# 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.
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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.