at each corresponding position (see Figure 2): N X 1 d2 = N en (i) i=1

(8)

Here N is the number of data points and en(i) = k~nd (i) ? ~nm (i)k2 : (9) In (5),  is a factor which makes the second term adapt the size of the parametric geons to the size of the object. It is de ned to be  = (ax + ay + az )=3, where ax ; ay and az are model size parameters. This factor also forces the selection of a model with a smaller size if the object data are t equally well by a model with dierent size parameter sets. This is similar to the volume factor used in [16]. In (5),  is a weighting constant, controlling the contribution of the second term to the objective function. There is no general rule for selecting it. In this paper, we choose  = 5 according to a heuristic based on the square dierence of normals between each pair of parametric geons [18]. If the object surface is very coarse, the computed normals are not very reliable, and thus  should be reduced.

Figure 4: The logarithm of the objective function in terms of two rotation parameters. The actual parameter space is in from nine to eleven dimensions. mainly involving transformation and deformation parameters, as well as the ratio of size parameters, will be clearly smaller than the entire parameter space. As the tting procedure progresses, the position, orientation and shape of the model will approach that of the object, and the contribution of the second term will gradually decrease. When the value of the rst term is similar to that of the second, both terms will contribute equally to the objective function and the search space becomes the full parameter space. Thus, a search in full parameter space without good initial estimations is automatically achieved by a subspace search followed by a full-space search with good initial estimations. In addition, by introducing the second measure, the objective function becomes steeper and the resulting tting residuals will be more discriminable than those using the distance measure alone.

3.1.3 Mixing dierent norms

We use an L1 (see (6)) and L2 norm (see (9)) to measure the dierence in the distance and the orientation, respectively. An L1 norm is denoted by (x) = jxj and an L2 norm by (x) = x2, where x is a residual measure. The sensitivity of the L2 norm to x increases linearly as the value of x increases [4]. In other words, the norm is relatively insensitive to small values and becomes increasingly more sensitive to outliers. On the other hand, the sensitivity of the L1 norm is constant for all values of x. It is also known that the absolute size of an object is independent of the measurement of the dierences between normals. Therefore, using this mixed objective function, we can create an ecient model parameter search procedure which makes the tting residuals more discriminative. When the tting procedure begins, the models and objects are not well aligned; hence, most of the data can be viewed as outliers. Thus the second term will be much larger than the rst and dominate the search. Obviously the exact size of the object has little eect on the second term. Thus the actual search space,

3.2 Very fast simulated re-annealing The procedure for tting parametric geons to range data is a search for a particular set of parameters which minimizes the objective function in (5). This function has a few deep local minima, caused by an inappropriate orientation of the model, and many shallow local minima, caused by noise and minor changes in object shape as shown in Figure 4. In order to obtain a best t of a model to an object, we need to nd the model parameters corresponding to the global minimum of the objective function. To accomplish this, we employ a stochastic optimization technique, Very Fast Simulated Re-annealing (VFSR) [11]. Motivated by an analogy to the statistical mechanics of annealing in solids, simulated annealing uses a 'temperature cooling' operation for non-physical optimization problems, thereby transforming a poor so4

l=2  ax  L=2; ?  rx  ; l=2  ay  L=2; ?  ry  ; l=2  az  L=2; ?  rz  ; cx ? d  tx  cx + d; 0  kx  1; cy ? d  ty  cy + d; 0  ky  1; cz ? d  tz  cz + d; 0    2=h: Table 1: Parameter constraints.

