This paper introduces an expectation-maximization (EM) algorithm for image restoration (deconvolution) based on a penalized likelihood formulated in the wavelet domain. Regularization is achieved by promoting a reconstruction with low-complexity, expressed in terms of the wavelet coecients, taking advantage of the well known sparsity of wavelet representations. Previous works have investigated wavelet-based restoration but, except for certain special cases, the resulting criteria are solved approximately or require very demanding optimization methods. The EM algorithm herein proposed combines the efficient image representation oered by the discrete wavelet transform (DWT) with the diagonalization of the convolution operator obtained in the Fourier domain. The algorithm alternates between an E-step based on the fast Fourier transform (FFT) and a DWT-based M-step, resulting in an ecient iterative process requiring O(N log N) operations per iteration. Thus, it is the rst image restoration algorithm that optimizes a wavelet-based penalized likelihood criterion and has computational complexity comparable to that of standard wavelet denoising or frequency domain deconvolution methods. The convergence behavior of the algorithm is investigated, and it is shown that under mild conditions the algorithm converges to a globally optimal restoration. Moreover, our new approach outperforms several of the best existing methods in benchmark tests, and in some cases is also much less computationally demanding.

. In this paper we show that the lagged diffusivity fixed point algorithm introduced by Vogel and Oman in [10] to solve the problem of Total Variation denoising, proposed by Rudin, Osher and Fatemi in [9], is a particular instance of a class of algorithms introduced by Eckhardt and Voss in [11], whose origins can be traced back to Weiszfeld's original work for minimizing a sum of Euclidean lengths [12]. There have recently appeared several proofs for the convergence of this algorithm [2], [3], [6]. Here we present a proof of the global and linear convergence using the framework introduced in [11] and give a bound for the convergence rate of the fixed point iteration that agrees with our experimental results. These results are also valid for suitable generalizations of the fixed point algorithm. 1. Introduction. Recently, a new class of nonlinear PDE based techniques has emerged for image restoration problems, primarily because they preserve sharp edges better. A particularly popular te...

We study the problem of coordinating a set of autonomous mobile robots that can freely move in a two-dimensional plane; in particular, we want them to gather at a point not fixed in advance (GATHERING PROBLEM). We introduce a model of weak robots (decentralized, asynchronous, no common knowledge, no identities, no central coordination, no direct communication, oblivious) which can observe the set of all points in the plane which are occupied by other robots. Based on this observation, a robot uses a deterministic algorithm to compute a destination, and moves there. We prove that these robots are too weak to gather at a point in finite time. Therefore, we strengthen them with the ability to detect whether more than one robot is at a point (multiplicity). We analyze the GATHERING PROBLEM for these stronger robots. We show that the problem is still unsolvable if there are only two robots in the system. For 3 and 4 robots, we give algorithms that solve the GATHERING PROBLEM. For more than 4 robots, we present an algorithm that gathers the robots in finite time if they are not in a specific symmetric configuration at the beginning (biangular configuration). We show how to solve such initial configurations separately. However, the general solution of the GATHERING PROBLEM remains an open problem.

A linear wirelength objective more e#ectively captures timing, congestion, and other global placement considerations than a squared wirelength objective. The GORDIAN-L cell placement tool #16# minimizes linear wirelength by #rst approximating the linear wirelength objectiveby a modi#ed squared wirelength objective, then executing the following loop # #1# minimize the current objective to yield some approximate solution, and #2# use the resulting solution to construct a more accurate objective#until the solution converges. In this paper, we #rst show that the GORDIAN-L loop can be viewed as a special case of a new algorithm that generalizes a 1937 iteration due to Weiszfeld #19#. Speci#- cally,we formulate the Weiszfeld iteration using a regularization parameter to control the tradeo# between convergence and solution accuracy; the GORDIAN-L iteration is equivalent to setting this regularization parameter to zero. Other novel numerical methods described in the paper, the Primal Newton it...

In this paper, we propose a method for robust kernel density estimation. We interpret a KDE with Gaussian kernel as the inner product between a mapped test point and the centroid of mapped training points in kernel feature space. Our robust KDE replaces the centroid with a robust estimate based on M-estimation [1]. The iteratively re-weighted least squares (IRWLS) algorithm for M-estimation depends only on inner products, and can therefore be implemented using the kernel trick. We prove the IRWLS method monotonically decreases its objective value at every iteration for a broad class of robust loss functions. Our proposed method is applied to synthetic data and network traffic volumes, and the results compare favorably to the standard KDE. Index Terms — kernel density estimation, M-estimator, outlier, kernel feature space, kernel trick 1.

Abstract. Consider a set of n> 2 simple autonomous mobile robots (decentralized, asynchronous, no common coordinate system, no identities, no central coordination, no direct communication, no memory of the past, deterministic) moving freely in the plane and able to sense the positions of the other robots. We study the primitive task of gathering them at a point not fixed in advance (Gathering Problem). In the literature, most contributions are simulation-validated heuristics. The existing algorithmic contributions for such robots are limited to solutions for n ≤ 4 or for restricted sets of initial configurations of the robots. In this paper, we present the first algorithm that solves the Gathering Problem for any initial configuration of the robots. 1

The design of the distribution system is a strategic issue for almost every company. The problem of locating facilities and allocating customers covers the core topics of distribution system design. Model formulations and solution algorithms which address the issue vary widely in terms of fundamental assumptions, mathematical complexity and computational performance. This paper reviews some of the contributions to the current state-of-the-art. In particular, continuous location models, network location models, mixed-integer programming models, and applications are summarized.

This article presents new heuristic methods for solving a class of hard centroid clustering problems including the f-median, the sum-of-squares clustering and the multi-source Weber problems. Centroid clustering is to partition a set of entities into a given number of subsets and to find the location of a centre for each subset in such a way that a dissimilarity measure between the entities and the centres is minimized. The first method proposed is a candidate list search that produces good solutions in a short amount of time if the number of centres in the problem is not too large. The second method is a general local optimization approach that finds very good solutions. The third method is designed for problems with a large number of centres; it decomposes the problem into subproblems that are solved independently. Numer- ical results show that these methods are efficient --- dozens of best solutions known to problem instances of the literature have been improved--- and fast, handling problem instances with more than 85'000 entities and 15'000 centres ---much larger than those solved in the literature. The expected complexity of these new procedures is discussed and shown to be comparable to that of an existing method which is known to be very fast.

We introduce a Locally Optimal Projection operator (LOP) for surface approximation from point-set data. The operator is parameterization free, in the sense that it does not rely on estimating a local normal, fitting a local plane, or using any other local parametric representation. Therefore, it can deal with noisy data which clutters the orientation of the points. The method performs well in cases of ambiguous orientation, e.g., if two folds of a surface lie near each other, and other cases of complex geometry in which methods based upon local plane fitting may fail. Although defined by a global minimization problem, the method is effectively local, and it provides a second order approximation to smooth surfaces. Hence allowing good surface approximation without using any explicit or implicit approximation space. Furthermore, we show that LOP is highly robust to noise and outliers and demonstrate its effectiveness by applying it to raw scanned data of complex shapes.

Good heuristic solutions for large Multisource Weber problems can be obtained by solving related p-median problems in which potential locations of the facilities are users locations and then solving Weber problems for the sets of users of each facility.