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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENTAL SEMINAR}
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\centerline{4:15 p.m., Tuesday, February 15, 2000}
\centerline{Sequoia Hall Rm. 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Emmanuel Candes}
\centerline{\sl Stanford University}
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\centerline{\bf Curvelets and Statistical Linear Inverse Problems} 
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The problem of recovering an input signal from noisy and linearly distorted
data arises in many different areas of scientific investigation; e.g.,
noisy Radon inversion (tomography) is a problem of special interest
and considerable practical relevance in medical imaging. We will argue that
traditional methods for solving inverse problems -- damping of the singular
value decomposition (SVD) or cognate methods -- behave poorly when the
object to recover has edges.

We apply a new system of representation, namely, {\it the curvelets} in
this setting. Curvelets provide near-optimal representations of otherwise
smooth objects with discontinuities along smooth $C^2$ edges. Inspired by
some recent work on nonlinear estimation, we construct a curvelet-based
biorthogonal decomposition of the Radon operator and build estimators based
on the shrinkage (or thresholding) of the noisy curvelet coefficients. We
prove that the shrinkage can be tuned so that the estimator will attain,
within logarithmic factors, the optimal estimation rate. In comparison,
linear procedures -- SVD included -- obtain markedly suboptimal rates of
convergence, as do `wavelet-like' shrinkage methods.

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