Abstract: We describe a new vector-based primitive for creating smooth-shaded images, called the diffusion curve. A diffusion curve partitions the space through which it is drawn, defining different colors on either side. These colors may vary smoothly along the curve. In addition, the sharpness of the color transition from one side of the curve to the other can be controlled. Given a set of diffusion curves, the final image is constructed by solving a Poisson equation whose constraints are specified by the set of gradients across all diffusion curves. Like all vector-based primitives, diffusion curves conveniently support a variety of operations, including geometry-based editing, keyframe animation, and ready stylization. Moreover, their representation is compact and inherently resolution-independent. We describe a GPU-based implementation for rendering images defined by a set of diffusion curves in realtime. We then demonstrate an interactive drawing system for allowing artists to create artworks using diffusion curves, either by drawing the curves in a freehand style, or by tracing existing imagery. The system is simple and intuitive: we show results created by artists after just a few minutes of instruction. Furthermore, we describe a completely automatic conversion process for taking an image and turning it into a set of diffusion curves that closely approximate the original image content.

Abstract: We present here a number of test cases and meshes which were designed to form a benchmark for finite volume schemes and give a summary of some of the results which were presented by the participants to this benchmark. We address a two-dimensional anisotropic diffusion problem, which is discretized on general, possibly non-conforming meshes. In most cases, the diffusion tensor is taken to be anisotropic, and at times heterogeneous and/or discontinuous. The meshes are either triangular or quadrangular, and sometimes quite distorted. Several methods were tested, among which finite element, discontinous Galerkin, cell centred and vertex centred finite volume methods, discrete duality finite volume methods, mimetic methods. The results given by the participants to the benchmark range from the number of unknowns, the errors on the fluxes or the minimum and maximum values and energy, to the order of convergence (when available).

Abstract: We propose and analyse a symmetric weighted interior penalty method to approximate in a discontinuous Galerkin framework advection―diffusion equations with anisotropic and discontinuous diffusivity. The originality of the method consists in the use of diffusivity-dependent weighted averages to better cope with locally small diffusivity (or equivalently with locally high Peclet numbers) on fitted meshes. The analysis yields convergence results for the natural energy norm that are optimal with respect to mesh size and robust with respect to diffusivity. The convergence results for the advective derivative are optimal with respect to mesh size and robust for isotropic diffusivity, as well as for anisotropic diffusivity if the cell Peclet numbers evaluated with the largest eigenvalue of the diffusivity tensor are large enough. Numerical results are presented to illustrate the performance of the proposed scheme.

Abstract: We give some diffusion models which can be used for restoration in image processing. The proposed model is a combination of fast growth with respect to low gradient and slow growth when the gradient is large. The existence theorem for solutions and some numerical results are given.

Abstract: Defocus can be modeled as a diffusion process and represented mathematically using the heat equation, where image blur corresponds to the diffusion of heat. This analogy can be extended to nonplanar scenes by allowing a space-varying diffusion coefficient. The inverse problem of reconstructing 3D structure from blurred images corresponds to an "inverse diffusion" that is notoriously ill posed. We show how to bypass this problem by using the notion of relative blur. Given two images, within each neighborhood, the amount of diffusion necessary to transform the sharper image into the blurrier one depends on the depth of the scene. This can be used to devise a global algorithm to estimate the depth profile of the scene without recovering the deblurred image using only forward diffusion.

Abstract: Compression is an important field of digital image processing where well-engineered methods with high performance exist. Partial differential equations (PDEs), however, have not much been explored in this context so far. In our paper we introduce a novel framework for image compression that makes use of the interpolation qualities of edge-enhancing diffusion. Although this anisotropic diffusion equation with a diffusion tensor was originally proposed for image denoising, we show that it outperforms many other PDEs when sparse scattered data must be interpolated. To exploit this property for image compression, we consider an adaptive triangulation method for removing less significant pixels from the image. The remaining points serve as scattered interpolation data for the diffusion process. They can be coded in a compact way that reflects the B-tree structure of the triangulation. We supplement the coding step with a number of amendments such as error threshold adaptation, diffusion-based point selection, and specific quantisation strategies. Our experiments illustrate the usefulness of each of these modifications. They demonstrate that for high compression rates, our PDE-based approach does not only give far better results than the widely-used JPEG standard, but can even come close to the quality of the highly optimised JPEG2000 codec.

