Showing papers by "Stanley Osher published in 2015"

Non-Local Retinex---A Unifying Framework and Beyond

Abstract: In this paper, we provide a short review of Retinex and then present a unifying framework. The fundamental assumption of all Retinex models is that the observed image is a multiplication between the illumination and the true underlying reflectance of the object. Starting from Morel's 2010 PDE model, where illumination is supposed to vary smoothly and where the reflectance is thus recovered from a hard-thresholded Laplacian of the observed image in a Poisson equation, we define our unifying Retinex model in two similar, but more general, steps. We reinterpret the gradient thresholding model as variational models with sparsity constraints. First, we look for a filtered gradient that is the solution of an optimization problem consisting of two terms: a sparsity prior of the reflectance and a fidelity prior of the reflectance gradient to the observed image gradient. Second, since this filtered gradient almost certainly is not a consistent image gradient, we then fit an actual reflectance gradient to it, subje...

Computational Aspects of Constrained L1-L2 Minimization for Compressive Sensing

Abstract: We study the computational properties of solving a constrained L 1-L 2 minimization via a difference of convex algorithm (DCA), which was proposed in our previous work [13,19] to recover sparse signals from a under-determined linear system. We prove that the DCA converges to a stationary point of the nonconvex L 1-L 2 model. We clarify the relationship of DCA to a convex method, Bregman iteration [20] for solving a constrained L 1 minimization. Through experiments, we discover that both L 1 and L 1-L 2 obtain better recovery results from more coherent matrices, which appears unknown in theoretical analysis of exact sparse recovery. In addition, numerical studies motivate us to consider a weighted difference model L 1-αL 2 (α > 1) to deal with ill-conditioned matrices when L 1-L 2 fails to obtain a good solution.

Nonlocal Structure Tensor Functionals for Image Regularization

Abstract: We present a nonlocal regularization framework that we apply to inverse imaging problems. As opposed to existing nonlocal regularization methods that rely on the graph gradient as the regularization operator, we introduce a family of nonlocal energy functionals that involves the standard image gradient. Our motivation for designing these functionals is to exploit at the same time two important properties inherent in natural images, namely the local structural image regularity and the nonlocal image self-similarity. To this end, our regularizers employ as their regularization operator a novel nonlocal version of the structure tensor. This operator performs a nonlocal weighted average of the image gradients computed at every image location and, thus, is able to provide a robust measure of image variation. Furthermore, we show a connection of the proposed regularizers to the total variation semi-norm and prove convexity. The convexity property allows us to employ powerful tools from convex optimization to design an efficient minimization algorithm. Our algorithm is based on a splitting variable strategy, which leads to an augmented Lagrangian formulation. To solve the corresponding optimization problem, we employ the alternating-direction methods of multipliers. Finally, we present extensive experiments on several inverse imaging problems, where we compare our regularizers with other competing local and nonlocal regularization approaches. Our results are shown to be systematically superior, both quantitatively and visually.

An L^1 Penalty Method for General Obstacle Problems

Abstract: We construct an efficient numerical scheme for solving obstacle problems in divergence form. The numerical method is based on a reformulation of the obstacle in terms of an $L^1$-like penalty on the variational problem. The reformulation is an exact regularizer in the sense that for a large (but finite) penalty parameter, we recover the exact solution. Our formulation is applied to classical elliptic obstacle problems as well as some related free boundary problems, for example, the two-phase membrane problem and the Hele--Shaw model. One advantage of the proposed method is that the free boundary inherent in the obstacle problem arises naturally in our energy minimization without any need for problem specific or complicated discretization. In addition, our scheme also works for nonlinear variational inequalities arising from convex minimization problems.

Defect-Tolerant Aligned Dipoles within Two-Dimensional Plastic Lattices

Abstract: Carboranethiol molecules self-assemble into upright molecular monolayers on Au{111} with aligned dipoles in two dimensions The positions and offsets of each molecule's geometric apex and local dipole moment are measured and correlated with sub-Angstrom precision Juxtaposing simultaneously acquired images, we observe monodirectional offsets between the molecular apexes and dipole extrema We determine dipole orientations using efficient new image analysis techniques and find aligned dipoles to be highly defect tolerant, crossing molecular domain boundaries and substrate step edges The alignment observed, consistent with Monte Carlo simulations, forms through favorable intermolecular dipole-dipole interactions

Normal Estimation of a Transparent Object Using a Video

Abstract: 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.

A Multiphase Image Segmentation Based on Fuzzy Membership Functions and L1-norm Fidelity

Abstract: In this paper, we propose a variational multiphase image segmentation model based on fuzzy membership functions and L1-norm fidelity. Then we apply the alternating direction method of multipliers to solve an equivalent problem. All the subproblems can be solved efficiently. Specifically, we propose a fast method to calculate the fuzzy median. Experimental results and comparisons show that the L1-norm based method is more robust to outliers such as impulse noise and keeps better contrast than its L2-norm counterpart. Theoretically, we prove the existence of the minimizer and analyze the convergence of the algorithm.

