Null space regularization and MAP reconstruction in the ill-posed inverse imaging problem

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Abstract
The ill-posed inverse problem is not an issue that is only restricted to optical tomography, but indeed a very common issue in image reconstruction problems in astronomy, geological surveying, and medical imaging in general. In this paper we investigate the consequences of ill-posed problems, and show that correct reconstruction is generally not possible using conventional linear inversion techniques because latter methods disregard contributions of the nullspace. We describe the rationale of a novel image reconstruction method that estimates the nullspace contribution using prior knowledge in a maximum-aposteriori- probability framework. We illustrate our concept by an example of optical tomographic reconstruction from simulated and experimental data.

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