Adaptive pixel resizing for multiscale recognition and reconstruction

Adaptive pixel resizing for multiscale recognition and reconstruction Marc Rodr´ıguez, Ga¨elle Largeteau-Skapin, Eric Andres Laboratory XLIM, SIC Department, University of Poitiers BP 30179, UMR CNRS 6712 86962 Futuroscope Chasseneuil Cedex, France

Abstract. This paper present an adaptive pixel resizing method based on a parameter space approach. The pixel resizing is designed for both multiscale recognition and reconstruction purposes. The general idea is valid in any dimension. In this paper we present an illustration of our method in 2D. Pixels are resized according to the local curvature of the curve to control the local error margin of the reconstructed Euclidean object. An efficient 2D algorithm is proposed.

1

Introduction

For several years now, the discrete geometry community works on an invertible reconstruction of a discrete object. A reconstruction is a transformation from the discrete space to the Euclidean space that associates a Euclidean object to a discrete one. The reconstruction is said to be invertible if the digitization of the reconstructed object is equal to the original discrete object. Recently, Isabelle Debled-Renesson et al. have started to consider reconstruction methods that allow a margin of error in the reconstruction [1, 2]. These methods are not invertible but allow a reconstruction of, for instance a discrete curve, where a pixel might be missing or out of place. This brought them to define the notion of blurred line segments and consider applications such as discrete curve denoising. Their method is based on thick arithmetical Reveill`es discrete straight lines. Contrary to an invertible reconstruction algorithm where all the pixels that verify the Reveill`es straight line equation 0 ≤ ax + by + c < max(|a|, |b|) belong to the discrete line, for blurred lines of thickness ω a pixel has to verify 0 ≤ ax + by + c < ω. The value ω defines in some way an error margin. A set of pixels might not, strictly, be a discrete straight line but as long as its pixels are not too far away from it, it will form a blurred line. The only problem with this approach is that the arithmetical thickness of those lines, and thus the error margin, is uniform for a given curve. This was the starting point of our research. Our goal in this paper is to propose an adaptive voxel resizing method based on a parameter space approach and give an illustration of this method with a 2D pixel resizing that fits the local curvature of a discrete curve. This example is a direct extension of the work of Debled-Renesson et al. and shows how our approach can generalize their work. Our long term goal however is to build a

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Marc Rodr´ıguez, Ga¨elle Largeteau-Skapin, Eric Andres

theoretical and practical framework for multiscale pattern recognition and object reconstruction. A resized voxel can be seen as a voxel at a different scale and the parameter space approach allows to work with voxels of various sizes, at various scales. The approach is based on a parameter space approach (see the J.Vittone approach [3, 4]). A Euclidean point, in nD, in the image space is associated to a Euclidean hyperplane, in nD, in the parameter space. The dual of a voxel (in the parameter space) will therefore be a sort of a butterfly figure that represents all the hyperplanes that cross the voxel. This is called the generalized preimage of the voxel. The intersection of all the generalized preimages of the voxels of a set is the generalized preimage of the voxel set. The preimage approach we propose here allows us to locally mix voxels of different scales. Having voxels of different scales means that voxels can overlap (see Fig.1). As an illustration of the voxel

Fig. 1. After resizing the pixels may overlap.

resizing approach, we propose a 2D application where pixels of a discrete curve are resized according to the maximal symmetric tangent [5, 6] (we can also use other curvature estimators such as the estimator defined by J.W. Bullard [7] which is defined in any dimension). This provides an adaptive pixel size according to the smoothness of the curve. Indeed, if the maximal symmetric tangent around a pixel is long it means that locally, around this pixel, the curve is flat and thus smooth. The curve does not need to be denoised much ( in the sense defined by Debled-Renesson et al. [1, 2]) and the pixel size will be left unchanged. On the contrary, a short maximal symmetric tangent around a pixel means that locally the curve is not flat (high curvature) and thus can be simplified by increasing the margin of error in the reconstruction by increasing the size of the pixel. The pixel resizing involves several modifications of the reconstruction algorithm proposed by M.Dexet [8]. Those modifications come directly from the fact that all pixel of a set do not share the same size. We state in this paper that the computation of the two-dimensional polytopes collecting all the straight lines crossing a pixel set can be reduced to the linear time computation of half-plane intersections. We use it to compute the generalized preimage of a resized pixel set. Further applications in discrete curve denoising and Euclidean curve simplification are considered. The paper is organized as follow: section two presents basic notions and notations, it describes the generalized preimage and its use. Section three presents our version of the reconstruction with resized pixels. We also propose in this section an iterative half-plane intersection algorithm in linear time for the preimage computation. Finally, some applications examples in simplification and denoising are presented.

