Brandt, J. and V.R. Algazi, 1992. Continuous skeleton computation by Voronoi diagram. Computer Vision, Graphics and Image Processing, v. 55, pp. 329-338.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the medial axis. For polyhedral input Etzion and Rappoport [16] suggest an approximation method based on octree subdivisions of space. Another scheme considered by many uses a set of sample points on the shape and then approximates the medial axis with the Voronoi diagram of these points [4, 5, 6, 11, 25, 29]. We follow the Voronoi diagram approach. It is particularly suitable for point cloud data, which are increasingly being used for geometric modeling over a wide range of applications. It is known that the Voronoi vertices approximate the medial axis of a curve in 2D. In fact, Schmitt [24] and ....

....it as the sample density approaches infinity. The convergence result of poles to the medial axis is a significant progress in the medial axis approximation in 3D. However, many applications require and often prefer a nondiscrete approximation rather than a discrete one. In 2D, Brandt and Algazi [11] achieve this by retaining a subset of Voronoi edges incident to the Voronoi vertices. In 3D, since poles lie close to the medial axis, Amenta, Choi and Kolluri [4] design an algorithm that connects them with a cell complex. They consider the Delaunay balls centering the poles and then compute the ....

J. W. Brandt and V. R. Algazi. Continuous skeleton computation by Voronoi diagram. Comput. Vision, Graphics, Image Process. 55 (1992), 329--338.

....bounded by straight line segments. The Voronoi diagram generated by a set of boundary points is such that the points at the intersection of two (or more) polygons are equally distant to two (or more) nearest boundary points. This property was used to generate skeletons from a set of boundary point [6, 18]. The Voronoi diagram of a discrete set of boundary points is called the discrete Voronoi medial axis (dvma) and can be regarded as the intersection of the vd with the generating shape. Once the dvma has been constructed, a skeleton can be extracted from it by pruning away undesirable Voronoi ....

J. W. Brandt and V. R. Algazi (1992), Continuous skeleton computation by Voronoi diagram. CVGIP: Image Understanding, 55(3): 329--337.

....union of the boundaries between the Voronoi regions de ned as the set of points at least as close to one site as to any other. Indeed, the following mathematical de nition of a VD is commonly found in the literature, as covered in Aurenhammer s survey [4] and more recently in the work of Brandt [7]: set of points at least as close to e as to any other site in E: V R(e) j d(p; e) d(p; e where d is the usual Euclidean distance between two sets. The Voronoi diagram of E, denoted V D(E) is usually de ned as the union of the boundaries of the Voronoi regions: V R(e i ) ....

.... Figure 13: Polygonal shape, its Voronoi diagram (following De nition 1) from edges and vertices as being the Voronoi sites (a) and its exact Medial axis through removing some particular branches of the VD (b) A novel approach to this subject came in 1991, with the work of Brandt and Algazi [7, 15]. Basically, they showed that the MA of a continuous regular shape can be approximated by a subset of the VD of points sampled along the border of the shape. They also showed that this approximation gets better as the sample density increases. This is illustrated in the following gure, where, as ....

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J.W. Brandt and V.R. Algazi, Continuous skeleton computation by voronoi diagram CVGIP: Image Understanding, 55(3):329-338, 1992.

....of the medial axis transform, invertibility, dimension reduction, and preservation of topological and geometric equivalence is of the most importance. The accuracy of the skeletons depends on user specified parameters as well as level of noise in the input image. Two di#erent categories [10] of approaches have been proposed. The majority of these algorithms are region based , where the input is usually an array of filled image data. The pixels of images are iteratively thinned, or equivalently, redundant pixels are successively deleted until a final skeleton is derived. The second ....

....from the set of discrete boundary points. These algorithms utilize Voronoi diagrams and Delaunay triangulation in generating discrete medial axis representation. Pruning procedures are then applied to handle errors in sampling and to represent hierarchical skeleton representation. Brandt [9, 10] used the regular set model to relate the boundary sampling density to the discrete skeleton for an accurate skeleton approximation. Once boundaries are sampled, the Voronoi diagram of the point set is generated and edges of the Voronoi diagram internal to the image are extracted. Pruning ....

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J. W. Brandt and V. R. Algazi. Continuous skeleton computation by voronoi diagram. CVGIP: Image Understanding, 55(3):329--338, May 1992.

....Related Work Our method for the construction of the roedial axis is one of a number of algorithms that use the Voronoi graph of a set of sample points regularly spaced along the object s boundary to form an approximation of the skeleton. The main idea of such algorithms, examples of which include [2,3,4], is to first compute the Voronoi graph of the points, then extract a subgraph to form the skeleton. For example, the subgraph can be extracted by taking only the Voronoi vertices that are inside the boundary of the object. Given a model or an image of an object, there are two main approaches for ....

