Showing papers by "Stanley Osher published in 2003"

Simultaneous structure and texture image inpainting

Abstract: An algorithm for the simultaneous filling-in of texture and structure in regions of missing image information is presented in this paper. The basic idea is to first decompose the image into the sum of two functions with different basic characteristics, and then reconstruct each one of these functions separately with structure and texture filling-in algorithms. The first function used in the decomposition is of bounded variation, representing the underlying image structure, while the second function captures the texture and possible noise. The region of missing information in the bounded variation image is reconstructed using image inpainting algorithms, while the same region in the texture image is filled-in with texture synthesis techniques. The original image is then reconstructed adding back these two sub-images. The novel contribution of this paper is then in the combination of these three previously developed components, image decomposition with inpainting and texture synthesis, which permits the simultaneous use of filling-in algorithms that are suited for different image characteristics. Examples on real images show the advantages of this proposed approach.

Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing

Abstract: This paper is devoted to the modeling of real textured images by functional minimization and partial differential equations. Following the ideas of Yves Meyer in a total variation minimization framework of L. Rudin, S. Osher, and E. Fatemi, we decompose a given (possible textured) image f into a sum of two functions u+v, where u∈BV is a function of bounded variation (a cartoon or sketchy approximation of f), while v is a function representing the texture or noise. To model v we use the space of oscillating functions introduced by Yves Meyer, which is in some sense the dual of the BV space. The new algorithm is very simple, making use of differential equations and is easily solved in practice. Finally, we implement the method by finite differences, and we present various numerical results on real textured images, showing the obtained decomposition u+v, but we also show how the method can be used for texture discrimination and texture segmentation.

Image Decomposition and Restoration Using Total Variation Minimization and the H1

Abstract: In this paper, we propose a new model for image restoration and image decomposition into cartoon and texture, based on the total variation minimization of Rudin, Osher, and Fatemi [Phys. D, 60 (1992), pp. 259--268], and on oscillatory functions, which follows results of Meyer [Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, Univ. Lecture Ser. 22, AMS, Providence, RI, 2002]. This paper also continues the ideas introduced by the authors in a previous work on image decomposition models into cartoon and texture [L. Vese and S. Osher, J. Sci. Comput., to appear]. Indeed, by an alternative formulation, an initial image f is decomposed here into a cartoon part u and a texture or noise part v. The u component is modeled by a function of bounded variation, while the v component is modeled by an oscillatory function, bounded in the norm dual to $|\cdot|_{H^1_0}$. After some transformation, the resulting PDE is of fourth order, envolving the Laplacian of the curvature of level lines. Fina...

Simultaneous structure and texture image inpainting

Abstract: An algorithm for the simultaneous filling-in of texture and structure in regions of missing image information is presented. The basic idea is to first decompose the image into the sum of two functions with different basic characteristics, and then reconstruct each one of these functions separately with structure and texture filling-in algorithms. The first function used in the decomposition is of bounded variation, representing the underlying image structure, while the second function captures the texture and possible noise. The region of missing information in the bounded variation image is reconstructed using image inpainting algorithms, while the same region in the texture image is filled-in with texture synthesis techniques. The original image is then reconstructed adding back these two sub-images. The novel contribution of the paper is then in the combination of these three previously developed components: image decomposition with inpainting and texture synthesis, which permits the simultaneous use of filling-in algorithms that are suited for different image characteristics. Examples on real images show the advantages of this proposed approach.

Fast Sweeping Algorithms for a Class of Hamilton--Jacobi Equations

Abstract: We derive a Godunov-type numerical flux for the class of strictly convex, homogeneous Hamiltonians that includes $H(p,q)=\sqrt{ap^{2}+bq^{2}-2cpq},$ $c^{2}

Level Set Methods

Abstract: The level set method for capturing moving fronts was introduced in 1987 by Osher and Sethian [401]. It has proven to be phenomenally successful as a numerical device. For example, as of June 2002, typing in “Level Set Methods” on Google’s search engine gives roughly 2800 responses and the original article has been cited over 530 times (according to web of science). Applications range from capturing multiphase fluid dynamical flows, to graphics, e.g. special effects in Hollywood, to visualization, image processing, control, epitaxial growth, computer vision and include many others. In this chapter we shall give an overview of the numerical technology and of applications in imaging science. These will include surface interpolation, solving PDE’s on manifolds, visibility, ray tracing, segmentation (including texture segmentation) and restoration.

