H. Blum and R. N. Nagel. Shape description using weighted symmetric axis features. PR, 10:167--180, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... and perception of continuous symmetry) Symmetry in 2D can be discussed as a global feature where all points in the object contribute to determining the symmetry [116, 11, 49, 64, 24, 103] or as a local feature where every symmetry element is supported locally by some subset of the object [14, 77, 19, 18, 23, 11, 85]. The global symmetry methods are much more efficient in run time, usually having a linear time complexity however they are generally sensitive to noise and occlusion. The local symmetry methods are more robust to noise and occlusion, and they are easily parallelized, however they have high time ....

....in terms of shape recognition. Shape description using local symmetry, termed axial description of shape is a region based description which involves spines and curves for representing 2D shapes. It is discussed in [87] and is briefly reviewed in the following. Symmetry Axis Transform SAT In [14], the Symmetry Axis Transform SAT is presented (also termed Medial axis transform MAT) The SAT of a given 2D shape is the loci of the centers of all maximal b. c. a. Figure 2.5: Symmetry Axis Transform (SAT) a c) The SAT is the loci of all maximal disks enclosed in the shape (dashed ....

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H. Blum and R.N. Nagel. Shape description using weighted symmetric axis features. Pattern Recognition, 10:167--180, 1978.

.... efficiently find good approximate solutions and there exist several classes of graphs for which the problem can be solved in polynomial time [5] In many computer vision problems, the graphs at hand have a peculiar structure: they are connected and acyclic, i.e. they are free trees (see, e.g. [3], 16] 18] 26] Other application domains where free trees arise quite frequently are pattern recognition [9] and biochemistry [1] Note that, unlike rooted trees, in free trees there is no distinguished node playing the role of the root and, hence, no hierarchy is imposed on them. Since in ....

H. Blum and R.N. Nagel, "Shape Description Using Weighted Symmetric Axis Features," Pattern Recognition, vol. 10, pp. 167-180, 1978.

.... can eliminate unstable branches and yield object relevant medial loci [18] 28] 23] 22] Nevertheless, the boundary to medial transformation is inherently unstable; the resulting branching topology is sensitive to slight boundary perturbations, especially at the regions known as ligatures [4], 1] Whereas the above methods start with a boundary description and yield the medial locus, synthetic medial representations, such as the one presented in this paper, use the medial loci themselves as a model for object representation. The model describes the medial branching topology and ....

....that both the medial surface and the vector field must satisfy in order to produce a continuous boundary without singularities. A. Medial Geometry In this work we describe a continuous medial representation based on the medial loci described by Blum as the symmetric axis transform (SAT)[4]. The SAT is constructed as a locus of centers of maximal disks inscribed into a geometric object. The local thickness of the object is described by the radii of the disks. A synthetic continuous medial representation (cm rep) defines the medial locus of an object as a set of connected ....

H. Blum and R.N. Nagel. Shape description using weighted symmetric axis features. Pattern Recognition, 10(3):167--180, 1978.

....concepts needs to be revised, as advocated in the present work. Small modi cations on the Voronoi diagram de nition are proposed to properly extend it to continuous connected shapes, while preserving the equidistance property discussed on Section 2. The medial axis, originally proposed by Blum [1, 2, 3], in turn, is extended to general objects including open curves, eliminating the necessity of well de ned internal and external regions, allowing Blum s idea to be applied to more general data sets. Several implications of such results are discussed and illustrated. Ricardo Fabbri is grateful ....

....be de ned. discrete cases (e.g. Figure 1) the same result is obtained as from the classical de nition. Further, it will be shown in Section 4 that the proposed rede nition also allows Voronoi diagrams and Medial axes to be properly related. 3 The Medial Axis The medial (or symmetry) axis [1, 2, 3] is most often understood as a skeleton, a geometrical concept of deep in uence in image analysis, being widely used for shape analysis and feature extraction. It corresponds to the points locally symmetric to a shape, in the sense of being a stick or central structure (see Figure 8) Figure ....

H. Blum and R.N. Nagel, Shape description using weighted symmetric axis features, Pattern Recognition, 10:167-180, 1978.

....starting with preattentive grouping and reaching up to knowledge driven construction. A classification scheme is proposed for sorting the instances of grouping by domain and abstraction level. 3. 2 Axial representation of planar shape Well known axial shape descriptions have been defined by Blum [5], Brooks [9] and Brady [7] A thorough analysis and review of these has been given by Rosenfeld [39] this analysis was extended by Ponce [34] Hierarchical decomposition of shape descriptions is discussed in a recent paper [38] Recovery of 3 D shape from 2 D contours is addressed in [44] and in ....

....for the edge contours, and then use the B spline representations for computing symmetry descriptions. Closed and complete boundaries of objects or object parts, binary images, or line drawings have been (implicitly) assumed in most of the early works dealing with axial shape descriptions [5, 7, 32, 39], as well as in the results reported later [3, 4, 33, 34, 38, 40, 44] Much of the analysis is general and equally valid for incomplete contours. Many of the methods presented can also be extended to apply to real (imperfect) edge maps. However, in the papers themselves, this important step has ....

