Level Set Regularization Using Geometric Flows

Numerische Mathematik, 2003

We introduce linear semi-implicit complementary volume numerical scheme for solving level set like nonlinear degenerate diffusion equations arising in image processing and curve evolution problems. We study discretization of image selective smoothing equation of mean curvature flow type given by Alvarez, Lions and Morel ([3]). Solution of the level set equation of Osher and Sethian ([26], ) is also included in the study. We prove L ∞ and W 1,1 estimates for the proposed scheme and give existence of its (generalized) solution in every discrete time-scale step. Efficiency of the scheme is given by its linearity and stability. Preconditioned iterative solvers are used for computing arising linear systems. We present computational results related to image processing and plane curve evolution.

2020

In this paper we propose a high-order accurate scheme for image segmentation based on the level-set method. In this approach, the curve evolution is described as the 0-level set of a representation function but we modify the velocity that drives the curve to the boundary of the object in order to obtain a new velocity with additional properties that are extremely useful to develop a more stable high-order approximation with a small additional cost. The approximation scheme proposed here is the first 2D version of an adaptive "filtered" scheme recently introduced and analyzed by the authors in 1D. This approach is interesting since the implementation of the filtered scheme is rather efficient and easy. The scheme combines two building blocks (a monotone scheme and a high-order scheme) via a filter function and smoothness indicators that allow to detect the regularity of the approximate solution adapting the scheme in an automatic way. Some numerical tests on synthetic and r...

2005

In this paper, we present a new variational formulation for geometric active contours that forces the level set function to be close to a signed distance function, and therefore completely eliminates the need of the costly re-initialization procedure. Our variational formulation consists of an internal energy term that penalizes the deviation of the level set function from a signed distance function, and an external energy term that drives the motion of the zero level set toward the desired image features, such as object boundaries. The resulting evolution of the level set function is the gradient flow that minimizes the overall energy functional. The proposed variational level set formulation has three main advantages over the traditional level set formulations. First, a significantly larger time step can be used for numerically solving the evolution partial differential equation, and therefore speeds up the curve evolution. Second, the level set function can be initialized with general functions that are more efficient to construct and easier to use in practice than the widely used signed distance function. Third, the level set evolution in our formulation can be easily implemented by simple finite difference scheme and is computationally more efficient. The proposed algorithm has been applied to both simulated and real images with promising results.

SIAM Journal on Applied Mathematics, 2002

A new morphological multiscale method in 3D image processing is presented which combines the image processing methodology based on nonlinear diffusion equations and the theory of geometric evolution problems. Its aim is to smooth level sets of a 3D image while simultaneously preserving geometric features such as edges and corners on the level sets. This is obtained by an anisotropic curvature evolution, where time serves as the multiscale parameter. Thereby the diffusion tensor depends on a regularized shape operator of the evolving level sets. As one suitable regularization local ¡ £ ¢ projection onto quadratic polynomials is considered. The method is compared to a related parametric surface approach and a geometric interpretation of the evolution and its invariance properties are given. A spatial finite element discretization on hexahedral meshes and a semi-implicit, regularized backward Euler discretization in time are the building blocks of the easy to code algorithm. Different applications underline the efficiency and flexibility of the presented image processing tool.

Journal of Differential Geometry, 1996

We develop a level set theory for the mean curvature evolution of surfaces with arbitrary co-dimension, thus generalizing the previous work [6, 13] on hypersurfaces. The main idea is to surround the evolving surface of co-dimension k in R d by a family of hypersurfaces (the level sets of a function) evolving with normal velocity equal to the sum of the (d − k) smallest principal curvatures. The existence and the uniqueness of a weak (level-set) solution, is easily established using mainly the results of [6] and the theory of viscosity solutions for second order nonlinear parabolic equations. The level set solutions coincide with the classical solutions whenever the latter exist. The proof of this connection uses a careful analysis of the squared distance from the surfaces. It is also shown that varifold solutions constructed by Brakke [5] are included in the level-set solutions. The idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of DeGiorgi. An introduction to the barriers and their connection to the level set solutions is also provided.

TELKOMNIKA, 2023

For image segmentation, level set models are frequently employed. It offer best solution to overcome the main limitations of deformable parametric models. However, the challenge when applying those models in medical images stills deal with removing blurs in image edges which directly affects the edge indicator function, leads to not adaptively segmenting images and causes a wrong analysis of pathologies wich prevents to conclude a correct diagnosis. To overcome such issues, an effective process is suggested by simultaneously modelling and solving systems’ two-dimensional partial differential equations (PDE). The first PDE equation allows restoration using Euler’s equation similar to an anisotropic smoothing based on a regularized Perona and Malik filter that eliminates noise while preserving edge information in accordance with detected contours in the second equation that segments the image based on the first equation solutions. This approach allows developing a new algorithm which overcome the studied model drawbacks. Results of the proposed method give clear segments that can be applied to any application. Experiments on many medical images in particular blurry images with high information losses, demonstrate that the developed approach produces superior segmentation results in terms of quantity and quality compared to other models already presented in previeous works.

Radio Science, 2002

Anisotropic mean curvature motion and in particular anisotropic surface diffusion play a crucial role in the evolution of material interfaces. This evolution interacts with conservations laws in the adjacent phases on both sides of the interface and are frequently expected to undergo topological chances. Thus, a level set formulation is an appropriate way to describe the propagation. Here we recall a general approach for the integration of geometric gradient flows over level set ensembles and apply it to derive a variational formulation for the level set representation of anisotropic mean curvature motion and anisotropic surface flow. The variational formulation leads to a semi-implicit discretization and enables the use of linear finite elements.