Multiplicative Denoising and Deblurring: Theory and Algorithms

Abstract: In [447, 449, 450], a constrained optimization type of numerical algorithm for restoring blurry, noisy images was developed and successfully tested. In this paper we present both theoretical and experimental justification for the method. Our main theoretical results involve constrained nonlinear partial differential equations. Our main experimental results involve blurry images which have been further corrupted with multiplicative noise. As in additive noise case of [447, 450] our numerical algorithm is simple to implement and is nonoscillatory (minimal ringing) and noninvasive (recovers sharp edges).

Variational Problems and Partial Differential Equations on Implicit Surfaces: Bye Bye Triangulated Surfaces?

Abstract: A novel framework for solving variational problems and partial differential equations for scalar and vector-valued data defined on surfaces is described in this chapter. The key idea is to implicitly represent the surface as the level set of a higher dimensional function, and solve the surface equations in a fixed Cartesian coordinate system using this new embedding function. The equations are then both intrinsic to the surface and defined in the embedding space. This approach thereby eliminates the need for performing complicated and not-accurate computations on triangulated surfaces, as it is commonly done in the literature. We describe the framework and present examples in computer graphics and image processing applications, including texture synthesis, flow field visualization, as well as image and vector field intrinsic regularization for data defined on 3D surfaces.

A Level Set Approach for the Numerical Simulation of Dendritic Growth

Abstract: In this paper, we present a level set approach for the modeling of dendritic solidification. These simulations exploit a recently developed second order accurate symmetric discretization of the Poisson equation, see l12r. Numerical results indicate that this method can be used successfully on complex interfacial shapes and can simulate many of the physical features of dendritic solidification. We apply this algorithm to the simulation of the dendritic crystallization of a pure melt and find that the dendrite tip velocity and tip shapes are in excellent agreement with solvability theory. Numerical results are presented in both two and three spatial dimensions.

Geometric surface processing via normal maps

Abstract: We propose that the generalization of signal and image processing to surfaces entails filtering the normals of the surface, rather than filtering the positions of points on a mesh. Using a variational strategy, penalty functions on the surface geometry can be formulated as penalty functions on the surface normals, which are computed using geometry-based shape metrics and minimized using fourth-order gradient descent partial differential equations (PDEs). In this paper, we introduce a two-step approach to implementing geometric processing tools for surfaces: (i) operating on the normal map of a surface, and (ii) manipulating the surface to fit the processed normals. Iterating this two-step process, we efficiently can implement geometric fourth-order flows by solving a set of coupled second-order PDEs. The computational approach uses level set surface models; therefore, the processing does not depend on any underlying parameterization. This paper will demonstrate that the proposed strategy provides for a wide range of surface processing operations, including edge-preserving smoothing and high-boost filtering. Furthermore, the generality of the implementation makes it appropriate for very complex surface models, for example, those constructed directly from measured data.

REVIEW ARTICLE: Level Set Methods and Their Applications in Image Science

Abstract: In this article, we discuss the question "What Level Set Methods can do for image science". We examine the scope of these techniques in image science, in particular in image segmentation, and introduce some relevant level set techinquies that are potnetially useful for this class of applications. We will show that image science demands multi-disciplinary knowledge and flexible but still robust methods. That is why the Level Set Method has become a thriving technique in this field.

Computational high-frequency wave propogation using the level-set method with applications to the semi-classical limit of the Schrödinger equations

Abstract: We introduce a level set method for computational high frequency wave propagation in dispersive media and consider the application to linear Schrodinger equation with high frequency initial data. High frequency asymptotics of dispersive equations often lead to the well-known WKB system where the phase of the plane wave evolves according to a nonlinear Hamilton-Jacobi equation and the intensity is governed by a linear conservation law. From the Hamilton-Jacobi equation, wave fronts with multiple phases are constructed by solving a linear Liouville equation of a vector valued level set function in the phase space. The multi-valued phase itself can be constructed either from an additional linear hyperbolic equation in phase space or an additional linear homogeneous equation and component to the level set function in an augmented phase space. This phase is in fact valid in the entire physical domain, but one of the components of the level set function can be used to restrict it to a wave front of interest. The use of the level set method in this numerical approach provides an Eulerian framework that automatically resolves the multi-valued wave fronts and phase from the superposition of solutions of the equations in phase space.