thresholding was performed to remove the supporting plane and other background data. Surface normals were computed by a least squares tting method. The view transformation, surface normals and residuals of the normal computation were used to remove redundant data appearing in more than one view and to transform range data in each camera coordinate system into a world coordinate frame [18]. The dense 3D data were subsampled at a 50 : 1 sampling rate. Before the search ensued, a constraint for each parameter to be searched was automatically determined. A rectangular region in a 3D space bounded by maximal and minimal x; y; z coordinates fXmax , Xmin , Ymax , Ymin , Zmax , Zmin g of the range data was computed. The maximum dimension in this space is de ned as the distance L from (Xmax ,Ymax ,Zmax ) to (Xmin ,Ymin ,Zmin). Constraints for the desired parameters are given in Table 1. l > 0 is the minimum possible length of the objects, (cx ; cy ; cz ) is the centroid of the data set, and d is the deviation from the centroid. l and d are free parameters determined according to a priori knowledge. Since the upper bound of the bending curvature can be set to the inverse of the minimum possible radius, we select h = min(Xmax ? Xmin ; Ymax ? Ymin ; Zmax ? Zmin ) as the minimum diameter of a bent sector. Thus h=2 is the minimum possible radius. tx ; ty and tz are translation and rx ; ry and rz are rotation parameters. Parameter search using VFSR terminates when any of the following conditions1, selected beforehand, is reached: 1. the smallest temperature value 2. the minimum value of the objective function 3. the maximum number of times the same point is sampled 4. the maximum number of times of state acceptance 5. the maximumnumber of evaluations of the objective function (we set it to 300,000). After parametric geons are tted to the 3D data, the best model for the object is selected according to the minimum tting residual. The execution time varies with respect to dierent data and models. The approximate average time taken for tting ellipsoid, cylinder or cuboid models (nine parameters) is about 30 minutes on an SGI(Personal Iris) R3000 workstation. The time taken for tting the other four models (ten or eleven parameters) is about two hours.

lution into a highly optimized, desirable solution[13]. The salient feature of this approach is that it statistically computes a global optimal solution. Employing an alternative strategy of sampling in the parameter space, VFSR permits an annealing schedule which decreases exponentially. This is much faster than traditional (Boltzmann) annealing whose annealing schedule decreases logarithmically. The re-annealing property permits adaptation to the changing sensitivities in the multidimensional parameter space. Recent research [12] has shown that VFSR is orders of magnitude more ecient than standard genetic algorithms, another popular contender for global optimization. Some researchers have already used nonlinear least squares minimization, adding random walks to escape local minima [16]. This is similar to simulated annealing but with an extremely fast annealing schedule. In some cases where a good initial estimation of the parameters can be obtained, this technique usually takes much less time than general global optimization methods. However, with an inappropriate initial guess and employing an extremely fast annealing schedule, this may trap the algorithm at a local minimum.

4 Experiments The following experiments were conducted to investigate the eciency of the objective function and the discriminative properties of parametric geons. We are interested in examining the residual dierences among all t models, especially when the object data contain noise and the object shapes do not exactly conform to the shapes of the parametric geons. All objects used in the experiments were single-part objects lacking sharp concavities. We used synthetic data, range data of geon-like objects, and range data of imperfect geon-like objects. We only illustrate the last two experiments in this paper. In these two cases, range image data was acquired from four viewpoints using a laser range nder. The registration of images taken from the dierent views was obtained by a method described in [7]. Simple

1 The de nition of the rst four conditions can be found in: [ftp.alumni.caltech.edu:/pub/ingber/ASA-2.20-shar.Z], 1994.

5

OBJECTS elli cyld cubd tcyld tcubd ccyld ccubd

MODELS

elli

cyld cubd 12.075 19.368 0.7968 12.976 17.819 1.313 20.740 27.190 28.241 14.256 21.986 22.148 26.664 23.213

1.206

22.535 36.156 17.993 34.276 24.153 30.735

tcyld 16.511 0:864 28.707

tcubd 26.449 12.834 1:327 2.339 15.203 20.156 1.667 21.197 47.987 21.595 16.291

ccyld 14.512 0:871 17.807 17.625 28.228

ccubd 24.162 14.432 1:338 25.740 14.242 3.300 22.197 14.341 2.818

Table 2:

Fitting models to range data of geon-like objects. The symbols elli, cyld, cubd, tcyld, tcubd, ccyld and ccuboid denote ellipsoid, cylinder, cuboid, tapered cylinder, tapered cuboid, curved cylinder and curved cuboid, respectively. 30.735 4.474

(a)

Figure 5:

26.664 5.911

(b)

23.213 3.523

(c)

21.595 3.715

(d)

16.291 5.693

(e)

14.341 2.466

(f)

2.818 1.945

(g)

Fitted models superimposed on range data of a curved cuboid object. The models are obtained by using Equation (5).

