Mathematical Sciences: Inverse Estimation Problems

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Many problems in science and engineering share the common form that an unknown "signal" has to be restored from blurred observations on a known transformation of the signal. Mathematically this requires solving a (linear) system of equations, typically in an infinite dimensional space, where the data only suffice to construct a noisy approximation of the transformed signal. Solution depends upon the inversion of the operator. Studying this inversion problem in a Hilbert space setting can provide a unifying approach for constructing solutions, by exploiting techniques from spectral and harmonic analysis. Then statistical aspects such as error analysis and cross-validation can be examined. A famous example of this kind of problem is the technique of computerized tomography used in medical practice to recover an image of internal structures in the body. Because in practice, measurements are blurred by errors, and because only a limited x- ray exposure is possible, measurement of the "signal" (in this case the attenuation of the x-rays passing through the body) is imperfect indeed. Still, very refined images can be reconstructed and are very useful. Other well-known applications are image reconstruction and all kinds of deconvolution problems.