Abstract: Ultrasound imaging systems provide the clinician with noninvasive, low-cost, and real-time images that can help them in diagnosis, planning, and therapy. However, although the human eye is able to derive the meaningful information from these images, automatic processing is very difficult due to noise and artifacts present in the image. The speckle reducing anisotropic diffusion filter was recently proposed to adapt the anisotropic diffusion filter to the characteristics of the speckle noise present in the ultrasound images and to facilitate automatic processing of images. We analyze the properties of the numerical scheme associated with this filter, using a semi-explicit scheme. We then extend the filter to a matrix anisotropic diffusion, allowing different levels of filtering across the image contours and in the principal curvature directions. We also show a relation between the local directional variance of the image intensity and the local geometry of the image, which can justify the choice of the gradient and the principal curvature directions as a basis for the diffusion matrix. Finally, different filtering techniques are compared on a 2-D synthetic image with two different levels of multiplicative noise and on a 3-D synthetic image of a Y-junction, and the new filter is applied on a 3-D real ultrasound image of the liver

Abstract: We present an image reconstruction method for diffuse optical tomography (DOT) by using the sparsity regularization and expectation-maximization (EM) algorithm. Typical image reconstruction approaches in DOT employ Tikhonov-type regularization, which imposes restrictions on the L(2) norm of the optical properties (absorption/scattering coefficients). It tends to cause a blurring effect in the reconstructed image and works best when the unknown parameters follow a Gaussian distribution. In reality, the abnormality is often localized in space. Therefore, the vector corresponding to the change of the optical properties compared with the background would be sparse with only a few elements being nonzero. To incorporate this information and improve the performance, we propose an image reconstruction method by regularizing the L(1) norm of the unknown parameters and solve it iteratively using the expectation-maximization algorithm. We verify our method using simulated 3D examples and compare the reconstruction performance of our approach with the level-set algorithm, Tikhonov regularization, and simultaneous iterative reconstruction technique (SIRT). Numerical results show that our method provides better resolution than the Tikhonov-type regularization and is also efficient in estimating two closely spaced abnormalities.

Abstract: Diffusion tensor imaging (DT-MRI) is very sensitive to corrupting noise due to the non linear relationship between the diffusion-weighted image intensities (DW-MRI) and the resulting diffusion tensor. Denoising is a crucial step to increase the quality of the estimated tensor field. This enhanced quality allows for a better quantification and a better image interpretation. The methods proposed in this paper are based on the Non-Local (NL) means algorithm. This approach uses the natural redundancy of information in images to remove the noise. We introduce three variations of the NL-means algorithms adapted to DW-MRI and to DT-MRI. Experiments were carried out on a set of 12 diffusion-weighted images (DW-MRI) of the same subject. The results show that the intensity based NL-means approaches give better results in the context of DT-MRI than other classical denoising methods, such as Gaussian Smoothing, Anisotropic Diffusion and Total Variation.

Abstract: In this paper we present an anisotropic filter for speckle reduction in ultrasound images and an adaptation of the geodesic active contours technique for the segmentation of breast tumors. The anisotropic diffusion we propose is based on a texture description provided by a set of Gabor filters and allows reducing speckle noise while preserving edges. Furthermore, it is used to extract an initial pre-segmentation of breast tumors which is used as initialization for the active contours technique. This technique has been adapted to the characteristics of ultrasonography by adding certain texture-related terms which provide a better discrimination of the regions inside and outside the nodules. These terms allow obtaining a more accurate contour when the gradients are not high and uniform.

Abstract: The axonal membrane and the myelin sheets around axons restrict water diffusion perpendicular to the fiber orientation in white matter but not along the axon’s main axis, which leads to anisotropic diffusion of water. Over the last decade, the quantitative measurement of this anisotropy by using what has been called diffusion tensor imaging (DTI) has become well established in research applications, revealing the macroscopic threedimensional architecture of white matter and the constituent white matter tracts. The depiction of white matter architecture in orientation-based color coding is a visualization approach in which the image brightness depicts diffusion anisotropy and a red/green/blue color scheme indicates tract orientation. Using a multiple region-of-interest (ROI) approach that exploits existing anatomic knowledge of tract trajectories, white matter bundles of interest can be reconstructed non-invasively in vivo. We have created a human white matter brain atlas, consisting of projection, association, and commissural fibers as well as brainstem fibers, examples of which are depicted here.

