Medial axis transform (MAT) is very sensitive to the noise, in the sense that, even if a shape is perturbed only slightly, the Hausdorff distance between the MATs of the original shape and the perturbed one may be large. But it turns out that MAT is stable, if we view this phenomenon with the one-sided Hausdorff distance, rather than with the two-sided Hausdorff distance. In this paper, we show that, if the original domain is weakly injective, which means that the MAT of the domain has no end point which is the center of an inscribed circle osculating the boundary at only one point, the one-sided Hausdorff distance of the original domain's MAT with respect to that of the perturbed one is bounded linearly with the Hausdorff distance of the perturbation. We also show by example that the linearity of this bound cannot be achieved for the domains which are not weakly injective. In particular, these results apply to the domains with the sharp corners, which were excluded in the past. One consequence of these results is that we can clarify theoretically the notion of extracting "the essential part of the MAT", which is the heart of the existing pruning methods.

Although useful in many applications, the medial axis transform (MAT) has a few fit-falls, one of which is its extreme sensitivity to the boundary perturbation. In this paper, we first summarizes the previous attempts to get around this by bounding the one-sided Hausdorff distance of the MAT with respect to the boundary perturbation. We illustrate these results and their optimality with various examples. Finally, we suggest an application of them in pruning. In particular, we discuss the advantage of the results for the domains which are not weakly injective, over those for the weakly injective ones.

In this thesis we aim to develop a framework for clustering trees and representing and learning a generative model of graph structures from a set of training samples. The approach is applied to the problem of the recognition and classification of shape abstracted in terms of its morphological skeleton. We make five contributions. The first is an algorithm to approximate tree edit-distance using relaxation labeling. The second is the introduction of the tree union, a representation capable of representing the modes of structural variation present in a set of trees. The third is an information theoretic approach to learning a generative model of tree structures from a training set. While the skeletal...

Summary. The medial axis of a geometric shape captures its connectivity. In spite of its inherent instability, it has found applications in a number of areas that deal with shapes. In this survey paper, we focus on results that shed light on this instability and use the new insights to generate simplified and stable modifications of the medial axis. 1

Abstract. Skeleton can be viewed as a compact shape representation in that the shape can be completely reconstructed form the skeleton. We present a novel method for skeleton pruning that is based on this fundamental skeleton property. We iteratively remove skeleton end braches with smallest relevance for shape reconstruction. The relevance of branches is measured as their contribution to shape reconstruction. The proposed pruning method allows us to overcome the instability of skeleton representation: a small boundary deformation leads to large changes in skeleton topology. Consequently, we are able to obtain very stable skeleton representation of planar shapes.

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.

Using the geometrical properties of medial axis of closed planar shapes, we propose a shape representation called chain of circles (CoCs), and show its use for shape matching and recognition. CoCs is directly extracted from a resulting axial configuration of shapes along boundary curves without any intermediate description like graph obtained from segmenting shapes, and has the form of vector, each of which components is related with the parallelness and relative thickness of local shapes, the structure of whole shapes, and so on, respectively. The dynamic programming technique is used to match two shapes represented by CoCs and produces relative distance vector between shapes. The distance vector facilitates recognition by efficient indexing and retrieval of shapes in image database. Through experiments we show that the proposed method can be effectively used not depending on shape class. 1 Introduction With increasing use of images by computer systems in the fie...

This thesis evaluates the suitability of Voronoi ball models (VBMs) as a multipurpose shape representation for applications in computer graphics, scientific visualization, and computer vision. The effectiveness of VBMs is judged with respect to six key properties, namely stability, flexibility, accuracy, complexity, efficiency, and intuitiveness. These properties have a significant impact on the range of applicability of a computational shape model. The ability of VBMs to support a number of core shape-driven operations, in particular shape extraction, simplification, matching, interpolation, manipulation, and surface reconstruction, is examined by determining the strength of the key properties in the representation. The general approach is to use VBMs in a number of representative applications, each requiring several of the shape operations being considered. These applications include image matching and interpolation, shape model extraction from image data, two and three-dimensional shape simplification, and polygonal surface reconstruc-tion. The performance of VBMs in these applications is indicative of the extent to which each key property is present. The results of the experiments are very positive. They indicate that a VBM-based shape similarity measure can be effectively applied to quantify 2D shape differences and solve the 2D/3D shape correspondence problem. The findings also show that the VBM and the medial axis can be used together to take advantage of their complementary properties; the VBM gives the medial axis greater stability, while the axis adds connectivity and topological information to the

(U) (U) In this paper, we present a semi-automatic scene analysis system. In most cases, object-related detectors are not able to find objects of interest reliably and efficiently on their own. We thus propose a scheme in which high-level knowledge may be introduced in an efficient way. This knowledge is used to steer the interpretation and to enforce a coherent result. The proposed scheme allows the user to intervene at any moment to help the interpretation by correcting errors, stopping the research in non-promising areas, etc. (U) The architecture of the presented system is generic and may be used for various applications. However, application related knowledge must be fed into the system. In this paper, we briefly describe the detectors, strategy and rules used in the scope of the SAHARA project. In this project, the interpretation of airports was considered and highresolution multi-sensor imagery was used (VIS 2m, SAR 5m, TIR 5m). In this scope objects of interest like runways, t...

ABSTRACT (U) (U) In this paper, we present a semi-automatic scene analysis system. In most cases, object-related detectors are not able to find objects of interest reliably and efficiently on their own. We thus propose a scheme in which high-level knowledge may be introduced in an efficient way. This knowledge is used to steer the interpretation and to enforce a coherent result. The proposed scheme allows the user to intervene at any moment to help the interpretation by correcting errors, stopping the research in non-promising areas, etc. (U) The architecture of the presented system is generic and may be used for various applications. However, application related knowledge must be fed into the system. In this paper, we briefly describe the detectors, strategy and rules used in the scope of the SAHARA project. In this project, the interpretation of airports was considered and highresolution multi-sensor imagery was used (VIS 2m, SAR 5m, TIR 5m). In this scope objects of interest like runways, taxiways, buildings and shelters are searched for. (U) The results presented are quite encouraging and show that, using the system, a sketch of the airport may easily be drawn. The most promising application of the system is probably in the field of change detection to which it can easily be adapted.