Abstract: The authors introduce several sets of time-efficient gradient waveforms for applying isotropic diffusion weighting in NMR experiments. This creates signal attenuation that depends on the trace of the diffusion tensor and is therefore rotationally invariant. Numerical methods for the calculation of such gradient sets are outlined, and results are shown for isotropic and anisotropic gradient hardware and first order flow moment nulled diffusion weighting gradients. Preliminary experimental results from the human brain validate this new technique.

Abstract: The ideas of texture analysis by means of the structure tensor are combined with the scale-space concept of anisotropic diffusion filtering. In contrast to many other nonlinear diffusion techniques, the proposed one uses a diffusion tensor instead of a scalar diffusivity. This allows true anisotropic behaviour. The preferred diffusion direction is determined according to the phase angle of the structure tensor. The diffusivity in this direction is increasing with the local coherence of the signal. This filter is constructed in such a way that it gives a mathematically well-funded scale-space representation of the original image. Experiments demonstrate its usefulness for the processing of interrupted one-dimensional structures such as fingerprint and fabric images.

Abstract: NMR measurements of anisotropic diffusion were studied using a three-dimensional random-walk model. It was found that the apparent diffusion coefficient can be expressed in a canonical form as the product of a diagonal matrix, an orthonormal rotation matrix, and a vector representing the encoding magnetic field gradient. The diffusion coefficient can be interpreted as the sum of the corresponding coefficients measured along the principal diffusion axes, weighted by the squares of the directional cosines of the encoding direction with respect to the principal axes. The analysis revealed that determining the orientation of anisotropy, in a cylindrically symmetric system, requires a minimum of four diffusion measurements. A special pulse sequence which minimized gradient cross-terms and possible restricted diffusion effects was used to characterize diffusion anisotropy in cut chicken gizzards. Diffusion coefficients parallel to the muscle fibers were found to be approximately two to three times larger than those in the transverse direction. Furthermore, the method was successful in detecting the angular change when the sample was rotated by 30°. Results indicate that the proposed approach to measure fiber orientation is valid and may be used to improve the time efficiency of diffusion anisotropy measurements.

Abstract: This paper introduces a discrete scheme for mean curvature motion using a morphological image processing approach. An axiomatic approach of image processing and the mean curvature partial differential equation (PDE) are briefly presented, then the properties of the proposed scheme are studied. In particular, consistency and convergence are proved. The applications of mean curvature motion in image denoising and form evolution are developed and experiences are presented.

Abstract: This paper deals with edge-preserving regularization The definition of the regularizing functions and properties such as edge modeling or stability are studied in the variational approach (by minimizing of a criterion) and in the anisotropic diffusion approach (by solving a PDE) We propose sufficient conditions to define an edge-preserving regularizing function, and analyze comparatively several usual functions We use the algorithm ARTUR based on the half-quadratic transform to solve the nonlinear equation The image and the edge map are simultaneously estimated

Abstract: A time-dependent multi-term solution of the Boltzmann equation is used to calculate the drift and diffusion coefficients of electron swarms in gases under the influence of a time varying electric field. Two model gases are considered and for a.c. electric fields results are presented for a wide range of applied frequencies. Of particular interest is the anomalous temporal behaviour of the longitudinal diffusion coefficient, which is discussed here for the first time.

Abstract: The techniques of a posteriori image restoration and iterative image feature extraction are described and compared. Image feature extraction methods known as graduated nonconvexity (GNC); variable conductance diffusion (VCD), anisotropic diffusion, and biased anisotropic diffusion (BAD), which extract edges from noisy images, are compared with a restoration/feature extraction method known as mean field annealing (MFA). All are shown to be performing the same basic operation: image relaxation. This equivalence shows the relationship between energy minimization methods and spatial analysis methods and between their respective parameters of temperature and scale. As a result of the equivalence, VCD is demonstrated to minimize a cost function, and that cost is specified explicitly. Furthermore, operations over scale space are shown to be a method of avoiding local minima. >

