Inverse Problems...
... is a fancy way of saying "study of imaging." In the context of
wave propagation, if you are given a source and some boundary conditions, you
can work out the wave structure and evolution in pretty much every location
in space-time. This is the "forward problem." Generally, forward problems
are solvable (albeit often with difficulty). The inverse problem is posed as
follows: if I am given some observations of a wave, say along one or several
surfaces, can I infer the source? As you can imagine, this is not always possible.
If the information given by the measurement is incomplete, the inverse problem
is, in general, "ill-posed." This means that there might be several
possible sources that would give rise to the same observation and we have no
way to determine which one among them is the real one. (The terms ill-posed
and incomplete are used in a loose sense here). Inverse problems is the field
of study that tries to quantify when a problem is ill-posed and to what degree,
and extract maximum information (again, in the loose, every-day sense of the
word) under practical circumstances. For instance, an astronomer observing the
sky with a telescope only might think that a blob of light originated from a
single star; but if she applies an algorithm called CLEAN (or its variants),
she might actually discern (the official term is "resolve") two or
more stars in a constellation. This is the typical mode in which imaging system
design benefits from knowledge of inverse problem theory. Note that this kind
of imaging system is hybrid (telescope+computer). In addition to astronomy,
these techniques benefit several application domains such as biomedical imaging
and industrial inspection. The most spectacular success in the field of inverse
problems was the invention of an inversion algorithm for Computed Tomography
by Cormack (1963) and its experimental demonstration by Hounsfield (1973). The
two shared the Nobel prize in Physiology or Medicine in 1979.
What do I gain by learning about inverse problems?
- Understand how the performance of imaging systems can be quantified; learn
how to properly use terms such as "resolution," and "space-bandwidth
product."
- Build intuition about the effects of noise on imaging, and how some of these
can be survived.
- Learn about popular imaging techniques such Computed Tomography (CT) and
functional Magnetic Resonance Imaging (fMRI) which rely heavily on the theory
of inverse problems. See the example below.
- Share the instructor's fascination of blending inverse problems with information
theory, through which information transfer from a physical object through
an imaging system can be precisely quantified.
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Magnetic Resonance Imaging (MRI) and functional
Magnetic Resonance Imaging (fMRI) are two non-invasive imaging techniques
that rely on data measured around a part of a subject's body (e.g. the
head) to acquire information about what's going on inside the body part
(e.g. the brain). The data are processed using a mathematical technique
called Filtered Backprojection, which is a computationally efficient
and stable way of computing the Inverse Radon transform. The MRI
machine shown on the left is made by GE Medical Systems. An example of
the kind of data that can be acquired via this technique is the image
of a patient's brain shown on the right.
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