A sequential iteration algorithm with non-monotoneous behaviour in the method of projections onto convex sets

In image coding, image information can often be transmitted with relatively little bandwidth requirement. This is the case in the pixel domain if there are large blocks of homogeneous intensity within a single frame, or when comparing adjacent frames there are large blocks of unchanged pixels. In the frequency domain, this is the case when only a few of the frequency coefficients are deemed significant. This paper presents an algorithm for improving the bandwidth compression of the error pattern generated by typical motion-compensated coding techniques. The algorithm involves two well established techniques; conventional transform coding and the method of successive Projection onto Convex Sets.

To design efficient parallel algorithms, some recent papers showed that many sequential iterative algorithms can be directly parallelized but there are still challenges in achieving work-efficiency and high-parallelism. Work-efficiency can be hard for certain problems where the number of dependences is asymptotically more than optimal sequential work bound. To achieve high-parallelism, we want to process as many objects as possible in parallel. The goal is to achieve $\tilde{O}(D)$ span for a problem with the deepest dependence length $D$. We refer to this property as round-efficiency. In this paper, we show work-efficient and round-efficient algorithms for a variety of classic problems and propose general approaches to do so. To efficiently parallelize many sequential iterative algorithms, we propose the phase-parallel framework. The framework assigns a rank to each object and processes them accordingly. All objects with the same rank can be processed in parallel. To enable work-efficiency and high parallelism, we use two types of general techniques. Type 1 algorithms aim to use range queries to extract all objects with the same rank, such that we avoid evaluating all the dependences. We discuss activity selection, unlimited knapsack, and more using Type 1 framework. Type 2 algorithms aim to wake up an object when the last object it depends on is finished. We discuss activity selection, longest increasing subsequence (LIS), and many other algorithms using Type 2 framework. All of our algorithms are (nearly) work-efficient and round-efficient. Many of them improve previous best bounds, and some of them are the first to achieve work-efficiency with round-efficiency. We also implement many of them. On inputs with reasonable dependence depth, our algorithms are highly parallelized and significantly outperform their sequential counterparts.

Robust Adaptive Control

The method of projection onto convex sets is extended to the problem of deconvolution in the presence of noise. A collection of convex sets is defined by using properties of the noise and of the ideal signal. The ideal signal is then a member of the intersection of these convex sets. A solution to the deconvolution problem is to choose a member of this intersection. Such a member is obtained by successive projections onto convex sets. If the intersection is small, then any member of the intersection should be a good estimate. One and two dimensional results have shown that this method is effective in producing estimates that are superior to conventional methods when sufficient a priori knowledge is available to define the convex sets.

This paper is devoted to the problem of solar image recovery in synthesizing radio brightness maps in the case of an incomplete filling of the spectral region (the Fourier data are bounded by a cone with an angle much smaller than 180 deg.) The image was recovered by the method of projection onto closed convex sets (PCCS). Brief theoretical postulates which formed the basis for this method are presented. A comparative characteristic with the CLEAN method is given. The results of using this method for the model data and real data (recovery of the solar radio image obtained in the knife-edge beams of the SSRT) are presented.

Binary steering in discrete tomography reconstruction with sequential and simultaneous iterative algorithms

We study some methods of subgradient projections for solving a convex feasibility problem with general (not necessarily hyperplanes or half-spaces) convex sets in the inconsistent case and propose a strategy that controls the relaxation parameters in a specific self-adapting manner. This strategy leaves enough user-flexibility but gives a mathematical guarantee for the algorithm's behavior in the inconsistent case. We present numerical results of computational experiments that illustrate the computational advantage of the new method.

Introduction. This paper gives counterexamples for two unrelated conjectures which pertain to convexity theory. Supposef is any function defined on a closed convex set C in finitedimensional Euclidean space. Its envelope function env f is defined as the pointwise supremum of all linear functions which f dominates everywhere. It is elementary and well known that env f is convex, and hence continuous in the relative interior of C. It is similarly known that env f is lower-semicontinuous on the boundary of C. Witsenhausen [14] uses and studies envelope functions. He suggested the following orally: CONJECTURE 1. Iff is continuous everywhere on C, then envf is also continuous everywhere on C. I provide a very simple counterexample in 3-space which shows that this is not true. Incidentally, though the proof is not on paper, I feel fairly certain that Witsenhausen's conjecture is correct in 2-space. If C is a closed convex set in finite-dimensional Euclidean space, then for any point x in the space there is a unique point P(x) in C which is nearest to x. The mapping P is called the projection onto C. (Many interesting results about this mapping are given in Phelps [9], [10].) It is easy to prove and well known that P is a contraction mapping, hence continuous. It is also easy to see that P need not have much in the way of differentiability properties, for if C is a polyhedron in 2-space, then it is easy to find points and directions at which P does not even have a two-sided directional derivative. The existence in general of a one-sided directional derivative, however, is not so easy to decide. This question does not seem to be raised in such natural places as Eggleston [4] and Valentine [11]. CONJECTURE 2. For all points and directions, P has a one-sided directional derivative. My mildly complicated counterexample is a set C (in 3-space), which is the convex hull of a countable infinity of points. I suspect that the conjecture is correct in 2-space. In n-space it would be interesting to know whether the set of points and directions at which the conjecture holds is dense in EnXSn-. If not, the convex set would have to be extremely jagged!

