Showing papers by "Stanley Osher published in 2007"

Fast Global Minimization of the Active Contour/Snake Model

Abstract: The active contour/snake model is one of the most successful variational models in image segmentation. It consists of evolving a contour in images toward the boundaries of objects. Its success is based on strong mathematical properties and efficient numerical schemes based on the level set method. The only drawback of this model is the existence of local minima in the active contour energy, which makes the initial guess critical to get satisfactory results. In this paper, we propose to solve this problem by determining a global minimum of the active contour model. Our approach is based on the unification of image segmentation and image denoising tasks into a global minimization framework. More precisely, we propose to unify three well-known image variational models, namely the snake model, the Rudin---Osher---Fatemi denoising model and the Mumford---Shah segmentation model. We will establish theorems with proofs to determine the existence of a global minimum of the active contour model. From a numerical point of view, we propose a new practical way to solve the active contour propagation problem toward object boundaries through a dual formulation of the minimization problem. The dual formulation, easy to implement, allows us a fast global minimization of the snake energy. It avoids the usual drawback in the level set approach that consists of initializing the active contour in a distance function and re-initializing it periodically during the evolution, which is time-consuming. We apply our segmentation algorithms on synthetic and real-world images, such as texture images and medical images, to emphasize the performances of our model compared with other segmentation models.

Nonlocal Linear Image Regularization and Supervised Segmentation

Abstract: A nonlocal quadratic functional of weighted differences is examined. The weights are based on image features and represent the affinity between different pixels in the image. By prescribing different formulas for the weights, one can generalize many local and nonlocal linear denoising algorithms, including the nonlocal means filter and the bilateral filter. In this framework one can easily show that continuous iterations of the generalized filter obey certain global characteristics and converge to a constant solution. The linear operator associated with the Euler–Lagrange equation of the functional is closely related to the graph Laplacian. We can thus interpret the steepest descent for minimizing the functional as a nonlocal diffusion process. This formulation allows a convenient framework for nonlocal variational minimizations, including variational denoising, Bregman iterations, and the recently proposed inverse scale space. It is also demonstrated how the steepest descent flow can be used for segmenta...

Iterative Regularization and Nonlinear Inverse Scale Space Applied to Wavelet-Based Denoising

Abstract: In this paper, we generalize the iterative regularization method and the inverse scale space method, recently developed for total-variation (TV) based image restoration, to wavelet-based image restoration. This continues our earlier joint work with others where we applied these techniques to variational-based image restoration, obtaining significant improvement over the Rudin-Osher-Fatemi TV-based restoration. Here, we apply these techniques to soft shrinkage and obtain the somewhat surprising result that a) the iterative procedure applied to soft shrinkage gives firm shrinkage and converges to hard shrinkage and b) that these procedures enhance the noise-removal capability both theoretically, in the sense of generalized Bregman distance, and for some examples, experimentally, in terms of the signal-to-noise ratio, leaving less signal in the residual

The Total Variation Regularized $L^1$ Model for Multiscale Decomposition

Abstract: This paper studies the total variation regularization with an $L^1$ fidelity term (TV‐$L^1$) model for decomposing an image into features of different scales. We first show that the images produced by this model can be formed from the minimizers of a sequence of decoupled geometry subproblems. Using this result we show that the TV‐$L^1$ model is able to separate image features according to their scales, where the scale is analytically defined by the G‐value. A number of other properties including the geometric and morphological invariance of the TV‐$L^1$ model are also proved and their applications discussed.

Topology Preserving Log-Unbiased Nonlinear Image Registration: Theory and Implementation

Abstract: In this paper, we present a novel framework for constructing large deformation log-unbiased image registration models that generate theoretically and intuitively correct deformation maps. Such registration models do not rely on regridding and are inherently topology preserving. We apply information theory to quantify the magnitude of deformations and examine the statistical distributions of Jacobian maps in the logarithmic space. To demonstrate the power of the proposed framework, we generalize the well known viscous fluid registration model to compute log-unbiased deformations. We tested the proposed method using a pair of binary corpus callosum images, a pair of two-dimensional serial MRI images, and a set of three-dimensional serial MRI brain images. We compared our results to those computed using the viscous fluid registration method, and demonstrated that the proposed method is advantageous when recovering voxel-wise maps of local tissue change.

