Abstract: In biological tissues, typical MRI voxels comprise multiple microscopic environments, the local organization of which can be captured by microscopic diffusion tensors. The measured diffusion MRI signal can, therefore, be written as the multidimensional Laplace transform of an intravoxel diffusion tensor distribution (DTD). Tensor-valued diffusion encoding schemes have been designed to probe specific features of the DTD, and several algorithms have been introduced to invert such data and estimate statistical descriptors of the DTD, such as the mean diffusivity, the variance of isotropic diffusivities, and the mean squared diffusion anisotropy. However, the accuracy and precision of these estimations have not been assessed systematically and compared across methods. In this article, we perform and compare such estimations in silico for a one-dimensional Gamma fit, a generalized two-term cumulant approach, and two-dimensional and four-dimensional Monte-Carlo-based inversion techniques, using a clinically feasible tensor-valued acquisition scheme. In particular, we compare their performance at different signal-to-noise ratios (SNRs) for voxel contents varying in terms of the aforementioned statistical descriptors, orientational order, and fractions of isotropic and anisotropic components. We find that all inversion techniques share similar precision (except for a lower precision of the two-dimensional Monte Carlo inversion) but differ in terms of accuracy. While the Gamma fit exhibits infinite-SNR biases when the signal deviates strongly from monoexponentiality and is unaffected by orientational order, the generalized cumulant approach shows infinite-SNR biases when this deviation originates from the variance in isotropic diffusivities or from the low orientational order of anisotropic diffusion components. The two-dimensional Monte Carlo inversion shows remarkable accuracy in all systems studied, given that the acquisition scheme possesses enough directions to yield a rotationally invariant powder average. The four-dimensional Monte Carlo inversion presents no infinite-SNR bias, but suffers significantly from noise in the data, while preserving good contrast in most systems investigated.

Abstract: Ultrasound (US) imaging can examine human bodies of various ages; however, in the process of obtaining a US image, speckle noise is generated. The speckle noise inhibits physicians from accurately examining lesions; thus, a speckle noise removal method is essential technology. To enhance speckle noise elimination, we propose a novel algorithm using the characteristics of speckle noise and filtering methods based on speckle reducing anisotropic diffusion (SRAD) filtering, discrete wavelet transform (DWT) using symmetry characteristics, weighted guided image filtering (WGIF), and gradient domain guided image filtering (GDGIF). The SRAD filter is exploited as a preprocessing filter because it can be directly applied to a medical US image containing speckle noise without a log-compression. The wavelet domain has the advantage of suppressing the additive noise. Therefore, a homomorphic transformation is utilized to convert the multiplicative noise into additive noise. After two-level DWT decomposition is applied, to suppress the residual noise of an SRAD filtered image, GDGIF and WGIF are exploited to reduce noise from seven high-frequency sub-band images and one low-frequency sub-band image, respectively. Finally, a noise-free image is attained through inverse DWT and an exponential transform. The proposed algorithm exhibits excellent speckle noise elimination and edge conservation as compared with conventional denoising methods.

Abstract: The bitonic filter was recently developed to embody the novel concept of signal bitonicity (one local extremum within a set range) to differentiate from noise, by use of data ranking and linear operators. For processing images, the spatial extent was locally constrained to a fixed circular mask. Since structure in natural images varies, a novel structurally varying bitonic filter is presented, which locally adapts the mask, without following patterns in the noise. This new filter includes novel robust structurally varying morphological operations, with efficient implementations, and a novel formulation of non-iterative directional Gaussian filtering. Data thresholds are also integrated with the morphological operations, increasing noise reduction for low noise, and enabling a multi-resolution framework for high noise levels. The structurally varying bitonic filter is presented without presuming prior knowledge of morphological filtering, and compared to high-performance linear noise-reduction filters, to set this novel concept in context. These are tested over a wide range of noise levels, on a fairly broad set of images. The new filter is a considerable improvement on the fixed-mask bitonic, outperforms anisotropic diffusion and image-guided filtering in all but extremely low noise, non-local means at all noise levels, but not the block-matching 3D filter, though results are promising for very high noise. The structurally varying bitonic tends to have less characteristic residual noise in regions of smooth signal, and very good preservation of signal edges, though with some loss of small scale detail when compared to the block-matching 3D filter. The efficient implementation means that processing time, though slower than the fixed-mask bitonic filter, remains competitive.

