Abstract: We present a unified anisotropic geometric diffusion PDE model for smoothing (fairing) out noise both in triangulated two-manifold surface meshes in IR3 and functions defined on these surface meshes, while enhancing curve features on both by careful choice of an anisotropic diffusion tensor. We combine the C1 limit representation of Loop's subdivision for triangular surface meshes and vector functions on the surface mesh with the established diffusion model to arrive at a discretized version of the diffusion problem in the spatial direction. The time direction discretization then leads to a sparse linear system of equations. Iteratively solving the sparse linear system yields a sequence of faired (smoothed) meshes as well as faired functions.

Abstract: We present a new, single-pass nonlinear filter for edge-preserving smoothing and visual detail removal for N dimensional signals in computer graphics, image processing and computer vision applications. Built from two modified forms of Tomasi and Manduchi's bilateral filter, the new "trilateral" filter smoothes signals towards a sharply-bounded, piecewise-linear approximation. Unlike bilateral filters or anisotropic diffusion methods that smooth towards piecewise constant solutions, the trilateral filter provides stronger noise reduction and better outlier rejection in high-gradient regions, and it mimics the edge-limited smoothing behavior of shock-forming PDEs by region nding with a fast min-max stack. Yet the trilateral filter requires only one user-set parameter, filters an input signal in a single pass, and does not use an iterative solver as required by most PDE methods. Like the bilateral filter, the trilateral filter easily extends to N-dimensional signals, yet it also offers better performance for many visual applications including appearance-preserving contrast reduction problems for digital photography and denoising polygonal meshes.

Abstract: Because the overall tumbling provides a major contribution to protein spectral densities measured in solution, the choice of a proper model for this motion is critical for accurate analysis of protein dynamics. Here we study the overall and backbone dynamics of the B3 domain of protein G using (15)N relaxation measurements and show that the picture of local motions is markedly dependent on the model of overall tumbling. The main difference is in the interpretation of the elevated R(2) values in the alpha-helix: the isotropic model results in conformational exchange throughout the entire helix, whereas no exchange is predicted by anisotropic models that place the longitudinal axis of diffusion tensor almost parallel to the helix axis. Due to small size (fast tumbling) of the protein, the T(1) values have low sensitivity to NH bond orientation. The diffusion tensor derived from orientation dependence of R(2)/R(1) is anisotropic (D(par)/D(perp)=1.4), with a small rhombic component. In order to distinguish the correct picture of motion, we apply model-independent methods that are sensitive to conformational exchange and do not require knowledge of protein structure or assumptions about its dynamics. A comparison of the CSA/dipolar cross-correlation rate constants with (15)N relaxation rates and the estimation of R(ex) terms from relaxation data at 9.4 and 14.1 T indicate no conformational exchange in the helix, in support of the anisotropic models. The experimentally derived diffusion tensor is in excellent agreement with theoretical predictions from hydrodynamic calculations; a detailed comparison with various hydrodynamic models revealed optimal parameters for hydrodynamic calculations.

Abstract: An anisotropic nonlinear diffusion equation for image restoration is presented. The model has two terms: the diffusion term and the forcing term. The balance between these terms is made in a selective way, in which boundary points and interior points of the objects that make up the image are treated differently. The optimal smoothing time concept, which allows for finding the ideal stop time for the evolution of the partial differential equation is also proposed. Numerical results show the proposed model's high performance.