Pattern Recognition Letters, 2007

Several computer vision problems, like segmentation, tracking and shape modeling, are increasingly being solved using level set methodologies. But the critical issues of stability and convergence have always been neglected in most of the level set implementations. This often leads to either complete breakdown or premature/delayed termination of the curve evolution process, resulting in unsatisfactory results. We present a generic convergence criterion and also a means of determining the optimal time-step involved in the numerical solution of the level set equation. The significant improvement in the performance of level set algorithms, as a result of the proposed changes, is demonstrated using object tracking and shape-contour extraction results.

Tatra Mountains Mathematical Publications, 2018

The aim of the paper is to study problem of image segmentation and missing boundaries completion introduced in [Mikula, K.—Sarti, A.––Sgallarri, A.: Co-volume method for Riemannian mean curvature flow in subjective surfaces multiscale segmentation, Comput. Vis. Sci. 9 (2006), 23–31], [Mikula, K.—Sarti, A.—Sgallari, F.: Co-volume level set method in subjective surface based medical image segmentation, in: Handbook of Medical Image Analysis: Segmentation and Registration Models (J. Suri et al., eds.), Springer, New York, 583–626, 2005], [Mikula, K.—Ramarosy, N.: Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing, Numer. Math. 89 (2001), 561–590] and [Tibenský, M.: VyužitieMetód Založených na Level Set Rovnici v Spracovaní Obrazu, Faculty of mathematics, physics and informatics, Comenius University, Bratislava, 2016]. We generalize approach presented in [Eymard, R.—Handlovičová, A.—Mikula, K.: Study of a finite volume scheme for regularised...

2006 9th International Conference on Control, Automation, Robotics and Vision, 2006

Accurate and fast image segmentation algorithms are of paramount importance for a wide range of medical imaging applications. Level set algorithms based on narrow band implementation have been among the most widely used segmentation algorithms. However, the accuracy of standard level set algorithms is compromised by the fact that their evolution schemes deteriorate the signed distance level set functions required for accurate computation of normals and curvatures. The most common remedy is to use an ad-hoc reinitialization step to rebuild the signed distance function frequently. Meanwhile, complex upwind finite difference schemes are required for stable evolution. They together make the overall computation expensive. In this paper, we propose a novel fast narrow band distance preserving level set evolution algorithm that eliminates the need for both reinitialization and complex upwind finite difference schemes. This is achieved by incorporating into a variational level set formulation with a signed distance preserving term that regularizes the evolution. As a result, stable, accurate, fast evolution could be obtained using a simple finite difference scheme within a very narrow band, defined as the union of all 3 × 3 pixel blocks around the zero crossing pixels. Also, our method allows the use of larger time step to speed up the convergence while ensuring accurate result, as well as the use of more general and computational efficient initial level set functions rather than the signed distance functions required by standard level set methods. The proposed algorithm has been applied on both synthetic and real images of different modalities with promising results.

Communications on Pure and Applied Mathematics, 2017

We showed earlier that the level set function of a monotonic advancing front is twice differentiable everywhere with bounded second derivative. We show here that the second derivative is continuous if and only if the flow has a single singular time where it becomes extinct and the singular set consists of a closed C 1 manifold with cylindrical singularities.

Computing and Visualization in Science, 1998

Numerical approximation of a nonlinear diffusion equation of mean curvature flow type is discussed. Computational results related to image processing are presented.

Lecture Notes in Computer Science, 2000

This paper is concerned with the simulation of the Partial Differential Equation (PDE) driven evolution of a closed surface by means of an implicit representation. In most applications, the natural choice for the implicit representation is the signed distance function to the closed surface. Osher and Sethian propose to evolve the distance function with a Hamilton-Jacobi equation. Unfortunately the solution to this equation is not a distance function. As a consequence, the practical application of the level set method is plagued with such questions as when do we have to "reinitialize" the distance function? How do we "reinitialize" the distance function? Etc... which reveal a disagreement between the theory and its implementation. This paper proposes an alternative to the use of Hamilton-Jacobi equations which eliminates this contradiction: in our method the implicit representation always remains a distance function by construction, and the implementation does not differ from the theory anymore. This is achieved through the introduction of a new equation. Besides its theoretical advantages, the proposed method also has several practical advantages which we demonstrate in three applications: (i) the segmentation of the human cortex surfaces from MRI images using two coupled surfaces [27], (ii) the construction of a hierarchy of Euclidean skeletons of a 3D surface, (iii) the reconstruction of the surface of 3D objects through stereo [13].

IEEE Visualization 2005 - (VIS'05), 2005

Level set methods, an important class of partial differential equation (PDE) methods, define dynamic surfaces implicitly as the level set (isosurface) of a sampled, evolving nD function. This course is targeted for researchers interested in learning about level set and other PDE-based methods, and their application to visualization. The course material will be presented by several of the recognized experts in the field, and will include introductory concepts, practical considerations and extensive details on a variety of level set/PDE applications. The course will begin with preparatory material that introduces the concept of using partial differential equations to solve problems in visualization. This will include the structure and behavior of several different types of differential equations, e.g. the level set, heat and reaction-diffusion equations, as well as a general approach to developing PDE-based applications. The second stage of the course will describe the numerical methods and algorithms needed to implement the mathematics and methods presented in the first stage, including information on implementing the algorithms on GPUs. Throughout the course the technical material will be tied to applications, e.g. image processing, geometric modeling, dataset segmentation, model processing, surface reconstruction, anisotropic geometric diffusion, flow field post-processing and vector visualization. Prerequisites Knowledge of calculus, linear algebra, computer graphics, visualization, geometric modeling and computer vision. Some familiarity with differential geometry, differential equations, numerical computing and image processing is strongly recommended, but not required.