Abstract: We propose an ultrasound speckle filtering method for not only preserving various edge features but also filtering tissue-dependent complex speckle noises in ultrasound images. The key idea is to detect these various edges using a phase congruence-based edge significance measure called phase asymmetry (PAS), which is invariant to the intensity amplitude of edges and takes 0 in non-edge smooth regions and 1 at the idea step edge, while also taking intermediate values at slowly varying ramp edges. By leveraging the PAS metric in designing weighting coefficients to maintain a balance between fractional-order anisotropic diffusion and total variation (TV) filters in TV cost function, we propose a new fractional TV framework to not only achieve the best despeckling performance with ramp edge preservation but also reduce the staircase effect produced by integral-order filters. Then, we exploit the PAS metric in designing a new fractional-order diffusion coefficient to properly preserve low-contrast edges in diffusion filtering. Finally, different from fixed fractional-order diffusion filters, an adaptive fractional order is introduced based on the PAS metric to enhance various weak edges in the spatially transitional areas between objects. The proposed fractional TV model is minimized using the gradient descent method to obtain the final denoised image. The experimental results and real application of ultrasound breast image segmentation show that the proposed method outperforms other state-of-the-art ultrasound despeckling filters for both speckle reduction and feature preservation in terms of visual evaluation and quantitative indices. The best scores on feature similarity indices have achieved 0.867, 0.844 and 0.834 under three different levels of noise, while the best breast ultrasound segmentation accuracy in terms of the mean and median dice similarity coefficient are 96.25% and 96.15%, respectively.

Abstract: A PDE model based on an image specific auto generated multi-well potential function and a 4th order anisotropic diffusion with good performance with respect to loss due to smoothing effects has been proposed for grayscale image inpainting. An unconditionally stable numerical scheme using the notion of convexity splitting in time and Fourier spectral method in space has been derived. The stable scheme is both consistent and convergent. Numerical computations are done for some standard test images and results are compared with the results of other existing models in the literature using image analysis specs such as PSNR, SNR, and SSIM.

Abstract: Medical image fusion can combine information from multi-modality images and express them through a single image. How to design a fusion method to preserve more information becomes a hot topic. In this paper, we propose a novel multi-modality medical image fusion method based on Synchronized-Anisotropic Diffusion Equation (S-ADE). First, the modified S-ADE model which is more suitable for Magnetic Resonance Imaging (MRI) and Computed Tomography (CT) images is employed to decompose two source images. We get the base layers and texture layers. Next, the “Maximum Absolute Value” rule is used for base layers fusion. On texture layers, the fusion decision map is calculated by New Sum of Modified Anisotropic Laplacian (NSMAL) algorithm which is designed using common decomposition coefficients by anisotropic diffusion. Furthermore, the consistency check is constructed on the decision map to mitigate the staircase effect. After that, the fused image is obtained by a simple linear combination of layers. Finally, the fused MR/CT image is obtained after image correction. Its aim is to eliminate redundant texture information which is from MRI images in the contour part. The extensive experimental results demonstrate that the proposed method preserves much information as well as guarantees image quality and visual effects. It outperforms other state-of-the-art methods in terms of subjective and objective evaluations.