Abstract: Diffusion-weighted steady-state free precession (DW-SSFP) accumulates signal from multiple echoes over several TRs yielding a strong sensitivity to diffusion with short gradient durations and imaging times. Although the DW-SSFP signal is well characterized for isotropic, Gaussian diffusion, it is unclear how the DW-SSFP signal propagates in inhomogeneous media such as brain tissue. This article presents a more general analytical expression for the DW-SSFP signal which accommodates Gaussian and non-Gaussian spin displacement probability density functions. This new framework for calculating the DW-SSFP signal is used to investigate signal behavior for a single fiber, crossing fibers, and reflective barriers. DW-SSFP measurements in the corpus callosum of a fixed brain are shown to be in good agreement with theoretical predictions. Further measurements in fixed brain tissue also demonstrate that 3D DW-SSFP out-performs 3D diffusion weighted spin echo in both SNR and CNR efficiency providing a compelling example of its potential to be used for high resolution diffusion tensor imaging. Magn Reson Med 60:405–413, 2008. © 2008 Wiley-Liss, Inc.

Abstract: We measure the expansion of an ultracold plasma across the field lines of a uniform magnetic field. We image the ion distribution by extracting the ions with a high-voltage pulse onto a position-sensitive detector. Early in the lifetime of the plasma ($l20\text{ }\text{ }\ensuremath{\mu}\mathrm{s}$), the size of the image is dominated by the time-of-flight Coulomb explosion of the dense ion cloud. For later times, we measure the 2D Gaussian width of the ion image, obtaining the transverse expansion velocity as a function of the magnetic field (up to 70 G). We observe that the expansion velocity scales as ${B}^{\ensuremath{-}1/2}$, explained by a nonlinear ambipolar diffusion model with anisotropic diffusion in two different directions.

Abstract: Edge-preserving filters such as local M-smoothers or bilateral filtering are usually designed for Gaussian noise. This paper investigates how these filters can be adapted in order to efficiently deal with Poissonian noise. In addition, the issue of photometry invariance is addressed by changing the way filter coefficients are normalized. The proposed normalization is additive, instead of being multiplicative, and leads to a strong connection with anisotropic diffusion. Experiments show that ensuring the photometry invariance leads to comparable denoising performances in terms of the root mean square error computed on the signal.

Abstract: In this work, we consider the dense reconstruction of specular objects. We propose the use of a specularity constraint, based on surface normal/depth consistency, to define a matching cost function that can drive standard stereo reconstruction methods. We discuss the types of ambiguity that can arise, and suggest an aggregation method based on anisotropic diffusion that is particularly suitable for this matching cost function. We also present a controlled illumination setup that includes a pair of cameras and one LCD monitor, which is used as a calibrated, variable-position light source. We use this setup to evaluate the proposed method on real data, and demonstrate its capacity to recover high-quality depth and orientation from specular objects.

Abstract: We report on the direct observations of the effect of quantum confinement of surface-state electrons on atomic diffusion. Confined electronic states induced by open nanoscale resonators [consisting of two parallel monatomic Cu chains on Cu(111)] are studied by means of scanning tunneling microscope measurements and first-principles calculations. Strongly anisotropic diffusion of adatoms around and inside resonators is revealed at low temperatures. The formation of diffusion channels and empty zones is demonstrated. We show that it is possible to engineer atomic diffusion by varying the distance between the resonator walls.

Abstract: Removing noise from data is often the first step in data analysis. De-noising technique should not be only reduce the noise, but do so without blurring or changing the location of the edges. Many approaches have been proposed to accomplish this; in this paper, we have compared three recently developed techniques for image enhancement and denoising. These methods are based on the use of partial differential equations, including second order, fourth order, and the complex partial differential. We consider various well-known measuring metrics used in image processing applied to standard images in this comparison. In this study, it is shown that the capability of the PDE-based approaches depends highly on the neighboring structure. Our investigations show that in an image where the energy of noise is low, the complex diffusion method offers a better result in image denoising compared to other methods. However, when the energy of the noise increases, performance of the complex diffusion method declines. In general, for the case when the energy of noise in an image is unpredictable, using the heat equation for image denoising is recommended.

Abstract: Conventional electrophotographic printers tend to produce Moire artifacts when used for printing images scanned from printed material such as books and magazines. Inspired by anisotropic diffusion, we propose a novel non-iterative, non-linear, and space-variant de- screening filter that removes a wide range of Moire-causing screen frequencies in a scanned document while preserving image sharpness and edge detail. The amount of diffusion of the image intensity resulting from applying the filter is governed by an estimate of the gradient that is robust under halftone noise. More precisely, the filter extracts a spatial feature vector comprising local intensity gradients estimated from a local window in a pre-smoothed version of the noisy input image. Tunable non-linear polynomial functions of this feature vector are then used to perform one iteration of a discrete diffusion controlled by the intensity gradient. We compare the performance of the proposed algorithm to other descreening solutions and demonstrate that the new algorithm improves quality over the existing methods while reducing computation.

