Abstract: The linear and nonlinear scale spaces, generated by the inherently real-valued diffusion equation, are generalized to complex diffusion processes, by incorporating the free Schrodinger equation. A fundamental solution for the linear case of the complex diffusion equation is developed. Analysis of its behavior shows that the generalized diffusion process combines properties of both forward and inverse diffusion. We prove that the imaginary part is a smoothed second derivative, scaled by time, when the complex diffusion coefficient approaches the real axis. Based on this observation, we develop two examples of nonlinear complex processes, useful in image processing: a regularized shock filter for image enhancement and a ramp preserving denoising process.

Abstract: We introduce a novel approach to the cerebral white matter connectivity mapping from diffusion tensor MRI DT-MRI is the unique non-invasive technique capable of probing and quantifying the anisotropic diffusion of water molecules in biological tissues We address the problem of consistent neural fibers reconstruction in areas of complex diffusion profiles with potentially multiple fibers orientations Our method relies on a global modelization of the acquired MRI volume as a Riemannian manifold M and proceeds in 4 majors steps: First, we establish the link between Brownian motion and diffusion MRI by using the Laplace-Beltrami operator on M We then expose how the sole knowledge of the diffusion properties of water molecules on M is sufficient to infer its geometry There exists a direct mapping between the diffusion tensor and the metric of M Next, having access to that metric, we propose a novel level set formulation scheme to approximate the distance function related to a radial Brownian motion on M Finally, a rigorous numerical scheme using the exponential map is derived to estimate the geodesics of M, seen as the diffusion paths of water molecules Numerical experimentations conducted on synthetic and real diffusion MRI datasets illustrate the potentialities of this global approach

Abstract: In this paper we describe a method for skeletonization of gray-scale images without segmentation. Our method is based on anisotropic vector diffusion. The skeleton strength map, calculated from the diffused vector field, provides us a measure of how possible each pixel could be on the skeletons. The final skeletons are traced from the skeleton strength map, which mimics the behavior of edge detection from the edge strength map of the original image. A couple of real or synthesized images will be shown to demonstrate the performance of our algorithm.

Abstract: In this paper we present a new diffusion-based method for the delineation of coastlines from space-borne polarimetric SAR imagery of coastal urban areas. Both polarimetric filtering and speckle reducing anisotropic diffusion (SRAD) are exploited to generate a base image where speckle is reduced and edges are enhanced. The primary edge information is then derived from the base image using the instantaneous coefficient of variation edge detector. Next, the resulting edge image is parsed by a watershed transform, which partitions the image into disjoint segments where the division lines between segments are collocated with detected edges. The over-segmentation problem associated with the watershed transform is solved by a region merging technique that combines neighbouring segments with similar radar brightness. As a result, undesired boundary segments are eliminated and true coastlines are correctly delineated. The proposed algorithm has been applied to a space-borne polarimetric SAR dataset, demonstrating ...

Abstract: Two-photon microscopy in combination with novel fluorescent labeling techniques enables imaging of three-dimensional neuronal morphologies in intact brain tissue. In principle it is now possible to automatically reconstruct the dendritic branching patterns of neurons from 3-D fluorescence image stacks. In practice however, the signal-to-noise ratio can be low, in particular in the case of thin dendrites or axons imaged relatively deep in the tissue. Here we present a nonlinear anisotropic diffusion filter that enhances the signal-to-noise ratio while preserving the original dimensions of the structural elements. The key idea is to use structural information in the raw data—the local moments of inertia—to locally control the strength and direction of diffusion filtering. A cylindrical dendrite, for example, is effectively smoothed only parallel to its longitudinal axis, not perpendicular to it. This is demonstrated for artificial data as well as for in vivo two-photon microscopic data from pyramidal neurons of rat neocortex. In both cases noise is averaged out along the dendrites, leading to bridging of apparent gaps, while dendritic diameters are not affected. The filter is a valuable general tool for smoothing cellular processes and is well suited for preparing data for subsequent image segmentation and neuron reconstruction.