Abstract: This research presents a machine vision approach to detect lesions in liver ultrasound as well as resolving some issues in ultrasound such as artifacts, speckle noise, and blurring effect. The anisotropic diffusion is modified using the edge preservation conditions which found better than traditional ones in quantitative evolution. To dig for more potential information, a learnable super-resolution (SR) is embedded into the deep CNN. The feature is fused using Gabor Wavelet Transform (GWT) and Local Binary Pattern (LBP) with a pre-trained deep CNN model. Moreover, we propose a Bayes rule-based informative patch selection approach to reduce the processing time with the selective image patches and design an algorithm to mark the lesion region from identified ultrasound image patches. To train this model, standard data ensures promising resolution. The testing phase considers generalized data with a varying resolution and test the performance of the model. Exploring cross-validation, it finds that a 5-fold strategy can successfully eradicate the overfitting problem. Experiment data are collected using 298 consecutive ultrasounds comprising 15,296 image patches. This proposed feature fusion technique confirms satisfactory performance compared to the current relevant works with an accuracy of 98.40%.

Abstract: Anisotropic-diffusion is a commonly used signal preprocessing technique that allows extracting meaningful local characteristics from a signal, such as edges in an image and can be used to support higher-level processing tasks, such as shape detection This paper presents a fully scalable CMOS-RRAM architecture of an edge-aware-anisotropic filtering algorithm aimed at computer vision applications The CMOS circuitry controls the scale-space image data to perform pseudo-parallel in-memory computing and nonlinear processing through RRAM crossbar The arithmetic operations for in-memory computation of brightness gradients are efficiently accumulated to produce the enhanced image in several iterations The proposed architecture uses single RRAM as a computing and storage element to perform both arithmetic operations and accumulations Thanks to the in-memory computation, memory accesses and arithmetic operations are reduced by 64% and 92%, respectively, compared to traditional digital implementations This, in turn, results in a potential reduction of power and area costs of about 75% and 85%, respectively The processing time is also reduced by 67%

Abstract: We propose an unsupervised real-time dense depth completion from a sparse depth map guided by a single image. Our method generates a smooth depth map while preserving discontinuity between different objects. Our key idea is a Binary Anisotropic Diffusion Tensor (B-ADT) which can completely eliminate smoothness constraint at intended positions and directions by applying it to variational regularization. We also propose an Image-guided Nearest Neighbor Search (IGNNS) to derive a piecewise constant depth map which is used for B-ADT derivation and in the data term of the variational energy. Our experiments show that our method can outperform previous unsupervised and semi-supervised depth completion methods in terms of accuracy. Moreover, since our resulting depth map preserves the discontinuity between objects, the result can be converted to a visually plausible point cloud. This is remarkable since previous methods generate unnatural surface-like artifacts between discontinuous objects.

Abstract: Haze and fog removing from videos and images has got massive concentration in the field of video and image processing because videos and images are severely affected by fog in tracking and surveillance system, object detection. Different defogging techniques proposed so far are based on polarisation, colour-line model, anisotropic diffusion, dark channel prior (DCP) etc. However, these methods are unable to produce output image with desirable quality in the presence of dense fog and sky region. In this study, the authors have proposed a novel fog removal technique where DCP is applied on the low-frequency component of empirical wavelet transformation coefficients of the foggy input image. They apply unsharp masking on wavelet coefficients of the embedded wavelet transformed image for improving the sharpness of the output image. Later contrast limited adaptive histogram equalisation technique is used as a post-processing task to the inverse transformed image for producing the sharp and high contrast output. Finally, the colour and intensity of the contrast-enhanced image are uplifted through S-channel and V-channel gain adjustment. The proposed method provides significant improvement to the overall quality of the output image compared to contemporary techniques. The quantitative and qualitative measurements confirm the claims.