Abstract: The anisotropic diffusion techniques are in general efficient to preserve image edges when they are used to reduce noise. However, they are not very effective to denoise those images that are corrupted by a high level of noise mainly for the lack of a reliable edge-stopping criterion in the partial differential equation (PDE). In this paper, a new algorithm is developed to tackle this problem. The main contribution of this paper is in the construction of a new regularization method for the PDE by using the over-complete dyadic wavelet transform (DWT). It proposes to perform anisotropic diffusion in the more stationary DWT domain rather than directly in the raw noisy image domain. In the DWT domain, since noise tends to decrease as the scale increases, at each scale, noise has less influence on the PDE than that in the raw noisy image domain. As a result, the edge-stopping criterion and other partial derivative measurements in the PDE become more reliable. Furthermore, there is no need to do Gaussian smoothing or any other smoothing operations. Experiment results show that the proposed algorithm can significantly reduce noise while preserving image edges.

Abstract: The inhomogeneous Poisson (Laplace) equation with internal Dirichlet boundary conditions has recently appeared in several applications ranging from image segmentation [1, 2, 3] to image colorization [4], digital photo matting [5, 6] and image filtering [7, 8]. In addition, the problem we address may also be considered as the generalized eigenvector problem associated with Normalized Cuts [9], the linearized anisotropic diffusion problem [10, 11, 8] solved with a backward Euler method, visual surface reconstruction with discontinuities [12, 13] or optical flow [14]. Although these approaches have demonstrated quality results, the computational burden of finding a solution requires an efficient solver. Design of an efficient multigrid solver is difficult for these problems due to unpredictable inhomogeneity in the equation coefficients and internal Dirichlet boundary conditions with unpredictable location and value. Previous approaches to multigrid solvers have typically employed either a data-driven operator (with fast convergence) or the maintenance of a lattice structure at coarse levels (with low memory overhead). In addition to memory efficiency, a lattice structure at coarse levels is also essential to taking advantage of the power of a GPU implementation [15,16,5,3]. In this work, we present a multigrid method that maintains the low memory overhead (and GPU suitability) associated with a regular lattice while benefiting from the fast convergence of a data-driven coarse operator.

Abstract: Recent variational stereo approaches suffer from at least one of the following drawbacks: Either they use an isotropic disparity-driven smoothness term that ignores the directional information of the disparity field, or they apply anisotropic imagedriven regularisation that suffers from oversegmentation artifacts. As a remedy, we present a novel anisotropic disparity-drivenapproach for stereo vision. It is designed as a highly adaptive anisotropic diffusion-reaction equation that incorporates a diffusion process which has been used successfully for image denoising and inpainting. Its directional adaptation allows to better control the smoothing w.r.t. the local structure of the disparity field. Experiments that compare our model to a recent isotropic variational method and a probabilistic graph cut approach demonstrate the superior quality of our approach. Moreover, a multigrid algorithm allows for moderate run times that do not depend on the disparity range.

Abstract: Partial differential equations (PDEs) have recently shown to be very promising for image interpolation and compression. Inspired from this work, we present a PDE based approach to interpolation of surfaces from scattered point sets using the geometric diffusion equation. Triangulated surfaces are considered in the discrete setting, and the geometric diffusion equation is discretized by the finite element method directly on the triangular mesh. Furthermore, a PDE based method for lossy compression of triangulated surfaces is presented. The idea is to store only a few relevant vertex coordinates in the encoding step. In the decoding step, the remaining vertices are reconstructed by solving the geometric diffusion equation. Finally, two modified reconstruction methods are proposed that are shown to improve the compression quality for both images and surfaces. These reconstruction methods approximate instead of interpolating, and have links to Hopscotch methods for the numerical solution of PDEs. Experiments are presented illustrating that results of high quality can be obtained using simple geometric diffusion without any information on surface normals.

Abstract: A new method to perform blind image deblurring is proposed. Very few assumptions are made on the blurring filter and on the original image: the blurring filter is assumed to have limited support and the original image is assumed to be a sharp natural image. A new prior is used, which gives higher probability to images with sharp edges. The estimation of both the deblurred image and the blurring filter is made in a progressive way, first taking into account the main features of the image, and then proceeding to smaller details. The results obtained with synthetically blurred images are good, even when the blur operator is rather ill-conditioned and the blurred image is noisy. The method also yields improvements in real-life photographs with focus and motion blurs.

