Kaleem Siddiqi

Shock graphs and shape matching

We have been developing a theory for the generic representation of 2-D shape, where structural descriptions are derived from the shocks (singularities) of a curve evolution process, acting on bounding contours. We now apply the theory to the problem of shape matching. The shocks are organized into a directed, acyclic shock graph, and complexity is managed by attending to the most significant (central) shape components first. The space of all such graphs is highly structured and can be characterized by the rules of a shock graph grammar. The grammar permits a reduction of a shock graph to a unique rooted shock tree. We introduce a novel tree matching algorithm which finds the best set of corresponding nodes between two shock trees in polynomial time. Using a diverse database of shapes, we demonstrate our systems performance under articulation, occlusion, and moderate changes in viewpoint.

TurboPixels: Fast Superpixels Using Geometric Flows

We describe a geometric-flow-based algorithm for computing a dense oversegmentation of an image, often referred to as superpixels. It produces segments that, on one hand, respect local image boundaries, while, on the other hand, limiting undersegmentation through a compactness constraint. It is very fast, with complexity that is approximately linear in image size, and can be applied to megapixel sized images with high superpixel densities in a matter of minutes. We show qualitative demonstrations of high-quality results on several complex images. The Berkeley database is used to quantitatively compare its performance to a number of oversegmentation algorithms, showing that it yields less undersegmentation than algorithms that lack a compactness constraint while offering a significant speedup over N-cuts, which does enforce compactness.

Hamilton-Jacobi Skeletons

The eikonal equation and variants of it are of significant interest for problems in computer vision and image processing. It is the basis for continuous versions of mathematical morphology, stereo, shape-from-shading and for recent dynamic theories of shape. Its numerical simulation can be delicate, owing to the formation of singularities in the evolving front and is typically based on level set methods. However, there are more classical approaches rooted in Hamiltonian physics which have yet to be widely used by the computer vision community. In this paper we review the Hamiltonian formulation, which offers specific advantages when it comes to the detection of singularities or shocks. We specialize to the case of Blums grassfire flow and measure the average outward flux of the vector field that underlies the Hamiltonian system. This measure has very different limiting behaviors depending upon whether the region over which it is computed shrinks to a singular point or a non-singular one. Hence, it is an effective way to distinguish between these two cases. We combine the flux measurement with a homotopy preserving thinning process applied in a discrete lattice. This leads to a robust and accurate algorithm for computing skeletons in 2D as well as 3D, which has low computational complexity. We illustrate the approach with several computational examples.

Flow-based fiber tracking with diffusion tensor and q-ball data: Validation and comparison to principal diffusion direction techniques

In this study, we evaluate the performance of a flow-based surface evolution fiber tracking algorithm by means of a physical anisotropic diffusion phantom with known connectivity. We introduce a novel speed function for surface evolution that is derived from either diffusion tensor (DT) data, high angular resolution diffusion (HARD) data, or a combined DT-HARD hybrid approach. We use the model-free q-ball imaging (QBI) approach for HARD reconstruction. The anisotropic diffusion phantom allows us to compare and evaluate the performance of different fiber tracking approaches in the presence of real imaging artifacts, noise, and subvoxel partial volume averaging of fiber directions. The surface evolution approach, using the full diffusion tensor as opposed to the principal diffusion direction (PDD) only, is compared to PDD-based line propagation fiber tracking. Additionally, DT reconstruction is compared to HARD reconstruction for fiber tracking, both using surface evolution. We show the potential for surface evolution using the full diffusion tensor to map connections in regions of subvoxel partial volume averaging of fiber directions, which can be difficult to map with PDD-based methods. We then show that the fiber tracking results can be improved by using high angular resolution reconstruction of the diffusion orientation distribution function in cases where the diffusion tensor model fits the data poorly.

Multiscale Medial Loci and Their Properties

Blums 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.

The Hamilton-Jacobi skeleton

The eikonal equation and variants of it are of significant interest for problems in computer vision and image processing. It is the basis for continuous versions of mathematical morphology, stereo, shape-from-shading and for recent dynamic theories of shape. Its numerical simulation can be delicate, owing to the formation of singularities in the evolving front, and is typically based or, level set methods. However there are more classical approaches rooted in Hamiltonian physics, which have received little consideration in computer vision. In this paper we first introduce a new algorithm for simulating the eikonal equation, which offers a number of computational and conceptual advantages over the earlier methods when it comes to shock tracking. Next, we introduce a very efficient algorithm for shock detection, where the key idea is to measure the net outward flux of a vector field per unit volume, and to detect locations where a conservation of energy principle is violated. We illustrate the approach with several numerical examples including skeletons of complex 2D and 3D shapes.

