@Article{fischl1997:learning,
author = {Bruce Fischl and Eric L. Schwartz},
title = {Learning an integral equation approximation to
nonlinear anisotropic diffusion in image processing},
journal = {IEEE Pattern Analysis and Machine Intelligence},
year = 1997,
volume = 19,
pages = {342-351},
INSPEC = 5606020,
month = April,
datestr = 199704,
abstract = {Multiscale image enhancement and representation is
an important part of biological and machine early
vision systems. The process of constructing this
representation must be both rapid and insensitive to
noise, while retaining image structure at all
scales. This is a complex task as small scale
structure is difficult to distinguish from noise,
while larger scale structure requires more
computational effort. In both cases good
localization can be problematic. Errors can also
arise when conflicting results at different scales
require crossscale arbitration. Broadly speaking,
multiscale image analysis has historically been
accomplished using two types of techniques: those
which are sensitive to image structure and those
which are not. Algorithms in the latter category
typically use a set of variously sized blurring
kernels to produce images each of which retain
structure at a different scale [Marr and Hildreth,
1980], [Burt and Adelson, 1983], [Witkin, 1983],
[Koenderink, 1984], [Hummel, 1986]. The kernels used
for the blurring are predefined and independent of
the content of the image. Koenderink showed that if
the kernels are Gaussian, then this process is
equivalent to the evolution of the linear heat (or
diffusion) equation. He thus transformed the
integral equation representing the convolution
process into the solution of a partial differential
equation (PDE). Structure sensitive multiscale
techniques attempt to analyze an image at a variety
of scales within a single image [Klinger, 1971],
[Perona and Malik, 1987], [Nitzberg and Shiota,
1992]. Klinger [Klinger, 1971] proposed the quad
tree, one of the earliest structuresensitive
multiscale image representations. In this approach,
a tree structure is built by recursively subdividing
an image based on pixel variance in subregions. The
final tree contains leaves representing image
regions whose variance is small according to some
measure. Recently [Perona and Malik, 1987], [Perona
and Malik, 1990], the PDE formalism introduced by
Koenderink has been extended to allow
structuresensitive multiscale analysis. Instead of
the uniform blurring of the linear heat equation
which destroys small scale structure as time
evolves, Perona and Malik use a spacevariant
conductance coefficient based on the magnitude of
the intensity gradient in the image, giving rise to
a nonlinear PDE. Like the quadtree, the intent is to
produce a single image representation which contains
information at all scales of interest. The Perona
and Malik approach produces impressive results, but
the numerical integration of a nonlinear PDE is a
costly and inherently serial process. In this paper
we present a technique which obtains an approximate
solution to the PDE for a specific time, via the
solution of an integral equation which is the
nonlinear analog of convolution. The kernel function
of the integral equation plays the same role that a
Green's function does for a linear PDE, allowing the
direct solution of the nonlinear PDE for a specific
time without requiring integration through
intermediate times. We then use a learning technique
to approximate the kernel function for arbitrary
input images. The result is an improvement in 3
speed and noisesensitivity, as well as providing a
means to parallelize an otherwise serial algorithm.},
keywords = {approximation theory, diffusion, image enhancement,
image representation, integral equations, learning
systems, partial differential equations}
}