Study Of The Signal Convex Optimization Reconstruction By The Magnitude Or Phase Of Transform Domain

The signal reconstruction technique is to recover the complete original signal by the partial information of the original signal. It is one of the important issues in the field of signal processing. Phase retrieval and magnitude recovery are special cases of signal reconstruction, where they reconstruct the original signal by leveraging only the magnitude or phase in the transform domain, respectively. With the rapid development of phase retrieval algorithm and magnitude recovery algorithm, signal reconstruction technology is widely used in the field of holography, wave-front phase detection, X-ray tomography, diffractive optical element design, filter design, target detection, optical encryption, image encryption, and image restoration, etc. At the beginning of this century, the presence of compressed sensing theory and matrix completion technology has made it available to accurately recover a sparse matrix and a low rank matrix, by solving a convex optimization problem with high probability under appropriate conditions. These techniques have made the phase retrieval and magnitude recovery promising research hotspots again for current research, which makes a potential new stage for the signal reconstruction technology.Firstly, this thesis gives a brief introduction about the origin, development and application of phase re-trieval and magnitude recovery techniques. Therefore, we have a good understanding about the background of the phase retrieval problem and the magnitude recovery problem and their developing directions in the literature. Secondly, we describe in detail the various phase retrieval algorithms and magnitude recovery algorithms, including GS (Gerchberg and Saxton) method, Conjugate gradient algorithm, Yang-Gu method, PhaseLift method, PhaseCut method, Iterative algorithm, Projection onto Convex sets method and Local phase algorithm. Because the PhaseCut algorithm is the optimal phase recovery algorithm based on matrix completion and convex optimization, we present a new magnitude recovery algorithm in this thesis, named MagnitudeCut algorithm. The third chapter focuses on the mathematical derivation process of the Magni-tudeCut algorithm. Finally, experimental results demonstrates the feasibility and effectiveness of the new algorithm.MagnitudeCut algorithm is a new magnitude recovery algorithm. It not only inherits the accurate proper-ties of the classical iterative algorithm, but also improves the quality of the reconstruction and is effective for symmetric signals. More surprisingly, our reconstruction result would not be influenced by the initial value of the algorithm. The process of this new method can be summarized as follows:firstly, we transform the magnitude recovery problem into a new convex optimization problem, which makes the original problem into a new one with more easily solved form and excludes the possibility of getting a local optimal solution. Then, we simplify the matrix optimization problem to some vector optimization problem in parallel by using the Block Coordinate Descent algorithm (BCD), which reduces the computational complexity. Next, the vector optimization problem is solved by the interior point method to get the core iterative formula of MagnitudeCut. Finally, we iteratively repeat the above steps to recover the original signal.In our experiments, on the one hand, we realize multiple signal/image reconstruction by leveraging our algorithm. Experimental results show that the proposed new algorithm can reconstruct multiple types of sig-nals and images. On the other hand, we compare the new algorithm with the traditional magnitude recovery methods and phase retrieval algorithms in depth. Compared with the greedy algorithm and the iterative algo-rithms, the proposed MagnitudeCut method can reconstruct the original signal with fewer sampling number of phase information under the same reconstruction error. In comparison with the PhaseCut algorithm, only the MagnitudeCut algorithm can reconstruct the original signal when the number of sampling information is equal to the number of original signal. This not only illustrates that the MagnitudeCut can achieve the same result with less information compared to PhaseCut algorithm, but also verifies that the phase domain contains more information than magnitude domain under the same number of samplings.