Discrete ScaleSpace Theory and the ScaleSpace Primal Sketch
Tony Lindeberg
PhD thesis, Department of Numerical Analysis and Computing Science,
Royal Institute of Technology, S100 44 Stockholm, Sweden, May 1991.
(ISRN KTH NA/P91/8SE)
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Abstract
This thesis, within the subfield of computer science known as computer
vision, deals with the use of scalespace analysis in early lowlevel
processing of visual information.
The main contributions comprise the following five subjects:

The formulation of a scalespace theory for discrete signals.
Previously, the scalespace concept has been expressed for continuous
signals only. We propose that the canonical way to construct a scalespace
for discrete signals is by convolution with a kernel called the
discrete analogue of the Gaussian kernel, or equivalently by solving a
semidiscretized version of the diffusion equation.
Both the onedimensional and twodimensional cases are covered.
An extensive analysis of discrete smoothing kernels is carried out for
onedimensional signals and the discrete scalespace properties of
the most common discretizations to the continuous theory are analysed.

A representation, called the scalespace 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 scalespace 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 bottomup datadriven way.

Examples demonstrating how such qualitative information extracted
from the scalespace 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 scalespace, comprising drift velocity estimates
under scalespace 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 twodimensional signals the generic bifurcation events are
annihilations and creations of extremumsaddle 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.
Keywords:
computer vision,
lowlevel processing,
scalespace,
diffusion,
Gaussian filtering,
discrete smoothing,
primal sketch,
segmentation,
descriptive elements,
scale detection,
image structure,
focusofattention,
tuning lowlevel processing,
blob detection,
edge detection,
edge focusing,
histogram analysis,
junction classification,
perceptual grouping,
texture analysis,
critical points,
classification of blob events,
bifurcations,
drift velocity,
density of local extrema,
multiscale representation,
digital signal processing
Further reading:
(Monograph on scalespace theory)
(Other publications on scalespace theory)
(Encyclopedia entry on scalespace theory)