Abstract: We propose and analyze a posteriori energy-norm error estimates for weighted interior penalty discontinuous Galerkin approximations to advection-diffusion-reaction equations with heterogeneous and anisotropic diffusion. The weights, which play a key role in the analysis, depend on the diffusion tensor and are used to formulate the consistency terms in the discontinuous Galerkin method. The error upper bounds, in which all the constants are specified, consist of three terms: a residual estimator which depends only on the elementwise fluctuation of the discrete solution residual, a diffusive flux estimator where the weights used in the method enter explicitly, and a non-conforming estimator which is nonzero because of the use of discontinuous finite element spaces. The three estimators can be bounded locally by the approximation error. For moderate advection, it is shown that full robustness with respect to diffusion heterogeneities is achieved owing to the specific design of the weights in the discontinuous Galerkin method, while diffusion anisotropies remain purely local and impact the constants through the square root of the condition number of the diffusion tensor. For dominant advection, it is shown, in the spirit of previous work by Verfurth on continuous finite elements, that the constants are bounded by the square root of the local Peclet number.

Abstract: Segmentation of images holds an important position in the area of image processing. It becomes more important while typically dealing with medical images where pre-surgery and post surgery decisions are required for the purpose of initiating and speeding up the recovery process (noodle.med.yale.edu 1993). Computer aided detection of abnormal growth of tissues is primarily motivated by the necessity of achieving maximum possible accuracy. Manual segmentation of these abnormal tissues cannot be compared with modern day’s high speed computing machines which enable us to visually observe the volume and location of unwanted tissues. A well known segmentation problem within MRI is the task of labeling voxels according to their tissue type which include White Matter (WM), Grey Matter (GM), Cerebrospinal Fluid (CSF) and sometimes pathological tissues like tumor etc. This chapter describes an efficient method for automatic brain tumor segmentation for the extraction of tumor tissues from MR images. It combines Perona and Malik (1990) anisotropic diffusion model for image enhancement and Kmeans clustering technique for grouping tissues belonging to a specific group. The proposed method uses T1, T2 and PD weighted gray level intensity images. The proposed technique produced appreciative results.

Abstract: In order to improve signal-to-noise ratio (SNR) and contrast-to-noise ratio, this paper introduces a local variance-controlled forward-and-backward (LVCFAB) diffusion algorithm for edge enhancement and noise reduction. In our algorithm, an alternative FAB diffusion algorithm is proposed. The results for the alternative FAB algorithm show better algorithm behavior than other existing diffusion FAB approaches. Furthermore, two distinct discontinuity measures and the alternative FAB diffusion are incorporated into a LVCFAB diffusion algorithm, where the joint use of the two measures leads to a complementary effect for preserving edge features in digital images. This LVC mechanism adaptively modifies the degree of diffusion at any image location and is dependent on both local gradient and inhomogeneity. Qualitative experiments, based on general digital images and magnetic resonance images, show significant improvements when the LVCFAB diffusion algorithm is used versus the existing anisotropic diffusion and the previous FAB diffusion algorithms for enhancing edge features and improving image contrast. Quantitative analyses, based on peak SNR, confirm the superiority of the proposed LVCFAB diffusion algorithm.

Abstract: The annulus fibrosus (AF) of the intervertebral disc (IVD) exhibits a fiber-organized structure which is responsible for anisotropic and inhomogeneous mechanical and transport properties. Due to its particular morphology, nutrient transport within AF is regulated by complex transport kinetics. This work investigates the diffusive transport of a small solute in the posterior and anterior regions of AF since diffusion is the major transport mechanism for low molecular weight nutrients (e.g., oxygen and glucose) in IVD. Diffusion coefficient (D) of fluorescein (332 Da) in bovine coccygeal AF was measured in the three major (axial, circumferential, and radial) directions of the IVD by means of fluorescence recovery after photobleaching (FRAP) technique. It was found that the diffusion coefficient was anisotropic and inhomogeneous. In both anterior and posterior regions, the diffusion coefficient in the radial direction was found to be the lowest. Circumferential and axial diffusion coefficients were not significantly different in both posterior and anterior regions and their values were about 130% and 150% the value of the radial diffusion coefficient, respectively. The values of diffusion coefficients in the anterior region were in general higher than those of corresponding diffusion coefficients in the posterior region. This study represents the first quantitative analysis of anisotropic diffusion transport in AF by means of FRAP technique and provides additional knowledge on understanding the pathways of nutritional supply into IVD.

Abstract: The extraction of "Level 2" detail -- ridge terminations, ridge bifurcations, bridges etc. -- from digitised images of fingerprints requires an accurate segmentation of the image into ridges and valleys. Small breaks and irregularities in the ridge pattern occur as a result of imperfections in the print capture process that, if not rectified, give rise to many false level 2 features at later stages of the analysis. We propose a method for enhancing the ridge pattern by applying a process of oriented diffusion, which is an adaptation of anisotropic diffusion. This acts to smooth the image only in the direction parallel to the ridge flow. The result is an image in which intensity varies smoothly as one traverses along the ridges or valleys, with most of the small irregularities and breaks removed, but with the identity of the individual ridges and valleys preserved. The method offers the advantage of requiring no prior estimate of the ridge frequency. Results show improved performance by comparison with the method of enhancement using frequency-tuned filters, which sometimes performs well but may produce erroneous results if the filter is tuned to a frequency that does not match the actual ridge frequency.