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3

Preliminaries Notation

Let Zn be the subset of the nD Euclidean space Rn containing all the integer coordinate points. A discrete (resp. Euclidean) point is an element of Zn (resp. Rn ). A discrete (resp. Euclidean) object is a set of discrete (resp. Euclidean) points. Discrete-Euclidean operations [9] are operations that operate partially in the discrete space Zn and partially in the continuous space Rn . Appropriate transforms are used to travel between both spaces. A transformation from the Euclidean to the Discrete space is called discretization. The transformation from the discrete space to the Euclidean space is classically called reconstruction. We denote pi the ith coordinate of a point or vector p. The voxel V(p) ⊂ Rn of a discrete nD point p is defined by V(p) = [p1 − 21 , p1 + 21 ]×...×[pn − 21 , pn + 21 ]. Standard hyperplanes [10] are a particular case of analytical discrete hyperplanes defined by J.-P. Reveilles [11, 12]. A set of successive points Ci,j of a digital curve C is called a Digital Straight Segment (DSS) if there exists a standard discrete line (a, b, µ) containing them. The predicate Ci,j is a DSS is denoted S(i, j). Around any point Ck of a digital curve C, the DSS Ck−l,k+l with S(k − l, k + l) and not S(k − l − 1, k + l + 1) is called Symmetric Tangent [5, 6]. Let h be the direct Hausdorff distance: A ⊂ Rn , B ⊂ Rn , h(A, B) = maxa∈A (minb∈B (d2 (a, b))), where d2 is the Euclidean distance. The Hausdorff distance between A and B is H(A, B) = max(h(A, B), h(B, A)). An homeomorphism or topological isomorphism is a bicontinuous function between two topological spaces. Two spaces are called homeomorphic if they are the same from a topological point of view. 2.2

Recognition and invertible reconstruction based on a preimage

In computer imagery, parameter spaces are often used to recognize shapes in a picture. Those spaces are defined with a transformation that associates a point in the image space to a geometrical object in the parameter space. In this paper, we use the generalized preimage definition [8]: any Euclidean point pE = (x1 , ..., xn ) ∈ Rn (the image space) is associated to a hyperplane (in the parameter space Rn ): n o Pn−1 DE (pE ) = (y1 , ..., yn ) ∈ Rn | yn = xn − i=1 (xi yi ) . Contrary to the Hough transform [13], the parameter space here is not digitized and the transform is analytical. The parametric image of a voxel will therefore be a sort of a butterfly figure (see Fig.2 for examples of pixel preimages) that represents all the hyperplanes that cross the voxel. This is called the generalized preimage of the voxel. Each point of the generalized preimage in the parameter space corresponds to one hyperplane that cuts the voxel in the image space. The generalized preimage of a set of voxels is defined as the intersection of

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Marc Rodr´ıguez, Ga¨elle Largeteau-Skapin, Eric Andres

