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Alan Johnson
UC Berkeley
(asj) at uclink dot berkeley dot edu
Title: Diffuse Tomography
Abstract:
Diffuse tomography is similar to classic tomography, except that the
wave being sent through the body can be diffracted and thus does not
travel in a straight line. The data available to the inverse
tomographer is a matrix of the probabilities of a photon entering a
given border site, and exiting a given border site. The process
is modelled using a rectangular grid, where the photon is able to move
in straight lines between the nodes. The probability that the
photon moves in a given direction is dependent on its previous positon,
and since there are four possible directions to travel due to the
rectangular model, there are 16 parameters per pixel. Simple
cases are examined. The forward problem of taking in the
parameters at each pixel and producing the data is solved, and is used
in an algorithm to numerically approximate solutions to two examlpes.
Full article
Slides from talk
Scott Meckler
SUNY at Geneseo
(sjm9) at geneseo dot edu
Title: Using Rays of Light and a Mirror to Determine the Makeup of the
Body of an Object
Abstract:
In physics, objects have a property known as an attenuation
coefficient. This is a number with the dimension 1/length that measures
the relative
change in intensity of light that passes through a given portion of the
object. If we know the attenuation coefficient of the object and
the portion of the object through which a light passes, we can derive
the relative change in intensity of the light. If we were to pass
a light through an object that does not have a uniform attenuation
coefficient (i.e. the attenuation coefficient at a given point in the
object is a function of its position), then we could take the sum of
all of the attenuation coefficients at each part of the object
multiplied by the lengths of the portions of the respective parts
through which the light passes. In this project, we consider a
cross-section $\Omega$ of an object that does not have a uniform
attenuation coefficient and that sits in a material $z$ that has a
uniform attenuation coefficient. On one side of $\Omega$ we place
a mirror and on the opposite side we shoot rays of light from a source
so that they will travel through $\Omega$ and reflect against the
mirror (assuming no absorption of the light by the mirror) and then
travel back through $\Omega$, where they will then be picked up and
their intensities will be measured by a detector on the same side as
the source. We explore the question of whether or not we can
determine what the attenuation coefficients of the subsections of
$\Omega$ are by knowing just the relative changes in intensity of a set
of reflected rays of light. If this is possible, then we can let the
subsections of $\Omega$ approach $\infty$ and thereby represent the
attenuation coefficients continuously throughout $\Omega$ as a function
of the position.
Full article
Adam Gully
University of Utah
(peakgully) at comcast dot net
Title: Digital Cleaning of Old Paintings
Abstract:
Many old paintings have lost their original color due to being exposed
to harmful chemicals over centuries. In this paper, several methods are
presented about ways one might digitally recover the lost color of old
paintings, with some promising results.
Full article
Slides from talk
Gregory Lanson
Colby College
(gslanson) at colby dot edu
Title: Recovering Exposure Coecients in the SEIR Model in Two and
Three Populations
Abstract:
The SEIR model, which is used by epidemiologists, requires
knowing the values of multiple coefficients in order to correctly model
the disease they are
studying. The forward problem when using this model is to
determine the spread of an epidemic over time. The goal of this
research is to study the inverse
problem and recover one of the coefficients in the SEIR model when
applied to multiple populations.
Full article
Christopher Calaway
University of Utah
(eladamri72) at yahoo dot com
Title: Inverse Problems in Additive Number Theory
Abstract:
Additive number theory is the study of sums of sets, or
sumsets. For example the sumset $A + B = \{a + b : a \in A, b \in
B\}$.
In inverse additive number theory problems information is know about
the sumset
and information about the original sets is deduced. One interesting
problem to
study is finding limits of sumsets; this is a direct problem. However,
finding
information about the sets which cause the extreme sumsets is an even
more
interesting inverse problem. This is what I will focus on.
Full article
David Groulx
University of Utah
(dgroulx) at math dot utah dot
edu
Title:
Abstract:
Despite the rapid increases in computational power, real time
visualization and rendering are still rarities. As such, small
changes in the efficency of rendering may yield large returns in
reduced rendering times. A scene is defined by it's lighting,
geometry, and material properties. The scene may be then be
turned into an image through many different rendering methods. While
the forward problem is deterministic, i.e. a scene will always produce
the same image, the inverse problem may not be so straitforward.
Could a different scene produce the same image, or at least a close
enough approximation? And if a family of scenes can produce the
same image, how do we find the scene that will render the
quickest? This paper presents an algorithm for optimizing the
lighting configuration of a scene through inverse rendering. A
scene with $m$ lights is raytraced. We then apply inverse
rendering methods to the image to solve for the optimal configuration
of $n$ lights, where $n<m$. The new scene should produce a
similar image while requiring less time to render.
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