Abstract: Computer graphics cameras lack the finite depth of field (DOF) present in real world ones. This results in all objects being rendered sharp regardless of their depth, reducing the realism of the scene. On top of that, real-world DOF provides a depth cue, that helps the human visual system decode the elements of a scene. Several methods have been proposed to render images with finite DOF, but these have always implied an important trade-off between speed and accuracy. We introduce a novel anisotropic diffusion partial differential equation (PDE) that is applied to the 2D image of the scene rendered with a pin-hole camera. In this PDE, the amount of blurring on the 2D image depends on the depth information of the 3D scene, present in the Z-buffer. This equation is well posed, has existence and uniqueness results, and it is a good approximation of the optical phenomenon, without the visual artifacts and depth inconsistencies present in other approaches. Because both inputs to our algorithm are present at the graphics card at every moment, we can run the processing entirely in the GPU. This fact, coupled with the particular numerical scheme chosen for our PDE, allows for real-time rendering using a programmable graphics card.

Abstract: In this paper we are interested in the joint reconstruction of geometry and photometry of scenes with multiple moving objects from a collection of motion-blurred images. We make simplifying assumptions on the photometry of the scene (we neglect complex illumination effects) and infer the motion field of the scene, its depth map, and its radiance. In particular, we choose to partition the image into regions where motion is well approximated by a simple planar translation. We model motion-blurred images as the solution of an anisotropic diffusion equation, whose initial conditions depend on the radiance and whose diffusion tensor encodes the depth map of the scene and the motion field. We propose an algorithm to infer the unknowns of the model by minimizing the discrepancy between the measured images and the ones synthesized via diffusion. Since the problem is ill-posed, we also introduce additional Tikhonov regularization terms.

Abstract: We present a morphological multi-scale method for image sequence processing, which results in a truly coupled spatio-temporal anisotropic diffusion. The aim of the method is not to smooth the level-sets of single frames but to denoise the whole sequence while retaining geometric features such as spatial edges and highly accelerated motions. This is obtained by an anisotropic spatio-temporal level-set evolution, where the additional artificial time variable serves as the multi-scale parameter. The diffusion tensor of the evolution depends on the morphology of the sequence, given by spatial curvatures of the level-sets and the curvature of trajectories (=acceleration) in sequence-time. We discuss different regularization techniques and describe an operator splitting technique for solving the problem. Finally we compare the new method with existing multi-scale image sequence processing methodologies.

Abstract: We propose a solution to the problem of inferring the depth map, radiance and motion of a scene from a collection of motion-blurred and defocused images. We model motion-blur and defocus as an anisotropic diffusion process, whose initial conditions depend on the radiance and whose diffusion tensor encodes the shape of the scene, the motion field and the optics parameters. We show that this model is well-posed and propose an efficient algorithm to infer the unknowns of the model. Inference is performed by minimizing the discrepancy between the measured blurred images and the ones synthesized via forward diffusion. Since the problem is ill-posed, we also introduce additional Tikhonov regularization terms. The resulting method is fast and robust to noise as shown by experiments with both synthetic and real data.

Abstract: We propose a new approach to reduce speckle noise and enhance structures in speckle-corrupted images. It utilizes a median-anisotropic diffusion compound scheme. The median-filter-based reaction term acts as a guided energy source to boost the structures in the image being processed. In addition, it regularizes the diffusion equation to ensure the existence and uniqueness of a solution. We also introduce a decimation and back reconstruction scheme to further enhance the processing result. Before the iteration of the diffusion process, the image is decimated and a subpixel shifted image set is formed. This allows a multichannel parallel diffusion iteration, and more importantly, the speckle noise is broken into impulsive or salt-pepper noise, which is easy to remove by median filtering. The advantage of the proposed technique is clear when it is compared to other diffusion algorithms and the well-known adaptive weighted median filtering (AWMF) scheme in both simulation and real medical ultrasound images.