Abstract: Portable image processing systems require algorithms which reduce the power consumption while maintaining video rate operation. This paper describes how splitting the data stream into multiple processing pipelines can reduce power consumption in contrast to the traditional spatial (pipeline) parallel processing technique. Real-time image processing system functions (Sobel filters and anisotropic diffusion) were implemented to show the principle of the technique. Our results show that partitioning the image into multiple sections gives increasing benefits with shorter processing times, and up to 45% drop in the power consumption.

Abstract: In this letter, we propose a structure-oriented multidirectional Wiener filter to reduce additive white Gaussian noise in image and video signals. A local activity profile based on second derivatives is used to restrict filtering to homogeneous directions to combat blurring. The proposed filter improves the Wiener estimate of denoised pixels to reduce the residual blurring of the conventional Wiener filter while achieving higher noise-reduction gains of up to 5.6 dB peak signal-to-noise-ratio (PSNR). The parameters of the proposed filter (block size, shape and coefficients) are adapted to image structure and noise level for optimization with respect to noise-reduction gain and structure preservation. The effectiveness of the proposed method is shown using both the PSNR and the modulation transfer function calculated for a range of spatial frequencies to measure the degradation in contrast due to blurring. Our results show that the proposed method achieves a higher contrast transfer ratio than the conventional Wiener filter indicating improved preservation of high frequency content. We also show the performance of the proposed filter relative to reference anisotropic diffusion and wavelet methods.

Abstract: Anisotropic diffusion has been widely used to reduce speckle noise from ultrasound images. However, in traditional anisotropic diffusion methods, usually four directions neighborhood are used, which have many disadvantages, such as the sensitivity to the noise, the loss of image details and false contours. In this paper, we modify the standard SRAD (speckle reducing anisotropic diffusion) algorithm in the discrete domain by considering larger neighborhoods in its computation; then we adopt a confine term to keep the closest equivalence between the filtered image and the original image. Experimental results show that, in the presence of speckle nose, this proposed method could effectively preserve edges and detailed structures while suppressing speckle noise These preliminary results indicate that the proposed speckle reduction method could improve image quality and the visibility of small structures and fine details in the medical ultrasound images.

Abstract: [1] Water molecules and contaminants migrate in water-saturated porous strata by diffusion in systems with small Peclet numbers. Natural porous rocks possess the anisotropy for diffusive transport along the percolated pore space. An X-ray computed tomography (CT) based approach is presented to quickly characterize anisotropic diffusion in porous rocks. High-resolution three-dimensional (3-D) pore images were obtained for a pumice and three sandstones by microfocus X-ray CT and synchrotron microtomography systems. The cluster-labeling process was applied to each image set to extract the 3-D image of a single percolated pore cluster through which diffusing species can migrate a long distance. The nonsorbing lattice random walk simulation was performed on the percolated pore cluster to obtain the mean square displacement. The self-diffusion coefficient along each direction in the 3-D space was calculated by taking the time derivative of the mean square displacement projected on the corresponding direction. A diffusion ellipsoid (i.e., polar representation of the direction-dependent normalized self-diffusivity) with three orthogonal principal axes was obtained for each rock sample. The 3-D two-point autocorrelation was also calculated for the percolated pore cluster of each rock sample to estimate the pore diameter anisotropy. The autocorrelation ellipsoids obtained by the ellipsoid fitting to the high correlation zone were prolate or oblate in shape, presumably depending on the eruption-induced deformation of magma and regional stress during sandstone diagenesis. The pore network anisotropy was estimated by calculating the diffusion ellipsoid for uniaxially elongated or compressed rock images. The degree and direction of the geological deformation of the samples estimated by the pore diameter anisotropy analysis agreed well with those estimated by the pore network anisotropy analysis. We found that the direction of the geological deformation coincided with the direction of the major (or minor) principal axis of the prolate (or oblate) diffusion ellipsoid for each sample. Thus, it can be concluded that the deformation-induced pore structure anisotropy is responsible for the anisotropy of the diffusive transport properties.

Abstract: A computer system for processing synthetic aperture radar (SAR) images includes a database for storing SAR images to be processed, and a processor for processing a SAR image from the database. The processing includes determining noise in a SAR image to be processed, selecting a noise threshold for the SAR image based on the determined noise, and mathematically adjusting an anisotropic diffusion algorithm based on the selected noise threshold. The adjusted anisotropic diffusion algorithm is applied to the SAR image.