Abstract: The Laplacian pyramid is ubiquitous for decomposing images into multiple scales and is widely used for image analysis. However, because it is constructed with spatially invariant Gaussian kernels, the Laplacian pyramid is widely believed as being unable to represent edges well and as being ill-suited for edge-aware operations such as edge-preserving smoothing and tone mapping. To tackle these tasks, a wealth of alternative techniques and representations have been proposed, e.g., anisotropic diffusion, neighborhood filtering, and specialized wavelet bases. While these methods have demonstrated successful results, they come at the price of additional complexity, often accompanied by higher computational cost or the need to post-process the generated results. In this paper, we show state-of-the-art edge-aware processing using standard Laplacian pyramids. We characterize edges with a simple threshold on pixel values that allows us to differentiate large-scale edges from small-scale details. Building upon this result, we propose a set of image filters to achieve edge-preserving smoothing, detail enhancement, tone mapping, and inverse tone mapping. The advantage of our approach is its simplicity and flexibility, relying only on simple point-wise nonlinearities and small Gaussian convolutions; no optimization or post-processing is required. As we demonstrate, our method produces consistently high-quality results, without degrading edges or introducing halos.

Abstract: In this paper we address the problem of producing a high-resolution image from a single low-resolution image without any external training set. We propose a framework for both magnification and deblurring using only the original low-resolution image and its blurred version. In our method, each pixel is predicted by its neighbors through the Gaussian process regression. We show that when using a proper covariance function, the Gaussian process regression can perform soft clustering of pixels based on their local structures. We further demonstrate that our algorithm can extract adequate information contained in a single low-resolution image to generate a high-resolution image with sharp edges, which is comparable to or even superior in quality to the performance of other edge-directed and example-based super-resolution algorithms. Experimental results also show that our approach maintains high-quality performance at large magnifications.

Abstract: In this paper, a detailed description and comparison of speckle reduction of medical ultrasound, and in particular echocardiography, is presented. Fifteen speckle reduction filters are described in a detailed fashion to facilitate implementation for research and evaluation. The filtering techniques considered include anisotropic diffusion, wavelet denoising, and local statistics. Common nomenclature and notation are adopted, to expedite comparison between approaches. Comparison of the filters is based on their application to simulated images, clinical videos, and a computational requirement analysis. The ultrasound simulation method provides a realistic model of the image acquisition process, and permits the use of a noise-free reference image for comparison. Application of objective quality metrics quantifies the preservation of image edges, overall image distortion, and improvement in image contrast. The computational analysis quantifies the number of operations required for each speckle reduction method. A speed-accuracy analysis of discretization methods for anisotropic diffusion is included. It is concluded that the optimal method is the OSRAD diffusion filter. This method is capable of strong speckle suppression, increasing the average SNRA of the simulated images by a factor of two. This method also shows favorable edge preservation and contrast improvement, and may be efficiently implemented.

Abstract: We present a number of test cases and meshes that were designed as a benchmark for numerical schemes dedicated to the approximation of three-dimensional anisotropic and heterogeneous diffusion problems. These numerical schemes may be applied to general, possibly non conforming, meshes composed of tetrahedra, hexahedra and quite distorted general polyhedra. A number of methods were tested among which conforming finite element methods, discontinuous Galerkin finite element methods, cell-centered finite volume methods, discrete duality finite volume methods, mimetic finite difference methods, mixed finite element methods, and gradient schemes. We summarize the results presented by the participants to the benchmark, which range from the number of unknowns, the approximation errors of the solution and its gradient, to the minimum and maximum values and energy. We also compare the performance of several iterative or direct linear solvers for the resolution of the linear systems issued from the presented schemes.

