Phase Retrieval of Color Images: Analysis, Algorithms and Applications

When phase information is lost in measurements, the recovery of the underlying object from the magnitudes of its Fourier transform is called phase retrieval problem. Phase retrieval is one of important problem in imaging science. There are many scientific and engineering applications involving phase retrieval problems, for example, astronomical imaging, microscopy imaging and optics. There are theoretical and computational challenges in phase retrieval problem. They include questions of uniqueness, regularization, optimization, efficient reconstruction algorithms and convergence analysis. The main aim of this proposal is to study and analyze phase retrieval of color images. It is interesting to note that there a strong correlation among red, green and blue channels of color images and such correlation can be used for solving phase retrieval problems more efficiently.

Our main focus will be proposing new mathematical analysis and developing efficient solution methods for phase retrieval of color images. Our contributions are to propose new mathematical models and algorithms, and analyze their theoretical and convergence properties. To the best of our knowledge, this is the first attempt to make use of correlated color channels to study and analyze phase retrieval problems. Our idea is to study phase retrieval of color images under the quaternion framework by encoding red, green and blue color pixel values on the three imaginary parts of a quaternion. Because of non-commutative nature of quaternion, we propose to use real-structure preserving tool to study and analyze the corresponding phase retrieval problem. Moreover, we develop and analyze algorithms for phase retrieval of color images by using transformbased regularization, and also study sparsity effect and support information in phase retrieval problem. The recovery guarantee and convergence properties of the proposed methods will be investigated.

On the other hand, we propose to apply our theoretical results and develop numerical algorithms in the application of wide-field, high-resolution Fourier ptychographic microscopy imaging. The main issue is to study high-to-low resolution matrix and the spectral multiplexing and coherent-state decomposition in the our mathematical models and optimization algorithms. We will develop iterative algorithms for solving such ptychographic microscopy setting. We can make use of experimental data sets to verify the proposed mathematical models for phase retrieval of color images and demonstrate the performance of the proposed algorithms for color image recovery.