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Gauss–Newton Optimization for Phase Recovery from the Bispectrum James L

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Gauss–Newton Optimization for Phase Recovery from the Bispectrum James L 1 Gauss–Newton Optimization for Phase Recovery from the Bispectrum James L. Herring, James Nagy, and Lars Ruthotto Abstract—Phase recovery from the bispectrum is a central This work focuses on the problem of phase recovery from problem in speckle interferometry which can be posed as an an object’s collected bispectrum. The broader problem of optimization problem minimizing a weighted nonlinear least- phase retrieval occurs in numerous engineering fields and squares objective function. We look at two different formu- lations of the phase recovery problem from the literature, the sciences including astronomy, electronmicroscopy, crys- both of which can be minimized with respect to either the tallography, and optical imaging. For a recent overview of recovered phase or the recovered image. Previously, strategies applications and challenges, we recommend [4]. The prob- for solving these formulations have been limited to gradient lem of phase recovery from the bispectrum has its origins descent or quasi-Newton methods. This paper explores Gauss– in astronomical imaging for low-light images in the visible Newton optimization schemes for the problem of phase recovery from the bispectrum. We implement efficient Gauss–Newton spectrum [2], [3], [5], [6], [7], [8]. More recently, the problem optimization schemes for all the formulations. For the two of has been of interest in multiple applications including long- these formulations which optimize with respect to the recovered range horizontal and slant path imaging [9], [10], [11] and image, we also extend to projected Gauss–Newton to enforce phase recovery in underwater imaging [12], [13]. The general element-wise lower and upper bounds on the pixel intensities problem of phase retrieval from Fourier measurements also of the recovered image. We show that our efficient Gauss– Newton schemes result in better image reconstructions with no continues to be an active area of research in the signal or limited additional computational cost compared to previously processing community [14], [15], [16], [17], [18], [19], [20], implemented first-order optimization schemes for phase recovery [21], [22]. Many phase retrieval applications considered in the from the bispectrum. MATLAB implementations of all methods literature are based on the relationship between an object’s and simulations are made publicly available in the BiBox phase and higher order statistical moments collected from the repository on Github. data (such as the bispectrum) and require solving constrained Index Terms—Phase recovery, bispectrum, bispectral imaging, nonlinear least-squares problems; see, e.g., [16], [21], [22]. Gauss–Newton method Thus while we consider only the problem of phase recovery from the bispectrum, the content of this paper is more broadly applicable to the phase retrieval problem in general. I. INTRODUCTION For phase recovery from the bispectrum, strategies can MAGE blurring due to turbulence poses a significant ob- be separated into two categories: recursive algorithms and I stacle in many applications. One approach to obtaining weighted least-squares problems. Recursive strategies fix a high spatial frequency images of an object through turbulent small set of known phase values near the center of the Fourier optical systems is speckle interferometry, which is built upon domain and use these known phase values along with the col- Labeyrie’s observation that high spatial frequency information lected bispectrum to recursively compute the remainder of the can be recovered from short-exposure images [1]. This work reconstructed phase. Such strategies are well explored in the provided the means to obtain diffraction-limited reconstruc- literature [5], [6], [10], [23], [24], [25], [26], [27], [28]. One tions of an object’s Fourier modulus, or power spectrum, using limitation of recursive strategies is poor performance when short-exposure, photon-limited data. attempting to reconstruct the phase values associated with high arXiv:1812.05152v3 [math.NA] 25 Jun 2019 While an object’s Fourier modulus may be sufficient in frequency information when using noisy data [7]. To improve the case of some simple objects, many applications also on this, the phase recovery problem can be reformulated as require recovery of the object’s Fourier phase in order to a weighted least-squares problem minimizing the mismatch produce high quality images. Thus, phase retrieval is often between the unknown object phase and the collected phase an essential subproblem when using speckle interferometry. of the bispectrum. This least-squares problem is typically Several methods have been developed to solve the phase nonlinear because the