Ligature instabilities in the perceptual organization of shape (1999)

by J August, K Siddiqi, S W Zucker
Venue:Comput. Vis. Image Underst
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Statistical Shape Analysis of Neuroanatomical Structures Based on Medial Models

by M. Styner, G. Gerig, J. Lieberman, D. Jones, D. Weinberger - Medical Image Analysis (MEDIA
"... Knowledge about the biological variability of anatomical objects is essential for statistical shape analysis and discrimination between healthy and pathological structures. This paper describes a novel approach that incorporates the variability of an object population into the generation of a charac ..."
Abstract - Cited by 64 (13 self) - Add to MetaCart
Knowledge about the biological variability of anatomical objects is essential for statistical shape analysis and discrimination between healthy and pathological structures. This paper describes a novel approach that incorporates the variability of an object population into the generation of a characteristic 3D shape model. The proposed shape representation is a coarse-scale sampled medial description derived from a fine-scale spherical harmonics (SPHARM) boundary description. This medial description is composed of a net of medial samples (m-rep) with fixed graph properties. The medial model is computed automatically from a predefined shape space using pruned 3D Voronoi skeletons. A new method determines the stable medial branching topology from the shape space. An intrinsic coordinate system and an implicit correspondence between shapes is defined on the medial manifold. Several studies of biological structures clearly demonstrate that the novel representation has the promise to describe shape changes in a natural and intuitive way. A new medial shape similarity study of group di#erences between Monozygotic and Dizygotic twins in lateral ventricle shape demonstrates the meaningful and powerful representation of local and global form.

Multiscale Medial Loci and Their Properties

by Stephen M. Pizer, Kaleem Siddiqi, Gabor Székely, James N. Damon, Steven W. Zucker
"... Blum's medial axes have great strengths, in principle, in intuitively describing object shape in terms of a quasihierarchy of figures. But it is well known that, derived from a boundary, they are damagingly sensitive to detail in that boundary. The development of notions of spatial scale has le ..."
Abstract - Cited by 60 (8 self) - Add to MetaCart
Blum's medial axes have great strengths, in principle, in intuitively describing object shape in terms of a quasihierarchy of figures. But it is well known that, derived from a boundary, they are damagingly sensitive to detail in that boundary. The development of notions of spatial scale has led to some definitions of multiscale medial axes different from the Blum medial axis that considerably overcame the weakness. Three major multiscale medial axes have been proposed: iteratively pruned trees of Voronoi edges [Ogniewicz 1993, Székely 1996, Näf 1996], shock loci of reaction-diffusion equations [Kimia et al. 1995, Siddiqi & Kimia 1996], and height ridges of medialness (cores) [Fritsch et al. 1994, Morse et al. 1993, Pizer et al. 1998]. These are different from the Blum medial axis, and each has different mathematical properties of generic branching and ending properties, singular transitions, and geometry of implied boundary, and they have different strengths and weaknesses for computing object descriptions from images or from object boundaries. These mathematical properties and computational abilities are laid out and compared and contrasted in this paper.

Path similarity skeleton graph matching

by Xiang Bai, Longin Jan Latecki - IEEE TRANS. PAMI , 2008
"... This paper proposes a novel graph matching algorithm and applies it to shape recognition based on object silhouettes. The main idea is to match skeleton graphs by comparing the geodesic paths between skeleton endpoints. In contrast to typical tree or graph matching methods, we do not consider the to ..."
Abstract - Cited by 53 (8 self) - Add to MetaCart
This paper proposes a novel graph matching algorithm and applies it to shape recognition based on object silhouettes. The main idea is to match skeleton graphs by comparing the geodesic paths between skeleton endpoints. In contrast to typical tree or graph matching methods, we do not consider the topological graph structure. Our approach is motivated by the fact that visually similar skeleton graphs may have completely different topological structures. The proposed comparison of geodesic paths between endpoints of skeleton graphs yields correct matching results in such cases. The skeletons are pruned by contour partitioning with Discrete Curve Evolution, which implies that the endpoints of skeleton branches correspond to visual parts of the objects. The experimental results demonstrate that our method is able to produce correct results in the presence of articulations, stretching, and contour deformations.