Signed Distance Functions

Abstract: In the last chapter we defined implicit functions with φ(x↦) ≤ 0 in the interior region Ω-, φ((x↦) > 0 in the exterior region Ω+, and φ((x↦) = 0 on the boundary ∂Ω. Little was said about φ otherwise, except that smoothness is a desirable property especially in sampling the function or using numerical approximations. In this chapter we discuss signed distance functions, which are a subset of the implicit functions defined in the last chapter. We define signed distance functions to be positive on the exterior, negative on the interior, and zero on the boundary. An extra condition of |∇φ(x↦)| = 1 is imposed on a signed distance function.

Precursor Tissue Analogs as a Tissue-Engineering Strategy

Abstract: Natural tissues are composed of functionally diverse cell types that are organized in spatially complex arrangements. Organogenesis of complex tissues requires a coordinated sequential transformation process, with individual stages involving time-dependent expression of cell-cell, cell-matrix, and cell-signal interactions in three dimensions. The common theme of temporal-spatial patterning of these cellular interactions is also observed in other physiological processes, such as growth and development, wound healing, and tumor migration. The "precursor tissue analog" (PTA) applies the temporal-spatial patterning theme to tissue engineering. The goal of PTA in tissue engineering is not to fabricate the final transplantable tissue but rather to guide the dynamic organization, maturation, and remodeling leading to the formation of normal and functional tissues. We describe the critical design principles of PTA. First, structural, mechanical, and physiological requirements of the PTA as a temporary scaffold must be met by a fabrication method with flexibility. The fabrication potential incorporating biological materials such as living cells and plasmid DNA has been addressed. Second, the PTA concept is considered suitable for future tissue engineering in light of the use of undifferentiated stem cells, and may possess a capability to guide stem cells toward diverse differentiation characteristics in situ. To this end, the behavior of the engineered cell and tissue must be monitored in detail. The development of a practical phenotype monitoring system such as a DNA microarray may be integral to the fabrication strategies of PTA. Third, the microtopographical and microenvironmental control on the liquid-solid interaction may lead to a critical design for PTA to provide soluble factors, nutrients, and gases to the cells embedded within the scaffold. We suggest that the level set numerical simulation method may be utilized to engineer the consistent circulation of bioactive liquid throughout the PTA microenvironment.

A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations

Abstract: We introduce two types of finite difference methods to compute the L-solution and the proper viscosity solution recently proposed by the second author for semi-discontinuous solutions to a class of Hamilton-Jacobi equations. By regarding the graph of the solution as the zero level curve of a continuous function in one dimension higher, we can treat the corresponding level set equation using the viscosity theory introduced by Crandall and Lions. However, we need to pay special attention both analytically and numerically to prevent the zero level curve from overturning so that it can be interpreted as the graph of a function. We demonstrate our Lax-Friedrichs type numerical methods for computing the L-solution using its original level set formulation. In addition, we couple our numerical methods with a singular diffusive term which is essential to computing solutions to a more general class of HJ equations that includes conservation laws. With this singular viscosity, our numerical methods do not require the divergence structure of equations and do apply to more general equations developing shocks other than conservation laws. These numerical methods are generalized to higher order accuracy using weighted ENO local Lax-Friedrichs methods as developed recently by Jiang and Peng. We verify that our numerical solutions approximate the proper viscosity solutions obtained by the second author in a recent Hokkaido University preprint. Finally, since the solution of scalar conservation law equations can be constructed using existing numerical techniques, we use it to verify that our numerical solution approximates the entropy solution.

Level set methods in image science

Abstract: In this article, we discuss the question "what level set methods can do for image science". We examine the presence of these and related methods in image science and introduce some relevant level set techniques that are potentially useful for this class of applications. We will show that image science demands multidisciplinary knowledge and flexible but still robust methods, and that is why the level set method has become a thriving technique in this field.

