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The illuminating probes a(r) used in ptychographic reconstructions of the cameraman and gold ball images.
Source publication
Ptychography promises diffraction limited resolution without the need for
high resolution lenses. To achieve high resolution one has to solve the phase
problem for many partially overlapping frames. Here we review some of the
existing methods for solving ptychographic phase retrieval problem from a
numerical analysis point of view, and propose alte...
Contexts in source publication
Context 1
... this section, we show the convergence behavior of different iterative algorithms we discussed in section 3 by numerical experiments. In the cameraman image reconstruc- tion experiment, we choose the illuminating probe a(r) to be a 64 × 64 binary probe shown in Figure 4(a). The pixels within the 32 × 32 square at the center of the probe assume the value of 1. ...
Context 2
... stack contains a set of 64 × 64 diffraction frames. These frames are generated by translating the probe shown in Figure 4(b) by different amount in horizontal and vertical directions. The larger the translation, the smaller the overlap is between two adjacent images. ...
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Citations
... Here η is a damping term that is set to 0.7 by default [47]. Similar to conjugate gradient solvers [48,55,106], the momentum term accelerates the search direction and prevents zigzag motion towards the optimum. We emphasize that O n+1,oFOV in this subsection denotes the entire probe field of view, while in other subsection O is an object box of the same size as the probe window. ...
Conventional (CP) and Fourier (FP) ptychography have emerged as versatile quantitative phase imaging techniques. While the main application cases for each technique are different, namely lens-less short wavelength imaging for CP and lens-based visible light imaging for FP, both methods share a common algorithmic ground. CP and FP have in part independently evolved to include experimentally robust forward models and inversion techniques. This separation has resulted in a plethora of algorithmic extensions, some of which have not crossed the boundary from one modality to the other. Here, we present an open source, cross-platform software, called PtyLab, enabling both CP and FP data analysis in a unified framework. With this framework, we aim to facilitate and accelerate cross-pollination between the two techniques. Moreover, the availability in Matlab, Python, and Julia will set a low barrier to enter each field.
... Phase retrieval is a non-linear inverse problem that has been studied for long time since it is of interest in a number of fields such as X-ray crystallography [15], microscopy [16], imaging [17], etc. Classically, it is addressed by non-convex approaches like iterated projection methods [18] and least squares methods [19] which may suffer from false solutions (local minima, saddle points) [20]. ...
The problem of detecting defective turned-off elements in antenna arrays from amplitude-only data is addressed. Commonly used antenna diagnostics methods, such as the back transformation method (BTM) and the matrix method (MM), exploit amplitude and phase field data. Here, instead, the diagnostics is cast as the recovery of a real (binary) signal from few amplitude measurements of the near-field. Therefore, it is basically a phase retrieval problem. Taking inspiration from the recently introduced PhaseMax algorithm, the phase retrieval is formulated as a convex optimization. In particular, PhaseMax is adapted to the faults detection problem by removing the need to estimate a reference solution. Moreover, it is shown that the convex optimization is equivalent to a sparse minimization problem which allows to employ all the powerful tools of compressive sensing realm. Preliminary numerical results are presented to assess the achievable performance as the number of faulty elements increase. Finally, a strategy that reduces the number of measurement points by employing steering diversities is presented and checked.
... In general, ER does not converge globally to {αx : |α| = 1} [12, p.830]. If the loss L 2 is differentiable at z t , the ER iteration will not increase the value of L 2 , i.e., L 2 (z t+1 ) ≤ L 2 (z t ) [12,38]. ...
In this paper, we consider two iterative algorithms for the phase retrieval problem: the well-known Error Reduction method and the Amplitude Flow algorithm, which performs minimization of the amplitude-based squared loss via the gradient descent. We show that Error Reduction can be interpreted as a scaled gradient method applied to minimize the same amplitude-based squared loss, which allows to establish its convergence properties. Moreover, we show that for a class of measurement scenarios, such as ptychography, both methods have the same computational complexity and sometimes even coincide.
... A comparison between the amplitude loss function and the intensity loss function is provided in [60]. Sometimes the function to be minimized is defined by considering a generic l p norm instead of the usual l 2 norm. ...
In the framework of inverse problems in electromagnetics, the inverse source problem plays an important role. The latter consists in recovering the source current from the knowledge of its radiated field and, from a mathematical point of view, it requires the inversion of a linear integral operator called radiation operator.