Automatic and robust computation of 3d medial models incorporating object variability

by Martin Styner, Guido Gerig, Sarang Joshi, Stephen Pizer - International Journal of Computer Vision , 2003
"... Abstract. This paper presents a novel processing scheme for the automatic and robust computation of a medial shape model, which represents an object population with shape variability. The sensitivity of medial descriptions to object variations and small boundary perturbations are fundamental problem ..."
Abstract - Cited by 35 (11 self) - Add to MetaCart
Abstract. This paper presents a novel processing scheme for the automatic and robust computation of a medial shape model, which represents an object population with shape variability. The sensitivity of medial descriptions to object variations and small boundary perturbations are fundamental problems of any skeletonization technique. These problems are approached with the computation of a model with common medial branching topology and grid sampling. This model is then used for a medial shape description of individual objects via a constrained model fit. The process starts from parametric 3D boundary representations with existing point-to-point homology between objects. The Voronoi skeleton of each sampled object boundary is partitioned into non-branching medial sheets and simplified by a novel pruning algorithm using a volumetric contribution criterion. Using the surface homology, medial sheets are combined to form a common medial branching topology. Finally, the medial sheets are sampled and represented as meshes of medial primitives. Results on populations of up to 184 biological objects clearly demonstrate that the common medial branching topology can be described by a small number of medial sheets and that even a coarse sampling leads to a close approximation of individual objects.
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Medial models incorporating object variability for 3D shape analysis

by Martin Styner, Guido Gerig - Proc. IPMI, UC Davis , 2001
"... Knowledge about the biological variability of anatomical objects is essential for statistical shape analysis and discrimination between healthy and pathological structures. This paper describes a novel approach that incorporates variability of an object population into the generation of a charac ..."
Abstract - Cited by 34 (2 self) - Add to MetaCart
Knowledge about the biological variability of anatomical objects is essential for statistical shape analysis and discrimination between healthy and pathological structures. This paper describes a novel approach that incorporates variability of an object population into the generation of a characteristic 3D shape model. The proposed shape representation is based on a fine-scale spherical harmonics (SPHARM) boundary description and a coarse-scale sampled medial description. The medial description is composed of a net of medial samples (m-rep) with fixed graph properties. The medial model is computed automatically from a predefined shape space using pruned 3D Voronoi skeletons to determine the stable medial branching topology. An intrinsic coordinate system and an implicit correspondence between shapes is defined on the medial manifold.
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Untangling the Blum Medial Axis Transform

by Robert A. Katz, Stephen M. Pizer - International Journal of Computer Vision , 2001
"... Submitted for IJCV special issue on UNC MIDAG For over 30 years, Blum’s Medial Axis Transform (MAT) has proven to be an intriguing tool for analyzing and computing with shape, but it is one that is notoriously difficult to apply in a robust and stable way. It is well documented how a tiny change to ..."
Abstract - Cited by 25 (3 self) - Add to MetaCart
Submitted for IJCV special issue on UNC MIDAG For over 30 years, Blum’s Medial Axis Transform (MAT) has proven to be an intriguing tool for analyzing and computing with shape, but it is one that is notoriously difficult to apply in a robust and stable way. It is well documented how a tiny change to an object’s boundary can cause a large change in its MAT. There has also been great difficulty in using the MAT to decompose an object into a hierarchy of parts reflecting the natural parts-hierarchy that we perceive. This paper argues that the underlying cause of these problems is that medial representations embody both the substance of each part of an object and the connections between adjacent parts. A small change in an object’s boundary corresponds to a small change in its substance but may involve a large change in its connection information. The problems with Blum’s MAT are generated because it does not explicitly represent this dichotomy of information. To use the Blum MAT to it’s full potential, this paper presents a method for separating the substance and connection information of an object. This provides a natural parts-hierarchy while eliminating instabilities due to small boundary changes. The method also allows for graded, fuzzy classifications of object parts to match the ambiguity in human perception of many objects.