Abstract: An extended diffusion equation is derived which takes into account anisotropic transport of mass in a shear flow. Analytical solutions for systems with spatially linear but time-dependent flow and sedimentation velocities, a time-dependent diffusion tensor as well as linear sinks and sources are determined. Considering only systems in spatially infinite domains, the solutions for Gaussian initial distributions are derived. Then, starting from this general case, the twoand three-dimensional solutions are deduced for a horizontal and linear shear flow which generalize the results of former Gaussian plume models. Further, it is shown that the extended diffusion equation can be treated by solution methods of the Fokker-Planck equation. Finally, it is discussed how the time-dependent diffusion tensor can be determined from the first and second moments of these solutions.

Abstract: Noise filtering of images is essentially a smoothing process and it is an issue that has been addressed for many years. The most commonly used low-pass filtering methods blur important image structures such as edges and lines thus reducing the image contrast and damaging image fidelity. The paper presents a structure adaptive anisotropic filtering method with its application to processing multidimensional magnetic resonance images.

Abstract: Effective 3-D image processing algorithms are presented for automatic counting and analysis of cells in anisotropic 3-D biological images that are collected by laser-scanning confocal microscopes. In these instruments, the x-y resolution is much better than the resolution along the z axis, hence the voxels (pixels in 3-D) are anisotropic. In this work, the images are pre-processed by a 3-D extension of an anisotropic diffusion algorithm, and the resulting images are binarized by a clustering based segmentation algorithm. As a result of binary segmentation, some regions consist of individual objects while others are multi-object clusters. An extension of Vincent and Soille's watershed algorithm (1991) to anisotropic 3D spaces is used to separate such cell clusters. The watershed algorithm is applied on marker functions that are generated using a combination of 3-D morphological inverse distance functions and 3-D image gradients. Cell measurements, such as volume, average intensity and locations, are calculated on the result of watershed segmentation. This algorithm has been successfully applied to the automated analysis of cell populations from a variety of biological studies involving large numbers of tissue samples.

Abstract: We have recently proposed the use of geometry in image processing by representing an image as a surface in 3-space. The linear variations in intensity (edges) were shown to have a nondivergent surface normal. Exploiting this feature we introduced a nonlinear adaptive filter that only averages the divergence in the direction of the surface normal. This led to an inhomogeneous diffusion (ID) that averages the mean curvature of the surface, rendering edges invariant while removing noise. This mean curvature diffusion (MCD) when applied to an isolated edge imbedded in additive Gaussian noise results in complete noise removal and edge enhancement with the edge location left intact. In this paper we introduce a new filter that will render corners (two intersecting edges), as well as edges, invariant to the diffusion process. Because many edges in images are not isolated the corner model better represents the image than the edge model. For this reason, this new filtering technique, while encompassing MCD, also outperforms it when applied to images. Many applications will benefit from this geometrical interpretation of image processing, and those discussed in this paper include image noise removal, edge and/or corner detection and enhancement, and perceptually transparent coding.

Abstract: An image pre-processing method and apparatus processes image data line-by-line using an error diffusion process to generate a halftoned image The method and apparatus varies the processing direction from line-to-line to minimise the presence of worm-type artifacts in a halftoned image where the processing direction of each line in the image is dependent on the content of the image More particularly, the processing direction of each line in the image is dependent upon the gray-scale value and/or the quantization error associated with one or more pixels of a previously processed line in the image

Abstract: An improved error diffusion method for incorporating into a halftoning algorithm for processing gray scale images to counterpart binary images. The improved error diffusion method adopts a tone scale function to process the input gray scale image before proceeding with the halftoning (error diffusion) processing in order to improve the image darkening problem caused by the conventional error diffusion method and a noise correction process to eliminate the background noises resulted from the accumulated error of the conventional error diffusion method. The proceeding of the tone scaling process and the halftoning process are parallel with the correction process so that errors generated in either one will not affect the other.