....operations typically use a global scale measure (e.g. size of smoothing kernel) to filter out noise, and smaller object features are often altered or destroyed during preprocessing. The other main approach is to start with the complete axis and prune the branches using some heuristic (e.g. [3,9]) The general idea is to have a significance neasure that assigns an importance value to each branch. During pruning, this value is compared to a user given threshold to determine how much of each branch gets cut. With an ideal significance measure and threshold, only the parts of the axis ....

Brandt, J., Algazi, V.: Continuous skeleton computation by voronoi diagram. CVGIP: Image Understanding 55 (1992) 329-337

....methods. Discrete methods work directly on binary images. The skeleton is a set of pixels that is computed using distance transforms [8,27] or morphological thinnings [19,20,22,23] Continuous methods are generally based on the Voronoi graph of a set of points located on the boundary of the object [4,7,9,24]. 10 simple point multiple point endpoint p 1 (s) s (s) p 0 Fig. 5. Terminology relative to skeletons. In this paper, we focus on the second family of methods. We use the approach proposed by Boissonnat [5] The input is a triangulated surface S which approaches the boundary of the ....

....of the skeleton. Di erent removing criteria have been proposed in the literature. One can for instance measure the di erence between the initial shape and the shape reconstructed from the simpli ed skeleton. Branches are shortened as long as this di erence remains smaller than a xed threshold [9]. More complex criteria may be found in [24] In the following, we will distinguish three types of skeleton points: simple points whose maximal sphere touches the object boundary on two contact points, endpoints whose maximal sphere touches the object boundary on one point and multiple points ....

J. W. Brandt and V. R. Algazi. Continuous skeleton computation by Voronoi diagram. CVGIP: Image Understanding, 55(3):329-337, 1992.

....X , we denote by Y e X one of its contact point other than X, and by j e X the minimum over all such points Y e X of kX GammaY e X k 2R e X . We borrow from Amenta and Bern [1] the notion of local feature size. A related notion is the r regularity introduced by Serra [18] see also [4, 7]) Denition 7 (Amenta Bern) The local feature size lfs(X) at a point X 2 S is the Euclidean distance from X to the medial axis of S. Lemma 9 For any X;Y 2 S, lfs(X) lfs(Y ) kX Gamma Y k: Proof. B(X; lfs(Y ) kX Gamma Y k) contains B(Y; lfs(Y ) Since, by denition of the local feature size, ....

....outside O. 4 Approximation of the medial axis In this section, we show that the centers of a subset of the Delaunay balls converge towards the medial axis when the sampling density increases. This is an extension to higher dimensions of a result proved in the plane by Schmitt [17] see also [7]) More precisely, Schmitt proved that, when tends to 0, the centers of all the Delaunay circles converge towards the medial axis of S. This result does not extend in higher dimensions. Indeed, Amenta, Bern and Eppstein have shown that, even in three dimensions, the centers of some Delaunay ....

J. W. Brandt and V. R. Algazi. Continuous skeleton computation by Voronoi diagram. CVGIP: Image Understanding, 55(3):329338, 1992.

....used, such general surface descriptions are not as useful in designing efficient exploratory procedures and obtaining easily segmented information. The skeletonization method that is used to extract the medial axis transform of feature data is taken from medial axis algorithms for 2D image data [5]. Traditionally, the boundaries are extracted from the image by edge detection and skeletons are generated directly from the boundary data. Much of the overhead of the boundary approach is in the detection of a boundary of the image and removal of excessive small branches due to noisy data [19] ....

....by tracing the feature boundary. Because the portion of local exploration, which is identified as a tracing phase, is well defined, the data taken during that phase can be separated from the rest of the tactile or position data. This creates a clean group of boundary samples. Brandt and Algazi [5] and Ogniewicz [31] have used approaches in which boundary points are used to find Voronoi diagrams in order to generate a discrete medial axis. The algorithm begins with the creation of a Voronoi diagram from the boundary data points. The Voronoi diagram is the partitioning of a plane with n ....

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J. W. Brandt and V. R. Algazi. Continuous skeleton computation by voronoi diagram. CVGIP: Image Understanding, 55(3):329--338, 1992.