Inverse Total Variation Flow

Abstract: In this paper we analyze iterative regularization with the Bregman distance of the total variation seminorm. Moreover, we prove existence of a solution of the corresponding flow equation as introduced in [M. Burger, G. Gilboa, S. Osher, and J. Xu, Commun. Math. Sci., 4 (2006), pp. 179–212] in a functional analytical setting using methods from convex analysis. The results are generalized to variational denoising methods with ${\rm L}^p$-norm fit-to-data terms and Bregman distance regularization terms. For the associated flow equations well-posedness is derived using recent results on metric gradient flows from [L. Ambrosio, N. Gigli, and G. Savare, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zurich, Birkhauser Verlag, Basel, 2005]. In contrast to previous work the results of this paper apply for the analysis of variational denoising methods with the Bregman distance under adequate noise assumptions. Aside from the theoretical results we introduce a ...

Image Inpainting Using a TV-Stokes Equation

Abstract: Based on some geometrical considerations, we propose a two-step method to do digital image inpainting. In the first step, we try to propagate the isophote directions into the inpainting domain. An energy minimization model combined with the zero divergence condition is used to get a nonlinear Stokes equation. Once the isophote directions are constructed, an image is restored to fit the constructed directions. Both steps reduce to the solving of some nonlinear partial differential equations. Details about the discretization and implementation are explained. The algorithms have been intensively tested on synthetic and real images. The advantages of the proposed methods are demonstrated by these experiments.

A TV-stokes denoising algorithm

Abstract: In this paper, we propose a two-step algorithm for denoising digital images with additive noise. Observing that the isophote directions of an image correspond to an incompressible velocity field, we impose the constraint of zero divergence on the tangential field. Combined with an energy minimization problem corresponding to the smoothing of tangential vectors, this constraint gives rise to a nonlinear Stokes equation where the nonlinearity is in the viscosity function. Once the isophote directions are found, an image is reconstructed that fits those directions by solving another nonlinear partial differential equation. In both steps, we use finite difference schemes to solve. We present several numerical examples to show the effectiveness of our approach.

Solving the Chan-Vese model by a multiphase level set algorithm based on the topological derivative

Abstract: In this work, we specifically solve the Chan-Vese active contour model by multiphase level set methods. We first develop a fast algorithm based on calculating the variational energy of the Chan-Vese model without the length term. We check whether the energy decreases or not when we move a point to another segmented region. Then we draw a connection between this algorithm and the topological derivative, a concept emerged from the shape optimization field. Furthermore, to include the length term of the Chan-Vese model, we apply a preprocessing step on the image by using nonlinear diffusion. We show numerical experiments to demonstrate the efficiency and the robustness of our algorithm.

Nonlocal evolutions for image regularization

Abstract: A nonlocal quadratic functional of weighted differences is examined. The weights are based on image features and represent the affinity between different pixels in the image. By prescribing different formulas for the weights, one can generalize many local and nonlocal linear denoising algorithms, including nonlocal means and bilateral filters. The steepest descent for minimizing the functional can be interpreted as a nonlocal diffusion process. We show state of the art denoising results using the nonlocal flow.

