Essential dimension as a well-defined number of degrees of freedom of finite-convolution operators appearing in optics

Abstract

Inversion of a finite-convolution operator is known to be an ill-posed problem. However, although the complete solution cannot be recovered to within any specified accuracy, certain components of the solution can be accurately determined. We present an estimate of the number of such components, termed here the essential dimension of the finite-convolution operator, that is dependent on the noise levels in the data, the desired accuracy in the solution, and the singular values of the finite convolution. We then show that the required singular values may be easily and accurately approximated so that the essential dimension is easily estimated and indicate its superiority over previously proposed measures of ill conditioning for this problem.