all the preimages of each voxel of the set. Each point of the generalized preimage of a set of voxels in the parameter space corresponds to a hyperplane that cuts all the voxels of the set in the image space. This corresponds to an invertible reconstruction if the standard [10] or the supercover discrete analytical model is considered [14]. The analytical reconstruction consists of two steps: firstly a recognition step is performed that divides the discrete object into discrete analytical hyperplanes pieces [11, 10] (subsets of discrete analytical hyperplanes) and computes their generalized preimage in the parameter space. A hyperplane recognition algorithm in dimension n based on the general preimage definition has been proposed by M.Dexet [8]. Secondly comes the reconstruction itself. It consists in picking a Euclidean solution from each generalized preimage of each hyperplane piece and group them to form the final Euclidean reconstructed object. The main difficulty here is not so much the recognition step than the reconstruction step. A largely unsolved problem is the problem of reconstructing a closed Euclidean surface from the different Euclidean hyperplane pieces. Invertible reconstructions have been proposed in dimension 2 by R.Breton et al. [15] for the standard analytical model for discrete objects [10] and in dimension 3 by I. Sivignon et al. [16]. The problem is still partly open in dimension 3 and largely open in dimension higher than three. In two dimensions, the preimage of a point is a Euclidean straight line. The preimage of a pixel is an unbounded polygon with edges that are the preimages of the four vertices of the pixel (see example of pixel preimages in Fig.2). The invertible reconstruction of a discrete curve goes basically as follows (see [15] for more details): the preimage of consecutive pixels are intersected as long as the result is not empty. This process is illustrated in fig.2. At the end, we obtain a polygon in the parameter space: for figure 2, the generalized preimage of the five pixels. Each point of this polygon corresponds to a Euclidean line that crosses all the five pixels of the pixel set. The standard or supercover discretization of each of these Euclidean straight lines contains the five pixels. At this point we have recognized a DSS. When handling a discrete curve, we start over with the last pixel of the previous DSS recognition in order to recognize a new DSS until we have dealt with all the pixels of the curve. Each DSS is then replaced by a Euclidean straight line segment corresponding to a point chosen inside the preimage of the DSS (the intersection of the preimage of its pixels). Some attention needs to be given to the intersections of the different Euclidean straight line segments in order to have an invertible reconstruction. There are two important advantages to this method. Firstly, it computes the exhaustive set of all the Euclidean straight lines (resp. hyperplanes) crossing all the pixels (resp. voxels) of a given pixel set. Secondly, nothing in the method imposes that the pixels or voxels we are considering are of the same size (i.e. at the same scale). The grid elements we are considering do not even have to be squares, rectangles, hypercubes, etc. as long as the preimage is known. Heterogeneous grids can therefore be handled with the generalized preimage approach.

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Fig. 2. Straight line recognition process: In dark, the current pixel and below the preimage of the current pixel. In grey the pixels already processed and the intersection of the preimages of those pixels. At the end, the intersection is not empty: the pixels belong to a discrete straight line segment.

It is this second property that will be used in what follows with pixels and voxels of different sizes scales.

3

Reconstruction with resized pixels

The aim of this section is to propose a new reconstruction method with resized pixels. The basic idea is to reproduce the principle described in section 2 with pixels that have different sizes (that come from grids at different scales). This generalizes the approach with blurred lines proposed by I. Debled-Rennesson et al. [1, 2] where the thickness of the blurred line is a fixed parameter. In our case the pixels may, independently, have different sizes. The general principle of the reconstruction method based on generalized preimages does indeed not change with pixels of different sizes, even if they overlap. This general idea is valid in dimension n as well. The algorithm proposed by M. Dexet [8] in dimension n needs however some adaptation which we present here in dimension 2 in a specific application: Adaptive Pixel Resizing according to the local curvature of the discrete curve. First however, let us show how these ideas are well adapted for recognition and reconstruction within multiscale and hereterogeneous grids. 3.1

Multiscale framework

In what follows (starting in section 3.2), we consider a curve in a grid where the individual pixels are resized according to a given criteria (in our case a local discrete curvature value). This is the basically the same point of view than the one proposed by I. Debled-Rennesson et al. [1, 2]. There is however another way of looking at this, not as pixels that are resized but as pixels of different grids at different scales. As we can see in Fig. 3, one can consider that the preimage method allows recognition and reconstruction in a multiscale setting where, contrary to many applications, the different scales (i.e. corresponding grids) do not necessarily have to match each other. The pixels from the different grids may overlap or not. Recognition and reconstruction within heterogeneous grids is another possible application field of general preimage based approaches. These