bispectrum is collected modulo-2π in retrieval problem, most using relationships defined by high- the range [−π; π], i.e., the bispectrum is “wrapped.” One can order statistical correlation measures such as an object’s triple solve this nonlinear least-squares problem [7], [25], [26], [29], correlation or its Fourier counterpart, the bispectrum [2], [3]. [30], [31], [32]. Alternatively, techniques have been proposed These measures can be collected from the short-exposure data to unwrap the collected bispectrum, e.g., [33], [34], resulting and used to reconstruct an object’s phase. in a linear least-squares problem, but these approaches have been shown to produce inferior results to solving the nonlinear J. Herring, Department of Mathematics, University of Houston, Houston, formulation of the problem [29]. Previous approaches to solv- TX 77004, USA email: [email protected] J. Nagy and L. Ruthotto, Department of Mathematics, Emory University, ing both the linear and nonlinear least-squares formulations Atlanta, GA 30322, email: jnagy,lruthotto @emory.edu have focused on gradient-based first-order and quasi-Newton f g 2 methods such as gradient descent and the limited memory nonlinear least-squares optimization problem. Section III de- BroydenFletcherGoldfarbShanno method (L-BFGS) [7], [29], tails our proposed iterative Gauss–Newton optimization for [30]. solving the phase recovery problem from the previous section. In this paper, we focus on the formulation of phase re- Specifically, we introduce expressions for the gradient and covery from the bispectrum as a nonlinear weighted least- Gauss–Newton approximations to the Hessians for four sepa- squares problem using the wrapped bispectrum. We present rate formulations of the phase recovery problem and discuss Gauss–Newton schemes as an alternative to the gradient-based efficient strategies for solving the linear system associated optimization approaches previously used in the literature. with the Gauss–Newton step within each iteration of the Specifically, we make the following contributions: optimization. Lastly, for two of the problem formulations, we extend our implementation to projected Gauss–Newton • We implement efficient Gauss–Newton optimization which allows for phase recovery while simultaneously im- methods for two nonlinear least-squares formulations of posing pixel-wise intensity bounds on the recovered image. the phase recovery problem from the literature. Both SectionIV presents numerical experiments. We compare our of these formulations can be minimized with respect proposed Gauss–Newton schemes with common first-order to the recovered phase or recovered image, resulting in optimization methods: gradient descent, projected gradient four possible formulations. Our implementations exploit descent, and quasi-Newton L-BFGS method. We also demon- sparsity, matrix reorderings, incomplete factorization, and strate the robustness of our proposed Gauss–Newton schemes the speed of the fast Fourier transform (FFT) to reduce for a range of problem parameters. We end with concluding the cost associated with solving the linear system to remarks in SectionV. calculate the Gauss–Newton step at each iteration of the optimization. The resulting schemes have per iteration costs on the same order of magnitude as gradient descent II. PHASE RECOVERY PROBLEM and quasi-Newton methods like NLCG and L-BFGS but Most techniques for recovering an object’s phase from benefit from faster convergence. This results in improved speckle image data rely on the object’s triple correlation and time-to-solution and lower overall computational cost its Fourier transform, the bispectrum [2], [3]. An object’s compared with previous approaches. triple correlation is a second-order statistical moment given by • For the two formulations that are minimized with respect measuring an object against two independently shifted copies to the resulting image, we also explore the constrained of itself. For a two dimensional object o(x) with x 2 R2, this problem with pixel-wise non-negativity constraints on is expressed by the pixel intensities of the recovered image. We use a ZZ 1 TC ∗ projected Gauss–Newton method to enforce these con- o (x1; x2) = o (x)o(x + x1)o(x + x2)dx: straints. This strategy improves the quality of the recov- −∞ ered image. Additionally, it eliminates the need for a Taking the Fourier transform of this and using the convolution regularizer enforcing non-negativity and offers the option property, we get the object’s bispectrum, to use other regularization options while still enforcing (3) ∗ non-negativity. We demonstrate this for two common O (u; v) = O(u)O(v)O (u + v): regularizers, a discrete gradient regularizer, and a total Here, O(u) is the Fourier transform of the object and u; v 2 variation regularizer. For comparison with constrained 2 R are spatial frequencies. It follows
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