Abstract: Diffusion anisotropy in diffusion tensor imaging (DTI) is commonly quantified with normalized diffusion anisotropy indices (DAIs). Most often, the fractional anisotropy (FA) is used, but several alternative DAIs have been introduced in attempts to maximize the contrast-to-noise ratio (CNR) in diffusion anisotropy maps. Examples include the scaled relative anisotropy (sRA), the gamma variate anisotropy index (GV), the surface anisotropy (UAsurf), and the lattice index (LI). With the advent of multidimensional diffusion encoding it became possible to determine the presence of microscopic diffusion anisotropy in a voxel, which is theoretically independent of orientation coherence. In accordance with DTI, the microscopic anisotropy is typically quantified by the microscopic fractional anisotropy (μFA). In this work, in addition to the μFA, the four microscopic diffusion anisotropy indices (μDAIs) μsRA, μGV, μUAsurf, and μLI are defined in analogy to the respective DAIs by means of the average diffusion tensor and the covariance tensor. Simulations with three representative distributions of microscopic diffusion tensors revealed distinct CNR differences when differentiating between isotropic and microscopically anisotropic diffusion. q-Space trajectory imaging (QTI) was employed to acquire brain in-vivo maps of all indices. For this purpose, a 15 min protocol featuring linear, planar, and spherical tensor encoding was used. The resulting maps were of good quality and exhibited different contrasts, e.g. between gray and white matter. This indicates that it may be beneficial to use more than one μDAI in future investigational studies.

Abstract: Conventional cosmic-ray propagation models usually assume an isotropic diffusion coefficient to account for the random deflection of cosmic rays by the turbulent interstellar magnetic field. Such a picture is very successful in explaining many observational phenomena related to the propagation of Galactic cosmic rays, such as broken power-law energy spectra, secondary-to-primary ratios, etc. However, the isotropic diffusion presupposition is facing severe challenges from recent observations. In particular, such observations on the large-scale anisotropy of TeV cosmic rays show that the dipole direction differs from the prediction of the conventional model. One possible reason is that the large-scale regular magnetic field, which leads to an anisotropic diffusion of cosmic rays, has not been included in the model provided by the public numerical packages. In this work, we propose two numerical schemes to solve the $3$-dimensional anisotropic transport equation: the pseudo source method and Hundsdorfer-Verwer scheme. Both methods are verified by reproducing the measured B/C and proton spectrum and the radial variation of spectral index expected by former 2D simulation. As a demonstration of the prediction capability, dipole anisotropy is also calculated by a toy simulation with a rough magnetic field.

Abstract: We prove a Harnack inequality for positive solutions of a parabolic equation with slow anisotropic spatial diffusion. After identifying its natural scalings, we reduce the problem to a Fokker-Planck equation and construct a self-similar Barenblatt solution. We exploit translation invariance to obtain positivity near the origin via a self-iteration method and deduce a sharp anisotropic expansion of positivity. This eventually yields a scale invariant Harnack inequality in an anisotropic geometry dictated by the speed of the diffusion coefficients. As a corollary, we infer H\"older continuity, an elliptic Harnack inequality and a Liouville theorem.

Abstract: A new nonlinear optimization approach is proposed for the sparse reconstruction of log-conductivities in current density impedance imaging. This framework comprises of minimizing an objective functional involving a least squares fit of the interior electric field data corresponding to two boundary voltage measurements, where the conductivity and the electric potential are related through an elliptic PDE arising in electrical impedance tomography. Further, the objective functional consists of a $$L^1$$ regularization term that promotes sparsity patterns in the conductivity and a Perona–Malik anisotropic diffusion term that enhances the edges to facilitate high contrast and resolution. This framework is motivated by a similar recent approach to solve an inverse problem in acousto-electric tomography. Several numerical experiments and comparison with an existing method demonstrate the effectiveness of the proposed method for superior image reconstructions of a wide variety of log-conductivity patterns.

Abstract: Image acquisition systems usually acquire images with distortions due to various factors associated with digitization processes. Poisson is one of the common types of noises present in the image, and it distorts the fine features. Hence, it is necessary to denoise the noisy image by smoothing it to extract the features with fine details. Among the denoising methods, anisotropic diffusion method provides more adequate results. In this chapter, the authors dealt with existing models such as Perona-Malik (PM), total variation, Tsai, Chao, Chao TFT, difference eigen value PM, adaptive PM, modified PM, and Maiseli models. The performances of the models were tested on synthetic image added with the Poisson noise. Quality metrics are used to quantify and to ensure the smoothness of the resultant images. However, in order to ensure the completeness of the denoising effect, the qualitative attributes such as sharpness, blurriness, blockiness, edge quality, and false contouring are considered on smoothened images. The analysis results are shown the completeness of the denoising effect of the models.

