Discrete Scale-Space Theory and the Scale-Space Primal Sketch
PhD thesis, Department of Numerical Analysis and Computing Science,
Royal Institute of Technology, S-100 44 Stockholm, Sweden, May 1991.
(ISRN KTH NA/P--91/8--SE)
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This thesis, within the subfield of computer science known as computer
vision, deals with the use of scale-space analysis in early low-level
processing of visual information.
The main contributions comprise the following five subjects:
The formulation of a scale-space theory for discrete signals.
Previously, the scale-space concept has been expressed for continuous
signals only. We propose that the canonical way to construct a scale-space
for discrete signals is by convolution with a kernel called the
discrete analogue of the Gaussian kernel, or equivalently by solving a
semi-discretized version of the diffusion equation.
Both the one-dimensional and two-dimensional cases are covered.
An extensive analysis of discrete smoothing kernels is carried out for
one-dimensional signals and the discrete scale-space properties of
the most common discretizations to the continuous theory are analysed.
A representation, called the scale-space primal sketch,
which gives a formal description of the hierarchical relations between
structures at different levels of scale. It is aimed at making information
in the scale-space representation explicit.
We give a theory for its construction and an
algorithm for computing it.
A theory for extracting significant image structures
and determining the scales of these structures
from this representation in a solely bottom-up data-driven way.
Examples demonstrating how such qualitative information extracted
from the scale-space primal sketch can be used
for guiding and simplifying other early visual processes.
Applications are given to edge detection, histogram analysis and
classification based on local features. Among other possible
applications one can mention perceptual grouping, texture
analysis, stereo matching, model matching and motion.
A detailed theoretical analysis of the evolution properties of critical
points and blobs in scale-space, comprising drift velocity estimates
under scale-space smoothing, a classification of the possible types
of generic events at bifurcation situations
and estimates of how the number of
local extrema in a signal can be expected to decrease as function
of the scale parameter.
For two-dimensional signals the generic bifurcation events are
annihilations and creations of extremum-saddle point pairs.
Interpreted in terms of blobs, these transitions correspond to
annihilations, merges, splits and creations.
Experiments on different types of real imagery demonstrate
that the proposed theory gives perceptually intuitive results.
tuning low-level processing,
classification of blob events,
density of local extrema,
digital signal processing
(Monograph on scale-space theory)
(Other publications on scale-space theory)
(Encyclopedia entry on scale-space theory)
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