Abstract: A unified approach for treating the scale selection problem in the anisotropic scale-space is proposed. The anisotropic scale-space is a generalization of the classical isotropic Gaussian scale-space by considering the Gaussian kernel with a fully parameterized analysis scale (bandwidth) matrix. The "maximum-over-scales" and the "most-stable-over-scales" criteria are constructed by employing the "L-normalized scale-space derivatives", i.e., response-normalized derivatives in the anisotropic scale-space. This extension allows us to directly analyze the anisotropic (ellipsoidal) shape of local structures. The main conclusions are (i) the norm of the /spl gamma/- and L-normalized anisotropic scale-space derivatives with a constant /spl gamma/ =1/2 are maximized regardless of the signal's dimension iff the analysis scale matrix is equal to the signal's covariance and (ii) the most-stable-over-scales criterion with the isotropic scale-space outperforms the maximum-over-scales criterion in the presence of noise. Experiments with 1D and 2D synthetic data confirm the above findings. 3D implementations of the most-stable-over-scales methods are applied to the problem of estimating anisotropic spreads of pulmonary tumors shown in high-resolution computed-tomography (HRCT) images. Comparison of the first- and second-order methods shows the advantage of exploiting the second-order information.

Abstract: In mathematical image processing we are often presented with amazing examples of image enhancement algorithms. Yet, when applied to different noisy images, they can produce unwanted effects. The analysis of such algorithms lags behind their intuitive development. Two essentially different models have found wide recognition: a variational approach due to Mumford and Shah, and an anisotropic diffusion approach leading to an evolution type equation by Perona and Malik. In this survey we shall explain a surprising connection between these two approaches. Copyright © 2004 John Wiley & Sons, Ltd.

Abstract: A Lattice Boltzmann model with two relaxation times for the 2D/3D advection and anisotropic diffusion equation (AADE) is introduced The method is applied to Richards' equation for variably saturated flow in isotropic homogeneous media by extending retention curves into the saturated zone in a linear manner The Darcy velocity is computed locally from the population solution The method possesses intrinsic mass conservation, it is explicit and especially suitable for parallel computations Designed for regular grids, the LB approach meets the boundary conditions accurately with an unified “multi-reflexion” technique, introduced to fit pressure head and/or specified flux conditions on static and seepage boundaries The physical space can assume an uniform rectangular discretization grid which is transformed into the cubic computational grid after proper rescaling [9] of the AADE The diffusion term is considered in two forms: the conventional one and the transformed one The integral transformation may avoid problems encountered with the unbounded diffusion coefficients at the residual and saturated limits An analytical expression for the transformed diffusion function is obtained for the original and modified VGM retention curves [14] Analytical instationary solutions for constant flux infiltration with non-linear models [2,15] are revised An exact unstationary solution [1] valid for unsaturated, saturated or variably saturated flow is constructed using the BCM hydraulic conductivity function [3,11], moisture tension being fixed at the surface Stationary infiltration profiles are generated for the BCM and the VGM conductivity functions The LB method is validated against these and other reference solutions

Abstract: Diffused expectation maximisation is a novel algorithm for image segmentation. The method models an image as a finite mixture, where each mixture component corresponds to a region class and uses a maximum likelihood approach to estimate the parameters of each class, via the expectation maximisation algorithm, coupled with anisotropic diffusion on classes, in order to account for the spatial dependencies among pixels.

Abstract: The common methods used to determine the diffusion coefficients of polymer composites are based on the solution of Fickian diffusion equation in one-dimensional (ID) rectangular domain. However, these diffusivities usually involve errors primarily due to finite sample dimensions and anisotropy introduced by fiber reinforcements. In this study, the solution of transient, three-dimensional (3D) anisotropic Fickian diffusion equation is nondimensionalized using six parameters. The solution is then used to analyze the combined contribution of finite sample dimensions and anisotropy to the errors involved in diffusion constants calculated by ID methods. The small time solution of the Fickian diffusion equation in 3D domain is used to analyze the slope used in diffusivity calculations. It is shown that the diffusion coefficient calculated by the ID approach is exact only if the correct slope of percent mass gain versus root square time curve at t=0 is used. However, it has also been shown that depending on the part geometry and degree of anisotropy, there might be considerable differences between the measured slope from the experimental data and the actual slope at t =0. The mismatch between the slopes results in as much as 50% errors in estimates of diffusion coefficients. Using the 3D solution in nondimensional form, the magnitudes of these errors are studied. A least-square curve-fit method, which yields accurate anisotropic diffusion coefficients, is proposed. The method is demonstrated on artificially generated experimental data for a polymer composite containing 50% unidirectional reinforcement. The anisotropic diffusion coefficients used to generate the data are recovered with less than 1% error.