A shock grammar for recognition

We confront the theoretical and practical difficulties of computing a representation for two-dimensional shape, based on shocks or singularities that arise as the shapes boundary is deformed. First, we develop subpixel local detectors for finding and classifying shocks. Second, to show that shock patterns are not arbitrary but obey the rules of a grammar, and in addition satisfy specific topological and geometric constraints. Shock hypotheses that violate the grammar or are topologically or geometrically invalid are pruned to enforce global consistency. Survivors are organized into a hierarchical graph of shock groups computed in the reaction-diffusion space, where diffusion plays a role of regularization to determine the significance of each shock group. The shock groups can be functionally related to the objects parts, protrusions and bends, and the representation is suited to recognition: several examples illustrate its stability with rotations, scale changes, occlusion and movement of parts, even at very low resolutions.

Indexing using a spectral encoding of topological structure

In an object recognition system, if the extracted image features are multilevel or multiscale, the indexing structure may take the form of a tree. Such structures are not only common in computer vision, but also appear in linguistics, graphics, computational biology, and a wide range of other domains. In this paper, we develop an indexing mechanism that maps the topological structure of a tree into a low-dimensional vector space. Based on a novel eigenvalue characterization of a tree, this topological signature allows us to efficiently retrieve a small set of candidates from a database of models. To accommodate occlusion and local deformation, local evidence is accumulated in each of the trees topological subspaces. We demonstrate the approach with a series of indexing experiments in the domain of 2-D object recognition.

Hippocampal shape analysis using medial surfaces

In magnetic resonance imaging (MRI) research, significant attention has been paid to the analysis of the hippocampus (HC) within the medial temporal lobe because of its importance in memory and learning, and its role in neurodegenerative diseases. Manual segmentation protocols have established a volume decline in the HC in conjunction with Alzheimers disease, epilepsy, post-traumatic stress disorder, and depression. Furthermore, recent studies have investigated age-related changes of HC volume which show an interaction with gender; in early adulthood, volume reduction of the HC is found in men but not in women. In this paper, we investigated gender differences in normal subjects in young adulthood by employing a shape analysis of the HC using medial surfaces. For each subject, the most prominent medial manifold of the HC was extracted and flattened. The flattened sheets were then registered using both a rigid and a non-rigid alignment technique, and the medial surface radius was expressed as a height function over them. This allowed for an investigation of the association between subject variables and the local width of the HC. With regard to the effects of age and gender, it could be shown that the previously observed gender differences were mostly due to volume loss in males in the lateral areas of the HC head and tail. We suggest that the analysis of HC shape using medial surfaces might thus serve as a complimentary technique to investigate group differences to the established segmentation protocols for volume quantification in MRI.

PARTS OF VISUAL FORM : PSYCHOPHYSICAL ASPECTS

Part-based representations allow for recognition that is robust in the presence of occlusion, movement, growth, and deletion of portions of an object, and play an important role in theories of object categorization and classification. A partitioning theory for visual form is proposed that is based on two types of parts: limb-based parts arise from a pair of negative curvature minima with evidence for ‘good continuation’ of boundaries on one side; neck-based parts arise from narrowings in shape. The motivation for this model is computational requirements for recognition. The psychophysical relevance of this model is addressed by measuring intrasubject and intersubject consistency in partitioning tasks and comparing perceived and computed parts. A series of experiments were performed in which subjects were required to partition a variety of biological and nonsense two-dimensional shapes into perceived components. Specifically, it was examined (1) whether a subject determines components consistently across different trials of the same partitioning task, (2) whether there is evidence for consistency between subjects for the same partitioning task, and (3) how the perceived parts compare with limbs and necks resulting from the computational model. The results are interpreted as suggesting that there are high levels of both intrasubject and intersubject consistency and that a large majority of the perceived parts do in fact correspond to the parts computed on the basis of our model. The implications of our model are discussed in relation to previous experimental results. Intuitive observations concerning the relationship between parts of visual form and their function are then presented. Finally, a role is envisioned for parts in figure/ground segregation; the notion of a ‘parts receptive field’ through which parts can serve as an intermediate representation between local image features, eg edges, and global object models, is suggested.