Fast evolution of image manifolds and application to filtering andsegmentation in 3D medical images (original) (raw)

A Level Set Method for Anisotropic Geometric Diffusion in 3D Image Processing

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.

Non-diffusive numerical scheme for regularized mean curvature flow level set equation in image processing

2010 3rd International Congress on Image and Signal Processing, 2010

We present new numerical scheme for solving regularised mean curvature flow level set equation and show its behavior in image filtering examples. The scheme is based on finite volume space discretization and semi-implicit time discretization [9], it is unconditionally stable and very weakly diffusive. Such properties are important in image filtering where they guarantee correct reconstruction of shapes deteriorated by high level of noise in stable and computationally efficient way. We compare the filtering capabilities of our new scheme with the standard explicit finite difference approximation of the mean curvature level set equation [15] and show appropriate behavior of the new method.

Medical surface smoothing via adaptive diffusion of differential fields

The 2011 IEEE/ICME International Conference on Complex Medical Engineering, 2011

Efficient visualization of vascular structures is essential for therapy planning and medical education. Existing techniques achieve high-quality visualization of vascular surfaces at the cost of low rendering speed and large size of resulting surface. In this paper, we present an approach for visualizing vascular structures by exploiting the local curvature information of a given surface. To handle complex topology of loop and multiple parents and/or multiple children, bidirectional adaptive sampling and modified normal calculations at joints are proposed. The proposed method has been applied to cerebral vascular trees, liver vessel trees, and aortic vessel trees. The experimental results show that it can obtain a high-quality surface visualization with fewer polygons in the approximation.

3D early embryogenesis image filtering by nonlinear partial differential equations

Medical Image Analysis, 2010

We present nonlinear diffusion equations, numerical schemes to solve them and their application for filtering 3D images obtained from laser scanning microscopy (LSM) of living zebrafish embryos, with a goal to identify the optimal filtering method and its parameters. In the large scale applications dealing with analysis of 3D+time embryogenesis images, an important objective is a correct detection of the number and position of cell nuclei yielding the spatio-temporal cell lineage tree of embryogenesis. The filtering is the first and necessary step of the image analysis chain and must lead to correct results, removing the noise, sharpening the nuclei edges and correcting the acquisition errors related to spuriously connected subregions. In this paper we study such properties for the regularized Perona-Malik model and for the generalized mean curvature flow equations in the level-set formulation. A comparison with other nonlinear diffusion filters, like tensor anisotropic diffusion and Beltrami flow, is also included. All numerical schemes are based on the same discretization principles, i.e. finite volume method in space and semi-implicit scheme in time, for solving nonlinear partial differential equations. These numerical schemes are unconditionally stable, fast and naturally parallelizable. The filtering results are evaluated and compared first using the Mean Hausdorff distance between a gold standard and different isosurfaces of original and filtered data. Then, the number of isosurface connected components in a region of interest (ROI) detected in original and after the filtering is compared with the corresponding correct number of nuclei in the gold standard. Such analysis proves the robustness and reliability of the edge preserving nonlinear diffusion filtering for this type of data and lead to finding the optimal filtering parameters for the studied models and numerical schemes. Further comparisons consist in ability of splitting the very close objects which are artificially connected due to acquisition error intrinsically linked to physics of LSM. In all studied aspects it turned out that the nonlinear diffusion filter which is called geodesic mean curvature flow (GMCF) has the best performance.

Structure-preserving smoothing of biomedical images

Pattern Recognition, 2011

Smoothing of biomedical images should preserve gray-level transitions between adjacent tissues, while restoring contours consistent with anatomical structures. Anisotropic diffusion operators are based on image appearance discontinuities (either local or contextual) and might fail at weak inter-tissue transitions. Meanwhile, the output of block-wise and morphological operations is prone to present a block structure due to the shape and size of the considered pixel neighborhood. In this contribution, we use differential geometry concepts to define a diffusion operator that restricts to image consistent level-sets. In this manner, the final state is a non-uniform intensity image presenting homogeneous inter-tissue transitions along anatomical structures, while smoothing intra-structure texture. Experiments on different types of medical images (magnetic resonance, computerized tomography) illustrate its benefit on a further process (such as segmentation) of images.

Algebraic Filtering of Surfaces from 3D Medical Images with Julia

CAD'20

In this paper we introduce a novel algebraic filter, based on algebraic topology methods, to extract and smooth the boundary surface of any subset of voxels arising from the segmentation of a 3D medical image. The input of the Linear Algebraic Representation (lar) Surface extraction filter (lar-surf) is defined as a chain, i.e., an element of a linear space of chains here subsets of voxels represented in coordinates as a sparse binary vector. The output is produced by a linear mapping between spaces of 3-and 2-chains, given by the boundary operator ∂ 3 : C 3 → C 2. The only data structures used in this approach are sparse arrays with one or two indices, i.e., sparse vectors and sparse matrices. This work is based on lar algebraic methods and is implemented in Julia language, natively supporting parallel computing on hybrid hardware architectures.

Geometric surface smoothing via anisotropic diffusion of normals

IEEE Visualization, 2002. VIS 2002., 2002

Figure 1: Processing results on the MRI head model: (a) original isosurface, (b) isotropic diffusion (intrinsic Laplacian of mean curvature ¤ow), and (c) anisotropic diffusion. The small protrusion under the nose is a physical marker used for registration.

Extending anisotropic operators to recover smooth shapes

Computer Vision and Image Understanding, 2005

Anisotropic differential operators are widely used in image enhancement processes. Recently, their property of smoothly extending functions to the whole image domain has begun to be exploited. Strong ellipticity of differential operators is a requirement that ensures existence of a unique solution. This condition is too restrictive for operators designed to extend image level sets: their own functionality implies that they should restrict to some vector field. The diffusion tensor that defines the diffusion operator links anisotropic processes with Riemmanian manifolds. In this context, degeneracy implies restricting diffusion to the varieties generated by the vector fields of positive eigenvalues, provided that an integrability condition is satisfied. We will use that any smooth vector field fulfills this integrability requirement to design line connection algorithms for contour completion. As application we present a segmenting strategy that assures convergent snakes whatever the geometry of the object to be modelled is.

Level Set Regularization Using Geometric Flows

SIAM J. Imaging Sci., 2018

In this paper we study a geometric partial differential equation including a forcing term. This equation defines a hypersurface evolution that we use for level set regularization. We study the shape of the radial solutions of the equation and some qualitative properties about the level set propagations. We show that under a suitable choice of the forcing term, the geometric equation has nontrivial asymptotic states and it represents a model for level set regularization. We show that by using a forcing term which is merely a bounded Holder continuous function, we can obtain finite time stabilization of the solutions. We introduce an explicit finite difference scheme to compute numerically the solution of the equation and we present some applications of the model to nonlinear two-dimensional image filtering and three-dimensional segmentation in the context of medical imaging.