Created on Wed Sep 9 21:30:50 2020
- ptychography40.stitching.stitching.stitch(objects, wtype='weighted', threshold=1e-12)¶
In paralelized ptychography, each node returns recovered part of the object up to a phase factor. Our goal is to stich those pieces by synchronizing phase factors. To do so, we solve the angular synchronization, a recovery of the unknown phase factors from their (noisy) pairwise differences To obtain these pairwise differences, we compute the inner product between two overlapping parts. Since on the overlap two objects are approximately the same, it gives us the phase difference. Mathematically, it is given by < e^(i a_r1) O_r1, e^(i a_r2) O_r2 > approx. = e^(i [a_r1 - a_r2]) norm(O_r1,r_2)^2, where O_r is recovered r-th part of the object, O_r1,r2 - part of the object corresponging to the overlap of two parts, a_r - unknown phase factor.
This inner products are stored in the ph_diff matrix. Then, the matrix is entrywise normalized to separate phases e^(i [a_r1 - a_r2]) and non-negative norm(O_r1,r_2)^2.
In angular synchronization we take phase differences as data and can use norms as weights to solve (un)weighted least squares problem. Problem itself is NP-hard and, thus, its relaxation instead is solved, which is simply a computation of the eigenvector corresponding to the smallest eigenvalue of the graph laplacian matrix.
Obtained eigenvector (v in code) is entrywise normalized to get phases. Each part of the object is then phase-rotated correspondingly. Finally, we join the pieces, avaraging on the overlaps.
Note: obtained solution is the original object up to a globabl phase factor.
For more details see Section 2 of https://arxiv.org/pdf/2005.02032.pdf
Possible speed up of the routine: avoid dot product calculation of between all parts, but only those which overlap.
objects (numpy.ndarray) – d1 X d2 X R array containing recovered pieces of object, required to be stitched
wtype (str, optional) – Selected type of weights to be used in the angular synchronization There are 2 possible choices: ‘unweighted’ or ‘weighted’, depending on usage or not of the weights for angular synchronization (explained above)
threshold (float, optional) – Minimum absolute value to be considered in finding the phase match for stitching.