MR image reconstruction from undersampled data by using the iterative refinement procedure

Abstract: Magnetic resonance imaging (MRI) reconstruction from sparsely sampled data has been a difficult problem in medical imaging field. We approach this problem by formulating a cost functional that includes a constraint term that is imposed by the raw measurement data in k-space and the L1 norm of a sparse representation of the reconstructed image. The sparse representation is usually realized by total variational regularization and/or wavelet transform. We have applied the Bregman iteration to minimize this functional to recover finer scales in our recent work. Here we propose nonlinear inverse scale space methods in addition to the iterative refinement procedure. Numerical results from the two methods are presented and it shows that the nonlinear inverse scale space method is a more efficient algorithm than the iterated refinement method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Level Set Based Simulations of Two-Phase Oil-Water Flows in Pipes

Abstract: We simulate the axisymmetric pipeline transportation of oil and water numerically under the assumption that the densities of the two fluids are different and that the viscosity of the oil core is very large. We develop the appropriate equations for core-annular flows using the level set methodology. Our method consists of a finite difference scheme for solving the model equations, and a level set approach for capturing the interface between two liquids (oil and water). A variable density projection method combined with a TVD Runge---Kutta scheme is used to advance the computed solution in time. The simulations succeed in predicting the spatially periodic waves called bamboo waves, which have been observed in the experiments of [Bai et al. (1992) J. Fluid Mech. 240, 97---142.] on up-flow in vertical core flow. In contrast to the stable case, our simulations succeed in cases where the oil breaks up in the water, and then merging occurs. Comparisons are made with other numerical methods and with both theoretical and experimental results.

Dynamic Visibility in an Implicit Framework

Abstract: We investigate the problem of determining visible regions in two or three dimensional space given a set of obstacles and a moving vantage point. This is of importance in several fields of study including rendering in computer graphics, etching in materials construction, and navigation. Our approach to this problem is through an implicit framework, where the obstacles are represented by a level set function. An efficient generic multiscale level set method is developed to generate the visible and invisible regions in space. Furthermore, we study the dynamics of shadow boundaries on the surfaces of the obstacles using special level set techniques when the vantage point moves with a given trajectory. In all of these situations, topological changes such as merging and breaking occur in the regions of interest. These are automatically handled by the level set framework here proposed. Finally, we obtain additional useful information through simple operations in the level set framework. Research supported by ONR N00014-97-1-0027, DARPA/NSF VIP grant NSF DMS 9615854 and ARO DAAG 55-98-1-0323 yDepartment of Mathematics, University of California Los Angeles, Los Angeles, California 90095, email:ytsai@math.ucla.edu zDepartment of Mathematics, UCSD, La Jolla, CA 92093-0112. Email: lcheng@math.ucsd.edu xDepartment of Mathematics, University of California Los Angeles, Los Angeles, California 90095, email:burchard@math.ucla.edu {Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095, email:sjo@math.ucla.edu kDepartment of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, email: guille@ece.umn.edu. Supported by ONR,NSF, PEC ASE, and CAREER.

Multiphase Segmentation of Deformation using Logarithmic Priors

Abstract: In [8], the authors proposed the large deformation log-unbiased diffeomorphic nonlinear image registration model which has been successfully used to obtain theoretically and intuitively correct deformation maps. In this paper, we extend this idea to simultaneously registering and tracking deforming objects in a sequence of two or more images. We generalize a level set based Chan-Vese multiphase segmentation model to consider Jacobian fields while segmenting regions of growth and shrinkage in deformations. Deforming objects are thus classified based on magnitude of homogeneous deformation. Numerical experiments demonstrating our results include a pair of two-dimensional synthetic images and pairs of two-dimensional and three-dimensional serial MRI images.

A fast multiphase level set algorithm for solving the Chan‐Vese model

Abstract: In this work, we specifically solve the C-V active contour model by multiphase level set methods. We first develop a fast algorithm based on calculating the variational energy of the C-V model without the length term. We check whether the energy decreases or not when we move a point to another segmented region. Then we draw a connection between this algorithm and the topological derivative, a concept emerged from the shape optimization field. Furthermore, to include the length term of the C-V model, a preprocessing step is taken by using nonlinear diffusion. Numerical experiments have demonstrated the efficiency and the robustness of our algorithm. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)