Abstract: In this paper a finite volume scheme for the heterogeneous and anisotropic diffusion equations is proposed on general, possibly nonconforming meshes. This scheme has both cell-centered unknowns and vertex unknowns. The vertex unknowns are treated as intermediate ones and are expressed as a linear weighted combination of the surrounding cell-centered unknowns, which reduces the scheme to a completely cellcentered one. We propose two types of new explicit weights which allow arbitrary diffusion tensors, and are neither discontinuity dependent nor mesh topology dependent. Both the derivation of the scheme and that of new weights satisfy the linearity-preserving criterion which requires that a discretization scheme should be exact on linear solutions. The resulting new scheme is called as the linearity-preserving cellcentered scheme and the numerical results show that it maintain optimal convergence rates for the solution and flux on general polygonal distorted meshes in case that the diffusion tensor is taken to be anisotropic, at times heterogeneous, and/or discontinuous. Copyright 2010 John Wiley & Sons, Ltd.

Abstract: A new anisotropic diffusion model is proposed for image denoising, which is based on reaction–diffusion systems with p ( x ) -growth. By Galerkin’s method, we establish the existence and uniqueness of weak solutions of the system for Neumann boundary conditions. Experimental results illustrate the effectiveness of the model in image restoration.

Abstract: In diagnosis of diseases Ultrasonic devices are frequently used by healthcare professionals. The main problem during diagnosis is the distortion of visual signals obtained which is due to the consequence of the coherent of nature of the wave transmitted. These distortions are termed as ‘Speckle Noise’. The present study focuses on proposing a technique to reduce speckle noise from ultrasonic devices. This technique uses a hybrid model that combines fourth order PDE based anisotropic diffusion, linked with SRAD filter and wavelet based BayesShrink technique. The proposed filter is compared with traditional filters and existing filters using anisotropic diffusion. Experimental results prove that the proposed method is efficient in reaching convergence quickly and producing quality denoised images.

Abstract: A Delaunay-type mesh condition is developed for a linear finite element approximation of two-dimensional anisotropic diffusion problems to satisfy a discrete maximum principle. The condition is weaker than the existing anisotropic non-obtuse angle condition and reduces to the well known Delaunay condition for the special case with the identity diffusion matrix. Numerical results are presented to verify the theoretical findings.

Abstract: Denoising with edge preservation is very important in digital x-ray imaging since it may allow us to reduce x-ray dose in human subjects without noticeable degradation of the image quality. In denoising filter design for x-ray imaging, edge preservation as well as noise reduction is of great concern not to lose detailed spatial information for accurate diagnosis. In addition to this, fast computation is also important since digital x-ray images are mostly comprised of large sized matrices. We have developed a new denoising filter based on the nonlinear diffusion filter model. Rather than employing four directional gradients around the pixel of interest, we use geometric parameters derived from the local pixel intensity distribution in calculating the diffusion coefficients in the horizontal and vertical directions. We have tested the filter performance, including edge preservation and noise reduction, using low dose digital radiography and micro-CT images. The proposed denoising filter shows performance similar to those of nonlinear anisotropic diffusion filters (ADFs), one Perona-Malik ADF and the other Weickert's ADF in terms of edge preservation and noise reduction. However, the computation time has been greatly reduced. We expect the proposed denoising filter can be greatly used for fast noise reduction particularly in low-dose x-ray imaging.

Abstract: Background Speckles in ultrasound imaging affect image quality and can make the post-processing difficult. Speckle reduction technologies have been employed for removing speckles for some time. One of the effective speckle reduction technologies is anisotropic diffusion. Anisotropic diffusion technology can remove the speckles effectively while preserving the edges of the image and thus has drawn great attention from image processing scientists. However, the proposed methods in the past have different disadvantages, such as being sensitive to the number of iterations or low capability of preserving the details of the ultrasound images. Thus a detail preserved anisotropic diffusion speckle reduction with less sensitive to the number of iterations is needed. This paper aims to develop this kind of technologies.

Abstract: Fluorescence diffuse optical tomography (fDOT) is an imaging modality that provides images of the fluorochrome distribution within the object of study. The image reconstruction problem is ill-posed and highly underdetermined and, therefore, regularisation techniques need to be used. In this paper we use a nonlinear anisotropic diffusion regularisation term that incorporates anatomical prior information. We introduce a split operator method that reduces the nonlinear inverse problem to two simpler problems, allowing fast and efficient solution of the fDOT problem. We tested our method using simulated, phantom and ex-vivo mouse data, and found that it provides reconstructions with better spatial localisation and size of fluorochrome inclusions than using the standard Tikhonov penalty term.