....in areas which may be as diverse as, e.g. robotics or medical diagnosis and therapy planning. Theories and methodologies for the analysis and representation of planar shapes have reached a certain level of maturity. We may distinguish between contour based [1, 2, 3, 4, 5] and region based methods [6, 7, 8, 9, 10, 11, 12, 13], and combinations of both approaches [14] Still, longstanding theoretical problems like combining different scales of contour smoothing remain unsolved [4] and applications, in particular to nontechnical shapes, leave room for critique. The problems are compounded in 3D. Developments in 3D ....

....regions partition continuous 2D space or 3D space into a net of convex polygons (2D) resp. polyhedra (3D) which is nothing but the Voronoi diagram (VD) generated by the set of border points S [18] Efficient algorithms exist for exact as well as approximate computation of the Voronoi diagram [11, 12, 13, 23, 24, 25]. The VD of S shows a multitude of branches in 2D or laminae in 3D not normally associated with the concept of the MAT. This appearance is due to resolving the continuous macroscopic boundary into its atomic generators and their associated microscopic wave fronts. In order to develop the VD into ....

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J. W. Brandt and V. R. Algazi, Continuous Skeleton Computation by Voronoi Diagram, CVGIP: Image Understanding 55(3) p. 329--338, 1992.

....less precision than the global test, which requires evaluation of polynomials of degree 12 in the input coordinates. A second idea was that Quad Edges that failed the crust criterion were part of the skeleton or anti crust . This term was mentioned briefly in the conclusions of [ABE98] citing [RCGH92, BA92, O94]. This is based on the idea that the dual of a crust edge is a Voronoi edge that intersects the crust and has been rejected. The remaining Voronoi edges form a tree structure that extends towards the crust but does not cross it. Indeed, with the Quad Edge structure, each leaf of the skeleton ....

Brandt, J. and V.R. Algazi, 1992. Continuous skeleton computation by Voronoi diagram. Computer Vision, Graphics and Image Processing, v. 55, pp. 329-338.

....embodied in the skeleton space to automatically determine scale and degree of pruning is shown in Section 7. Results of the novel pruning scheme are depicted in Section 8. 2. Skeletons from Voronoi Diagrams Substantially distinct skeletonization algorithms have been proposed in literature [29, 26, 24, 17, 6]) The approach [15, 29, 26, 30] employed in this paper is based on the close relationship between the medial axis transform, distance transforms [41] DT) and the Voronoi diagram (VD) 34] of a shape s boundary points. Voronoi tessellation: Given a set of points (sites) Omega = fp i g in the ....

....= p i 2 Omega V (p i ) 2) Medial Axis Approximation from Voronoi Diagram. Let us assume that the (planar) shape S is sufficiently approximated by a polygon B(S) whose vertices b B(S) fp i g have been obtained by equidistant sampling of S with sampling density Delta. It can be shown [6] that as Delta 0, Vor( b B) becomes an increasingly precise approximation of the medial axis transform of S, and eventually becomes equivalent to the continuous MAT(S) In addition, Vor( b B) represents the one to one analogue of the grassfire [3] implemented for a discrete set of boundary ....

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J. W. Brandt and V. R. Algazi, Continuous skeleton computation by Voronoi diagram, Computer Vision, Graphics, and Image Processing, 55(3):329--338, 1992.

....the same class of homotopy. Different removing criteria have been proposed. One can for instance measure the difference between the initial shape and the shape reconstructed from the simplified skeleton. Branches are shortened as long as this difference remains smaller than a fixed threshold [2, 3]. More complex criteria may be found in [4] Existing methods left unresolved some crucial aspect of the simplification problem. They do not study the effect of noise on the skeleton. They depend on thresholds that are difficult to find automatically as they change with the objects. In this paper, ....

....The skeleton is a set of pixels that is computed using distance transforms or morphological thinnings [5, 6] Continuous methods are derived from computational geometry. They are generally based on the computation of the Voronoi graph of a set of points located on the boundary of the object [2, 7, 8]. In this paper, the skeleton is computed, using the continuous approach described in [2] The input is a set of points fp i g n i=1 located on the boundary of a smooth object X. The first step of the method consists in computing the Voronoi graph of the boundary points p i . The skeleton is ....

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J. W. Brandt and V. R. Algazi. Continuous skeleton computation by Voronoi diagram. CVGIP: Image Understanding, 55(3):329--337, 1992.

....is O(n log n) for n = jSj. 3 Previous Work Our work draws on a variety of sources. The closest line of research concerns shape recognition for computer vision. The emphasis there is on the closely related problem of estimating the medial axis from a set of boundary points. Brandt and Algazi [BA92] showed that the Delaunay triangulation of a sufficiently dense set of samples contains a reconstruction of the boundary as a subset of its edges (a slightly weaker version of our Theorem 12) Robinson, Colchester, Griffin and Hawkes [RCGH92] propose selecting the boundary reconstruction edges by ....

