Partitioned Separable Paraboloidal Surrogate Coordinate Ascent Algorithm for Image Restoration

We introduce a new fast converging parallelizable algorithm for image restoration. This algorithm is based on paraboloidal surrogate functions to simplify the optimization problem and a concavity technique developed by De Pierro (1995) to simultaneously update a set of pixels. To obtain large step sizes which affect the convergence rate, we choose the paraboloidal surrogate functions that have small curvatures. The concavity technique is applied to separate pixels into partitioned sets so that parallel processors can be assigned to each set. The partitioned separable paraboloidal surrogates are maximized by using coordinate ascent (CA) algorithms. Unlike other existing algorithms such EM and CA algorithms, the proposed algorithm not only requires less time per iteration to converge, but is guaranteed to monotonically increase the objective function and intrinsically accommodates nonnegativity constraints as well.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85847/1/Fessler161.pd

Image Recovery Using Partitioned-Separable Paraboloidal Surrogate Coordinate Ascent Algorithms

Iterative coordinate ascent algorithms have been shown to be useful for image recovery, but are poorly suited to parallel computing due to their sequential nature. This paper presents a new fast converging parallelizable algorithm for image recovery that can be applied to a very broad class of objective functions. This method is based on paraboloidal surrogate functions and a concavity technique. The paraboloidal surrogates simplify the optimization problem. The idea of the concavity technique is to partition pixels into subsets that can be updated in parallel to reduce the computation time. For fast convergence, pixels within each subset are updated sequentially using a coordinate ascent algorithm. The proposed algorithm is guaranteed to monotonically increase the objective function and intrinsically accommodates nonnegativity constraints. A global convergence proof is summarized. Simulation results show that the proposed algorithm requires less elapsed time for convergence than iterative coordinate ascent algorithms. With four parallel processors, the proposed algorithm yields a speedup factor of 3.77 relative to single processor coordinate ascent algorithms for a three-dimensional (3-D) confocal image restoration problem.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86024/1/Fessler72.pd

Reconstruction from Digital Holograms by Statistical Methods

Conventional numerical reconstruction for digital holography using a filter applied in the spatial frequency domain to extract the primary image may yield suboptimal image quality because of the loss in high-frequency components and interference from other undesirable terms of a hologram. In this paper, we propose a new numerical reconstruction approach using a statistical technique. This approach reconstructs the complex field of the object from the real-valued hologram intensity data. Because holographic image reconstruction is an ill-posed problem, our statistical technique is based on penalized-likelihood estimation. We develop a Poisson statistical model for this problem and derive an optimization transfer algorithm that monotonically decreases the cost function each iteration. Simulation results show that our statistical technique has the potential to improve image quality in digital holography relative to conventional reconstruction techniques.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85863/1/Fessler191.pd

Relaxed Ordered Subsets Algorithm for Image Restoration of Confocal Microscopy

The expectation-maximization (EM) algorithm for maximum-likelihood image recovery converges very slowly. Thus, the ordered subsets EM (OS-EM) algorithm has been widely used in image reconstruction for tomography due to an order-of-magnitude acceleration over the EM algorithm. However, OS-EM is not guaranteed to converge. The recently proposed ordered subsets, separable paraboloidal surrogates (OS-SPS) algorithm with relaxation has been shown to converge to the optimal point while providing fast convergence. In this paper, we develop a relaxed OS-SPS algorithm for image restoration. Because data acquisition is different in image restoration than in tomography, we adapt a different strategy for choosing subsets in image restoration which uses pixel location rather than projection angles. Simulation results show that the order-of-magnitude acceleration of the relaxed OS-SPS algorithm can be achieved in restoration. Thus the speed and the guarantee of the convergence of the OS algorithm is advantageous for image restoration as well.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85875/1/Fessler174.pd

Penalized-Likelihood Image Reconstruction for Digital Holography

Conventional numerical reconstruction for digital holography using a filter applied in the spatial-frequency domain to extract the primary image may yield suboptimal image quality because of the loss in high-frequency components and interference from other undesirable terms of a hologram. We propose a new numerical reconstruction approach using a statistical technique. This approach reconstructs the complex field of the object from the real-valued hologram intensity data. Because holographic image reconstruction is an ill-posed problem, our statistical technique is based on penalized-likelihood estimation. We develop a Poisson statistical model for this problem and derive an optimization transfer algorithm that monotonically decreases the cost function at each iteration. Simulation results show that our statistical technique has the potential to improve image quality in digital holography relative to conventional reconstruction techniques.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85917/1/Fessler60.pd

Relaxed Ordered-Subset Algorithm for Penalized-Likelihood Image Restoration

The expectation-maximization (EM) algorithm for maximum-likelihood image recovery is guaranteed to converge, but it converges slowly. Its ordered-subset version (OS-EM) is used widely in tomographic image reconstruction because of its order-of-magnitude acceleration compared with the EM algorithm, but it does not guarantee convergence. Recently the ordered-subset, separable-paraboloidal-surrogate (OS-SPS) algorithm with relaxation has been shown to converge to the optimal point while providing fast convergence. We adapt the relaxed OS-SPS algorithm to the problem of image restoration. Because data acquisition in image restoration is different from that in tomography, we employ a different strategy for choosing subsets, using pixel locations rather than projection angles. Simulation results show that the relaxed OS-SPS algorithm can provide an order-of-magnitude acceleration over the EM algorithm for image restoration. This new algorithm now provides the speed and guaranteed convergence necessary for efficient image restoration.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85918/1/Fessler68.pd

Statistical Image Recovery Techniques for Optical Imaging Systems

Statistical techniques are very attractive for image recovery because they can incorporate the physical model of imaging systems, thus improving the quality of recovered images. To overcome the ill-posed nature of image recovery, one often uses penalized-likelihood estimation. Since closed-form solutions for these statistical techniques are unavailable, iterative algorithms are needed. However, existing algorithms lack one or more desirable properties, such as the guarantee of convergence, rapid convergence, and efficient computation. In the first part of the dissertation, we present a new, fast-converging algorithm called partitioned-separable paraboloidal surrogate coordinate ascent (PPCA). This algorithm captures the fast convergence of iterative coordinate ascent algorithms, while remaining parallelizable to reduce computation time. The PPCA algorithm is based on paraboloidal surrogate functions and a concavity technique. It is most beneficial when applied to space-variant systems for which the fast Fourier transform (FFT) is inapplicable. Because our primary applications are confocal microscopy and image plane holography, for which space-invariance of the systems is usually assumed, in the second part of the dissertation, we develop another algorithm that can be used with the FFT for fast computation time. We adapt the relaxed ordered-subset separable paraboloidal surrogate (OS-SPS) algorithm, which was originally invented for projection-based tomographic reconstruction, to pixel-based image restoration. The relaxed OS-SPS algorithm provides very fast initial convergence and is guaranteed to converge to the optimal solution. Furthermore, we develop different strategies for choosing subsets and efficient implementation. Both the PPCA and relaxed OS-SPS algorithms can be applied to many imaging problems; here we demonstrate their use for confocal microscopy problems. In the third and last part of the dissertation, we develop a new statistical image reconstruction technique for digital holography including image plane holography. This approach reconstructs the complex object field from real-valued hologram intensity data. We develop a Poisson statistical model for this problem and derive an optimization transfer algorithm that monotonically decreases the cost function at each iteration and ensures convergence to a local minimum. Our statistical technique is shown to improve image quality in simulated digital holography relative to conventional numerical reconstruction using a filter applied in the frequency domain.Ph.D.Applied SciencesElectrical engineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/123709/2/3096206.pd