Abstract: Reactive diffusion in nanomaterials differs widely from that in bulk materials. Reviewed in this paper are the basic models and experimental data on how diffusion and phase transformations occur in multilayer nanosystems as these are being prepared and subsequently thermally annealed. The growth kinetics of amorphous silicide phases in Sc/Si and Mo/Si multilayer periodic systems are studied using the combination of high-resolution transmission electron microscopy and small-angle X-ray diffraction. A model is proposed for silicon diffusion through amorphous silicide that undergoes structural relaxation and crystallization as it grows. Anisotropic diffusion and growth of the silicide phase at adjacent interfaces are studied, and the diffusion parameters are measured for the earliest stages of diffusion annealing.

Abstract: Preservation of fine feature of an image is essential during the process of noise removal, especially via some types of smoothing such as using diffusion process-based methods to enhance images. In this paper, we present a new edge indicator called difference eigenvalue to measure image gradient magnitude in the diffusion process. Based on the eigenvalues of the Hessian matrix, the difference eigenvalue manifest itself in terms of structural information of an image. We adapt the new edge indicator to a diffusion model to achieve a better balance between noise removal and detail preservation. Experiments on both synthetic and real images show that the new model can obtain good results and outperforms existing methods.

Abstract: Conventional Canny edge detector can detect edges in image with additive noise effectively but not ultrasound image that are corrupted by multiplicative speckle noise which alleviates image resolution resulting in inaccurate characterization of object features. In this paper, we proposed to incorporate the modified SRAD into the Canny edge detector to replace the Gaussian blurring in the conventional Canny edge detector in order to suppress the multiplicative noise effectively while preserving the edge of the object in ultrasound image. The result shows that the proposed method can provide better result than conventional method in a much wider range of parameter values. The proposed method through experimental result indicates that it is capable of producing promising edge detection result in ultrasound image.

Abstract: In this work we proposed a lattice Boltzmann model for the nonlinear convection-diffusion equation (NCDE) with anisotropic diffusion. The constraints on the model for correctly recovering macroscopic equation are also carefully analyzed, which are ignored in some existing work. Detailed simulations of some 1D/2D NCDEs, including the nonlinear Schrodinger equation (NLSE), Buckley-Leverett equation with discontinuous initial data, NCDE with anisotropic diffusion, and generalized Zakharov system, are performed. The numerical results obtained by the proposed model agree well with the analytical solutions and/or the numerical solutions reported in previous studies. It is also found that, for complex-valued NLSE, the model using a complex distribution function is superior to that using two real distribution functions for the real and imaginary parts of the NLSE separately.

Abstract: In this paper, we consider anisotropic diffusion with decay, which takes the form �(x)c(x) div(D(x)grad(c(x))) = f(x) with decay coefficient�(x) � 0, and diffusivity coefficient D(x) to be a second-order symmetric and positive definite tensor. It is well-known that this partic- ular equation is a second-order elliptic equation, and satisfies a maximum principle under certain regularity assumptions. However, the finite element implementation of the classical Galerkin for- mulation for both anisotropic and isotropic diffusion with decay does not respect the maximum principle. Put differently, the classical Galerkin formulation violates the discrete maximum princi- ple for diffusion with decay even on structured computational meshes. We first show that the numerical accuracy of the classical Galerkin formulation deteriorates dramatically with an increase infor isotropic media and violates the discrete maximum prin- ciple. However, in the case of isotropic media, the extent of violation decreases with the mesh refinement. We then show that, in the case of anisotropic media, the classical Galerkin formulation for anisotropic diffusion with decay violates the discrete maximum principle even at lower values of decay coefficient and does not vanish with mesh refinement. We then present a methodology for enforcing maximum principles under the classical Galerkin formulation for anisotropic diffusion with decay on general computational grids using optimization techniques. Representative numer- ical results (which take into account anisotropy and heterogeneity) are presented to illustrate the performance of the proposed formulation. In this paper we consider heterogeneous anisotropic diffusion with decay, which takes the form: �(x)c(x)−div(D(x)grad(c(x))) = f(x) with �(x) � 0 and D(x) is a symmetric and positive definite second-order tensor. This equation is a linear second-order elliptic partial differential equation (21). There are many important problems in Mathematical Physics which give rise to this equation (60). Also, this equation arises in numerical methods and mathematical analysis of transient problems (35). Some of these cases include:

Abstract: An edge-preserving diffusion filter maintains the sharp edges in images while smoothing out image noise. An edge-preserving diffusion filter applies an edge-preserving smoothing filter to an image to form a filtered image. The modified image is blurred by a blurring filter to form a blurred image. The modified image and the blurred image are blended together to form an output image based on an error metric associated with each pixel. The edgepreserving diffusion filter may be utilized to perform a multilevel decomposition of the image. The edge-preserving diffusion filter may be applied to an unfiltered image to produce a base image. The difference between the unfiltered image and the base image defines a detail image. The detail image may be used as the input for recursively generating additional levels of detail. The multilevel decomposition may utilize filter kernels associated with different contrast levels for each iteration.

Abstract: We consider left-invariant diffusion processes on DTI data by embedding the data into the space $\mathbb{R}^3\rtimes S^2$ of 3D positions and orientations. We then define and solve the diffusion equation in a moving frame of reference defined using left-invariant derivatives. The diffusion process is made adaptive to the data in order to do Perona-Malik-like edge preserving smoothing, which is necessary to handle fiber structures near regions of large isotropic diffusion such as the ventricles of the brain. The corresponding partial differential systems are solved using finite difference stencils. We include experiments both on synthetic data and on DTI-images of the brain.

Abstract: Nonlinear partial differential equation (PDE) models are established approaches for image/signal processing, data analysis and surface construction. Most previous geometric PDEs are utilized as low-pass filters which give rise to image trend information. In an earlier work, we introduced mode decomposition evolution equations (MoDEEs), which behave like high-pass filters and are able to systematically provide intrinsic mode functions (IMFs) of signals and images. Due to their tunable time-frequency localization and perfect reconstruction, the operation of MoDEEs is called a PDE transform. By appropriate selection of PDE transform parameters, we can tune IMFs into trends, edges, textures, noise etc., which can be further utilized in the secondary processing for various purposes. This work introduces the variational formulation, performs the Fourier analysis, and conducts biomedical and biological applications of the proposed PDE transform. The variational formulation offers an algorithm to incorporate two image functions and two sets of low-pass PDE operators in the total energy functional. Two low-pass PDE operators have different signs, leading to energy disparity, while a coupling term, acting as a relative fidelity of two image functions, is introduced to reduce the disparity of two energy components. We construct variational PDE transforms by using Euler-Lagrange equation and artificial time propagation. Fourier analysis of a simplified PDE transform is presented to shed light on the filter properties of high order PDE transforms. Such an analysis also offers insight on the parameter selection of the PDE transform. The proposed PDE transform algorithm is validated by numerous benchmark tests. In one selected challenging example, we illustrate the ability of PDE transform to separate two adjacent frequencies of sin(x) and sin(1.1x). Such an ability is due to PDE transform's controllable frequency localization obtained by adjusting the order of PDEs. The frequency selection is achieved either by diffusion coefficients or by propagation time. Finally, we explore a large number of practical applications to further demonstrate the utility of proposed PDE transform.

Abstract: In this paper, we present a novel scheme for anisotropic diffusion driven by the image autocorrelation function. We show the equivalence of this scheme to a special case of iterated adaptive filtering. By determining the diffusion tensor field from an autocorrelation estimate, we obtain an evolution equation that is computed from a scalar product of diffusion tensor and the image Hessian. We propose further a set of filters to approximate the Hessian on a minimized spatial support. On standard benchmarks, the resulting method performs favorable in many cases, in particular at low noise levels. In a GPU implementation, video real-time performance is easily achieved.