Despite the performances of electronic instrumentation are continuously improved, especially at high frequencies, accurate phase measurements of the radiated field may become difficult. For such reason, the need of retrieving the source current from the intensity only of the radiated field arises. Such a task falls into the mathematical problem of phase retrieval. Although the inverse source problem is linear, the absence of phase information makes the problem nonlinear; hence, the task of finding a solution becomes more difficult.
With aim to find the solution to the problem two steps are required. The first is to discretize the continuous model efficiently. The second one is to choose a solution algorithm. Here, a solution procedure based on the minimization of a quartic functional given by the correspondent least square problem is considered. Despite such a method is feasible also for problems of large scale, the solution procedure may trap into false solution which can be divided into local minima and saddle points with null gradient.
In this thesis, with reference to 2D scalar geometries, both the issue of discretizing the continuous model and the question of false solutions are addressed by underlying the key role played by the dimension of data space. The latter can be defined as the minimum number of independent functions required to represent the phaseless data with a given degree of accuracy.
As concerns the discretization issue, at first, the minimum number of phaseless measurements required to discretize the continuous model without loss of information is found by analitically computing the dimension of data space. After, a discrete model that shares the same mathematical properties of the continuous one is found by exploiting the lifting technique and the sampling theory approach.
With regard to the traps issue, at first, it has been recalled from the literature that when the dimension of data increases, the manifold of data flattens, and the false solutions in the functional tend to disappear. Then, with reference to some classes of unknown sequences, the minimum value of the data space dimension for which no false solutions should exist in the functional has been found. In such conditions, the traps issue is overcome and the converge of phase retrieval via quartic functional minimization is ensured also from a totally random
starting point.
... For this case, researchers have proposed a variety of iterative solution methods, with most of them consisting of first-order gradient-based iterations. First-order strategies [4,19,[21][22][23][24][25][26] provide updates that are relatively easy to compute, but they are limited in their convergence speed [27,28]. While recent second-order iterative methods [27][28][29] address this convergence speed issue, their use in ptychography is strongly impaired by the high computational cost per iteration required to evaluate the second-order derivative matrix at each iterative update step [30]. ...
... First-order strategies [4,19,[21][22][23][24][25][26] provide updates that are relatively easy to compute, but they are limited in their convergence speed [27,28]. While recent second-order iterative methods [27][28][29] address this convergence speed issue, their use in ptychography is strongly impaired by the high computational cost per iteration required to evaluate the second-order derivative matrix at each iterative update step [30]. Furthermore, knowledge of the true nature of the probe as it illuminates the sample is difficult and in some cases impossible to ascertain with sufficient accuracy for the unconstrained ptychography problem. ...
... For the sake of clarity, we call Eq. (6) the "standard ptychographic reconstruction" problem (SPR). The problem above, equipped with the functional f g , is an example of the "coded diffraction pattern" problem that has been extensively analyzed in recent phase retrieval literature [23,24,27]. If the structure of the probing field is not completely known, we need to retrieve the object as well as the probe from the diffraction dataset. ...
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.
... where T stands for the radiation operator. One of the most popular algorithms to find a solution consists in the minimization of a cost functional obtained by the least-squares problem associated to Equation (1) (see [21,22]). Since the functional to be minimized is quartic, such a method may suffer from the local minima problem. ...
In this article, the question of how to sample the square amplitude of the radiated field in the framework of phaseless antenna diagnostics is addressed. In particular, the goal of the article is to find a discretization scheme that exploits a non-redundant number of samples and returns a discrete model whose mathematical properties are similar to those of the continuous one.
To this end, at first, the lifting technique is used to obtain a linear representation of the square amplitude of the radiated field.
Later, a discretization scheme based on the Shannon sampling theorem is exploited to discretize the continuous model. More in detail, the kernel of the related eigenvalue problem is first recast as the Fourier transform of a window function, and after, it is evaluated. Finally, the sampling theory approach is applied to obtain a discrete model whose singular values approximate all the relevant singular values of the continuous linear model.
The study refers to a strip source whose square magnitude of the radiated field is observed in the Fresnel zone over a 2D observation domain.
... For this case, researchers have proposed a variety of iterative solution methods, with most of them consisting of first-order gradient-based iterations. First-order strategies [4,19,21,22,23,24,25,26] provide updates that are relatively easy to compute, but they are limited in their convergence speed [27,28]. While recent second-order iterative methods [27,28,29] address this convergence speed issue, their use in ptychography is strongly impaired by the high computational cost per iteration required to evaluate the second-order derivative matrix at each iterative update step [30]. ...