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Marc Rodr´ıguez, Ga¨elle Largeteau-Skapin, Eric Andres

approaches work in every dimension as long as the grid cells are convexes, that we know what their preimages are (and know how to intersect them with each other) and that we define an order with which one adds cells into the recognition process. If the grid cells are not convex then it requires that we know how to divide the cell into convex pieces or, if we do not consider invertible reconstruction, that we know how to compute the cells convex hulls. As we can see, preimage methods are very flexible and allow recognition in a wide range of multiscale frameworks. Note that, contrary to classical line stabbing/transversal problems [17], we are not interested in simply knowing if there is a hyperplane that cuts a set of objects or proposing one hyperplane as solution. In order to tackle the much more complicated “reconstruction” problem in discrete geometry, we are looking for all the solutions (all the hyperplanes that cut a set of grid cells represented by the generalized preimage).

Fig. 3. Pixel are given at different scales.

3.2

Adaptive Pixel Resizing according to the curvature

From here on we consider an illustration of the preimage approach where pixels of a given discrete curve are resized according to a local criteria. In this particular case, with resized pixels, their center does not change (see Fig.1). This is different from what happens when the grid scale is simply changed since then, usually, the center of the pixels change. The preimage approach allows pixel resizing. One can imagine many ways of considering resized pixels/voxels or pixels/voxels with different scales. We are going to focus here on one such idea where we allow a bigger margin of error for the reconstruction in high curvature parts of a discrete curve. It is meant as a direct extension of the work of Debled-Rennesson et al. [1, 2]. In this paper however, in order to simplify the computations, we do not compute the actual curvature at a given discrete point of the curve but only a first order approximation. An ongoing work with an other curvature estimator (the estimator defined by J.W. Bullard [7] which is defined in any dimensions) is briefly illustrated in 3D in Fig. 5. Here, we consider the length of the longest discrete straight line included in the curve and centered on the discrete point (i.e. the length of the maximal discrete symmetric tangent). That means, for instance, that discrete cusps [15] will have their size increased much more than pixels in the middle of long straight lines (flat parts of the curve). We want to focus the simplification on uneven parts of curve where the curve is not flat. An increase in the size of the pixels there means that we will be able to smooth/denoise the discrete curve (see Fig.4). In other words, the greater the curvature of the discrete curve centered on the considered pixel,

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the greater the new size of this pixel. Our recognition method will smooth the curve and (small) oscillations will disappear. Of course, this is not meant as a realistic simplification method as it smoothes out points of interest in curves. It illustrates however well how the recognition and reconstruction with pixels of various sizes/scales works. To evaluate the curvature at a pixel, the length of 2 2 2 3 3 2 3 3 3 3 3 2 3 3 3 2 2 2

3 2 1 2 4 4 4 7 6 5 4 3 3 4 4 4 3 2 1 2 8 9 10 9 8 4 3 2 1 2 3 4 3 3 4 5 6 7 4 4 2 1 2 4 4 3 2 3 2 3 3 3 2 2 1 3 4 3 4 4 3 2 2 2 1 2 5 5 3 1 2 3 4 3 4 4 3 5 6 6 5 2 4 4 4 5 4 4 6 6 3 3 5 4 3 2 1 2

Fig. 4. Pixel resizing using the maximal symmetric tangent.

the maximal symmetric tangent [5, 6] centered on this pixel is considered. The maximal symmetric tangent centered on a discrete point p for a discrete curve, is the longest discrete line centered on p such that there are the same number of points on the discrete line on each side of p. It is a very simple criteria that gives a measure of the flatness of the curve around this pixel. It can be computed in linear time according to the number of discrete points of the curve. On the left of the figure 4, each pixel is labeled with p ∈ N where 2p + 1 is the size of the maximal discrete symmetric tangent. On the right of the figure 4, pixels are resized according to the resizing function r = max(1, 3 − 12 p).

Fig. 5. Voxel resizing using the bullard curvature estimator in dimension three.