Abstract: Noise reduction is an important image processing method which has wide applications in different fields. The key to noise reduction is to reduce the noise without deteriorating the important features in the images. Anisotropic diffusion filter is one of the methods which satisfy this need and draws much attention from researchers in the past. However, traditional anisotropic diffusion filter has many disadvantages, such as the sensitivity to the noise. In this paper, an improved anisotropic diffusion filter for preprocessing noised MR images is presented. The improved anisotropic diffusion filter uses adaptive threshold selection and a new gradient computation method, which is robust to noise. The proposed method was applied to real MR images and the results are impressive.

Abstract: Lateral diffusion on curved biological membranes has been studied theoretically and experimentally. However, how membrane geometries influence the diffusion process remains unclear. Here we show the significance of Gaussian curvature by numerically solving the diffusion equation in a geodesic polar coordinate system with regard to several types of surfaces including elliptic and hyperbolic paraboloids. On surfaces where Gaussian curvature has positive and negative values, diffusion is slower and faster than on the plane, respectively. The deviation from the normal diffusion on the plane tends to get larger as the absolute value of Gaussian curvature increases. Diffusion is anisotropic at a surface region where the normal curvature is anisotropic and Gaussian curvature has nonzero values. The anisotropy can be classified into several types according to whether diffusion is the fastest or the slowest in the principal directions. In the case of diffusion on spheroids, the limited area of a closed surface reduces the diffusion rate so greatly that the slowdown effects of positive values of Gaussian curvature are concealed. Analysis of the diffusion equation suggests that Gaussian curvature causes slowed or accelerated diffusion and anisotropic diffusion in any type of surface. Furthermore, it is discussed the degree to which Gaussian curvature influences diffusive phenomena taking place in real membranes through such effects. These results provide a different image of biological membranes that lateral diffusion of membrane molecules is usually anisotropic and the diffusion rate kaleidoscopically changes according to place.

Abstract: In this article we design the semiimplicit finite volume scheme for coherence enhancing diffusion in image processing and prove its convergence to the weak solution of the problem. The finite volume methods are natural tools for image processing applications since they use piecewise constant representation of approximate solutions similarly to the structure of digital images. They have been successfully applied in image processing, e.g., for solving the Perona-Malik equation or curvature-driven level set equations, where the nonlinearities are represented by a scalar function dependent on a solution gradient. Design of suitable finite volume schemes for tensor diffusion is a nontrivial task here we present the first such scheme with a convergence proof for the practical nonlinear model used in coherence-enhancing image smoothing. We provide basic information about this type of nonlinear diffusion including a construction of its diffusion tensor, and we derive a semiimplicit finite volume scheme for this nonlinear model with the help of covolume mesh. This method is well known as the diamond-cell method owing to the choice of covolume as a diamond-shaped polygon. Further, we prove a convergence of a discrete solution given by our scheme to the weak solution of the problem. The proof is based on Kolmogorov's compactness theorem and a bounding of a gradient in the tangential direction by using a gradient in the normal direction. Finally computational results illustrated in figures are discussed.

Abstract: We investigate a generalization of the model of Solc and Stockmayer to describe the diffusion-controlled reactions between chemically anisotropic reactants taking into account the partially reflecting conditions on two parts of the reaction surface. The exact solution of the relevant mixed boundary-value problem was found for different ratios of the intrinsic rate constants. The results obtained may be used to test numerical programs that describe diffusion-controlled reactions in real systems of particles with anisotropic reactivity.

Abstract: We propose a nonlinear image interpolation method, based on an anisotropic diffusion PDE and designed for the general case of vector-valued images. The interpolation solution is restricted to the subspace of functions that can recover the discrete input image, after an appropriate smoothing and sampling. The proposed nonlinear diffusion flow lies on this subspace and its strength and anisotropy effectively adapt to the local variations and geometry of image structures. The derived model efficiently reconstructs the real image structures, leading to a natural interpolation, with reduced blurring, staircase and ringing artifacts of classic methods. This method also outperforms other existing PDE-based interpolation methods. We present experimental results that prove the potential and efficacy of the method as applied to graylevel and color images.

Abstract: Linear theory of the ablative Rayleigh-Taylor instability in anisotropic diffusive materials is presented. This analysis indicates that enhancing diffusion in a plane transverse to the mean longitudinal flow can strongly reduce the growth of the instability. In the context of inertial confinement fusion, it is shown that anisotropic diffusion can be achieved using a laminated ablator made of successive layers of different diffusive properties. Numerical simulations confirm the theoretical predictions and indeed exhibit a significant stabilization of the ablation front for laminated ablators.

