Shape complexity is a hard-to-quantify quality, mainly due to its relative nature. Biased by Euclidean thinking, circles are commonly considered as the simplest. However, their constructions as digital images are only approximations to the ideal form. Consequently, complexity orders computed in reference to circle are unstable. Unlike circles which lose their circleness in digital images, squares retain their qualities. Hence, we consider squares (hypercubes in Zn) to be the simplest shapes relative to which complexity orders are constructed. Using the connection between L∞ norm and squares we effectively encode squareness-adapted simplification through which we obtain multi-scale complexity measure, where scale determines the level of interest to the boundary. The emergent scale above which the effect of a boundary feature (appendage) disappears is related to the ratio of the contacting width of the appendage to that of the main body. We discuss what zero complexity implies in terms of information repetition and constructibility and what kind of shapes in addition to squares have zero complexity.

We present a novel perspective on shape characterization using the screened Poisson equation. We discuss that the effect of the screening parameter is a change of measure of the underlying metric space. Screening also indicates a conditioned random walker biased by the choice of measure. A continuum of shape fields is created by varying the screening parameter or, equivalently, the bias of the random walker. In addition to creating a regional encoding of the diffusion with a different bias, we further break down the influence of boundary interactions by considering a number of independent random walks, each emanating from a certain boundary point, whose superposition yields the screened Poisson field. Probing the screened Poisson equation from these two complementary perspectives leads to a high-dimensional hyperfield: a rich characterization of the shape that encodes global, local, interior, and boundary interactions. To extract particular shape information as needed in a compact way from the hyperfield, we apply various decompositions either to unveil parts of a shape or parts of a boundary or to create consistent mappings. The latter technique involves lower-dimensional embeddings, which we call screened Poisson encoding maps (SPEM). The expressive power of the SPEM is demonstrated via illustrative experiments as well as a quantitative shape retrieval experiment over a public benchmark database on which the SPEM method shows a high-ranking performance among the existing state-of-the-art shape retrieval methods.

This paper presents a new image segmentation framework which employs a shape prior in the form of an edge strength function to introduce a higher-level influence on the segmentation process. We formulate segmentation as the minimization of three coupled functionals, respectively, defining three processes: prior-guided segmentation, shape feature extraction and local deformation estimation. Particularly, the shape feature extraction process is in charge of estimating an edge strength function from the evolving object region. The local deformation estimation process uses this function to determine a meaningful correspondence between a given prior and the evolving object region, and the deformation map estimated in return supervises the segmentation by enforcing the evolving object boundary towards the prior shape.

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 stable properties the shape instead of inaccurately measured secondary details. The new representation does not suffer from the common instability problems of the 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 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.

We present a new axis-based shape representation scheme along with a matching framework to address the problem of generic shape recognition. The main idea is to define the relative spatial arrangement of local symmetry axes and their metric properties in a shape centered coordinate frame. The resulting descriptions are invariant to scale, rotation, small changes in viewpoint and articulations. Symmetry points are extracted from a surface whose level curves roughly mimic the motion by curvature. By increasing the amount of smoothing on the evolving curve, only those symmetry axes that correspond to the most prominent parts of a shape are extracted. The representation does not suffer from the common instability problems of the traditional connected skeletons. It captures the perceptual qualities of shapes well. 'Therefore finding the similarities and the differences among shapes becomes easier. The matching process gives highly successful results on a diverse database of 2D shapes.

An original technique, based on ridge point detection directly from gray scale fingerprint images, is proposed. Our method avoids serious problems that algorithms which perform binarization of fingerprint images have. Each step can be easily hardware implemented, allowing a relevant speed up of the whole process.

Motivated by a need to define an object-centered reference system determined by the most salient characteristics of the shape, many methods have been proposed, all of which directly or indirectly involve an axis about which the shape is locally symmetric. Recently, a function /spl upsi/, called "the edge strength function", has been successfully used to determine efficiently the axes of local symmetries of 2-d shapes. The level curves of /spl upsi/ are interpreted as successively smoother versions of the initial shape boundary. The local minima of the absolute gradient /spl par//spl nabla//spl upsi//spl par/ along the level curves of /spl upsi/ are shown to be a robust criterion for determining the shape skeleton. More generally, at an extremal point of /spl par//spl nabla//spl upsi//spl par/ along a level curve, the level curve is locally symmetric with respect to the gradient vector /spl nabla//spl upsi/. That is, at such a point, the level curve is approximately a conic section whose one of the principal axes coincides with the gradient vector. Thus, the locus of the extremal points of /spl par//spl nabla//spl upsi//spl par/ along the level curves determines the axes of local symmetries of the shape. In this paper, we extend this method to shapes of arbitrary dimension.