Abstract: Segmentation of anatomical regions of the brain is one of the fundamental problems in medical image analysis. It is tradition- ally solved by iso-surfacing or through the use of active contours/ deformable models on a gray-scale magnetic resonance imaging (MRI) data. We develop a technique that uses anisotropic diffusion properties of brain tissue available from diffusion tensor (DT)-MRI to segment brain structures. We develop a computational pipeline starting from raw diffusion tensor data through computation of invari- ant anisotropy measures to construction of geometric models of the brain structures. This provides an environment for user-controlled 3-D segmentation of DT-MRI datasets. We use a level set approach to remove noise from the data and to produce smooth, geometric models. We apply our technique to DT-MRI data of a human subject and build models of the isotropic and strongly anisotropic regions of the brain. Once geometric models have been constructed they can be combined to study spatial relationships and quantitatively ana- lyzed to produce the volume and surface area of the segmented regions. © 2003 SPIE and IS&T. (DOI: 10.1117/1.1527628)

Abstract: We present a new framework for the hierarchical segmentation of color images. The proposed scheme comprises a nonlinear scale-space with vector-valued gradient watersheds. Our aim is to produce a meaningful hierarchy among the objects in the image using three image components of distinct perceptual significance for a human observer, namely strong edges, smooth segments and detailed segments. The scale-space is based on a vector-valued diffusion that uses the Additive Operator Splitting numerical scheme. Furthermore, we introduce the principle of the dynamics of contours in scale-space that combines scale and contrast information. The performance of the proposed segmentation scheme is presented via experimental results obtained with a wide range of images including natural and artificial scenes.

Abstract: Image segmentation is an essential processing step for many image analysis applications. We propose a novel pixon-based adaptive scale method for image segmentation. The key idea of our approach is that a pixon-based image model is combined with a Markov random field (MRF) model under a Bayesian framework. We introduce a new pixon scheme that is more suitable for image segmentation than the "fuzzy" pixon scheme. The anisotropic diffusion equation is successfully used to form the pixons in our new pixon scheme. Experimental results demonstrate that our algorithm performs fairly well and computational costs decrease dramatically compared with the pixel-based MRF algorithm.

Abstract: Many sensing techniques and image processing applicationsare characterized by noisy, or corrupted, image data.Anisotropic diffusion is a popular, and theoretically wellunderstood, technique for denoising such images. Diffusionapproaches however require the selection of an "edgestopping" function, the definition of which is typically adhoc. We exploit and extend recent work on the statisticsof natural images to define principled edge stopping functionsfor different types of imagery. We consider a varietyof anisotropic diffusion schemes and note that they computespatial derivatives at fixed scales from which we estimatethe appropriate algorithm-specific image statistics. Goingbeyond traditional work on image statistics, we also modelthe statistics of the eigenvalues of the local structure tensor.Novel edge-stopping functions are derived from these imagestatistics giving a principled way of formulating anisotropicdiffusion problems in which all edge-stopping parametersare learned from training data.

Abstract: To enlarge a digital image from a single frame preserving the perceptive cues is a relevant research issue. The best known algorithms take into account the presence of edges in the luminance channel, to interpolate correctly the samples/pixels of the original image. This approach allows the production of pictures where the interpolated artifacts (aliasing blurring effect,...) are limited but where high frequencies are not properly preserved. The zooming algorithm proposed in this paper on the other hand reduces the noise and enhances the contrast to the borders/edges of the enlarged picture using classical anisotropic diffusion improved by a smart heuristic strategy. The method requires limited computational resources and it works on gray-level images, RGB color pictures and Bayer data. Our experiments show that this algorithm outperforms in quality and efficiency the classical interpolation methods (replication, bilinear, bicubic).

Abstract: PDE-based, non-linear diffusion techniques are an effective way to denoise images. In a previous study, we investigated the effects of different parameters in the implementation of isotropic, non-linear diffusion. Using synthetic and real images, we showed that for images corrupted with additive Gaussian noise, such methods are quite effective, leading to lower mean-squared-error values in comparison with spatial filters and wavelet-based approaches. In this paper, we extend this work to include anisotropic diffusion, where the diffusivity is a tensor valued function which can be adapted to local edge orientation. This allows smoothing along the edges, but not perpendicular to it. We consider several anisotropic diffusivity functions as well as approaches for discretizing the diffusion operator that minimize the mesh orientation effects. We investigate how these tensor-valued diffusivity functions compare in image quality, ease of use, and computational costs relative to simple spatial filters, the more complex bilateral filters, wavelet-based methods, and isotropic non-linear diffusion based techniques.