Abstract: Image fusion is one of the important embranchments of data fusion. Its purpose is to synthesis multi-image information in one scene to one image which is more suitable to human vision and computer vision or more adapt to further image processing, such as target identification. This paper mainly discusses the image fusion method based on wavelet transformation. Firstly, the article gives the basic concept of multi-focus image fusion. On top of this, the paper gives the theory of wavelet analyses and its fast arithmetic, hereon gives the image fusion method based on singe wavelet. Getting on with the single wavelet, the paper presents some improved wavelet as multi-wavelet, multi-band multi-wavelet, including their theories and their arithmetic of decomposition and reconstruction. At the same time, the article applies the multi-band multi wavelet in the image fusion with the wavelet fusion thought. In the side of selecting fusion arithmetic operators, the paper compares the methods based on pictures, windows and regions, and adopts the fusion norm based on grads and characteristic measurement of regional energy. Besides it compares images which is based on different fusion norms and different wavelets in the aspects of entropy, peak value Signal-to-Noise, square root error and standard error in the experimentation. By using Matlab as experimental platform, we approved the feasibility and validity of the method mentioned in the article through a lot of experiments. The result indicates that multi-band multi-wavelet is very effective in image fusion. Furthermore, the article does some post processing to the fusion image. The method is based on anisotropic diffusion arithmetic based on partial differential equations. The experiments show that the brim diffusion enhanced the PSNR of image with the selective brim diffusion to the fusion image and depressed the image block domino effects caused by wavelet fusion method.

Abstract: A large number of applications in image processing and computer vision depend on image quality. In this paper, main concerns are image denoising and deblurring simultaneously in a restoration task by three types of methodologies: non-convex regularization, inverse diffusion and shock filter. We discuss their relations in the context of image deblurring: the inverse diffusion implied by the non-convex regularization, and the superior ability of deblurring edge of the shock filter to that of the inverse diffusion, both in 1D and 2D cases. Finally, we propose a region-based adaptive anisotropic diffusion with shock filter method, which shows advantages of deblurring edges, denoising and smoothing contours in experiments, compared with some related methods. Therein an idea of “divide and rule” is introduced.

Abstract: Existing Nonlinear Anisotropic Diffusion (NAD) methods in image smoothing cannot obtain satisfied results near singularities and isolated points because of the discretization errors. In this paper, we propose a new scheme, named Enclosed Laplacian Operator of Nonlinear Anisotropic Diffusion (ELONAD), which allows us to provide a unified framework for points in flat regions, edge points and corners, even can delete isolated points and spurs. ELONAD extends two diffusion directions of classical NAD to eight or more enclosed directions. Thus it not only performs NAD according to modules of enclosed directions which can reduce the influence of traction errors greatly, but also distinguishes isolated points and small spurs from corners which must be preserved. Smoothing results for test patterns and real images using different discretization schemes are also given to test and verify our discussions.

Abstract: Images are produced to record or display useful information. Due to imperfections in the imaging and capturing process, however, the recorded image invariably represents a degraded version of the original scene. The field of image restoration (sometimes referred to as image deblurring or image deconvolution) is concerned with the reconstruction or estimation of the uncorrupted image from a blurred and noisy one. The purpose of image restoration is to produce best estimate of source image, given the recorded data and some apriori knowledge. In this paper, technique is presented which attempts to use two algorithms for image restorations: Wiener filter and Fourier inverse filter including further work as implementation of Lucy Richardson algorithm. Inverse filtering is the process of recovering the degraded image. Inverse filters are useful for precorrecting an input signal in anticipation of the degradations caused by the system. This approach also suffers from problems that in most cases produce unacceptable results, assumes no noise, only blurring. The preferred approach is therefore to use methods based on least squares. The so-called Wiener filter is the classic solution to the problem of minimizing the mean squared restoration error, the difference between the original and restored images.

Abstract: In this paper we present a novel anisotropic diffusion model targeted for 3D scalar field data. Our model preserves material boundaries as well as fine tubular structures while noise is smoothed out. One of the major novelties is the use of the directional second derivative to define material boundaries instead of the gradient magnitude for thresholding. This results in a diffusion model that has much lower sensitivity to the diffusion parameter and smoothes material boundaries consistently compared to gradient magnitude based techniques. We empirically analyze the stability and convergence of the proposed diffusion and demonstrate its de-noising capabilities for both analytic and real data. We also discuss applications in the context of volume rendering.