## Complex domain (phase/amplitude) sparsity and data processing in coherent optics

Vladimir Katkovnik (Technological University of Tampere, Finland)

In optics, a monochromatic wavefront is represented as a complex-valued signal, where amplitude and phase are unknown variables of interest. This wavefront phase cannot be measured directly because all measurement instruments are sensitive with respect to the intensity but not to phase. Accordingly, one of the main targets of data processing is to extract phase information from measured intensities. For instance, in interferometry and holography the phase is retrieved using special reference wavefronts. Under modeling of wavefront, we understand any mathematical tools for prediction, interpolation, denoising, etc., installing links between the values of wavefronts at different coordinates.

In computational imaging, sparse and redundant representations (sparsity) have been successfully developed the last years as a general modeling instrument. It is based on the assumption that there exists a small number of basic functions such that image can be represented exactly or approximately with a good accuracy. In term of statistics, a sparse representation can be thought as a low-order parametric approximation. In this classical form, the sparsity concept is used in parametric approximations just zeroing small amplitude components. A specific point of the sparsity is that the sparse basis is unknown in advance and should be designed. In this lecture, we consider methods and algorithms for sparse modeling of wavefronts and their applications for phase imaging.

In computational imaging, sparse and redundant representations (sparsity) have been successfully developed the last years as a general modeling instrument. It is based on the assumption that there exists a small number of basic functions such that image can be represented exactly or approximately with a good accuracy. In term of statistics, a sparse representation can be thought as a low-order parametric approximation. In this classical form, the sparsity concept is used in parametric approximations just zeroing small amplitude components. A specific point of the sparsity is that the sparse basis is unknown in advance and should be designed. In this lecture, we consider methods and algorithms for sparse modeling of wavefronts and their applications for phase imaging.