Abstract: Three-dimensional ultrasound machines based on matrix phased-array transducers are gaining predominance for real-time dynamic screening in cardiac and obstetric practice. These transducers array acquire three-dimensional data in spherical coordinates along lines tiled in azimuth and elevation angles at incremental depth. This study aims at evaluating fast filtering and scan conversion algorithms applied in the spherical domain prior to visualization into Cartesian coordinates for visual quality and spatial measurement accuracy. Fast 3d scan conversion algorithms were implemented and with different order interpolation kernels. Downsizing and smoothing of sampling artifacts were integrated in the scan conversion process. In addition, a denoising scheme for spherical coordinate data with 3d anisotropic diffusion was implemented and applied prior to scan conversion to improve image quality. Reconstruction results under different parameter settings, such as different interpolation kernels, scaling factor, smoothing options, and denoising, are reported. Image quality was evaluated on several data sets via visual inspections and measurements of cylinder objects dimensions. Error measurements of the cylinder's radius, reported in this paper, show that the proposed fast scan conversion algorithm can correctly reconstruct three-dimensional ultrasound in Cartesian coordinates under tuned parameter settings. Denoising via three-dimensional anisotropic diffusion was able to greatly improve the quality of resampled data without affecting the accuracy of spatial information after the modification of the introduction of a variable gradient threshold parameter.

Abstract: Diffusion tensor MRI probes and quantifies the anisotropic diffusion of water molecules in biological tissues, making it possible to non-invasively infer the architecture of the underlying structures. In this article, we present a set of new techniques for the estimation and regularization of diffusion tensors MRI datasets as well as a novel approach to the cerebral white matter connectivity mapping. Numerical experimentations conducted on real diffusion weighted MRI will exhibit promising results.

Abstract: The enhanced backscattering cone displaying a strong anisotropy from a material with anisotropic diffusion is reported. The constructive interference of the wave is preserved in the helicity preserving polarization channel and completely lost in the nonpreserving one. The internal reflectivity at the interface modifies the width of the backscatter cone. The reflectivity coefficient is measured by angular-resolved transmission. This interface property is found to be isotropic, simplifying the backscatter cone analysis. The material used is a macroporous semiconductor, gallium phosphide, in which pores are etched in a disordered position but with a preferential direction.

Abstract: Segmenting echographic images is a difficult task due to their low contrast and to the presence of speckle. This paper presents an original robust B-Spline snake based on a novel external energy. Our approach combines anisotropic diffusion with the process of curve evolution by using a local coefficient of variation and the Tukey's error norm. This makes the snake robust to speckle. At each iteration the algorithm computes an edge image, and derives a new external energy based on the amplitude and direction of the gradient of the upper mentioned coefficient. The method has been tested on echocardographic images of a 12-week old foetus. The results, presented with different parameters, show a significant improvement.

Abstract: This article proposes a novel, partial derivatives based filter, for oriented patterns filtering and enhancement. We propose a filtering and restoration process with a strong anisotropic behaviour. Locally, our filter uses a superposition of one-dimensional diffusion processes; in each pixel the diffusion directions are given by a principal component analysis, on these directions the processes behaviour is modulated by non-linear functions depending on the absolute values of the directional derivatives. Selectively, depending on these measures, the filter can act like a classical diffusion filter, simplifying the image, or it can produce edge and corner enhancement. The equation we propose is very general and it can be modified for specific image restoration tasks. Through applications samples we would show the efficiency of our method both on synthetic images and on real ones.

Abstract: Level set methods, an important class of partial differential equation (PDE) methods, define dynamic surfaces implicitly as the level set (iso-surface) of a sampled, evolving nD function. The course begins with preparatory material that introduces the concept of using partial differential equations to solve problems in computer graphics, geometric modeling and computer vision. This will include the structure and behavior of several different types of differential equations, e.g. the level set equation and the heat equation, 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 actually implement the mathematics and methods presented in the first stage. The course closes with detailed presentations on several level set/PDE applications, including image/video inpainting, pattern formation, image/volume processing, 3D shape reconstruction, image/volume segmentation, image/shape morphing, geometric modeling, anisotropic diffusion, and natural phenomena simulation.

