A matrix-free Levenberg-Marquardt algorithm forefficient ptychographic phase retrieval
Version 2 2021-07-07, 19:16
Version 1 2021-07-07, 19:15
Posted on 2021-07-07 - 19:15
The phase retrieval problem, where one aims to recover a
complex-valued image from far-field intensity
measurements, is a classic problem encountered in a range of imaging applications. Modern phase
retrieval approaches usually rely on gradient descent methods in a nonlinear minimization framework.
Calculating closed-form gradients for use in these methods is tedious work, and formulating second order
derivatives is even more laborious. Additionally, second order techniques often require the storage and
inversion of large matrices of partial derivatives, with memory requirements that can be prohibitive for data-rich imaging modalities.
We use a reverse-mode automatic differentiation (AD) framework to
implement an efficient matrix-free version of the Levenberg-Marquardt (LM) algorithm, a
longstanding method that finds popular use in nonlinear least-square
minimization problems but which has seen little use in
phase retrieval.
Furthermore, we extend the basic LM algorithm so that it can be applied for more
general constrained optimization problems (including phase retrieval problems) beyond just the least-square
applications. Since we use AD, we only need to specify the physics-based forward model
for a specific imaging application; the first and second-order derivative terms are calculated
automatically through matrix-vector products, without explicitly forming the large Jacobian
or Gauss-Newton matrices typically required for the LM method.
We demonstrate that this algorithm can be used to solve both the unconstrained ptychographic object retrieval
problem and the constrained "blind" ptychographic object and probe retrieval problems, under the popular Gaussian noise
model as well as the Poisson noise model. We compare this algorithm to state-of-the-art first order
ptychographic reconstruction methods to demonstrate empirically that this method outperforms best-in-class first-order methods: it provides
excellent convergence guarantees with (in many cases) a superlinear
rate of convergence, all with a computational cost comparable to, or lower than, the tested first-order algorithms.
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Kandel, Saugat; Maddali, Siddharth; Nashed, Youssef; Hruszkewycz, Stephan; Jacobsen, Chris; Allain, Marc (2021). A matrix-free Levenberg-Marquardt algorithm forefficient ptychographic phase retrieval. Optica Publishing Group. Collection. https://doi.org/10.6084/m9.figshare.c.5453577.v1
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AUTHORS (6)
SK
Saugat Kandel
SM
Siddharth Maddali
YN
Youssef Nashed
SH
Stephan Hruszkewycz
CJ
Chris Jacobsen
MA
Marc Allain