4.1 Using geon-like objects

demonstrate that the seven parametric geons are very distinctive when based on the de ned objective function. Figure 5 shows the results of tting the seven parametric geons to the range data of a curved cuboid object. The lighter shaded volumes are the models obtained by the tting procedure and the darker sparse spots indicate the input data. (a) through (g) illustrate models of the ellipsoid, the cylinder, the cuboid, the tapered cylinder, the tapered cuboid, the curved cylinder and the curved cuboid superimposed on the 3D data, respectively. The algorithm selected the curved cuboid shown in (g) as the best model for the wooden object. This result is consistent with our expectations. We show, at the top right corner in each image, the tting residuals (top row) obtained by using Equation (5) so that they can be compared with the residuals (bottom row) obtained by just using Equation (6). Although models are tted to data much better in both (f) and (g) than others, we still can see the signi cant dierence in residuals (top row). This shows that the normal measure indeed makes the tting residuals more discriminative.

In this experiment, range data of seven machinemade wooden objects were used. The shape of each object is similar to one of parametric geons. We note that the object data are corrupted by sensor noise, data on the bottom of objects are missing, and view registration parameters may not be perfect. Table 2 lists the tting residuals resulting from the experiments. The types of test objects are shown in the rst column. Each row indicates the residuals obtained when tting the seven dierent models to a particular test object given in the rst column. The bold gures on the diagonal are the residuals computed by tting a model to its own speci c data. The underlined gures are residuals produced by tting tapered and curved models to a cylinder or cuboid. When this is done, kx; ky , or  take on values which are very close to 0. Thus the deformed models will appear to have the same shape as a regular cylinder or cuboid. In model selection, if two residuals are very close and much smaller than all the others, the algorithm selects the simplest of the two shapes. These results show that the correct models, denoted by bold gures, are selected for the given objects. Although noise and object self-occlusion aect the tting residuals, the values obtained by tting models to their own object type are still much smaller than those found by tting models to other object types. These results

4.2 Using imperfect geon-like objects This experiment was designed to examine the uniqueness of the shape approximation using parametric geons, given a set of real objects whose shapes 6

42.822

(a)

38.229

(b)

41.395 42.822

(c)

37.876 38.229 (d)

39.840

11.397

(e) 41.395

(f)

40.727

(g)

Figure 6:

Fitted models superimposed on the range data obtained from a banana. The models and tting residuals are obtained by using Equation (5).

elli 3.255

cyld 2.889

cubd 3.851

MODELS tcyld 3.324

tcubd 3.611

ccyld 1.000

ccubd 2.987

Mean Standard deviation 0.118 0.092 0.152 0.232 0.139 0.000 0.149 Maximum residual 4.00111 3.48881 4.71736 5.0178 4.32779 1.000 3.80155 Minimum residual 2.65569 2.45822 3.10204 2.46402 3.07318 1.000 2.38523 Table 3: Fitting models to range data of eleven bananas. varied slightly. Eleven real bananas were used as objects in this experiment. Figure 7 shows two of them. Their shapes cannot be simply depicted by any of the parametric geons. Some bananas had stems at their ends and some had relatively sharp surface variations. In some bananas, the curvature of the main axes changed slightly at the top and signi cantly at the bottom. None of the bananas had a perfectly symmetrical cross-section. The apparently noisy surfaces shown in the gure were due to the range nder's sampling error. This was because the bananas had to be placed far from the range nder in order for them to t within the its scanning eld-of-view. Figure 6 shows the results of tting the seven parametric geons to the range data. The algorithmselected the curved cylinder shown in (f) as the best model for all of the bananas. Clearly this result is consistent with our intuition of the banana's actual shape. Table 3 presents the average tting residuals, standard deviations, and maximum and minimum tting residuals for all of the bananas. Since absolute tting residuals are aected by the banana size which, of course,

Figure 7:

was not uniform, we cannot compare the tting residuals for dierent bananas. Thus, these residuals were normalized by the minimum residual obtained from the same banana as follows: Eijn = Eij =Emin;j ; i = 1; :::; 7 (10) Emin;j = min fE g i ij E n is the relative (normalized) residual value, while i and j are the indices of models and objects, respectively. The minimum relative residual is equal to one and the rest of the residuals are greater than one. Thus, Table 3 indicates how dierent, on average, the minimum relative residual is from the other residuals. The results show that the best model for all of the bananas was the curved cylinder, since it gives the smallest average residual value. Thus parametric geons and the described recovery procedure demonstrate robust behavior by uniquely representing the dierent banana shapes.

5 Conclusion We have described an approach to deriving shape approximations of object parts using parametric geons. The strength of parametric geons is that they (i) provide a global shape constraint which prevents model recovery from the in uence of noise and minor shape variations and (ii) result in both qualitative

Two bananas used in the experiments.