3D shape from anisotropic diffusion

Abstract: We cast the problem of inferring the 3D shape of a scene from a collection of defocused images in the framework of anisotropic diffusion We propose an algorithm that can estimate the shape of a scene by inferring the diffusion coefficient of a heat equation The method is optimal, as we pose it as the minimization of a certain cost functional based on the input images, and fast Furthermore, we also extend our algorithm to the case of multiple images, and derive a 3D scene segmentation algorithm that can work in the presence of pictorial camouflage

Hamilton-Jacobi Equations

Abstract: In this chapter we discuss numerical methods for the solution of general Hamilton-Jacobi equations of the form $${\phi _t} + H\left( { abla \phi } \right) = 0$$ (5.1) where H can be a function of both space and time. In three spatial dimensions, we can write $${\phi _t} + H\left( {{\phi _x},{\phi _y},{\phi _z}} \right) = 0$$ (5.2) as an expanded version of equation (5.1). Convection in an externally generated velocity field (equation (3.2)) is an example of a Hamilton-Jacobi equation where H(∇φ) = 0056;↦ ·∇φ. The level set equation (equation (4.4)) is another example of a Hamilton-Jacobi equation with H(∇φ) = V n |∇φ| Here V n can depend on 0078;↦, t, or even ∇φ /|∇φ|.

Image filling-in in a decomposition space

Abstract: An algorithm for the simultaneous filling-in of texture and structure in regions of missing image information is presented in this paper. The basic idea is to first decompose the image into the sum of two functions with different basic characteristics, and then reconstruct each one of these functions separately with structure and texture filling-in algorithms. The first function used in the decomposition is of bounded variation, representing the underlying image structure, while the second function captures the texture and possible noise. The region of missing information in the bounded variation image is reconstructed using image inpainting algorithms, while the same region in the texture image is filled-in with texture synthesis techniques. The original image is then reconstructed adding back these two subimages. The novel contribution of this paper is then in the combination of these three previously developed components, image decomposition with in-painting and texture synthesis, which permits the simultaneous use of filling-in algorithms that are suited for different image characteristics. The novelty in the approach is to perform filling-in in a domain different from the original given image space. Examples on real images show the advantages of this proposed approach.

Computational Anatomy and Implicit Object Representation: A Level Set Approach

Abstract: In this paper, we look at the fundamental problem of object matching in computational anatomy. We present a new framework for warping pairs of overlapping and non-overlapping shapes, open curves, and landmarks based on the level set approach. When implemented in 3-D, the same framework could be used to warp 3-D objects with minimal modification. Our approach is to use the level set functions to represent the objects to be matched. Using this representation, the problem becomes an energy minimization problem. Cost functions for warping overlapping, non-overlapping, open curves, and landmarks are proposed. Euler-Lagrange equations are applied and gradient descent is used to solve the corresponding partial differential equations. Moreover, a general framework for linking the level set approach and the infinite dimensional group actions is discussed.

Visualization, Analysis and Shape Reconstruction of Sparse Data

Abstract: In this chapter mathematical models and efficient algorithms are developed for the visualization, analysis and shape reconstruction for an arbitrary data set that can include unorganized points or continuous manifolds of any codimension, such as pieces of curves and surface patches. The distance function to the data set and its contours are used for fast visualization and analysis of the data set. A minimal surface and a convection model are used for shape reconstruction from the data set. All formulations and numerical algorithms are based on implicit representations on simple rectangular grids which extend to any number of dimensions and which also can easily be combined with the level set method for dynamic shape deformation and other manipulations.

Particle Level Set Method

Abstract: The great success of level set methods can in part be attributed to the role of curvature in regularizing the level set function such that the proper vanishing viscosity solution is obtained. It is much more difficult to obtain vanishing viscosity solutions with Lagrangian methods that faithfully follow the characteristics. For these methods one usually has to delete (or add) characteristic information “by hand” when a shock (or rarefaction) is detected. This ability of level set methods to identify and delete merging characteristics is clearly seen in a purely geometrically driven flow where a curve is advected normal to itself at constant speed, as shown in Figures 9.1 and 9.2. In the corners of the square, the flow field has merging characteristics that are appropriately deleted by the level set method. We demonstrate the difficulties associated with a Lagrangian calculation of this interface motion by initially seeding some marker particles interior to the interface, as shown in Figure 9.3 and passively advecting them with \( {\overrightarrow x_t} = \overrightarrow V \left( {\overrightarrow x, t} \right) \) where the velocity field V↦(x↦ t) is determined from the level set solution. Figure 9.4 illustrates that a number of particles incorrectly escape from inside the level set solution curve in the corners of the square where the characteristic information (represented by the particles themselves) needs to be deleted so that the correct vanishing viscosity solution can be obtained.