The pixel resizing is function of the maximal symmetric tangent length. To increase the pixel size, the resizing function has to be greater than one at any point. This function can be modified to control where the error margin is located. 3.3

Error measure

In order to evaluate the reconstruction transform, we propose an upper bound on the Hausdorff distance between a Euclidean object and the reconstruction of its digitization. It is indeed not very easy to measure otherwise the impact of a reconstruction transform. The reconstruction method certifies that every resized voxel is crossed by at least one edge of the boundary of the reconstructed Euclidean object. The error is therefore directly proportional to the size of the biggest resized voxel.

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Marc Rodr´ıguez, Ga¨elle Largeteau-Skapin, Eric Andres

Theorem 1. The Hausdorff distance between a Euclidean object A and the simplified object B (the reconstruction of its digitization) √depends on the maximal n voxel resizing factor R = maxi (r(i)): H(A, B) ≤ (R+1) where n is the dimen2 sion. Proof. The Hausdorff distance is controlled by the voxel resizing. We know that, for each reconstruction, a Euclidean hyperplane crosses the original voxel and the resized voxel. The biggest distance between those two hyperplanes for a resized voxel corresponds to two hyperplanes crossing the voxels at opposite corners. The biggest distance over all the voxels is obtained for the biggest resized voxel. This leads to the result. ⊓ ⊔ If many such operations are chained, the Hausdorff distance is not equal to the sum of the resizing factors because additional errors are added by the discretizations between each reconstruction. The error bound can diverge when chaining such operations. This reflects the fact that we can have geometrical deformations [9]. One way to avoid this problem is to apply every successive operation (with different resizing function for instance) on the original object. 3.4

Half-plane intersection algorithm for a 2D reconstruction

Let us now take a look at the algorithmic aspects of the preimage computation. In general, we can determine in θ(n) that there is a hyperplane intersecting all the elements of a set of n voxels independently of the dimension [17]. However, as already mentioned, in our case we are not looking for a single solution but for all the Euclidean straight lines that cut the set of all the (resized) pixels. For an orthonormal grid with pixels that all have the same size, the preimage of a set of pixels has at most four edges [18]. With pixels of various sizes, the number of edges of the preimage polytope can be infinite if the number of different pixel sizes is not bounded. Without an hypothesis on the number of vertices of the preimage, the iterative preimage construction can be computed in Θ(n2 ) with n the number of pixels. However, we know that the center of the resized pixels are discrete points in Z2 which means that the number of vertices of the preimage depends only on the number of different possible pixel sizes. Here we propose a limited number of possible pixel sizes which leads to a generalized preimage construction algorithm in Θ(n) where n is the number of (resized) pixels. During the reconstruction process, the first important step is the analytical primitive recognition in the parameter space. For orthonormal grids, a very efficient algorithm was proposed by M.Dexet in [8] but with resized pixels it is no longer working since pixels may now overlap. The preimage of a pixel with vertices (vi )i∈[1,4] = ((xi , yi ))i∈[1,4] is an unbounded polygon with edges defined by the preimage of the four vertices: the straight lines of equation di : ai x + bi = 0, i ∈ [1, 4]. A point p = (x, y) is inside the preimage if: (a1 x + b1 ≤ 0 ∪ a2 x + b2 ≤ 0) ∩ (a3 x + b3 ≥ 0 ∪ a4 x + b4 ≥ 0). This is equivalent to:

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(a1 x + b1 ≤ 0 ∩ a3 x + b3 ≥ 0) ∪ (a2 x + b2 ≤ 0 ∩ a4 x + b4 ≥ 0). The problem can be divided into two groups: the straight line solutions with positive slopes corresponding to any point p = (xp , yp ) verifying (a1 x + b1 ≤ 0 ∩ a4 x + b2 ≥ 0), (x, y) ∈ R+ × R and the solutions with negative slopes corresponding to any point q = (xq , yq ) verifying (a2 x + b2 ≤ 0 ∩ a4 x + b4 ≥ 0), (x, y) ∈ R− × R. The preimage of a pixel (α, β) can therefore be described by four constraints: (1) −(α − 21 )x − y + (β + 12 ) ≥ 0, (x, y) ∈ R+ × R. (2) (α + 12 )x + y − (β − 21 ) ≥ 0, (x, y) ∈ R+ × R. (3) −(α + 12 )x − y + (β + 12 ) ≥ 0, (x, y) ∈ R− × R. (4) (α − 12 )x + y − (β − 21 ) ≥ 0, (x, y) ∈ R− × R. Proposition 1. The solution polytope computation problem can be reduced to a half-plane intersection problem. The solution polytope is the union of two convexes, one in R+ × R and one in R− × R. Let us now describe our half-plane intersection algorithm. Without any hypothesis on the half-planes, the algorithms suppose that the half-planes are ordered according to, for example, the slope. The complexity of these algorithms is therefore in O(n ln(n)) [19]. In our case however, there are several hypothesis that allow us to compute the half-space intersections, and thus construct the preimage, in linear time. Firstly, the intersections of the half-planes are performed respectively in the two halves of the space x ≤ 0 and x ≥ 0. The first half-space to be considered in our intersection algorithm is therefore x ≤ 0 and x ≥ 0 respectively. Secondly, the parameter space definition ensures that no constraint (half-plane) can be vertical. This hypothesis is important because we know that the first constraint intersects the vertical axis x = 0. Lastly, we know that the number of edges of the preimage in each half of the space is bounded because the center of the resized pixels are integer coordinate points and the number of possible different pixel sizes is bounded. Algorithm main idea (see p.11): Four constraints (half planes) are associated to each pixel. Therefore 4n constraints are associated to a set of n pixels. Each constraint is a half-space in the parameter space bounded by a straight line. We add a first constraint which is x ≤ 0 and x ≥ 0 respectively. For each half-space constraint, we have two Boolean markers that are set to false at start. A marker indicates if a vertex of the preimage is on the half-plane border. Two markers at true means that we have a straight line segment. When we add a new constraint, we check all the vertices that we have already identified and determine like that if the constraint cuts the polytope into two pieces (and then the intersected lines are immediately identified), or if the vertices all verify or do not verify the new constraint. There is a slight difference in treatment here if the preimage at that point is still unbounded. When all the half-spaces have two markers at true then the preimage is

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bounded (it is not a half-space homeomorphism anymore). When a constraint is not marked (both at false) anymore then the constraint is dropped. A constraint with only one marker at true corresponds to a half line. The algorithm works in linear time because we know that the number of vertices of the preimage that needs to be checked is bounded. Figure 6 shows an example of the half-space intersection algorithm at work on the half-space x ≤ 0. d1

d1

d1 d2

d2

d3

d1 and d2 are accepted, d1 is now marked twice, both are marked once. the edge is no longer infinite. d4

d1

d4

d6

d1

d2

d2

d3

d3

d4

d1 d2

d5

the polytope is no longer an d3 is unmarked, half space homeomorphism. the contraint is dropped.

d5

d1 and d2 are unmarked both are dropped.

Fig. 6. Example of the iterative half plane intersection algorithm.

Theorem 2. The algorithm that computes iteratively the intersection of halfplanes works in linear time in the number of pixels of the border in the case of an Adaptive Pixel Size Reconstruction where the number of different possible new pixel sizes is bounded and if the center of each pixel has integer coordinates. Proof. The algorithm complexity is O(k × n) where n is the number of pixels of the digital curve and k the number of vertices of the solution polytope. If the center of pixels are discrete points of Zn and the number of different pixel sizes is bounded, then the number of vertices of the preimage is bounded. Indeed, the number of support points and the number of support lines is bounded (see [1, 2] for more details on support lines and support points). Each support point corresponds to an edge of the solution polytope in the parameter space. The number of vertices k of the preimage only depends on the number of different pixel sizes and not on the length of the curve. ⊓ ⊔

4

Example of Applications

Let us now present two short applications of our Adaptive pixel resizing reconstruction method: discrete curve denoising and Euclidean curve simplification. The discrete curve denoising is a discrete-Euclidean transform (see [9] for details). A discrete-Euclidean transform is a transform in the discrete world that operates partially in the Euclidean world. In this case, the starting point is a discrete curve that we reconstruct with our new Adaptive pixel resizing