Abstract: We investigate the asymptotic behavior of some anisotropic diffusion problems and give some estimates on the rate of convergence of the solution toward its limit. We also relate this type of elliptic problems to problems set in cylinder becoming unbounded in some directions and show how some information on one type leads to information for the other type and conversely.

Abstract: Linear and nonlinear diffusion equations are usually considered on an image, which is in fact a function on the translation group. In this paper we study diffusion on orientation scores, i.e. on functions on the Euclidean motion group SE(2). An orientation score is obtained from an image by a linear invertible transformation. The goal is to enhance elongated structures by applying nonlinear left-invariant diffusion on the orientation score of the image. For this purpose we describe how we can use Gaussian derivatives to obtain regularized left-invariant derivatives that obey the non-commutative structure of the Lie algebra of SE(2). The Hessian constructed with these derivatives is used to estimate local curvature and orientation strength and the diffusion is made nonlinearly dependent on these measures. We propose an explicit finite difference scheme to apply the nonlinear diffusion on orientation scores. The experiments show that preservation of crossing structures is the main advantage compared to approaches such as coherence enhancing diffusion.

Abstract: Self-diffusion coefficients of a fluid confined to slab and cylindrical geometries at the nanoscale with varying widths have been studied by considering different density profiles. The average second frequency sum rule of a velocity autocorrelation function in a direction parallel and perpendicular to the wall has been evaluated. It is found that confinement to the nano length scale results in anisotropic diffusion and the diffusion coefficient is directly dependent on the width of the channel and the density profile.

Abstract: It is said that knowledge doubles every 2.5 years. In the field of diagnostic imaging the doubling time is probably shorter due to the rapid pace of technological advances. Diffusion tensor imaging (DTI) and its potential application to neurologic disease represents one such advance. DTI is a special form of diffusion-weighted imaging that allows the assessment and visualization of large white matter fibers on a millimeter-level multidimensional scale.1 Although we can visualize and differentiate white and gray matter with standard MRI pulse sequences, they do not allow us to examine the integrity or directionality of white matter tracts. DTI takes advantage of the diffusivity of water and the restrictions imposed on the diffusion of water by white matter fiber tracts. When fiber tracts are dense, for example, the restriction imposed by their density leads to directionally dependent or anisotropic diffusion. By analogy, if an ink drop is placed in a narrow or oval tube the diffusion of that ink drop will adjust to the shape of the tube. In contrast, if a drop of ink is placed in a large bowl of water the drop of ink will be more spherical …

Abstract: Many algorithms have been developed to recognize regions, edges, color, and objects in images and videos For applications like surveillance or object-based video coding, it is important to segment the foreground objects from the background The task is very challenging in the case of a moving camera We present a foreground segmentation approach that is designed for sprite coding as well as other applications, eg video surveillance Accurate frame-to- frame image registration and sprite generation build the pre-processing step The segmentation algorithm operates on error images, which are produced by the image registration and subtraction from reconstructed background frames It is processed in several steps including low-pass filtering using anisotropic diffusion Experiments show excellent results with single- and multi-view test sequences

Abstract: Segmentation of molecular images is a difficult task due to the low signal-to-noise ratio of images. A novel two-dimensional fuzzy C-means (2DFCM) algorithm is proposed for the molecular image segmentation. The 2DFCM algorithm is composed of three stages. The first stage is the noise suppression by utilizing a method combining a Gaussian noise filter and anisotropic diffusion techniques. The second stage is the texture energy characterization using a Gabor wavelet method. The third stage is introducing spatial constraints provided by the denoising data and the textural information into the two-dimensional fuzzy clustering. The incorporation of intensity and textural information allows the 2DFCM algorithm to produce satisfactory segmentation results for images corrupted by noise (outliers) and intensity variations. The 2DFCM can achieve 0.96 ± 0.03 segmentation accuracy for synthetic images under different imaging conditions. Experimental results on a real molecular image also show the effectiveness of the proposed algorithm.

Abstract: The preprocessing step of most computer-aided diagnosis (CAD) systems for identifying the lung diseases is lung segmentation. We present a novel lung segmentation technique based on anisotropic diffusion and morphological operation which is performed fast and accurately. The proposed method consists of three steps. At first step, gray image is produced by the input image. And then anisotropic diffusion is preformed to blur the gray image. The second step is that morphological operation is performed to remove the airway and mediastinum and get the right and left lung area. At the third step, the binary image of the right and left lung area obtained in the second step is generated and is matched to the original to segment the lung part. The proposed method eliminates the tasks of finding an optimal threshold and separating the attached left and right lungs. We have applied our new approach on several pulmonary CT images and the results reveal the speed, robustness and accuracy of this method.