In the first part we study general properties of the metrics obtained by isometrically identifying a generic metric space with a subset of a Banach space; we obtain a rigidity result. We then discuss the Hausdorff distance, proposing some less-known but important results: a closed-form formula for geodesics; generically two compact sets are connected by a continuum of geodesics. In the second part we present and study a family of distances on the space of compact subsets of $\mathbb{R}^N$ (that we call “shapes''). These distances are “geometric,” that is, they are independent of rotation and translation, and the resulting metric spaces enjoy many interesting properties, as, for example, the existence of geodesics. We view our metric space of shapes as a subset of Banach (or Hilbert) spaces, so we can define a “tangent manifold” to shapes and (in a very weak form) talk of a “Riemannian geometry” of shapes. Some of the metrics that we propose are topologically equivalent to the Hausdorff distance.

In this paper we present a novel theory to analyze defocused images of a volume density by exploiting well-known results in Fourier analysis and the singular value decomposition. This analysis is fundamental in two respects: First, it gives a deep insight into the basic mechanisms of image formation of defocused images, and second, it shows how to incorporate additional a-priori knowledge about the geometry and photometry of the scene in restoration algorithms. For instance, we show that the case of a scene made of a single surface results in a simple constraint in the Fourier domain. We derive two basic types of algorithms for volumetric reconstruction: One based on a dense set of defocused images, and one based on a sparse set of defocused images. While the first one excels in simplicity, the second one is of more practical use. Both algorithms are tested on real and synthetic data.

We present a novel representation of shape for closed planar contours explicitly designed to possess a linear structure. This greatly simplifies linear operations such as averaging, principal component analysis or differentiation in the space of shapes. The representation relies upon embedding the contour on a subset of the space of harmonic functions of which the original contour is the zero level set.

Occlusions are commonplace in man-made and natural environments; they often result in photometric features where a line terminates at an occluding boundary, resembling a "T". We show that the 2-D motion of such T-junctions in multiple views carries nontrivial information on the 3-D structure of the scene and its motion relative to the camera. We show how the constraint among multiple views of T-junctions can be used to reliably detect them and differentiate them from ordinary point features. Finally, we propose an integrated algorithm to recursively and causally estimate structure and motion in the presence of T-junctions along with other point-features.