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There are several natural ways to take measurements of an unknown object.
One is by means of the support function,
which gives for any direction the (signed) distance from some fixed point
(usually the origin) to the hyperplane supporting the object orthogonal to that direction.
Another is by the distance between the two supporting hyperplanes orthogonal to a given direction.
There is also the possibility of measuring the brightness function,
which for an n-dimensional object gives the (n-1)-dimensional volumes of its orthogonal projections onto hyperplanes;
in other words, the areas of its shadows.
For convenience we refer to all these as "support-type functions".
The talk will survey some algorithms that reconstrut an approximation
to a shape from a finite number of noisy (that is, corrupted)
measurements of one of the above types. In the case of brightness
functions, the algorithms are the result of joint work with an
electrical engineer, Peyman Milanfar. These algorithms have been
implemented, and some sample reconstructions will be shown. We will
also describe some very recent work with Milanfar and Markus Kiderlen
in which we establish the convergence of some of the algorithms,
even estimating the rates of convergence.
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