Here we analyze three dimensional analogues of the classic Crofton formula for planar compact convex sets. In this formula a fundamental role is played by the visual angle of the convex set from an exterior point. A generalization of the visual angle to convex sets in the Euclidean space is the visual solid angle. This solid angle, being an spherically convex set in the unit sphere, has perimeter, area and other geometric quantities to be considered. The main goal of this note is to express invariant quantities of the original convex set depending on volume, surface area and mean curvature integral by means of integrals of functions related to the solid angle.

This paper considers the problem of limited angle tomography in which a complete sinogram is not available and proposes a linear extrapolator to extrapolate the missing part of the sinogram. Once the complete sinogram is obtained via extrapolation, standard reconstruction techniques can be used to generate artifact free reconstructions. The linear extrapolator is designed using the sampling theory of signals with non-compact spectral support, the knowledge that complete sinograms have bowtie-shaped spectral support, and regularization. Once the extrapolator is designed, it can be used to extrapolate any incomplete sinogram. Since the sinogram is non-iterative, it is much more efficient than other iterative algorithms such as the method of projection onto convex sets, the Papoulis-Gerchberg's algorithm and the Clark-Palmer-Lawrence interpolation method.

The rate of convergence for the cyclic projections algorithm III: Regularity of convex sets

Diffraction, Fourier Optics and Imaging

We study the convergence of an iterative projection/reflection algorithm originally proposed for solving what are known as phase retrieval problems in optics. There are two features that frustrate any analysis of iterative methods for solving the phase retrieval problem: nonconvexity and infeasibility. The algorithm that we developed, called relaxed averaged alternating reflections (RAAR), was designed primarily to address infeasibility, though our strategy has advantages for nonconvex problems as well. In the present work we investigate the asymptotic behavior of the RAAR algorithm for the general problem of finding points that achieve the minimum distance between two closed convex sets in a Hilbert space with empty intersection, and for the problem of finding points that achieve a local minimum distance between one closed convex set and a closed prox-regular set, also possibly nonintersecting. The nonconvex theory includes and expands prior results limited to convex sets with nonempty intersection. To place the RAAR algorithm in context, we develop parallel statements about the standard alternating projections algorithm and gradient descent. All of the various algorithms are unified as instances of iterated averaged alternating proximal reflectors applied to a sum of regularized maximal monotone mappings.

Image reconstruction can be formulated by the Fredholm equation of the first kind. The method of projections onto convex sets (POCS) is an iterative algorithm for solving the equation. Multiplicative algebraic reconstruction techniques (MART) is one of POCS for solving a system of simultaneous equation. By discretizing the image reconstruction problem, we applied the MART to the problems and evaluate the image quality in computer simulations. We also investigated the normalized mean square error of reconstructed images with respect to the variations of the number of detectors and views, the relaxation parameters.

We present a rank reduction algorithm that permits simultaneous reconstruction and random noise attenuation of seismic records. We based our technique on multichannel singular spectrum analysis (MSSA). The technique entails organizing spatial data at a given temporal frequency into a block Hankel matrix that in ideal conditions is a matrix of rank k, where k is the number of plane waves in the window of analysis. Additive noise and missing samples will increase the rank of the block Hankel matrix of the data. Consequently, rank reduction is proposed as a means to attenuate noise and recover missing traces. We present an iterative algorithm that resembles seismic data reconstruction with the method of projection onto convex sets. In addition, we propose to adopt a randomized singular value decomposition to accelerate the rank reduction stage of the algorithm. We apply MSSA reconstruction to synthetic examples and a field data set. Synthetic examples were used to assess the performance of the method in two reconstruction scenarios: a noise-free case and data contaminated with noise. In both cases, we found extremely low reconstructions errors that are indicative of an optimal recovery. The field data example consists of a 2D prestack volume that depends on common midpoint and offset. We use the MSSA reconstruction method to complete missing offsets and, at the same time, increase the signal-to-noise ratio of the seismic volume.