Abstract: Polarization diffusion in communication fibers is studied Diffusion of the states of polarization in an optical fiber is found to be anisotropic on the surface of the Poincare sphere The predicted anisotropy has significant implications for nonlinear evolution in long-distance communication systems

Abstract: Anisotropic diffusion has been extensively used as an efficient nonlinear filtering technique for simultaneously performing contrast enhancement and noise reduction, and for deriving consistent scale-space image descriptions. In this paper, we present a general study of anisotropic diffusion schemes based on differential group-invariant representations of local image structure. We show that the local geometry (i.e., shape and scale) of the photometric surface is intrinsically specified by two dual families of curves, respectively consisting of isophotes and stream lines, which remain invariant under isometries in the image domain. Within this framework, anisotropic diffusive processes induce a deformation flow on the network of isophotes and stream lines. Deriving the general expression of this flow leads to identifying canonical forms for admissible conduction functions, that yield an optimal and stable preservation of significant image structures. Moreover, relating scale to directional variations of isophote density results in controlling the diffusion dynamics by means of a heterogeneous damping density which allows us to adaptively reduce diffusion speed in the vicinity of high gradient lines while increasing it within stationary intensity domains. Finally, these results are extended to arbitrary image dimensions.

Abstract: This paper describes a class of ill-posed anisotropic diffusion models of the type presented by Perona and Malik (1990). The analysis is based on a previous result that anisotropic diffusion is a steepest descent motion on an energy surface and its behavior is thus determined by the shape of this energy surface. We show that the class of diffusion models are ill-posed because the energy surface is discontinuous at all continuous images, and all step images, which are dense in the space of piecewise images, are global minima of the energy surface.

Abstract: 3D-Echo provides more complete information about cardiac structures with respect to traditional 2D-Echo. A correct interpretation of 3D data is often made difficult by the high level of noise necessarily linked to the procedure of acquisition. Traditional filtering techniques do not enable either the noise to be eliminated or the edges to be maintained. It is therefore essential that processing is able to distinguish the noise from the edges of the various cardiac structures; this is possible by means of the filtering technique based on 3D multiscale anisotropic diffusion. Since in echocardiography the time sequence of the frames provides clinically significant information, the filtering technique has been extended to 3D+time to process sequences of 3D data.

Abstract: The behavior of Si adsorbates evaporated on a Si(001) surface is studied by tracing their diffusion caused by radiative heating. A reflection electron microscope (REM) is used to observe denuded zones that are created at the terrace edge and grow with heating time. Diffusion constants of Si adsorbates are determined using the denuded zone widths on a Si(001) surface. The diffusion constants on the 2×1 terrace have directions parallel to a surface dimer connected to two nearest-neighbor atoms on the surface. Similarly, the diffusion constants on the 1×2 terrace have directions perpendicular to the dimer. Diffusion constants in the opposite directions are the same on both structures, so the isotropic diffusions in the opposite directions are observed on a Si(001) surface: uD2×1= dD2×1=D2×1 and uD1×2= dD1×2=D1×2. The index u indicates diffusion from the down-side to the up-side and d vice versa. However, a difference in diffusion constants between D2×1 and D1×2 is observed. It is concluded that D1×2 is about 5-6 times as large as D2×1 on the Si(001) surface.

Abstract: This dissertation presents some signal recovery algorithms which were developed based on certain nonlinear optimization principles. The specific problems of interest are image enhancement, blind image restoration, and blind channel equalization. A systematic method is established for analyzing the behavior of anisotropic diffusion, a recent tool for image enhancement, by posing the anisotropic diffusion equation as resulting from a certain optimization problem. Principles for designing well-behaved anisotropic diffusion schemes are established based on the analysis and an orthogonal decomposition of the diffusion equation. A class of well-behaved anisotropic schemes is developed. Space-adaptive regularization methods for blind restoration of shift-invariantly degraded images are developed by posing the problem as minimizing a cost function, and alternating minimization is proposed to minimize the cost function efficiently. Well-behaved anisotropic diffusion is incorporated into this method to arrive at orientation-selective regularization so that ringing artifacts in the restored images are reduced. This method is then extended to restore shift-variantly blurred images. Alternating minimization is proposed as a general framework to accomplish joint data estimation and channel identification. Under this framework, a cost function is minimized through alternating two minimization steps which turn out to be data estimation and channel identification. The algorithms derived from this scheme are guaranteed to converge. A simple blind equalization algorithm is derived based on the steepest descent method. This algorithm degenerates into a simple sequence estimator if the channel response is known.