Abstract: The accuracy of the control-volume finite-element method for simulating strongly anisotropic diffusion problems on rectangular parallelepiped domains meshed with triangular prismatic elements is investigated. It is shown that a novel Gauss–Green gradient reconstruction technique provides significant improvements in accuracy over the classical finite-element shape function technique. Numerical investigations carried out on relatively coarse meshes highlight that the new Gauss–Green method offers a reduction of 80 % in solution error over the shape function method. In fact, a mesh with 27 times more elements would be required if the shape function method was to achieve the same accuracy as the new method, resulting in a substantial increase in computational overheads.

Abstract: Two-photon excited flash photolysis (TPEFP) was used to photorelease caged fluorescein in test solutions and inside fiber cells of the eye lens. Accurate alignment between the focus of the IR beam and the probe beam from the confocal microscope was achieved with an accessory focussing lens and computer models of diffusion were fit to experimental data to extract apparent diffusion coefficients. Inside a fiber cell, the diffusion coefficient for fluorescein was 4 10 7 cm 2 /s at 21°C, a value an order of magnitude lower than observed in free solution. Fluores- cence also diffused between fiber cells via gap junctions. In the periphery, diffusion between cells occurred mainly in a radial direction while deep in the lens the diffusion between cells appeared more isotropic. Diffusion between cells was slower than inside cells and corresponded to less than 1% of the area between cells being available for diffusion. This value is in good agreement with that expected from measurements of gap junction structure and packing density if a 1-1.5-nm aqueous gap junction pore is nearly always open. Microsc. Res. Tech. 63:50 -57, 2004. © 2003 Wiley-Liss, Inc.

Abstract: For surface reconstruction problems with noisy and incomplete range data, a Bayesian estimation approach can improve the overall quality of the surfaces. The Bayesian approach to surface estimation relies on a likelihood term, which ties the surface estimate to the input data, and the prior, which ensures surface smoothness or continuity. This paper introduces a new high-order, nonlinear prior for surface reconstruction. The proposed prior can smooth complex, noisy surfaces, while preserving sharp, geometric features, and it is a natural generalization of edge-preserving methods in image processing, such as anisotropic diffusion. An exact solution would require solving a fourth-order partial differential equation (PDE), which can be difficult with conventional numerical techniques. Our approach is to solve a cascade system of two second-order PDEs, which resembles the original fourth-order system. This strategy is based on the observation that the generalization of image processing to surfaces entails filtering the surface normals. We solve one PDE for processing the normals and one for refitting the surface to the normals. Furthermore, we implement the associated surface deformations using level sets. Hence, the algorithm can accommodate very complex shapes with arbitrary and changing topologies. This paper gives the mathematical formulation and describes the numerical algorithms. We also show results using range and medical data.

Abstract: In this paper, we propose a particle filtering approach for tracking applications in image sequences. The system we propose combines a measurement equation and a dynamic equation which both depend on the image sequence. Taking into account several possible observations, the likelihood is modeled as a linear combination of Gaussian laws. Such a model allows inferring an analytic expression of the optimal importance function used in the diffusion process of the particle filter. It also enables building a relevant approximation of a validation gate. We demonstrate the significance of this model for a point tracking application.

Abstract: Computing the transformation between two views of a planar scene is an important step in many computer vision applications. Spatial approaches to solve this problem need corresponding sets of primitives-points, lines, conies, etc. Identification of corresponding primitives in two images is non-trivial, limiting the applicability of such approaches. In this paper, we present a novel Fourier domain based approach that makes use of image intensities for computing the image-to-image transformation. Our approach transforms the images to the Fourier domain and then represents them in a coordinate system in which the affine transformation is reduced to an anisotropic scaling. The anisotropic scale factors can be computed using cross correlation methods, and working backwards from this, we compute the entire transformation. It does not require any correspondences thereby making it practically very useful. Applications to registration and recognition are discussed.