In this tutorial I will discuss two key image analysis components: metrics,
measuring differences between images and regularizations, critical to a
whole host of image analysis tasks.
Image comparisons are performed all the time. Is the person in this picture
the same as the person in that picture? Are the vortices in this fluid
simulation believably the same as those in this image from an experiment? Does
this denoised image differ from the data by a noise component? How significant
is the difference? Can we find this shape in that picture? These are a few of
the questions which require defensible image metrics. Construction of
such metrics requires a combination of understanding of the source of the image
data and a facility with the supporting mathematics.
Equally important to image analysis applications is the notion of
regularization, the (typically variational) enforcement of some generalized
smoothness, regularity or prior constraints. Without additional information, a
measured image is just a matrix of numbers, each between 0 and N. (A typical N
might be 255 and in the case of color images we have three such matrices, one
for each color). But in fact we always have additional information. This
information, introduced by way of a regularization term or a prior, permits
inference of image properties or features not directly observed. Such tasks as
image segmentation, inpainting, reconstruction from projections,
super-resolution and denoising all depend on our ability to regularize.
A brief schedule is as follows:
Tuesday) Introduction to metrics and regularization: basic concepts and
connection to statistical modeling
Wednesday) Metrics: examples, data fidelity terms, warping, and face
recognition
Thursday) Regularization: examples, denoising and total variation based
methods, and geometric analysis
I now list some references which cover some of what I will talk about. These
references are really only for the more ambitious since I will not assume you
have read them. A brief look at references 1 or 3 and references 4,5, and 6
should help you to get more from the tutorial.
[1] Chapter 3 of "Mathematical Problems in Image Processing" (2002) by
Aubert and Kornprobst is a nice introduction to image denoising, a
topic I will use to illustrate ideas in my tutorial.
[2] Chapter 5 of "Measure Theory and Fine Properties of Functions" (1999)
by Evans and Gariepy is a solid (nontrivial) introduction to functions of
bounded variation which will come up in a significant way in the third
hour of the tutorial. This will only be accessible to someone who has
already had a course in analysis.
[3] Chapter 4 of "Geometric Partial Differential Equations and Image
Analysis" (2001) by Sapiro is a nice exposition of a link between
statistical estimation and variational approaches to denoising.
[4] A
tutorial I gave in 2002, has some ideas that I will present again in
this tutorial.
[5] Our paper on face recognition using a novel, nonlinear metric can be
found by following this
link.
[6] The paper by Esedoglu and Chan, "Aspects of total variation
regularized
L1 function approximation", which can be
downloaded, is
relevant for the
second and third hours of the tutorial.
[7] For a large number of interesting and pertinent papers, see the
CAM reports here.
Kevin R. Vixie
is a member of the Theoretical Division (T-7)
at Los Alamos National Laboratory.
Presentation (PDF File)
Video of Talk (RealPlayer File)
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