H. Blum and R.N. Nagel, Shape description using weighted symmetric axes features, Pattern Recognition, 10, 1978, 167-180.

....escape from local optima, they always returned a globally optimal solution. 1 Introduction Graph matching is a classic problem in computer vision and pattern recognition, instances of which arise in areas as diverse as object recognition, motion and stereo analysis [1] In many problems (e.g. [2, 11, 19]) the graphs at hand have a peculiar structure: they are connected and acyclic, i.e. they are free trees. Note that, unlike rooted trees, in free trees there is no distinguished node playing the role of the root, and hence no hierarchy is imposed on them. Standard graph matching techniques, such ....

H. Blum and R. N. Nagel. Shape description using weighted symmetric axis features. Pattern Recognition, 10:167--180, 1978.

.... proposed his Medial Axis Transform (MAT) as a representation that embodies the skeleton of an object as well as the width of the object at every point on the skeleton [Blum67] This work has spawned a tremendous amount of research into the use of the Blum MAT and other skeleton representations [Blum78] [Brady84] Bruce85] Ogniewicz92] Pizer87] Pizer98] Szkely96] A goal of much of this work has been to create a form representation that defines a natural decomposition of an object into a set of basic parts that mirrors the object parts we perceive. At the same time, these representations are ....

Blum, Harry, and Roger N. Nagel, Shape Description Using Weighted Symmetric Axis Features, Pattern Recognition, 10:167-180 (1978)

....do not assume a global structure to the tubular network being extracted. The topologies of brain, lung, and liver vascular networks have too much interpatient variability to be fit by global models.Explicit centerline extraction has existed for decades as a basis of tubular object modeling [1] [10], 11] 1) Explicit centerline extraction is the basis of our approach to tubular object modeling [12] Our method starts from an initial point on or near a vessel and subsequently performs a multiscale extraction of the vessel s centerline via ridge traversal and radius estimation. The ....

H. Blum and R. N. Nagel, "Shape description using weighted symmetric axis features," Pattern Recogn., vol. 10, pp. 167--180, 1978.

....for the figure, inducing a magnificationinvariant geometry that provides helpful local shape characterizations. 2 Shape Representation with M rep Models The medial axis transform was first introduced in 2D and 3D by Blum [2] and its mathematical properties were developed in 2D by Blum Nagel [3] and in 3D by Nackman [7] A continuous medial axis, defined precisely from the boundary of an object, is complex and di#cult to represent or manipulate exactly. Also, small boundary perturbations can result in changes in the topology of the medial axis. M rep models instead describe shape through ....

Blum, H., Nagel, R.N.: Shape Description Using Weighted Symmetric Axis Features. Pattern Recognition, 10 (1978) 167--180.

....relevant features of the character for optical character recognition [5, 6] The objective of skeletonization is to find the medial axis of a character. Ideally, the medial axis is defined as a smooth curve (or set of curves) that follows the shape of a character equidistantly from its contours [7]. In case of hand written characters, one can also define the medial axis as the trajectory of the penstroke that created the letter. Most skeletonization algorithms approximate the medial axis by a unit width binary image. In one of the most widely used strategies, this binary image is obtained ....

....we found that if two penstrokes cross each other at a sharp angle, the thinning procedure tends to create two star3 vertices connected by a short simple path rather then a star4vertex. To detect these situations, we analyzed the behavior of the theoretical medial axis defined by Blum and Nagel [7] at the crossing point of two ideal strokes (Figure 12) It can be determined 16 Figure 11: Removing short loops. Skeleton graphs before (top row) and after (bottom row) the removal. analytically that the length of the short segment is (a;t) 1 if a 60 cosa sinacos if ....

H. Blum and R. Nagel, "Shape description using weighted symmetric axis features," Pattern Recognition, vol. 10, pp. 167--180, 1978.

....its symmetry. The results could be applied in object recognition, visual inspection, shape representation, etc. These studies have initially focused on the basic reflectional and rotational symmetry types, then on generalized and refined concepts such as the medial axis transform, skeletons [6, 8, 10, 11, 16, 29, 33, 34], ribbons and skewed symmetry [9, 15, 36, 44] 1 The input for algorithms that quantify or describe symmetry in shapes has normally been assumed to be the output of a successful segmentation procedure. However, a general purpose segmentation algorithm that consistently provides reliable results ....

H. Blum and R.N. Nagel, "Shape Description Using Weighted Symmetric Axis Features", Pattern Recognition, Vol. 10, pp. 167-180, 1978.

....617] Second, the normal velocity may become singular as well. To understand this, note that g 1 = g 2 implies 1 = 2 , using (34) From Theorem 1, we then find that 0 and 0 when g 1 = g 2 0. While this is not a singularity in the skeleton, it is in the boundary: 1 = 2 Gamma1. Blum [3] called such portions of skeleton full ligature: a non zero length of skeleton corresponds to exactly two concave corners in the boundary. Full ligature can occur when 0 as well, as seen in Fig. 4 and Movie 3. However, if only one of g 1 or g 2 approach 0, then jj 1. This new pathology ....