7

shape information and quantitative size and deformation information which support ecient object recognition. Multiview data, an objective function measuring both distance and normal dierences, and global optimization (VFSR) are used to obtain unique descriptions of single-part objects. Experimental results demonstrate the ability to approximate single-part objects using parametric geons with range data of perfect and imperfect geon-like objects. This suggests that parametric geons can be used as a coarse description of object parts for qualitative object recognition.

[9]

[10]

Acknowledgements

[11]

The authors would like to thank Dr. Lester Ingber for the VFSR computer code and Gerard Blais and Gilbert Soucy for technical help. M. D. Levine would like to thank the Canadian Institute for Advanced Research and PRECARN for its support. This work was partially supported by a Natural Sciences and Engineering Research Council of Canada Strategic Grant and an FCAR Grant from the Province of Quebec.

[12] [13] [14]

References

[15]

[1] A. H. Barr. Superquadrics and angle-preserving transformations. IEEE Computer Graphics Applications, 1:11{23, 1981. [2] A. H. Barr. Global and local deformations of solid primitives. Computer Graphics, 18(3):21{30, 1984. [3] R. Bergevin and M. D. Levine. Generic object recognition: Building and matching coarse descriptions from line drawings. IEEE Transactions on Pattern Analysis and Machine Intelligence, 15(1):19{36, January 1993. [4] P. J. Besl, J. B. Birch, and L. T. Watson. Robust window operators. In Proceedings of the 2nd International Conference on Computer Vision, pages 591{ 600, Tampa, FL, 1988. [5] I. Biederman. Human image understanding: Recent research and a theory. Computer Vision, Graphics, and Image Processing, 32:29{73, 1985. [6] T. O. Binford. Visual perception by computer. In IEEE Conference on Systems and Control, Miami, FL, 1971. [7] G. Blais and M. D. Levine. Registering multiview range data to create 3D computer objects. Technical Report TR-CIM-93-16, Center for Intelligent Machines, McGill University, Montreal, Quebec, Canada, October 1993. [8] S. J. Dickinson, A. P. Pentland, and A. Rosenfeld. 3D shape recovery using distributed aspect matching.

[16]

[17] [18] [19]

[20]

8

IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(2):174{198, February 1992. A. D. Gross and T. E. Boult. Error of t measures for recovering parametric solids. In Proceedings, 2nd International Conference on Computer Vision, pages 690{694, Tampa, Florida, 1988. Computer Society of the IEEE, IEEE Computer Society Press. S. Han, D. B. Goldgof, and K. Bowyer. Using hyperquadrics for shape recovery from range data. In Proceedings of Fourth International Conference on Computer Vision, pages 492{496, Berlin, Germany, May 1993. IEEE Computer Society Press. L. Ingber. Very fast simulated re-annealing. Mathematical and Computer Modelling, 12(8):967{973, 1989. L. Ingber and B. E. Rosen. Genetic algorithms and very fast simulated reannealing: A comparison. Mathematical and Computer Modelling, 16(11):87{ 100, 1992. S. Kirkpatrick, Jr. C. D. Gelatt, and M. P. Vecchi. Optimization by simulated annealing. Science, 220(4598):671{680, May 1983. A. P. Pentland. Recognition by parts. In The First International Conference on Computer Vision, pages 8{11, London, June 1987. N. S. Raja and A. K. Jain. Recognizing geons from superquadrics tted to range data. Image and Vision Computing, 10(3):179{190, April 1992. F. Solina and R. Bajcsy. Recovery of parametric models from range images: the case for superquadrics with global deformations. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(2):131{ 147, 1990. P. Whaite and F. P. Ferrie. From uncertainty to visual exploration. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(10):1038{1049, October 1991. K. Wu and M. D. Levine. 3-D object representation using parametric geons. Technical Report TR-CIM93-13, Center for Intelligent Machines, McGill University, Montreal, Quebec, Canada, September 1993. K. Wu and M. D. Levine. Parametric geons: A discrete set of shapes with parameterized attributes. In SPIE International Symposium on Optical Engineering and Photonics in Aerospace Sensing: Visual Information Processing III, Orlando, FL, April 1994. The International Society for Optical Engineering. W. Zorach. Zorach Explains Sculpture: What It Means and How It Is Made. Tudor Publishing Company, New York, 1960.