Last update: June 26, 2002

BACK

Shape is multifaceted in that seemingly conflicting views are possible, i.e., as a contour vs a region, as a composition of parts vs a result of growth and deformations, as a global percept vs a composition of local features, etc.

This fascinating collage of views has led to a varied set of representations for shape. Yet, none of the singular approaches have proved sufficient for recognition and perception. We have argued that in a world of change, deformations play a key role in shape representation and to capture a rich space of deformations all of these views must be simultaneously captured [1].

We have developed such a framework by representing shape as the set of singularities (shocks) that arise in a rich space of shape deformations as classified into four types:

1. first-order shocks are orientation discontinuities (corners) and arise from protrusions and indentations;
2. second-order shocks are formed when a shape breaks into two parts during a deformation;
3. third-order shocks represent bends, and
4. fourth-order shocks are the seeds for each component of a shape.

Shocks are grouped and represented in a hierarchy of scale and suggest a perceptual organization for the shape space in terms of three components: parts, protrusions and bends, see Figure 1. We have shown that the perception of parts is consistent with the computational framework for computing them [2].

We are currently investigating how such representations may be captured from real gray scale images where figure-ground segregation is not trivial. Our approach is a simultaneous propagation of contour-based and region-based information. Orientation elements (edges) propagate and carry local information by launching wavefronts and form shocks during a collision with other wavefronts. Occlusions, spurious edges, parts, and gaps affect the underlying figure symmetries. Such transformations are detected by a classification of waves depending on the reliability of the orientation source, leading to three types of shocks, regular, degenerate and semi-degenerate. The labeling of shocks allows for the recovery of parts, removal of spurious edges, grouping of edge elements and bridging across gaps [3].

Figure 2 illustrates regular, semi-degenerate, and degenerate shocks in a Kanizsa figures, as depicted in green, yellow, and red, respectively. Regular shocks depict true symmetries while degenerate and semi-degenerate shocks necessitate transformations, e.g. observe six suggested groupings in this case.

The resulting shock hierarchies are represented as graphs and matched against a database of objects for image indexing. The computational framework provides a solid substrate to formulate and examine figure-ground segregation, perceptual grouping, notion of similarity, and formation of basic level categories.

1. B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker. "Shapes, shocks, and deformations, I: The components of shape and the reaction-diffusion space". International Journal of Computer Vision, 15:189-224, 1995.
    
2. K. Siddiqi, K. J. Tresness, and B. B. Kimia. "Parts of visual form: Ecological and psychophysical aspects". Perception, 25:399-424, 1996.
    
3. P. Stoll, H. Tek, and B. B. Kimia. "Shocks from images: Propagation of orientation elements". In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Puerto Rico, June 15-16 1997, IEEE Computer Society Press.
    
4. H. Tek, F. F. Leymarie and B. B. Kimia, "Interpenetrating Waves and Multiple Generation Shocks via the CEDT." In "Advances in Visual Form Analysis," World Scientific, pp. 582-593, 1997.
    
5. H. Tek, "The role of symmetry maps in representing objects in images," PhD thesis, Division of Engineering, Brown University, 2000.

More references under:

• http://www.lems.brown.edu/~tek/publications.html

Last updated: June 26, 2002
by F. Leymarie

Back to research topics.