Talk
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Material
from Talks
Madjid Allili
(Center for Dynamical Systems and Nonlinear Studies School of
Mathematics, Skiles 115, Georgia Institute of Technology)
Cubical Homology Theory and Applications in Image Processing
and Computer Vision
Homology theory is a well known concept in algebraic topology
related to the notion of connectivity in multi-dimensional shapes.
We will discuss the cubical homology theory that is very suited
to the study of topological properties of images since by essence
an image is a cubic grid in the plane. We will also report the
recent progress made in designing algorithms and computer programs
computing homology of spaces and maps as well as some examples
of application of homology in Image Processing and Computer
Vision.
Marcelo
Bertalmio
(University of Minnesota)
Image Inpainting and High Order PDE's in Image Processing
Inpainting, the technique of modifying an image in an undetectable
form, it is as ancient as art itself. The goals and applications
of inpainting are numerous, from the restoration of damaged
paintings and photographs to the removal/replacement of selected
objects. We introduce a novel algorithm for digital inpainting
of still images that attempts to replicate the basic techniques
used by professional restorators. The algorithm automatically
fills-in regions with information surrounding them. The fill-in
is done in such a way that isophote lines arriving at the regions
boundaries are completed inside. The technique does not require
the user to specify where the novel information comes from.
This is automatically done (and in a fast way), thereby allowing
to simultaneously fill-in numerous regions containing completely
different structures and surrounding backgrounds. In addition,
no limitations are imposed on the topology of the region to
be inpainted. Applications of this technique include the restoration
of old photographs and damaged film; removal of superimposed
text like dates, subtitles, or publicity; and the removal of
entire objects from the image like microphones or wires in special
effects. This work also shows the importance of moving toward
high order PDE's in image processing and the relations of those
with other exciting areas of mathematical physics. Joint work
with G. Sapiro, V. Caselles, and C. Ballester
Tony
F. Chan (UCLA)
A Level Set Framework for Active Contours and Mumford-Shah
Segmentation
(co-author: Luminita A. Vese, UCLA).
In this talk, I will present a common framework for active contours
and Mumford-Shah segmentation, based on the level set method
of S. Osher and J. Sethian. First I will introduce an active
contour model "without" edges, based on segmentation and level
sets. By this model, we can detect objects whose boundaries
are not necessarily defined by gradient, as well as interior
contours automatically. Then I will show how this level set
model can be generalized, in order to minimize the Mumford-Shah
energy for segmentation, for piecewise-constant and piecewise-smooth
approximations. We represent the set of edges via one or more
level set functions, and we propose a new multiphase level set
representation, which has some advantages: we use only $n$ level
set functions to represent $2^n$ phases, and in addition, we
do not have the problems of vacuum and overlap, naturally arising
in multiphase problems. Also, we will see that triple junctions
can be detected and represented. Finally, I will show numerical
results on various images, in order to validate the algorithm.
Ingrid
Daubechies (Princeton University)
Tree Approximation and Image Compression
Joint
work with A. Cohen, W. Dahmer and R. DeVore studied tree approximations
and used the results to build a practical coder that gives a
performance similar to existing state-of-the-art coders. We
derive certain optimality results for our concrete tree approximation
algorithms. On the other hand, because the tree structure is
known not to be optimal for images, this also points to shortcomings
of state-of-the-art coders, which use a similar tree 'philosophy'.
In addition, we discuss some adaptivity properties of our tree
coder, useful when parts of very large images need to be examined.
David
Donoho (Stanford University)
Harmonic Analysis Perspective on Geometric Diffusions and
Low level Vision
In recent years, two very important trends have emerged which
are of compelling interest to the mathematically-trained who
are thinking of applications in image processing. On the one
hand, there is the widespread use of PDEs to process images,
for example with the use of geometry-driven diffusions to remove
noise from images and perform segmentation. On the other hand,
there are intensive studies of computational vision, and interesting
speculations and investigations about mathematical structures
which might be involved in biological vision.
In my talk, I will take a completely different discipline --
Harmonic Analysis -- and consider some recent developments in
this field. With constructions such as wavelets, time-frequency
analysis, and other more exotic schemes, there is a wealth of
ideas which can be compared and contrasted with recent developments
in both geometry-driven diffusions and in computational vision.
In my talk I will focus on two topics:
[1] Existing geometry-driven diffusions go in the right direction
-- smoothing anisotropically in the vicinity of edges. But this
is only qualitatively correct. Do they really do the quantitatively
correct thing? Does it matter?
[2] Many existing studies relating phenomena in natural images
to computational structures that might be relevant to the visual
cortex and computational analogs take Fourier and Gabor analysis,
and more recently wavelet analysis, as models for the possible
underlying structures which are best-adapted for image analysis.
Are these the right ideas? How do other developments in harmonic
analysis (e.g. brushlets, beamlets) compare?
I hope to convey both the spirit and some of the specific mathematical
ideas of recent developments in applied harmonic analysis.
Parts of my talk will describe joint work with Emmanuel Candes
(CalTech), with Drs. Georgina Flesia and Arne Stoschek (Stanford).