Density matrix minimization with $\ell_1$ regularization

Abstract: We propose a convex variational principle to find sparse representation of low-lying eigenspace of symmetric matrices. In the context of electronic structure calculation, this corresponds to a sparse density matrix minimization algorithm with $\ell_1$ regularization. The minimization problem can be efficiently solved by a split Bergman iteration type algorithm. We further prove that from any initial condition, the algorithm converges to a minimizer of the variational principle.

Hyperspectral unmixing by the alternating direction method of multipliers

Abstract: We have developed a method for hyperspectral image data unmixing that requires neither pure pixels nor any prior knowledge about the data. Based on the well-established Alternating Direction Method of Multipliers, the problem is formulated as a biconvex constrained optimization with the constraints enforced by Bregman splitting. The resulting algorithm estimates the spectral and spatial structure in the image through a numerically stable iterative approach that removes the need for separate endmember and spatial abundance estimation steps. The method is illustrated on data collected by the SpecTIR imaging sensor.

Space-Time Regularization for Video Decompression

Abstract: We consider the problem of reconstructing frames from a video which has been compressed using the video compressive sensing (VCS) method. In VCS data, each frame comes from first subsampling the original video data in space and then averaging the subsampled sequence in time. This results in a large linear system of equations whose inversion is ill-posed. We introduce a convex regularizer to invert the system, where the spatial component is regularized by the total variation seminorm, and the temporal component is regularized by enforcing sparsity on the difference between the spatial gradients of each frame. Since the regularizers are $L^1$-like norms, the model can be written in the form of an easy-to-solve saddle point problem. The saddle point problem is solved by the primal-dual algorithm, whose implementation calls for nearly pointwise operations (i.e., no direct linear inversion) and has a simple parallel version. Results show that our model decompresses videos more accurately than other popular mod...

A Dirichlet Energy Criterion for Graph-Based Image Segmentation

Abstract: We consider a graph-based approach for image segmentation We introduce several novel graph construction models which are based on graph-based segmentation criteria extending beyond -- and bridging the gap between -- segmentation approaches based on edges and homogeneous regions alone The resulting graph is partitioned using a criterion based on the sum of the minimal Dirichlet energies of partition components We propose an efficient primal-dual method for computing the Dirichlet energy ground state of partition components and a rearrangement algorithm is used to improve graph partitions The method is applied to a number of example segmentation problems We demonstrate the graph partitioning method on the five-moons toy problem, and illustrate the various image-based graph constructions, before successfully running a variety of region-, edge-, hybrid, and texture-based image segmentation experiments Our method seamlessly generalizes region-and edge-based image segmentation to the multi-phase case and can intrinsically deal with image bias as well as more interesting image features such as texture descriptors

Vectorial non-local total variation regularization for calibration-free parallel MRI reconstruction

Abstract: In this work we present a calibration-free parallel magnetic resonance imaging (pMRI) reconstruction approach by exploiting the fact that image structures typically tend to repeat themselves in several locations in the image domain. We use this prior information along with the correlation that exists among the different MR images, which are acquired from multiple receiver coils, to improve reconstructions from under-sampled data with arbitrary k-space trajectories. To accomplish this, we follow a variational approach and cast the pMRI reconstruction problem as the minimization of an energy functional that involves a vectorial non-local total variation (NLTV) regularizer. Further, to solve the posed optimization problem we propose an iterative algorithm which is based on a variable splitting strategy. To assess the reconstruction quality of the proposed method, we provide comparisons with alternative techniques and show that our results can be very competitive.

Determining Fluid Reservoir Connectivity Using Nanowire Probes

Abstract: Systems and methods of fabricating and functionalizing patterned nanowire probes that are stable under fluid reservoir conditions and have imageable contrast are provided. Optical imaging and deconstruction methods and systems are also provided that are capable of determining the distribution of nanowires of a particular pattern to determine the mixing between or leakage from fluid reservoirs.

A time continuation based fast approximate algorithm for compressed sensing related optimization

Abstract: Recently, significant connections between compressed sensing problems and optimization of a particular class of functions relating to solutions of Hamilton-Jacobi equation was discovered. In this paper we introduce a fast approximate algorithm to optimize this particular class of functions and subsequently find the solution to the compressed sensing problem. Although we emphasize that the methodology of our algorithm finds an approximate solution, numerical experiments show that our algorithm perfectly recovers the solution when the solution is relatively sparse with respect to the number of measurements. In these scenarios, the recovery is extremely fast compare to other methods available. Numerical experiments also demonstrate that the algorithm exhibits a sharp phase transition in success rate of recovery of the solution to compressed sensing problems as sparsity of solution varies. The algorithm proposed here is parameter free (except a tolerance parameter due to numerical machine precision), and very easy to implement.