Invariancy Methods for Points, Curves and Surfaces in Computational Vision

Author: Kalle Åström

Supervisor: Gunnar Sparr

Bibtex: @PHDTHESIS{kalle-avhandling, AUTHOR = {{\AA}str{\"o}m, K.}, TITLE = {Invariancy Methods for Points, Curves and Surfaces in Computational Vision}, SCHOOL = {Department of mathematics, Lund University, Sweden}, YEAR = {1996} }

Abstract: Many issues in computational vision can be understood from the interplay between camera geometry and the structure of images and objects. Typically, the image structure is available and the goal is to reconstruct object structure and camera geometry. This is often difficult due to the complex interdependence between these three entities. % The theme of this thesis is to use invariants to solve these and other problems of computational vision. Two types of invariancies are discussed; view-point invariance and object invariance. A view-point invariant does not depend on the camera geometry. The classical cross ratio of four collinear points is a typical example. A number of invariants for planar curves are developed and discussed. View-point invariants are useful for many purposes, for example to solve recognition problems. This idea is applied to navigation of laser guided vehicles and to the recognition of planar curves. An object invariant does not depend on the object structure. The epipolar constraint is a typical example. The epipolar constraint is generalised in several directions. Multilinear constraints are derived for both continuous and discrete time motion. Similar constraints are used to solve navigation problems. Generalised epipolar constraints are derived for curves and surfaces. The invariants are based on pure geometrical properties. To apply these ideas to real images it is necessary to consider practical issues such as noise. Stochastic properties of low-level vision are investigated to give guidelines for design of practical algorithms. A theory for interpolation and scale-space smoothing is developed. The resulting low-level algorithms, for example edge-detection and correlation, are invariant with respect to the position of the discretisation grid. The ideas are useful in order to understand existing algorithms and to design new ones.