Olivier
Faugeras (INRIA/MIT)
Dynamic
Shapes of Arbitrary Dimension: The Vector Distance Functions
We
present a novel method for representing and evolving objects
of arbitrary dimension. The method, called the Vector Distance
Function (VDF) method, uses the vector that connects any point
in space to its closest point on the object. It can deal with
smooth manifolds with and without boundaries and with shapes
of different dimensions. It can be used to evolve such objects
according to a variety of motions, including mean curvature.
If discontinuous velocity fields are allowed the dimension of
the objects can change. The evoluti on method that we propose
guarantees that we stay in the class of VDFs and therefore that
the intrinsic properties of the underlying shapes such as their
dimension, curvatures can be read off easily from the VDF and
its spatial derivatives at each time instant. The main disadvantage
of the method is its red undancy: the sizeof the representation
is always that of the ambient space even though the object we
are representing may be of a much lower dimension. This disadvantage
is also one of its strengths since it buys us flexibility.
Steven Haker (Yale University
School of Medicine)
Non-distorting Flattening for Virtual Colonoscopy
In this talk, we consider a novel 3D visualization technique
based on conformal surface flattening for virtual colonoscopy.
Such visualization methods could be important in virtual colonoscopy
since they have the potential for non-invasively determining
the presence of polyps and other pathologies. Further, we demonstrate
a method which presents a surface scan of the entire colon as
a cine, and affords a viewer the opportunity to examine each
point on the surface without distortion. From a triangulated
surface representation of the colon, we indicate how the flattening
procedure may be implemented using a finite element technique.
We give a simple example of how the flattening map can be composed
with other maps to enhance certain mapping properties. Finally,
we show how the use of curvature based colorization and shading
maps can be used to aid in the inspection process.
Stephane
Mallat (Ecole Polytechnique,
Courant Institute)
Geometrical Image Representations with Bandelets
Following the "2nd generation image coding'' dream, improving
current image transform codes will require to represent images
with features that are meaningful for the scene analysis. Such
an approach would allow us to build compact representations
that can also be used for search in large data-bases of images.
To achieve this goal, the first issue is to extract and efficiently
represent the image geometry. An approach is presented with
foveal wavelets and bandelets, with compression examples.
Donald
E. McClure (Division of Applied Mathematics, Brown
University) Donald.McClure@Brown.edu
Restoration
and Reformatting of Motion Images
Postproduction
processing of film and video is a source of a wide variety of
image processing and analysis problems for motion images. I
shall describe the formulation of problems in the areas of digital
repair of damage to film, conversion between different digital
video formats, and compression. Approaches to these problems
used in a current workstation-based system will be described
and illustrated with examples. Other contributors to the design
of this system are D. Geman, S. Geman, K. Manbeck and C. Yang.
David
Mumford (Brown University)
Modeling
the Full Statistics of Local Image Patches
I
will discuss the work of my group, Ann Lee and Jinggang Huang
, and that of Ulf Grenander on seeking a full description of
the joint probability model for all pixels in small image patches,
e.g. 2x2 to 8x8. I will also compare the statistics of optical
images with those of range images.
Stanley
Osher
Level
Set/PDE Based Algorithms for Image restoration, Surface Interpolation,
and PDE's on General Manifolds
We shall present new, fast level set based algorithms for image
restoration and for interpolating unorganized points, curves
and surface patches in 3D. We shall also present a new framework
for computing the solution of PDE'S and variational problems
on general manifolds, (in particular 3D surfaces) and apply
this to image processing problems. The work is joint with many
people including P. Burchard, L-T Cheng, M. Bertalmio, R. Fedkiw,
M. Kang, B. Merriman, H.K Zhao and A. Marquina
Ilya
Pollak (Purdue University)
Some
New Results on Optimality and Complexity of PDE-Based Segmentation
Algorithms
We
will present a very simple nonlinear diffusion equation and
show its utility for image segmentation. We will show that it
may be interpreted both as a variant of the Perona-Malik equation,
and as the steepest descent equation for the total variation.
Its analysis in 1-D will reveal it to be an exact solver of
certain maximum likelihood detection/estimation problems. The
major advantage over other methods to solve these problems is
O(N log N) computational complexity in one spatial dimension.
Finally, we will show our method to be a robust estimator (in
the spirit of H-infinity estimation) for a restricted class
of 1-D problems. Experiments suggest that the 2-D version of
our algorithm retains robustness properties of the 1-D version.
A remaining challenge is to extend our 1-D theoretical results
to 2-D, designing fast and optimal image segmentation algorithms.
Stefano
Soatto (University of California-Los Angeles/Washington
University) soatto@ucla.edu http://www.cs.ucla.edu/~soatto
Accommodation
as a Low-Level Visual Cue
I
call accommodation cues measurable properties of images of a
given scene that are associated with a change in the geometry
of the imaging device. For instance, in the human eye the shape
of the lens is controlled so as to bring the scene into focus
at the fovea; in a video camera the lens translates for the
same purpose.