Abstract: The gradient vector flow (GVF) is an effective external force to deform the active contours. However, it suffers from high computational cost. For efficiency, the virtual electric field (VEF) model has been proposed, which can be implemented in real time thanks to fast Fourier transform (FFT). The VEF model has large capture range and low computation cost, but it has the limitations of sensitivity to noise and leakage on weak edge. The recently proposed CONVEF (Convolutional Virtual Electric Field) model takes the VEF model as a convolutional operation and employed another convolution kernel to overcome the drawbacks of the VEF model. In this paper, we employ an edge stopping function similar to that in the anisotropic diffusion to further improve the CONVEF model, and the proposed model is referred to as MCONVEF (Modified CONVEF) model. In addition, a piecewise constant approximation algorithm is borrowed to accelerate the calculation of the MCONVEF model. The proposed MCONVEF model is compared with the GVF and VEF models, and the experimental results are presented to demonstrate its superiority in terms of noise robustness, weak edge preserving and deep concavity convergence.

Abstract: We derive the anisotropic diffusion equation for optical random media from the radiative transfer equation by correcting errors or eliminate simplifications in previous studies. The derived equation makes explicit the relation between microscopic (extinction cross-section and phase function) and macroscopic (transport mean free path tensor) parameters. We discover that transmittance and reflectance for a film made of an anisotropic material are determined from a dimensionless factor, which we define as anisotropy tensor, in addition to scattering/transport mean free paths and extrapolation length ratio. According to our derived equations, a direction component of the diagonalized transport mean free path tensor can be smaller than that of the diagonalized scattering mean free path tensor for forward scattering, which is physically impossible. We demonstrate that a direction component of the diagonalized inner product of transport mean free path tensor and the anisotropy tensor preserves the physical meaning of transport mean free path.

Abstract: This paper proposes a novel and efficient algorithm for defogging of color (RGB) images. The fog in a scene is mostly due to the attenuation and airlight map, which decrease the quality of the imag...

Abstract: Diffusion of particles has wide repercussions ranging from particle-based soft matter systems to solid state systems with particular electronic properties. Recently, in the field of magnetism, diffusion of magnetic skyrmions, topologically stabilized quasi-particles, has been demonstrated. Here we show that by applying a magnetic in-plane field and therefore breaking the symmetry of the system, the skyrmion diffusion becomes anisotropic with faster diffusion parallel to the field axis and slower diffusion perpendicular to it. We furthermore show that the absolute value of the applied field controls the absolute values of the diffusion coefficients so that one can thereby uniquely tune both the orientation of the diffusion and its strength. Based on the stochastic Thiele equation, we can explain the observed anisotropic diffusion as a result of the elliptical deformation of the skyrmions by the application of the in-plane field.

Abstract: We consider elliptic diffusion problems with a random anisotropic diffusion coefficient, where, in a notable direction given by a random vector field, the diffusion strength differs from the diffusion strength perpendicular to this notable direction. The Karhunen–Loeve expansion then yields a parametrisation of the random vector field and, therefore, also of the solution of the elliptic diffusion problem. We show that, given regularity of the elliptic diffusion problem, the decay of the Karhunen–Loeve expansion entirely determines the regularity of the solution’s dependence on the random parameter, also when considering this higher spatial regularity. This result then implies that multilevel quadrature methods may be used to lessen the computation complexity when approximating quantities of interest, like the solution’s mean or its second moment, while still yielding the expected rates of convergence. Numerical examples in three spatial dimensions are provided to validate the presented theory.