....an alternative derivation of T i = t C i s i . This requires that we study s i and its derivatives. To ensure that the arc length parameter s i runs counter clockwise along the boundary, we note: s i = Upsilon Z s 0 jC 0 i (oe)jdoe; and so we define the boundary axis ratio [3] along the boundary at C i as: g i : Upsilon s i s = jC 0 i j 0: 25) To compute the time derivative of the boundary axis ratio, observe that: g 2 i t = 2C 0 i Delta C 0 i = Upsilon2g i T i Delta ( i N i ) 0 = Upsilon2g i T i Delta ( 0 i N i i N 0 i ) ....

H. Blum and R. Nagel, "Shape description using weighted symmetric axis features," Patt. Recogn. 10 (1978), pp. 167--180.

.... using transform based techniques similar to those used in block based methods [110] 113] Typically, the shape information is represented by bitmap coding [11] 16] 91] chain coding of the contour information [95] quad tree shape representation [96] 97] or the medial axis transform [94] [98], 99] Simulation results show that shape coding requires an important portion of the global bit rate. One solution to reduce this cost is to use more efficient techniques for shape representation, such as the geodesic morphological skeleton as proposed by Brigger et al. 99] and or to perform a ....

H. Blum and R. N. Nagel, "Shape description using weighted symmetric axis features," Pattern Recognition, vol. 3, pp. 167--180, 1978.

....never perfect, one should focus on appearance; on the other hand, the organizational question seeks abstractions that can be organized into hierarchies to support search. Appearance representations [22] it is therefore widely held, should be intensity based, while abstractions such as skeletons [6, 24], shocks [17] and part primitives [4] from which organizational hierarchies can be built, are in practice not computable. A consequence of this tension is the shape dilemma: computable representations do not support shape based indexing abstractions; and structural abstractions are not ....

....skeleton in this example, and that this principle is fundamentally related to abstractions of shape. The basic idea is illustrated in Fig. 2, where we introduce the notion of ligature 1 as the glue that holds the limb onto the rectangle. While the formal definitions 1 After discovery by Blum [6], ligature was next used in [1] Tek et al. 32] have also used ligature, although 2 = Figure 2: Shape decomposition via ligature. A horizontal sausage and a vertical rectangle with their skeletons (top left) are combined to form a single whole (top right) Observe the formation of the ....

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H. Blum and R. N. Nagel. Shape description using weighted symmetric axis features. PR, 10:167--180, 1978.

....The Axial Shape Graph is based on an axial object description composed of linear axis segments that correspond to the fundamental parts of the object. There is a vast amount of literature on axial representations (often called skeletons) of 2D images, originating with the work of Blum [3]. In this section we will therefore not attempt to give a complete review, but instead describe the work most relevant to our approach. Many of the skeleton construction methods are based on one of several general approaches: iterative or parallel thinning [23] analytical calculation of the ....

.... skeleton construction methods are based on one of several general approaches: iterative or parallel thinning [23] analytical calculation of the medial axis (Delaunay triangulation and calculation of Voronoi regions [26, 21] and calculation of a Distance Map and its Medial Axis Transform (MAT) [3, 13, 1, 25, 36, 30, 22, 33, 12, 27]. Our implementation is of the last type, and thus this section focuses on MAT based methods. MAT based methods belong to a class of approaches that represent a shape by a spine and a geometric primitive [31] that moves along the spine sweeping out the shape, possibly changing its size as it ....

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H. Blum and R.N. Nagel. Shape description using weighted symmetric axis features. Pattern Recognition, 10:167--180, 1978.

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H. J. Blum and R. N. Nagel, Shape description using weighted symmetric axis features, Pattern Rec. 10 (1978) 167--180.

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H. Blum and R. N. Nagel. Shape description using weighted symmetric axis features. PR, 10:167--180, 1978.

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H. Blum and R.N. Nagle, "Shape Description using Weighted Symmetric Axis Features", Pattern Recognition, Vol. 10, pp. 167-180, 1978.

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H. Blum and R.N. Nagle, "Shape Description using Weighted Symmetric Axis Features", Pattern Recognition, Vol. 10, pp. 167-180, 1978.

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H. Blum and R. N. Nagel, Shape description using weighted symmetric axis features. Pattern Recognition, 10:167--180, 1978.

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H. Blum and R. Nagel, Shape description using weighted symmetric axis features, Pattern Recognition 10, 167--180 (1978).

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H. Blum and R.N. Nagel. Shape description using weighted symmetric axis features. Pattern Recognition, 10:167180, 1978.

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H. Blum and R.N. Nagel. Shape description using weighted symmetric axis features. Pattern Recognition, 10:167--180, 1978.

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H. Blum and R. Nagel, Shape description using weighted symmetric axis features, Pattern Recognition, vol. 10, pp. 167180, 1978.

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#33# H. Blum and R. Nagel. Shape description using weighted symmetric axis features. Pattern Recognition,

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