A new method of solving noisy Abel-type equations

A new approach to solving noisy integral equations of the first kind is app lied to the family of Abel equations. Such equations play a role in stereol ogy (Wicksell's unfolding problem), medicine, engineering, and astronomy. T he method is based on an expansion in an arbitrary orthonormal basis, coupl ed with exact inversion of the integral operator. The inverse appears in th e Fourier coefficients of the expansion, where it can be carried over to th e usually well-behaved basis elements in the form of the adjoint. This meth od is an alternative to Tikhonov regularization, regularization of the inve rse of the operator itself, or a wavelet-vaguelette/ singular-value decompo sition. The method is particularly interesting in irregularity of the kerne l, the input, or both. Because knowledge of the spectral properties of the operator is not required, the method is also of interest in regular cases w here these spectral properties are not sufficiently known or are hard to de al with. For smooth input functions, the simple basis of trigonometric func tions yields input estimators whose mean integrated squared error converges at the optimal rate for the entire family of Abel operators. This can be s hown when smooth wavelets are used for Abel operators with index smaller th an 1/2, and when the Haar wavelet is used for operators with index larger t han 1/2. (C) 2001 Academic Press.