基于正交字典学习的多像面相位恢复算法

Multi-Plane Phase Retrieval Algorithm Based on Orthogonal Dictionary Learning

相位恢复(Phase Retrieval,PR)是指利用信号的菲涅尔变换或其它线性变换的强度观测值恢复原始信号.由于相位信息的丢失,相位恢复问题是一个不适定问题.为解决该问题,可利用包含更多图像信息的多像面强度观测值和图像在字典下的稀疏性进行相位恢复.正交字典学习以速度快、效果好的优点得到了成像领域的关注.该文提出利用图像在正交字典下的稀疏表示进行多像面相位恢复的算法.首先,利用多像面强度观测值构造数据保真项,并结合图像在正交字典表示下的稀疏正则项构造多像面相位恢复优化问题.然后,利用分裂Bregman方法(Split Bregman Method)对该非线性优化问题进行求解;此外,针对含有数据保真项的非凸子问题,利用极大极小化(Majorization-Minimization,MM)算法将这一子问题转化为易处理的优化问题,并对其进行有效求解;该文提出的算法能够仅利用强度观测值重建图像同时学习字典,得到与之匹配的正交字典.正交字典学习通过阈值处理和奇异值分解两步训练字典.由于自适应正交字典能够捕获图像的结构信息,该文算法在像面个数较少时仍能得到高质量的重建图像.仿真实验表明,该算法无论从客观标准还是从主观视觉都优于现有算法,并对噪声鲁棒.

Phase retrieval(PR)refers to recover the original signal from the intensity measurements of its Fresnel transform,or recover the image from the intensities of some other linear transform of the signal.Due to the loss of the phase information,the phase retrieval problem is an ill-posed problem.To address this issue,the intensity measurements at multiple observations can be utilized for phase retrieval.Moreover,the sparsity of the underlying image under an adaptive dictionary is often utilized for phase retrieval.Multiple intensity measurements imply that the measured data contains much known information about the underlying image.Exploiting these multiple intensity measurements enables an improvement of the reconstruction quality.When orthogonal dictionary learning is applied to the image reconstruction field,training an orthogonal dictionary has the advantages of fast speed and good reconstruction quality.Based on this fact,orthogonal dictionary learning is focused by the researchers in the imaging field.In this paper,we propose a multi-plane phase retrieval algorithm.The algorithm exploits the sparse representation model of the underlying image over an adaptive orthogonal dictionary.Firstly,we exploit the intensity measurements of multiple planes to construct a data fidelity term.We combine the data fidelity term and the sparse induced regularization term of the underlying image over an orthogonal dictionary to formulate the phase retrieval optimization problem.Secondly,the split Bregman method is utilized to solve the corresponding nonlinear optimization problem.For the nonconvex optimization sub-problem involving the data fidelity term,we exploit the Majorization-Minimization(MM)method to recast this sub-problem into a tractable optimization problem.Moreover,we propose an efficient method to tackle this problem.MM algorithm constructs a surrogate function for the original complicated optimized problem,the sequence of points generated by solving the surrogate function by basic matrix multiplications is proved to converge to a stationary point of the original problem.The proposed algorithm in this paper can recover the image and learn an orthogonal dictionary simultaneously only from the intensity measurements.The learnt orthogonal dictionary can capture the structure of the underlying image efficiently,and its atoms can match with the structure of the recovered image.The orthogonal dictionary learning problem is often attacked in two steps:the sparse coding step and the dictionary updating step.Since the mutual incoherence constraints for the atoms of the orthogonal dictionary are imposed on the learnt dictionary,the hard thresholding operator can be used for the sparse coding step.For the dictionary updating step,the SVD decomposition is used to update an orthogonal dictionary.Through the two steps,we can train an orthogonal dictionary effectively from the training samples.As the result of the fact that the adaptive orthogonal dictionary can capture the structure of the underlying image,the proposed algorithm can recover the high-quality image only from measurements,even the measured planes are few.Simulated experimental results indicate that the proposed phase retrieval method outperforms the previous multi-plane PR algorithms in terms of both the objective criteria and the subjective visual evaluation.Meanwhile,our algorithm is robust to noise.