Continuous medial representations for geometric object modeling

by Paul Yushkevich, P. Thomas Fletcher, Sarang Joshi, Andrew Thall, Stephen M. Pizer - in 2D and 3D”, Image and Vision Computing , 2003
"... We describe a novel continuous medial representation for object geometry and a deformable templates method for fitting the representation to images. Our representation simultaneously describes the boundary and medial loci of geometrical objects, always maintaining Blum’s symmetric axis transform (SA ..."
Abstract - Cited by 23 (6 self) - Add to MetaCart
We describe a novel continuous medial representation for object geometry and a deformable templates method for fitting the representation to images. Our representation simultaneously describes the boundary and medial loci of geometrical objects, always maintaining Blum’s symmetric axis transform (SAT) relationship. Cubic b-splines define the continuous medial locus and the associated thickness field, which in turn generate the object boundary. We present geometrical properties of the representation and derive a set of constraints on the b-spline parameters. The 2D representation encompasses branching medial loci; the 3D version can model objects with a single medial surface, and the extension to branching medial surfaces is a subject of ongoing research. We present preliminary results of segmenting 2D and 3D medical images. The representation is ultimately intended for use in statistical shape analysis.
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From Skeletons to Bone Graphs: Medial Abstraction for Object Recognition

by Diego Macrini
"... Medial descriptions, such as shock graphs, have gained significant momentum in the shape-based object recognition community due to their invariance to translation, rotation, scale and articulation and their ability to cope with moderate amounts of within-class deformation. While they attempt to deco ..."
Abstract - Cited by 21 (1 self) - Add to MetaCart
Medial descriptions, such as shock graphs, have gained significant momentum in the shape-based object recognition community due to their invariance to translation, rotation, scale and articulation and their ability to cope with moderate amounts of within-class deformation. While they attempt to decompose a shape into a set of parts, this decomposition can suffer from ligature-induced instability. In particular, the addition of even a small part can have a dramatic impact on the representation in the vicinity of its attachment. We present an algorithm for identifying and representing the ligature structure, and restoring the nonligature structures that remain. This leads to a bone graph, a new medial shape abstraction that captures a more intuitive notion of an object’s parts than a skeleton or a shock graph, and offers improved stability and within-class deformation invariance. We demonstrate these advantages by comparing the use of bone graphs to shock graphs in a set of view-based object recognition and pose estimation trials. 1.
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On the Evolution of the Skeleton

by Jonas August, Allen Tannenbaum, Steven W. Zucker , 1999
"... It is commonly held that skeletons are sensitive to noise. It is also believed that smoothing, typically invoked to combat noise, obeys the causality principle that no new structures are created via smoothing. We demonstrate that both views are incorrect. We characterize how smooth points of the ske ..."
Abstract - Cited by 21 (2 self) - Add to MetaCart
It is commonly held that skeletons are sensitive to noise. It is also believed that smoothing, typically invoked to combat noise, obeys the causality principle that no new structures are created via smoothing. We demonstrate that both views are incorrect. We characterize how smooth points of the skeleton evolve under a general boundary evolution, with the corollary that, when the boundary is smoothed by a geometric heat equation, the skeleton evolves according to a related geometric heat equation. The surprise is that, while certain aspects of the skeleton simplify, as one would expect, others can behave wildly, including the creation of new skeleton branches. Fortunately such sections can be flagged as ligature, or those portions of the skeleton related to shape concavities. Our analysis also includes junctions and an explicit model for boundary noise. Provided a smoothness condition is met, the skeleton can often reduce noise. However, when the smoothness condition is violated, the skeleton can change violently, which, we speculate, corresponds to situations in which "parts" are created, e.g., when the handle appears on a rotating cup.

Disconnected Skeleton: Shape at its Absolute Scale

by Cagri Aslan, Aykut Erdem, Erkut Erdem, Sibel Tari , 2007
"... We present a new skeletal representation along with a matching framework to address the deformable shape recognition problem. The disconnectedness arises as a result of excessive regularization that we use to describe a shape at an attainably coarse scale. Our motivation is to rely on the stable pro ..."
Abstract - Cited by 20 (5 self) - Add to MetaCart
We present a new skeletal representation along with a matching framework to address the deformable shape recognition problem. The disconnectedness arises as a result of excessive regularization that we use to describe a shape at an attainably coarse scale. Our motivation is to rely on the stable properties of the shape instead of inaccurately measured secondary details. The new representation does not suffer from the common instability problems of traditional connected skeletons, and the matching process gives quite successful results on a diverse database of 2D shapes. An important difference of our approach from the conventional use of the skeleton is that we replace the local coordinate frame with a global Euclidean frame supported by additional mechanisms to handle articulations and local boundary deformations. As a result, we can produce descriptions that are sensitive to any combination of changes in scale, position, orientation and articulation, as well as invariant ones.