Abstract: We utilize an anisotropic diffusion model, which we call the anti-geometric heat flow, for adaptive thresholding of bimodal images and for segmentation of more general greyscale images In a departure from most anisotropic diffusion techniques, we select the local diffusion direction that smears edges in the image rather than seeking to preserve them In this manner, we are able rapidly to detect and discriminate between entire image regions that lie near, but on opposite sides of, a prominent edge The detection of such regions occurs during the diffusion process rather than afterward, thereby side-stepping the most notorious problem associated with diffusion methods, namely, when diffusion should stop We initially outline a procedure for adaptive thresholding, but ultimately show how this model may be used in a region splitting procedure which, when combined with energy based region merging procedures, provides a general framework for image segmentation We discuss a fast implementation of one such framework and demonstrate its effectiveness in segmenting medical, military, and scene imagery

Abstract: Many sensing techniques and image processing applications are characterized by noisy, or corrupted, image data Anisotropic diffusion is a popular, and theoretically well understood, technique for denoising such images Diffusion approaches however require the selection of an "edge stopping" function, the definition of which is typically ad hoc We exploit and extend recent work on the statistics of natural images to define principled edge stopping functions for different types of imagery We consider a variety of anisotropic diffusion schemes and note that they compute spatial derivatives at fixed scales from which we estimate the appropriate algorithm-specific image statistics Going beyond traditional work on image statistics, we also model the statistics of the eigenvalues of the local structure tensor Novel edge-stopping functions are derived from these image statistics giving a principled way of formulating anisotropic diffusion problems in which all edge-stopping parameters are learned from training data

Abstract: An MR method and system of determining elements of the apparent diffusion coefficient tensor in a material with plurality of anisotropic structural units that can be too small to be resolved by direct imaging. MR data is acquired with MR system including pulse sequences, the sequences including imaging or spectroscopy pulse sequences with a series of embedded diffusion-sensitizing gradient waveforms with different gradient strength applied to the material. A nonlinear function of a b-value corresponding to the pulse sequence is defined and the acquired MR data is processed according to defined nonlinear function. Images/maps of the components of the tensor of apparent diffusion coefficients, corresponding to anisotropic structural units, based on the processed MR data, are created. A method of evaluating of the geometrical parameters of lung airways is also described.

Abstract: Analytical expressions for transverse electron spin relaxation induced by small angle motion were derived for the first time within an anisotropic model for rotational diffusion by using an approximation of the spin Hamiltonian and its variation during reorientation that is valid close to the canonical orientations. The dependence of the decay of the stimulated echo on such motion was studied by extensive Monte Carlo simulations and regimes were identified in which the time constant of this decay is related to parameters of the anisotropic diffusion model by simple equations. For testing these theoretical findings and obtaining insight into hierarchical cage dynamics in soft matter, high-field electron paramagnetic resonance (EPR) measurements were performed at a frequency of 94 GHz where the canonical orientations for nitroxide spin labels are well resolved. A combination of continuous wave EPR, saturation recovery measurements, and measurements of the decay of primary and stimulated electron spin echoes...

Abstract: The methodology of NMR experiments intended to measure anisotropic diffusion is reviewed. Experiments of this kind preferably require oriented samples and/or orientation-dependent spin coupling and ...

