Kularia Muskan, Banerjee Manidipa, Khare Kedar
J Opt Soc Am A Opt Image Sci Vis. 2024 Jun 1;41(6):1166-1174. doi: 10.1364/JOSAA.516339.
The recovery of a complex-valued exit wavefront from its Fourier transform magnitude is challenging due to the stagnation problems associated with iterative phase retrieval algorithms. Among the various stagnation artifacts, the twin-image stagnation is the most difficult to address. The upright object and its inverted and complex-conjugated twin correspond to the identical Fourier magnitude data and hence appear simultaneously in the iterative solution. We show that the twin stagnation problem can be eliminated completely if a coherent beam with charge-1 vortex phase is used for illumination. Unlike the usual plane wave illumination case, a charge-1 vortex illumination intentionally introduces an isolated zero near the zero spatial frequency region, where maximal energy in the Fourier space is usually concentrated for most natural objects. The early iterations of iterative phase retrieval algorithms are observed to develop a clockwise or anti-clockwise vortex in the vicinity of this isolated zero. Once the Fourier transform of the solution latches onto a specific vortex profile in the neighborhood of this intentionally introduced intensity zero in early iterations, the solution quickly adjusts to the corresponding twin (upright or inverted) and further iterations are not observed to bring the other twin into the reconstruction. Our simulation studies with the well-known hybrid input-output (HIO) algorithm show that the solution always converges to one of the twins within a few hundred iterations when vortex phase illumination is used. Using a clockwise or anti-clockwise vortex phase as an initial guess is also seen to deterministically lead to a solution consisting of the corresponding twin. The resultant solution still has some faint residual artifacts that can be addressed via the recently introduced complexity guidance methodology. There is an additional vortex phase in the final solution that can simply be subtracted out to obtain the original test object. The near guaranteed convergence to a twin-stagnation-free solution with vortex illumination as described here is potentially valuable for deploying practical imaging systems that work based on the iterative phase retrieval algorithms.
从其傅里叶变换幅度恢复复值出射波前具有挑战性，这是由于与迭代相位恢复算法相关的停滞问题。在各种停滞伪像中，双像停滞最难解决。直立物体及其倒立且复共轭的孪生像对应于相同的傅里叶幅度数据，因此会在迭代解中同时出现。我们表明，如果使用具有单位电荷涡旋相位的相干光束进行照明，则可以完全消除双停滞问题。与通常的平面波照明情况不同，单位电荷涡旋照明会在零空间频率区域附近故意引入一个孤立的零值，对于大多数自然物体，傅里叶空间中的最大能量通常集中在此处。观察到迭代相位恢复算法的早期迭代会在这个孤立零值附近形成顺时针或逆时针涡旋。一旦解的傅里叶变换在早期迭代中锁定到这个故意引入的强度零值附近的特定涡旋轮廓上，解就会迅速调整到相应的孪生像（直立或倒立），并且在进一步的迭代中不会出现另一个孪生像进入重建的情况。我们使用著名的混合输入输出（HIO）算法进行的模拟研究表明，当使用涡旋相位照明时，解总是在几百次迭代内收敛到其中一个孪生像。使用顺时针或逆时针涡旋相位作为初始猜测也被证明可以确定性地导致由相应孪生像组成的解。所得解仍然有一些微弱的残余伪像，可以通过最近引入的复杂度引导方法来解决。最终解中还有一个额外的涡旋相位，可以简单地减去以获得原始测试物体。如本文所述，使用涡旋照明几乎可以保证收敛到无双停滞解，这对于部署基于迭代相位恢复算法工作的实际成像系统可能具有潜在价值。