Constructing Signed Distance Functions

Abstract: As we have seen, a number of simplifications can be made when φ is a signed distance function. For this reason, we dedicate this chapter to numerical techniques for constructing approximate signed distance functions. These techniques can be applied to the initial data in order to initialize φ to a signed distance function.

New framework for object warping: semi-Lagrangian level set approach

Abstract: In this paper, we present a new framework for object matching between two images. This method could handle multiple pairs of overlapping and non-overlapping shapes, open curves, and landmarks. When implemented in 3-D, the same framework could be used to warp 3-D objects with minimal modification. Our approach is to use the level set formulation to represent the objects to be matched. Using this representation, the problem becomes an energy minimization problem. Cost functions for warping overlapping, non-overlapping, open curves, and landmarks are proposed. Euler-Lagrange equations are applied and gradient descent is used to solve the corresponding partial differential equations. Moreover, a general framework for linking the level set approach and the infinite dimensional group actions is discussed.

Extrapolation in the Normal Direction

Abstract: In the last chapter we constructed signed distance functions by following characteristics that flow outward from the interface. Similar techniques can be used to propagate information in the direction of these characteristics. For example, $$ {S_t} + \overrightarrow N \cdot abla S = 0 $$ (8.1) is a Hamilton-Jacobi equation (in S) that extrapolates S normal to the interface, i.e. so that S is constant on rays normal to the interface. Since \( H\left( { abla S} \right) = \overrightarrow N \cdot abla S \), we can solve this equation with the techniques presented in Chapter 5 using H 1 = n1, H 2 = n2, and H 3 = n 3 .

Motion in an Externally Generated Velocity Field

Abstract: Suppose that the velocity of each point on the implicit surface is given as V↦(V↦); i.e., assume that V↦(V↦) is known for every point (V↦ with φ(V↦) = 0. Given this velocity field V↦ = (u, v, w), we wish to move all the points on the surface with this velocity. The simplest way to do this is to solve the ordinary differential equation (ODE) $$ \frac{{d\overrightarrow x }}{{dt}} = \overrightarrow V \left( {\overrightarrow x } \right) $$ for every point V↦ on the front, i.e., for all V↦ with φ(V↦) = 0. This is the Lagrangian formulation of the interface evolution equation. Since there are generally an infinite number of points on the front (except, of course, in one spatial dimension), this means discretizing the front into a finite number of pieces. For example, one could use segments in two spatial dimensions or triangles in three spatial dimensions and move the endpoints of these segments or triangles. This is not so hard to accomplish if the connectivity does not change and the surface elements are not distorted too much. Unfortunately, even the most trivial velocity fields can cause large distortion of boundary elements (segments or triangles), and the accuracy of the method can deteriorate quickly if one does not periodically modify the discretization in order to account for these deformations by smoothing and regularizing inaccurate surface elements.

Motion Involving Mean Curvature

Abstract: In the last chapter we discussed the motion of an interface in an externally generated velocity field V↦(x↦, t). In this chapter we discuss interface motion for a self-generated velocity field (x↦ that depends directly on the level set function φ. As an example, we consider motion by mean curvature where the interface moves in the normal direction with a velocity proportional to its curvature; i.e., V↦ = -bκN↦, where b > 0 is a constant and κ is the curvature. When b > 0, the interface moves in the direction of concavity, so that circles (in two dimensions) shrink to a single point and disappear. When b < 0, the interface moves in the direction of convexity, so that circles grow instead of shrink. This growing-circle effect leads to the growth of small perturbations in the front including those due to round-off errors. Because b < 0 allows small erroneous perturbations to incorrectly grow into 0(1) features, the b < 0 case is ill-posed, and we do not consider it here. Figure 4.1 shows the motion of a wound spiral in a curvature-driven flow. The high-curvature ends of the spiral move significantly faster than the relatively low curvature elongated body section. Figure 4.2 shows the evolution of a star-shaped interface in a curvature-driven flow. The tips of the star move inward, while the gaps in between the tips move outward.