Abstract: Linear systems theory is widely used to describe the principles of computed tomography (CT), radiography and other medical imaging systems. Using this approach, image blur can be represented as a convolution. However, it is shown that when image blur is a consequence of the spreading of quanta used to represent the image (e.g. X-rays, light or electron-hole pairs), use of the convolution integral underestimates noise in the blurred image. A "stochastic" convolution is introduced which solves this problem by accounting for the statistical properties of the image quanta. It can be used to correctly describe both image blur and image noise within a linear-systems framework.

Abstract: A space-variant filter was designed to improve signal-to-noise ratio (SNR) in individual positron emission tomography (PET) "activated minus control" images. The aim of this study was to avoid averaging of signals from grey matter (GM) with those from surrounding white matter (WM) occurring after the usual Gaussian smoothing. The filtered images were obtained by solving the heat diffusion equation with a non-constant diffusion coefficient, chosen in order to restrict inter-region smoothing. Several diffusion coefficient maps calculated from a segmented magnetic resonance (MR) image were tested. Monte-Carlo simulations were used to characterize anisotropic diffusion filtering (ADF), in terms of GM signal preservation and statistical properties.

Abstract: We consider a two-grid method based on approximation of the Schur complement. We study the dependence of the two-grid convergence rate on certain problem parameters. As test problems we take the rotated anisotropic diffusion equation and the convection-diffusion equation. Using Fourier analysis we show that for both test problems the two-grid method is robust w.r.t. variation in the relevant problem parameters. For the multigrid method we use a standardW-cycle on coarse grids. This multigrid method then has the same algorithmic structure as a standard multigrid method and is fairly efficient. Moreover, when applied to the two test problems then, as in the two-grid method, we have a strong robustness w.r.t. variation of the problem parameters.

Abstract: In (linear or nonlinear) diffusive scale-space representations, local variations of the luminance field with respect to infinitesimal scale transitions are described via a first-order parabolic partial differential equation modeling a generalized diffusion process. A geometric characterization of the scale-space structure is then classically derived by analyzing the properties of the deformation flow induced by scale transitions along specific geometric structures embedded on the photometric surface. In particular, studying the simultaneous deformation of the dual families of curves consisting of isophotes and stream lines of the luminance field yields a Euclidean-invariant geometric description of generalized diffusion processes. In this paper, the generalized diffusion equation is interpreted within the framework of the relativistic electromagnetic (EM) theory as a Lorentz gauge condition expressing the trace-invariance of an EM quadripotential with covariant scalar and contravariant vector components respectively related to luminance and geometric properties of the image. This gauge condition determines an EM quadrifield and quadricharge which satisfy Maxwell equations. Deriving the general expressions of these quadrivectors as functions of Euclidean characteristics of isophotes and stream lines leads to identifying Lorentz-invariants which synthetize under an extremely compact form intrinsic multiscale image properties. In addition, weak formulations of diffusive scale-spaces are consistently re-expressed in terms of Em energy density. The specific cases of linear scale-spaces, corresponding to purely electric fields, and of classical anisotropic diffusion models are studied in detail, providing a significant insight in the understanding of the deep structure of diffusive scale-spaces.

Abstract: We present a study of some image resoration techniques based on partial differential equations. We study separately the denoising problem and the restoration of discontinuities. We analyze the capabilities of the differential operators to restore images. In particular, we analyze a number of models present in the literature, and we present comparative results. Finally, we present a model based in the combination of the anisotropic diffusion of Alvarez, Lions, and Morel and the shock filters of Osher and Rudin.

Abstract: In this paper we discuss the initial boundary value problem for anisotropic diffusion equations.After establishing the existence and uniqueness,we obtain some results on the extinction properties of generalized solutions.