... First-order strategies [4,19,21,22,23,24,25,26] provide updates that are relatively easy to compute, but they are limited in their convergence speed [27,28]. While recent second-order iterative methods [27,28,29] address this convergence speed issue, their use in ptychography is strongly impaired by the high computational cost per iteration required to evaluate the second-order derivative matrix at each iterative update step [30]. Furthermore, knowledge of the true nature of the probe as it illuminates the sample is difficult and in some cases impossible to ascertain with sufficient accuracy for the unconstrained ptychography problem. ...
... For the sake of clarity, we call Eq. (2.6) the "standard ptychographic reconstruction" problem (SPR). The problem above, equipped with the functional f g , is an example of the "coded diffraction pattern" problem that has been extensively analyzed in recent phase retrieval literature [23,24,27]. If the structure of the probing field is not completely known, we need to retrieve the object as well as the probe from the diffraction dataset. ...
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 general constrained optimization 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 derivative terms are calculated automatically through matrix-vector products, without explicitly forming any large Jacobian or Gauss-Newton matrices. 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 both the Gaussian and Poisson noise models, and that this method outperforms best-in-class first-order ptychographic reconstruction 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.
... However, the reconstruction quality of iterative phase retrieval algorithms degrades as the overlap between the successively captured images in the Fourier domain decreases [9]. Reconstruction under less overlap becomes even more challenging for incomplete amplitude and phase information. ...
Fourier ptychography is a recently explored imaging method for overcoming the diffraction limit of conventional cameras with applications in microscopy and yielding high-resolution images. In order to splice together low-resolution images taken under different illumination angles of coherent light source, an iterative phase retrieval algorithm is adopted. However, the reconstruction procedure is slow and needs a good many of overlap in the Fourier domain for the continuous recorded low-resolution images and is also worse under system aberrations such as noise or random update sequence. In this paper, we propose a new retrieval algorithm that is based on convolutional neural networks. Once well trained, our model can perform high-quality reconstruction rapidly by using the graphics processing unit. The experiments demonstrate that our model achieves better reconstruction results and is more robust under system aberrations.
... where a H m indicates the conjugate transpose of the vector a m . Phase retrieval arises not only in inverse electromagnetic problems [4]- [13] but also in optics [14], [15], in electron microscopy [16], in X-ray crystallography [17], [18], in quantum mechanics, in astronomy [19], and in many other fields of sciences and engineering [20], [21]. In general, this matter is tackled when phase measurements cannot be performed as happens at high frequencies. ...
The reconstruction of the field radiated by a source
from square amplitude only data falls into the realm of phase
retrieval. In this paper we tackle phase retrieval with two different
approaches.
The first one is based on a convex optimization problem called
Phaselift. The latter exploits the lifting technique to recast phase
retrieval as a linear problem with an increased number of
unknowns and then, because the linear problem is highly undetermined, it adds further constraints (based on the mathematical
properties of the solution) to estimate it.
The second approach formulates phase retrieval as a least squares
problem therefore it requires to tackle the minimization of a
quartic functional which will be carried out by applying a
gradient descent method.
In the second part of the paper, in order to corroborate the
effectiveness of both approaches, we present numerical results.
Afterwards, we provide a comparison between the two methods
and finally we emphasize how the ratio between the number of
independent data and the number of unknowns impacts on the
performance in both approaches.
... This term introduces a penalty to the minimization function to yield a solution consistent with the one suggested by the prior information. The likelihood function in Eq. (3) is commonly approximated by a Gaussian because, by doing so, Eq. (5) reduces to a penalized least-squares (LS) problem [40], ...
We introduce a Bayesian framework for the ptychotomographic imaging of 3D objects under photon-limited conditions. This approach is significantly more robust to measurement noise by incorporating prior information on the probabilities of the object features and the measurement process into the reconstruction problem, and it can improve both the temporal and spatial resolution of current and future ptychography instruments. We use the alternating direction method of multipliers to solve the proposed optimization problem, where the ptychography and tomography subproblems are both solved using the conjugate-gradient method. The effectiveness of the framework is demonstrated by reconstructing a synthetic 3D integrated chip with a significantly reduced photon-budget for the simulated experiment.
