@PhdThesis{bonmassar1997:exponential_phd,
author = {Bonmassar, Giorgio},
title = {The exponential chirp transform for log-polar
sampled images},
school = {Boston University},
year = 1997,
address = {Boston, MA},
abstract = {Vision architectures which are space-variant, i.e.
the local image resolution is a function of space,
are of great importance in both biological and
machine vision. In the present study we focus on a
particular map, the log-polar map. According to this
map function, objects located in the fovea are
sampled with higher frequency, compared to those
present in the periphery. When such a mapping is
applied directly to a traditional image, the
resulting image contains a very small number of
pixels, compared to the original one, while
maintaining {\em locally} the initial resolution and
visual field. In this study we show that using such
a map-function, we are able to design a pattern
recognition system capable of recognizing an object
despite its position in space, rotation and scaling.
This is achieved by introducing a new linear
transform, called the {\em Exponential Chirp
Transform} (ECT), which provides frequency domain
estimation of log-polar warped images. The need for
such a tool derives from the extremely distorted
representation of images and performing even the
simplest object recognition task, such as detection
of the presence and position of an object in an
image already segmented, becomes a very complicated
task. The ECT can be computed very efficiently by
introducing a log-polar coordinate transformation in
frequency, as done for the Mellin-Fourier Transform,
through a cross-correlation with complexity $O(N_1
N_2 \log(N_1 N_2))$. \par We demonstrate the use of
the fast ECT with many examples, and show its
ability to perform traditional Fourier-based
processing: cross-correlation (template matching),
autocorrelation and filtering which are now {\em
space-variant}. In this study also a biological
model based on space-variant cross-correlation is
proposed as a basic pattern recognition algorithm
biologically plausible for vertebrate's visual
system. Finally, we outline a one-dimensional
transform, along with examples of logarithmic
sampling in time, which could be used in
applications of acoustic environment simulation.
\par This work provides, for the first time, a
conceptual basis for combining global spatial
frequency methods with space-variant mappings (i.e.
non-uniform sampling) in a way which is
generalizable for any type of mapping.},
datestr = {199701},
}