....r sampled, with r 1. 2 Theorem 12 Let F be an r sampled smooth curve in the plane, r 1. The Delaunay triangulation of the set S of samples contains an edge between every adjacent pair of samples. Proof: Implied immediately by Lemma 11 and the Empty Circle Property. 2 Note: Brandt and Algazi [BA92] also show that adjacent points on a densely sampled curve are separated by a Voronoi edge (the dual statement of Theorem 12) Let d be the minimum, over all points p 2 F , of LFS(p) Their sampling condition is that every point p must have a sample within distance d . The polygonal ....

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Brandt, J. and Algazi V.R., Continuous skeleton computation by Voronoi diagram, Computer Vision, Graphics and Image Processing 55 (3), (1992), pp. 329-338.

....feature size under which they show two graphs, crust and fi skeleton, coincide with G if the points are sufficiently sampled. Some of the other effective approaches include ff shapes by [6] which is analyzed later by [3] r regular shapes by [2] A shapes by [7] and a Delaunay based method by [4]. A survey of these methods appear in [5] In this paper we show that a modified nearest neighbor graph also coincides with G. The algorithm and its analysis are simple. Nevertheless, it improves the sampling density to 1=3 from 0:252 as required by [1] More importantly, the algorithm generalizes ....

J. Brandt and V. R. Algazi. Continuous skeleton computation by Voronoi diagram. Comput. Vision, Graphics, Image Process, 55 (1992), 329--338.

....an example of an original binary image and its outline loop. 3. 2 Scale space dominant points Scale space filtering 7 is used to construct a scale space map of the outline contour, by smoothing the outline curvatures with a digital Gaussian filter window 8 (with k=5) which is given by h[1] h[5] = 0.029412, h[2] h[4] 0.235294, h[3] 0.470588, where h[3] is the centre value and S k h[k] 1. The dominant points at each scale, that characterise the shape of the outline, are considered to lie at extrema of outline curvatures. These are extracted from the smoothed outlines by estimating ....

J. W. Brandt & V. R. Algazi, "Continuous Skeleton Computation by Voronoi Diagram", CVGIP:Image Understanding, Vol.55, No.3, pp.329-338, May 1992.

.... (X Phi rB0) Psi rB0 B(x; 2r) x r Figure 4: r regular shape The concept of r regular shapes was first introduced in the field of mathematical morphology [10] Since the parameter r can be arbitrarily small, r regular shapes offer a relevant model to represent objects arising in image analysis [11]. r regular shapes have many nice geometric properties (Fig. 4) 1. The boundary of an r regular shape has at each point a tangent and a radius of curvature greater or equal to r. 2. The boundary of an r regular shape divides any ball with radius 2r and center on the boundary in exactly two ....

J. W. Brandt and V. R. Algazi. Continuous skeleton computation by Voronoi diagram. CVGIP: Image Understanding, 55(3):329--337, 1992.

....The skeleton is redefined in the discrete space, and only discrete objects are handled [1, 7, 12, 15] ffl Semi continuous methods. The shape is discerned to us through a sampling of its boundary. The skeleton is approached by taking a subgraph of the Voronoi graph of the sampling boundary points [5, 6, 10]. ffl Exact methods. Unlike the other methods, the continuous shape is known and the exact skeleton is searched. So far, this problem has been solved for very few objects (polygons [9] ellipses) This work concerns the two last points. Firstly, the problem of computing the exact skeleton is ....

....computed. A collection of balls, circumscribed about the Delaunay simplexes, can be derived from this computation. Let Yw be the polyball formed by the above mentioned balls whose centers belong to X. The convergence of Yw to X is ensured if X is r regular. A proof in 2D space may be found in [6]. 3.2 Approximating continuous skeletons Assuming X to be regular enough (the boundary is C 3 ) Schmitt [14] has demonstrated that Delaunay balls tend to maximal balls, and thus, the skeleton of Yw tends to the skeleton of X. Note that the polyball Yw and its skeleton can easily be derived ....

J.W. Brandt and V.R. Algazi, "Continuous Skeleton Computation by Voronoi Diagram", CVGIP : Image Understanding, Vol. 55, No 3, pp. 329-337, 1992.

....in the case of Figure 7) and the Delaunay edge was a portion of the crust according to the criterion of [ABE98] A second idea was that Quad Edges that failed the crust criterion were part of the skeleton or anti crust . This term was mentioned briefly in the conclusions of [ABE98] citing [RCGH92, BA92, O94]. This is based on the idea that the dual of a crust edge was a Voronoi edge that intersected the crust and had been rejected. The remaining Voronoi edges formed a tree structure that extended towards the crust but did not cross it. Indeed, with the Quad Edge structure, each leaf of the ....