Abstract: We present an approach for significantly improving the quantitative analysis of motion in noisy fluorescence microscopic im- age sequences. The new partial differential equation based method is a general extension of a 2-D nonlinear anisotropic diffusion filtering scheme to a specially adapted 3-D nonlinear anisotropic diffusion filtering scheme, with two spatial image dimensions and the time t in the image sequence as the third dimension. Motion in image se- quences is considered as oriented, line-like structures in the spa- tiotemporal x,y,t domain, which are determined by the structure ten- sor method. Image enhancement is achieved by a structure adopted smoothing kernel in three dimensions, thereby using the full 3-D in- formation inherent in spatiotemporal image sequences. As an ex- ample for low signal-to-noise ratio (SNR) microscopic image se- quences we have applied this method to noisy in vitro motility assay data, where fluorescently labeled actin filaments move over a surface of immobilized myosin. With the 3-D anisotropic diffusion filtering the SNR is significantly improved (by a factor of 3.8) and closed object structures are reliably restored, which were originally degraded by noise. Generally, this approach is very valuable for all applications where motion has to be measured quantitatively in low light level fluorescence microscopic image sequences of cellular, subcellular, and molecular processes. © 2003 Society of Photo-Optical Instrumentation Engi-

Abstract: We cast the problem of inferring the 3D shape of a scene from a collection of defocused images in the framework of anisotropic diffusion We propose an algorithm that can estimate the shape of a scene by inferring the diffusion coefficient of a heat equation The method is optimal, as we pose it as the minimization of a certain cost functional based on the input images, and fast Furthermore, we also extend our algorithm to the case of multiple images, and derive a 3D scene segmentation algorithm that can work in the presence of pictorial camouflage

Abstract: Artists pictures rarely have photo-realistic detail. Tools to create pictures from digital photographs might, therefore, include methods for removing detail. These tools such as Gaussian and anisotropic diffusion filters and connected-set morphological filters (sieves) remove detail whilst maintaining scale-space causality, in other words new detail is not created using these operators. Non-photorealistic rendering is, therefore, a potential application of these vision techniques. Sieves, in particular, preserve the appropriate edges of retained segments of interest. The resulting image has fewer extrema and is perceptually simpler than the original and is a step towards an artistic nonphotorealistic rendering of the origina. By increasing the amount of simpli- fication towards the margins of the image, the picture composition can be modulated to direct attention to centre of interest of the image. A second artistic goal. The process also removes that detail that provides perceptual cues about texture. This allows the 'eye' to readily accept alternative, artistic, textures introduced to further create an artistic impression. Moreover, the edges bounding segments accurately represent shapes in the original image and so provide a starting point for sketches.

Abstract: The two-dimensional speckle reducing anisotropic diffusion (SRAD) algorithm developed for despeckling ultrasound imagery to obtain a SRAD algorithm capable of enhancing volumetric ultrasound data is presented in this paper. First, an instantaneous coefficient of variation edge detector appropriate for 3D data set presented. Then, the 3D SRAD partial differential equation is formulated by replacing the gradient operator in the traditional diffusion mechanism with the derived 3D ICOV operator. We demonstrate the performance of the 3D SRAD using synthetic 3D ultrasound data acquired from an ellipsoid and real data acquired from the left ventricle (LV) of a murine heart.

Abstract: The Monte Carlo method is the most accurate method for radiotherapy dose calculations, if used correctly. However, any Monte Carlo dose calculation is burdened with statistical noise. In this paper, denoising of Monte Carlo dose distributions with a three-dimensional adaptive anisotropic diffusion method was investigated. The standard anisotropic diffusion method was extended by changing the filtering parameters adaptively according to the local statistical noise. Smoothing of dose distributions with different noise levels in an inhomogeneous phantom, a conventional and an IMRT treatment case is shown. The resultant dose distributions were analysed using several evaluating criteria. It is shown that the adaptive anisotropic diffusion method can reduce statistical noise significantly (two to five times, corresponding to the reduction of simulation time by a factor of up to 20), while preserving important gradients of the dose distribution well. The choice of free parameters of the method was found to be fairly robust.

Abstract: Single molecule wide field imaging is applied to study the diffusion in ultrathin liquid films on solid surfaces. The results show a broad distribution of diffusion coefficients. This is tentatively ascribed to an anisotropy of the diffusion coefficient perpendicular to the surface and a slow exchange of molecules between regions of different diffusion coefficients. We have evidence, that these changes as well as the slow motion perpendicular to the surface are related to the molecular layering of the liquid close to the surface.