Is
accommodation an unambiguous cue? (i.e. is it possible to distinguish
two arbitrary shapes solely from accommodation cues?) Can a
surface be reconstructed uniquely from accommodation? Such conditions
clearly depend upon geometry (the shape of the surface) as well
as photometry (its radiance distribution). Is it possible to
characterize the set of "sufficiently exciting" distributions?
In
this talk I will present some preliminary results and partial
answers to the above questions, as well as describe two optimal
algorithms (in the sense of L2 and Information-divergence) to
reconstruct shape from accommodation cues, under certain assumptions.
This amounts to solving a blind deconvolution problem with some
special features. I will discuss some open problems and potential
applications of accommodation in visualization and endoscopic
surgery.
Bart
M. ter Haar Romeny (Utrecht University)
Front-End
Vision and Multiscale Image Analysis (a new tutorial book).
Both
linear and nonlinear scale-space theory have booked much progress,
and shown good performance in many compter vision algorithms.
The field however sees a relatively small growth when compared
to e.g. neural networks and wavelets. A new upcoming tutorial
book on linear and nonlinear scale-space theory is presented,
completely written in Mathematica 4. Computer algebra software
has now reached the level that even for large datasets as multidimensional
images complex mathematics can be done and presented in a fast
prototyping way. Every topic is illustrated with (typically
very short) code to carry out and modify all experiments. The
red wire through the book is the mathematics of apertures. A
thorough treatment of the front-end visual system is included.
Demo's will be presented of differential invariants to 4th order,
gauge coordinate systems, multiscale optic flow, steerable filters,
time-scale, color differential invariants, deep structure, geometry-driven
diffusion equations, winding numbers, edge focusing etc. The
book and CD-ROM will appear end this year with Kluwer Academic
Publishers.
Alan
S. Willsky (MIT)
Multiresolution Stochastic Models and Their Use in Modeling
and Analysis of Random Processes and Fields
In this talk we describe a body of research concerned with the
building and exploitation of multiresolution statistical models
of random phenomena and imagery. We begin with a discussion
of linear stochastic models on multiresolution trees, in particular
discussing the very efficient algorithms these models admit,
the realization of random phenomena using such models, some
relationships to wavelet decompositions of signals and images,
and applications of this formalism, and some of its critical
limitations. Motivated by two of these limitations, we describe
two research directions that use this basic formalism as a point
of departure. The first of these is a class of nonlinear models
we refer to as wavelet cascades, which maintain much of the
exploitable structure of our linear-tree models but allow us
to capture distinctive nonlinear characteristics of natural
imagery. The second is the examination of stochastic models
on graphical structures other than trees, a topic of considerable
interest in a number of quite different domains.
Anthony
Yezzi (Georgia Institute of Technology) ayezzi@ece.gatech.edu
Variational
Methods for Image Segmentation, Smoothing, Interpolation, Magnification,
Stereo Matching, and Shape from Shading
Partial Differential Equations have been used extensively to
derive geometric active contour models for the purpose of image
segmentation and to derive anisotropic diffusion models for
image smoothing. They have also been employed in low level vision
problems of inferring 3D structure from one or more 2D images
(e.g. stereo-matching and shape-from-shading).
In the first part of this talk we present a class of statistically
driven active contour models based upon deterministic energy
functionals designed to maximally separate the values of selected
statistics inside and outside each evolving contour. We follow
this with a less restrictive model, based upon the Mumford-Shah
functional, which simultaneously diffuses the image while evolving
a set of active contours towards the boundaries of objects.
A straightforward generalization of this model allows us to
treat images with regions of missing data and to create a unified
framework for simultaneous image segmentation, smoothing, and
magnification. In the second part of this talk, we will present
a novel approach to multiframe shape-from-shading which is stronly
motivated by the multiframe stereo-matching work of Faugeras
and Keriven.
Haomin
Zhou (UCLA)
Adaptive
ENO-wavelets for Image Compression
We have designed an adaptive ENO-wavelet transform for approximating
discontinuous functions without oscillations near the discontinuities.
Our approach is to apply the main idea from Essentially Non-Oscillatory
(ENO) schemes for numerical shock capturing to standard wavelet
transforms. The crucial point is that the wavelet coefficients
are computed without differencing function values across jumps.
However, we accomplish this in a different way than in the standard
ENO-schemes. Whereas in the standard ENO schemes, the stencils
are adaptively chosen, in the ENO-wavelet transforms, we adaptively
change the function and use the same uniform stencils. The ENO-wavelet
transform retains the essential properties and advantages of
standard wavelet transforms such as concentrating the energy
to the low frequencies, obtaining arbitrary high order accuracy
uniformly and having a multiresolution framework and fast algorithms,
all without any edge artifacts. We have obtained a rigorous
approximation error bound which shows that the error in the
ENO-wavelet approximation depends only on the size of the derivative
of the function away from the discontinuities. We will show
some numerical examples to illustrate this error estimate.
Material
from Talks
Image Processing and Low Level Vision
2000-2001
Program: Mathematics in Multimedia
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