Research On Phase Retrieval Method Via Low Rank Matrix Completion

Light is one of the most important media for information transmission and in essence a vector. The light transmitted in free space can be described by their complex amplitude:intensity and phase. Phase is a kind of inherent characteristics of optical information. About 75% of the information of the geometric and physical properties of objects is in the phase. But the oscillation frequency of light is very high (close to 1015Hz). Existing optical detection devices (e.g., CCD cameras etc.) cannot measure the phase information of a light directly. Therefore, It needs to measure the light intensity for calculating the phase. The technique is called phase retrieval.At present, the phase retrieval technology based on the intensity measurement mainly includes:the iterative phase retrieval, phase retrieval based on transport of intensity equation and phase retrieval via low rank matrix completion etc. Iterative phase retrieval method is back and forth between the space and requency domains to recover phase distribution of optical field on the input and output plane. But the method is not necessarily able to converge to the global optimal solution. The phase retrieval method based on transport of intensity equation recovers the phase information to solve transport of intensity equation which needs a focused image, at least two defocus images and the image spacing and other informations in the calculation. So the special experiment devices are needed to complete the image acquisition in practice. A new method for phase retrieval is called phase retrieval via low rank matrix completion which develops from the matrix completion theory. The method can converge to the global optimal solution up to global phase.The phase retrieval method via low rank matrix completion is researched in the thesis. Two methods named PhaseLift and PhaseCut are discussed in detail. The imaging setting, the design of masks and the phase retrieval algorithms are deeply studied about them.The main research works and innovations of the thesis are outlines as follows:(1) The basic matrix completion problem is analysed. Several typical low matrix completion algorithms such as CVX, SVT, ALM, FPCA, OptSpace, SET, GROUSE and LMaFit are summarized, The experiments compares and analyses their recovery performances with different ranks, sampling ratio, and added noise.(2) A new method named PhaseLift is studied. The method Lifts the classical problem recovering phase from intensity to low rank matrix completion problem, combines multiple illuminations and convex programming tools, and records multiple diffraction patterns. Then a rank-one matrix is obtained by semidefinite programming. Finally, original phase is recovered. The PhaseLift method can be guaranteed to converge to the global optimal solution up to global phase and has a higher robustness. But the PhaseLift method do not consider the specific imaging settings. Two more structured masks-Toeplitz mask and circulant mask are introduced to specific imaging settings in the thesis. Compared with Gaussian mask, so the number of random elements is reduced. Thereby, the difficulty and the cost of design of masks are reduced. Compared with binary mask, they need collect fewer diffraction patterns.(3) A new phase retrieval method-GSPhaseLift which combines GS algorithm and PhaseLift is proposed and solved by TFOCS package. The improved algorithm first constructs the measurement matrix. Then the one-dimensional signal and the two-dimensional complex amplitude are reconstructed respectively. The experiments show that on the one hand, the reconstruction sucess rate of improved algorithm is higher than that of GS algorithm. On the other hand, the convergent speed of improved algorithm is faster than the original PhaseLift, and can achieve phase retrieval quickly for complex amplitude of larger size. The quality of reconstructed image can also be improved.(4) Another method named PhaseCut is discussed by transforming phase retrieval problem to convex programming problem. The tightness and stability of PhaseCut are compared with that of PhaseLift when there is noiseless and noisy. A PURE-RBR-M algorithm suitable for MaxCut problems is proposed to solve PhaseCut. Compared with the original interior point algorithm, the new algorithm has much faster speed, and more robust to noisy measurement. Especially, the new algorithm has obvious advantages on computation speed when the dimension of signal is high. On the other hand, the design method of masks in phase retrieval via low rank completion is considered. The thesis proposes to modulate the signal with more structured ternary mask and octonary mask in PhaseCut. Compared with binary mask, the recovery results of them is better.