We propose an effective framework for multi-phase image segmentation and semi-supervised data clustering by introducing a novel region force term into the Potts model. Assume the probability that a pixel or a data point belongs to each class is known a priori. We show that the corresponding indicator function obeys the Bernoulli distribution and the new region force function can be computed as the negative log-likelihood function under the Bernoulli distribution. We solve the Potts model by the primal-dual hybrid gradient method and the augmented Lagrangian method, which are based on two different dual problems of the same primal problem. Empirical evaluations of the Potts model with the new region force function on benchmark problems show that it is competitive with existing variational methods in both image segmentation and semi-supervised data clustering.

Rafael Molina Departamento de Ciensas de la Computacion e I. A., Universidad de Granada, Granada, Spain e-mail: rms@decsai.ugr.esImages are ubiquitous and indispensable in science and everyday life. Mirroring the abilities of our own human visual system, it is natural to displayobservations of the world in graphical form. Images are obtained in areas ranging from everyday photography to astronomy, remote sensing, medical imaging, and microscopy. In each case, there is an underlying object or scene we wish to observe; the original or true image is the ideal representation of the observed scene. Yet the observation process is never perfect: there is uncertainty in the mea-surements, occurring as blur, noise, and other degradations in the recorded images. Digital image restoration aims to recover an estimate of the original image from the degraded observations. The key to being able to solve this ill-posed inverse problem is proper incorporation of prior knowledge about the original image into the restoration process. Classical image restoration seeks an estimate of the true image assumingthe blur is known. In contrast, blind image restoration tackles the much more difficult, but realistic, problem where the degradation is unknown. In general, the degradation is nonlinear (including, for example, saturation and quantization) and spatially varying (non-uniform motion, imperfect optics); however, for most of the work, it is assumed that the observed image is the output of a Linear Spatially Invariant (LSI) system to which noise is added. Therefore it becomes a Blind Deconvolution (BD) problem, with the unknown blur represented as a Point Spread Function (PSF). Classical restoration has matured since its inception, in the context of spaceexploration in the 1960s, and numerous techniques can be found in the literature (for recent reviews see [1, 2]). These differ primarily in the prior information about the image they include to perform the restoration task. The earliest algorithms to tackle the BD problem appeared as long ago as the mid-1970s [3,4], and attempted to identify known patterns in the blur; a small but dedicated effort followed through the late 1980s (see for instance [5-9]), and a resurgence was seen in the 1990s (see the earlier reviews in [10, 11]). Since then, the area has been extensively explored by the signal processing, astronomical, and optics communities. Many of the BD algorithms have their roots in estimation theory, linear algebra, and numerical analysis. An important question one may ask is why is BD useful? Could we notsimply use a better observation procedure in the first place? Perhaps, but there always exist physical limits, such as photonic noise, diffraction, or an observation channel outside of our control, and often images must be captured in suboptimal conditions. Also there are existing images of unique events that cannot be retaken that we would like to be able to recover (for instance with forensics or archive footage); furthermore in these cases it is often infeasible to measure properties of the imaging system directly. Another reason is that of cost. High-quality optics and sensing equipment are expensive. However, processing power is abundant today and opens the door to the application of increasingly sophisticated models. Thus BD represents a valuable tool that can be used for improving image quality without requiring complicated calibrations of the real-time image acquisition and processing system (i.e., in medical imaging, video conferencing, space exploration, x-ray imaging, bio-imaging, and so on). The BD problem is encountered in many different technical areas, such asastronomical imaging [12, 13], remote sensing [14], microscopy [15], medical imaging [16], optics [17,18], photography [19,20], superresolution applications [21], and motion tracking applications [22], among others. For example, astronomical imaging is one of the primary applications of BDalgorithms [12,13]. Ground-based imaging systems are subject to blurring due to the rapidly changing index of refractions of the atmosphere. Extraterrestrial observations of the Earth and the planets are degraded by motion blur as a result of slow camera shutter speeds relative to the rapid spacecraft motion.

In tandem with the rise in prominence of university rankings and the internationalization of higher education over the past decade, institutions globally have had to respond to increasing competition for faculty, students, funding and international standing.