Brandt, J. and V.R. Algazi, 1992. Continuous skeleton computation by Voronoi diagram. Computer Vision, Graphics and Image Processing, v. 55, pp. 329-338.

....2 P initiate a concentric fire front. The locus where the fronts meet is then nothing but the Voronoi diagram (VD) of P , Vor(P ) Henceforth, the Voronoi diagram of a (discrete) set of boundary points will be equivalently termed the discrete Voronoi medial axis(DVMA) It has been shown formally [11] that the DVMA will become an increasingly precise approximation of the continuous MAT as the density of boundary samples is increased. However, also the DVMA is characterized by the typical deficiencies of the medial axis transform such as sensitivity to noise effects upon the boundary and ....

J. W. Brandt and V. R. Algazi, "Continuous skeleton computation by voronoi diagram," Computer Vision, Graphics, and Image Processing, vol. 55, no. 3, pp. 329--338, 1992.

....topologically, as well as geometrically correct skeletons is the reformulation of Blum s concept in a semi continuous way. Objects on the raster plane can be faithfully represented by their discrete boundary points, providing the densest possible sampling of their (unknown) continuous outline [16, 4, 19, 14]. The Voronoi Diagram of these boundary points is a superset of the MAT [4] and can be generated very efficiently [14] The Voronoi Diagram (VD) is a fundamental structure in computational geometry. It is defined on a set S of n points in d dimensional Euclidean space E d . The points of S are ....

....Blum s concept in a semi continuous way. Objects on the raster plane can be faithfully represented by their discrete boundary points, providing the densest possible sampling of their (unknown) continuous outline [16, 4, 19, 14] The Voronoi Diagram of these boundary points is a superset of the MAT [4] and can be generated very efficiently [14] The Voronoi Diagram (VD) is a fundamental structure in computational geometry. It is defined on a set S of n points in d dimensional Euclidean space E d . The points of S are called sites (sometimes other basic elements, e.g. polygons, are used as ....

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Brandt J.W. and Algazi V.R. Continuous Skeleton Computation by Voronoi Diagram. CVGIP:IU, 55(3):329--338, 1992.

....information. The next section explains how the geometric skeleton, which may initially include surface elements, is approximated by a set of polylines. 2. 2 Computation of the geometric skeleton The geometric skeleton of an object is computed using the continuous approach described in [BA92] The input is a set of points p i n i=1 located on the surface of the object. The output is a subgraph of the Voronoi graph of the boundary points p i . The Voronoi graph is a well known data structure in computational geometry [Aur91] In summary, the Voronoi graph of a set of points ....

....of the boundary points p i . The Voronoi graph is a well known data structure in computational geometry [Aur91] In summary, the Voronoi graph of a set of points (called seeds) divides the space into regions. Each region is the set of points closer to a particular seed than to any other seed. In [BA92] the geometric skeleton is approximated with the Voronoi elements completely included in the object (see figure 2) a) b) Figure 2: Relation between the Voronoi graph (a) and the geometric skeleton (b) of a set of point. The result is a thin shape made up of straight line segments in 2D and ....

J. W. Brandt and V. R. Algazi. Continuous skeleton computation by Voronoi diagram. CVGIP: Image Understanding, 55(3):329--337, 1992.

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Brandt, J. and V.R. Algazi, 1992. Continuous skeleton computation by Voronoi diagram. Computer Vision, Graphics and Image Processing, v. 55, pp. 329-338.

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J. W. Brandt and V. R. Algazi, Continuous skeleton computation by voronoi diagram, Computer Vision, Graphics, and Image Processing 55(3), 329--338 (1992).

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J. Brandt and V. R. Algazi. Continuous skeleton computation by Voronoi diagram. Comput. Vision, Graphics, Image Process, vol. 55 (1992), 329-338.

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J. W. Brandt and V. R. Algazi, Continuous skeleton computation by Voronoi diagram, Computer Vision, Graphics, and Image Processing 55(3), 329--338 (1992).

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J. Brandt and V. R. Algazi. Continuous skeleton computation by Voronoi diagram. Comput. Vision, Graphics, Image Process, vol. 55 (1992), 329-338.

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J. W. Brandt and V. R. Algazi, Continuous skeleton computation by voronoi diagram, Computer Vision, Graphics, and Image Processing 55(3), 329--338 (1992).

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