Last update: Feb. 8, 2004

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Publications by Stephen M. Pizer* et al. on shape symmetry elicitation :

• Multiscale Medial Loci and Their Properties, 2003.
• Untangling the Blum Medial Axis Transform, 2003.
• Deformable M-Reps for 3D Medical Image Segmentation, 2003.
• Automatic and Robust Computation of 3D Medial Models Incorporating Object Variability, 2003.
• M-Reps: A New Object Representation for Graphics, 1999.
• Medial Node Models to identify and Measure Objects in Real-Time 3D Echocardiography, 1999.
• Marching optimal-parameter ridges: An algorithm to extract shape loci in 3D images, MICCAI'98.
• 3D/2D Registration via Skeletal Near Projective Invariance in Tubular Objects, MICCAI'98.
• Volume Rendering Guided by Multiscale Medial Models, 1997.
• Marching Cores: A Method for Extracting Cores from 3D Medical Images, 1996.
• Intensity Ridge & for Tubular Object Segmentation & Description, 1996.

BibTeX references.

Blum's medial axes have great strengths, in principle, in intuitively describing object shape in terms of a quasi-hierarchy 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 and 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.

Keywords: medial loci, multiscale, shape

For over 30 years, Blum's Medial Axis Transform (MAT) has proven to be an intriguing tool for analyzing and computing with form, 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.

Keywords: medial, medial analysis, form analysis, Blum MAT, object part hierarchies

M-reps (formerly called DSLs) are a multiscale medial means for modeling and rendering 3D solid geometry. They are particularly well suited to model anatomic objects and in particular to capture prior geometric information effectively in deformable models segmentation approaches. The representation is based on figural models, which define objects at coarse scale by a hierarchy of figures-each figure generally a slab representing a solid region and its boundary simultaneously. This paper focuses on the use of single figure models to segment objects of relatively simple structure. A single figure is a sheet of medial atoms, which is interpolated from the model formed by a net, i.e., a mesh or chain, of medial atoms (hence the name m-reps), each atom modeling a solid region via not only a position and a width but also a local figural frame giving figural directions and an object angle between opposing, corresponding positions on the boundary implied by the m-rep. The special capability of an m-rep is to provide spatial and orientational correspondence between an object in two different states of deformation. This ability is central to effective measurement of both geometric typicality and geometry to image match, the two terms of the objective function optimized in segmentation by deformable models. The other ability of m-reps central to effective segmentation is their ability to support segmentation at multiple levels of scale, with successively finer precision. Objects modeled by single figures are segmented first by a similarity transform augmented by object elongation, then by adjustment of each medial atom, and finally by displacing a dense sampling of the m-rep implied boundary. While these models and approaches also exist in 2D, we focus on 3D objects. The segmentation of the kidney from CT and the hippocampus from MRI serve as the major examples in this paper. The accuracy of segmentation as compared to manual, slice-by-slice segmentation is reported.

Keywords: segmentation, medial, deformable model, object, shape, medical image

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.

Keywords: medical imaging, shape analysis, Voronoi skeleton, medial shape description, skeleton pruning

A method for identifying image loci that can be used as a basis for object segmentation & image registration. The focus is on 1D & 2D shape loci in 3D images. This method, called marching ridges, uses generalized height ridges, oriented medialness measures and a marching cubes like algorithm to extract optimal scale-orientation cores. It can aslo be used of other tasks such as finding intensity skeletons of objects and identifying object boundaries.

Maximum Convexity Height Ridge (Eberly 1996):

• An m-D maximum convexity height ridge of a function M: X -> IR, or m-ridge of M, is an m-D locus in the n-D (n > m) space embedding M. In general, this definition involves:
  
  1. Given the Hessian matrix of 2nd derivatives H(M), re-arrange the Hessian's eigenvalues Li, with associated eigenvector Vi, in increasing order. Then, the first n-m such vectors are transverse directions.
     
  2. M has a relative maximum in each ot the transverse directions, i.e.:
     • < Vi , Grad(M) > (x) = 0 for i = 1,..., n-m
     • Ln-m < 0 at x.

Expanded upon the above, the authors propose the following - more general - new definition.

Height Ridge:

• An m-D height ridge of a function M: X -> IR, or m-ridge of M, is an m-D locus in the n-D (n > m) space embedding M. In general, this definition involves:
  
  1. A rule for choosing n-m linearly independent directions, Vi, transverse to the putative ridge at a location x .
     
  2. The requirement that M has a relative maximum in each ot the (1D) transverse directions, i.e.:
     • < Vi , Grad(M) > (x) = 0 for i = 1,..., n-m , and
     • (Vi)^t . H(M)(x) . Vi < 0 for i = 1,..., n-m .

Skeletons & edges finding require a choice of directions different than that of the maximal convexity definition (??? unclear why ??? not explained ... ).

Optimal Parameter Height Ridge (OPHR):

• An m-D optimal parameter height ridge of a function M: X × S -> IR, or m-ridge of M, is an m-D locus in the n-D (n > m) space embedding M, such that:
  
  1. Given the Hessian matrix of M restricted to the subspace S, Hs(M), let each of its eigenvectors be a transverse function, and have a rule for choosing the remaining transverse directions from X in a way possibly dependent on the eigenvectors of Hs(M).
     
  2. Require that M has a relative maximum in each ot the (1D) transverse directions, i.e.:
     • < Vi , Grad(M) > (x) = 0 for each Vi in X, the gradient of M restricted to S vanishes, and
     • (Vi)^t . H(M)(x) . Vi < 0 for each Vi in X, and Hs(M) is negative definite at x.

N.B.: For OPHR, each ridge point is a maximum of the function M when restricted to the space spanned by the directions transverse to the ridge.

Marching Ridges Algorithm

Marching ridge incorporates the strategies of both marching lines and cubes. Furthermore, beside identifying the zero-x of 1st derivatives, 2nd derivatives must also be checked (negative).

The skeletons of tubular anatomical structures (e.g. intracerebral blood vessels) are used as registration primitives. The method is based on the property of near projective invariance (of 3D space curves into 2D planar curves) found in tubular objects.

Projective invariance of a 3D tubular object

• A 3D tube is said to preserve projective invariance if the projection of the 3D skeleton is tha same as the 2D skeleton of the tube's projection (and vice-versa).

In practice, the tubular shape does not preserve this property all along its length, due to overlaps in the projection. There are 2 such types of overlaps:

1. Local overlap or self-occlusion: a contiguous portion of the same tube overlaps under projection.
2. Non-local overlap: protions of 2 distinct tubes overlap.

Registration algorithm

1. 3D skeletons are extracted from volume data (3D MRA), using core centers.
2. 2D skeletons are extracted from X-ray angiograms, also using cores.
3. Disparity/correspondence fields are computed between these 2 sets of skeletons, where the 3D skeletons are first projected

David Chen's PhD dissertation (html) on the same topic.

The Marching Cores algorithm generates cores of 3D medical images and also generalizes to finding implicitly defined manifolds of codimension greater than one. As we march along the core, we use medialness kernels to generate new medialness values and then find ridges in the extended medial space using the geometric definition of height ridges and mathematical models of manifold intersections. Results from both a test image and a CT image illustrate the algorithm.

Some illustrative results here.

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