Abstract: We prove that a suitable rescaling of biased Perona–Malik energies, defined in the discrete setting, Γ-converges to an anisotropic version of the Mumford–Shah functional. Numerical results are discussed.

Abstract: The self diffusion coefficient of methane in silicalite-1 is influenced by the flexibility of the lattice unlike the self diffusion coefficient of methane in the cation-free zeolite of type A. In the present paper, besides the influence of lattice vibrations on this process, the influence of internal vibrations of the methane molecule and the applicability of several spherical models of this molecule are examined. The method of moments [Chem. Phys. Lett. 198 (1992) 283] is generalized to anisotropic diffusion.

Abstract: Despite common practice, inhomogeneous and/or anisotropic diffusion cannot be considered without regarding the microscopic details breaking the translational and/or angular symmetry. The macroscopic diffusion equation and the stationary solution are determined by the microscopic model and depend in general on all the microscopic parameters and not simply on the combination in the diffusion tensor. The traditional diffusion equation is only valid under special conditions and it cannot, in general, be used for anisotropic diffusion. An alternative form of the diffusion equation has a wider range of applicability. It is shown that for isotropic diffusion all variants of the diffusion equation are mathematically (but not physically) equivalent and can be transformed into each other by introduction of effective potentials. This is not the case for anisotropic diffusion where the traditional diffusion equation in most cases will give incorrect results. Two examples illustrate the differences between the two dyn...

Abstract: Many nano-scale sensing techniques and image processing applications are characterized by noisy, or corrupted, image data. Unlike typical camera-based computer vision imagery where noise can be modeled quite well as additive, zero-mean white or Gaussian noise, nano-scale images suffer from low intensities and thus mainly from Poisson-like noise. In addition, noise distributions can not be considered symmetric due to the limited gray value range of sensors and resulting truncation of over- and underflows. In this paper we adapt B-spline channel smoothing to meet the requirements imposed by this noise characteristics. Like PDE-based diffusion schemes it has a close connection to robust statistics but, unlike diffusion schemes, it can handle non-zero-mean noises. In order to account for the multiplicative nature of Poisson noise the variance of the smoothing kernels applied to each channel is properly adapted. We demonstrate the properties of this technique on noisy nano-scale images of silicon structures and compare to anisotropic diffusion schemes that were specially adapted to this data.

Abstract: For 3D surface reconstruction problems with noisy and incomplete range data measured from complex scenes with arbitrary topologies, a low-level representation, such as level set surfaces, is used. Such surface reconstruction is typically accomplished by minimizing a weighted sum of datamodel discrepancy and model smoothness terms. We introduce a new nonlinear model smoothness term for surface reconstruction based on variations of the surface normals. A direct solution requires solving a fourth-order partial differential equation (PDE), which is very difficult with; conventional numerical techniques. Our solution is based on processing the normals separately from the surface, which allows us to separate the problem into two second-order PDEs. The proposed method 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.

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. We introduce a novel approach to the cerebral white matter connectivity mapping from diffusion tensor MRI. 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:1. We establish the link between Brownian motion and diffusion MRI by using the Laplace-Beltrami operator on M.2. 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.3. Having access to that metric, we propose a novel level set formulation to approximate the distance function related to a radial Brownian motion on M.4. On that basis, 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: This paper introduces a new method to register images that are rotated and translated with respect to each other. The method works by transforming each image to a gradient distribution space. This space represents the likelihood of finding a particular gradient in the image and is invariant to translation. Once transformed the rotation between the images is efficiently found using correlation. Unlike Fourier based methods, phase information is retained in the gradient distribution space, thus a larger class of images can be accurately registered. The method is computationally efficient and does not require nonlinear optimization or iterative methods. Furthermore, large rotations and translations can easily be handled.