Lecture Notes in Computer Science

Algorithm 1: Iterative computation of a preimage polytope using halfplane intersections Data: A polytope P defined by a constraint list CL, a vertex list V L where a vertex is defined with two constraints v( C1) and v(C2). A constraint C: αX + βY + γ ≥ 0 Result: a polytope P ′ defined by a constraint list CL′ and a vertex list V L (if P = P ′ the constraint C is redundant). begin if CL.size() = 0 then add the edge C to CL else if CL.size() = 1 then add the vertex C ∩ CL(0) to V L; add the edge C to CL ; mark C and CL(0) one time else for all v ∈ V L do if v do not verify C then unmark vC1 and vC2 one time ; remove v from V L if The polytope is an half-space homeomorphism then for all c ∈ CL do if c is not marked then if c is not an infinite edge then remove c from CL else add the vertex C ∩ c to V L ; mark C and c one time else if c is marked one time and c is not an infinite edge then add the vertex C ∩ c to V L ; mark C and c one time if the intersection between C and an infinite edge e verify all the CL constraints then add the vertex C ∩ e to V L ; mark e and C one time ; //if C has two marks, the polytope is no longer an half-space homeomorphism ; //else, C become an infinite edge else for all c ∈ CL do if c is not marked then remove c from CL else if c is marked one time then add the vertex C ∩ c to V L ; mark C and c one time end

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reconstruction method. The reconstructed Euclidean curve is then discretized to obtain a new discrete curve that is smoother. Missing information or misplaced pixels can also be replaced with this method as long as the pixels are ordered in a sequence on the discrete curve (see Fig. 7).

Fig. 7. Example of discrete curve denoising.

In the case of a Euclidean-discrete curve simplification, the original object is a Euclidean curve. The adaptive pixel size reconstruction is applied to the discretization of the Euclidean curve. This transform is therefore a Euclideandiscrete transform (see [9] for details) that transforms a Euclidean curve into a Euclidean curve. The transform operates partially in the discrete world. On

Fig. 8. Discrete-Euclidean simplification.

figure 8, we can see the discretization and the reconstruction with resized pixels. Those operations transform the original Euclidean object into an other one with less vertices depending on the resizing function.

5

Conclusion and Perspectives

In this paper we have presented a new two-dimensional reconstruction based on the computation of the generalized preimage with an adaptive pixel size. This extends the recognition and reconstruction based on blurred lines proposed by Isabelle Debled-Rennesson et al. [1, 2]. In our method, each pixel can be individually resized according to a given criteria. We have proposed to resize each pixel according to the length of the maximal discrete symmetric tangent at that pixel. Extensions to more precise curve estimators is one of the immediate perspectives we are working on. We have already started to work with the Bullard curvature estimator that is defined for all dimensions. First voxel resizing according to the curvature have been done in 3D (See Fig. 5). An efficient algorithm was implemented that provides a two-dimensional reconstruction in linear time if the number of different possible pixel sizes is bounded and if the centers of the pixels belong to a predefined regular grid. One of the major perspectives of this work is its extension to higher dimensions. The Bullard curvature estimator [7] can be computed very quickly at the fly in any dimension. High dimensional primitive recognition in a multiscale, heterogeneous grid setting seems now to be achievable in a very general framework. Questions about algorithmic complexity of the generic recognition algorithms and heuristics behind the way voxels are added into the recognition algorithm

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(there is no natural order anymore on surfaces) is however still an open and difficult problem. Reconstruction in dimensions higher than two is still quite open, even on regular grids, if you consider global techniques such as the ones we discussed in this paper rather than local techniques such as, for instance, the Marching Cubes technique. This is one of the long term goals we have. A more immediate perspective is the topological control of the reconstruction in 2D. Right now, the method does not guaranty that the reconstructed curve does not modify the topology of the object when pixel sizes are increased. Features of the simplified curve might disappear and the different edges might intersect each other elsewhere than on the common vertices. This represents right now the short term focus of our research. A first look at these topological problems has been proposed at IWCIA2009 [20]. A last perspective comes from the comparison with the work of Isabelle Debled-Rennesson et al. There are obvious links between both approaches on a theoretical level that need to be explored. This is certainly an interesting perspective especially when considering higher dimensions.

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