We establish theoretical recovery guarantees of a family of Riemannian optimization algorithms for low rank matrix recovery, which is about recovering an $m\times n$ rank $r$ matrix from $p < mn$ number of linear measurements. The algorithms are first interpreted as iterative hard thresholding algorithms with subspace projections. Based on this connection, we show that provided the restricted isometry constant $R_{3r}$ of the sensing operator is less than $C_\kappa /\sqrt{r}$, the Riemannian gradient descent algorithm and a restarted variant of the Riemannian conjugate gradient algorithm are guaranteed to converge linearly to the underlying rank $r$ matrix if they are initialized by one step hard thresholding. Empirical evaluation shows that the algorithms are able to recover a low rank matrix from nearly the minimum number of measurements necessary.

Reconstructing transparent objects is a challenging problem. While producing reasonable results for quite complex objects, existing approaches require custom calibration or somewhat expensive labor to achieve high precision. When an overall shape preserving salient and fine details is sufficient, we show in this paper a significant step toward solving the problem when the object's silhouette is available and simple user interaction is allowed, by using a video of a transparent object shot under varying illumination. Specifically, we estimate the normal map of the exterior surface of a given solid transparent object, from which the surface depth can be integrated. Our technical contribution lies in relating this normal estimation problem to one of graph-cut segmentation. Unlike conventional formulations, however, our graph is dual-layered, since we can see a transparent object's foreground as well as the background behind it. Quantitative and qualitative evaluation are performed to verify the efficacy of this practical solution.

Recently, many variational models using high order derivatives have been proposed to accomplish advanced tasks in image processing. Even though these models are effective in fulfilling those tasks, it is very challenging to minimize the associated high order functionals. In [33], we focused on a recently proposed mean curvature based image denoising model and developed an efficient algorithm to minimize it using augmented Lagrangian method, where minimizers of the original high order functional can be obtained by solving several low order functionals. Specifically, these low order functionals either have closed form solutions or can be solved using FFT. Since FFT yields exact solutions to the associated equations, in this work, we consider to use only approximations to replace these exact solutions in order to reduce the computational cost. We thus employ the Gauss-Seidel method to solve those equations and observe that the new strategy produces almost the same results as the previous one but needs less computational time, and the reduction of the computational time becomes salient for images of large sizes.

this paper with a large-scale example problem that highlights the advantagesof FETI-DP. The following problem is a shell model of a wheel rim composedof over 313856 elements, 156017 nodes, and containing 936, 102 degrees of freedom.Three points were fixed along the inner rim, e#ectively constraining the model. Thena gravity load was applied to the model which was decomposed into 500 subdomains.A DUAL-PRIMAL FETI METHOD 427Figure 4: Finite element model of wheel rimAs one can see, the...

The active contour segmentation model of Chan and Vese has been widely used and generalized in different contexts in the literature. One possible modification is to employ Euler’s elastica as the regularization of active contour. In this paper, we study the new effects of this modification and validate them numerically using the augmented Lagrangian method.

We introduce a framework for outdoor photometric stereo utilizing natural environmental illumination. Our framework extends beyond existing photometric stereo methods intended for laboratory environments to encompass robust outdoor operation in the real world. In this paper, we motivate our framework, describe the components of its processing pipeline, and assess its performance in synthetic experiments as well as in natural experiments including objects in outdoor environments with complex real-world illuminations.

this paper we continue development of the DirichletDirichletpreconditioner [DA99, KV99]. We extend the method to the case of arbitrarytype of Lagrange multiplier space and large jumps of coe#cients. The proposed algorithmpossesses natural parallelism. It is illustrated on a set of numerical experiments.The mortar element method with Lagrange multipliers

this paper is trying to find an e#ective way to remove as many ofthe geologic non-matching grids as possible from the interfaces. The operator splitmethod, not a very new technique, proposed by Douglas and Dupont[JD71], can solvethis problem. Because, for quite a few reservoir problems, thereservoir#can be takento be unions of right prisms. Or,mathematically,#=

In this paper we shall consider the construction of the solution by the method of lines coupled with a fictitious domain for the following Dirichlet problem (1) in a bounded domain Omega of R&amp;sup2;...