## What is/are Isotropic Diffusion?

Isotropic Diffusion - This is followed by fusion of these generated multi-exposure images using nonlinear Anisotropic Diffusion (AD).^{[1]}The speckle reducing anisotropic diffusion (SRAD) filter is extensively used to process the ultrasound image.

^{[2]}The objective of the work is to compare the performance of two filters, one isotropic and another one of anisotropic diffusion based on gradient.

^{[3]}Nonlinear anisotropic diffusion (NAD) filtering is a procedure based on nonlinear evolution PDEs which seeks to improve images qualitatively by removing noise while preserving details and even enhancing edges.

^{[4]}The proposed methodology includes a combinational approach of anisotropic diffusion and Laplacian pyramid leading to image enhancement.

^{[5]}Both oblate and prolate particles exhibit an anisotropic diffusion with respect to the orientation of the nematic director.

^{[6]}This study proposes a speckle noise reduction algorithm while using the speckle reducing anisotropic diffusion (SRAD) filter, discrete wavelet transform (DWT), soft threshold, improved guided filter (IGF), and guided filter (GF), with the aim of removing speckle noise.

^{[7]}The images were normalized through gray scaling and anisotropic diffusion.

^{[8]}We propose the use of anisotropic diffusion to include environmental factors of the evaluated geographic area in the traditional crime prediction model, thereby aiming to predict crime occurrence at a finer scale regarding spatiotemporal aspects and environmental similarity.

^{[9]}The proposed method works with the Speckle Reducing Anisotropic Diffusion (SRAD) in combination with the Discrete Wavelet Transform (DWT) using Bayesian Threshold.

^{[10]}First, anisotropic diffusion is used to decompose the source images into base and detail layers.

^{[11]}Compared to the other smoothing methods, such as anisotropic diffusion, comparative filters, guided and rolling guidance filters, the benefit of our method is that it natively works with the image structure – thickness map, so it does not depend on the various levels of image noise, lightning conditions and effects.

^{[12]}Advanced methods are targeted: wavelet transform-based algorithms with Daubechies, symlets, curvelets, contourlets, patch-based methods such as BM3D, NL-means algorithms and deep learning approaches; in addition, classical spatial filtering methods are considered, such as Wiener, median, Gauss filtering, and adaptive filtering approaches such as anisotropic diffusion and synthetic aperture radar filtering.

^{[13]}Enhancement of mammograms is done by anisotropic diffusion.

^{[14]}Natural liquids can contain self-aligned molecules (such as liquid crystals and biological membranes) which give them unique properties of anisotropic diffusion, coupling between the molecular orientation and flow, etc.

^{[15]}These two sources are effectively and efficiently fused by combining ideas from anisotropic diffusion and semi-global matching.

^{[16]}We present a suite of high-resolution cosmological simulations, using the FIRE-2 feedback physics together with explicit treatment of magnetic fields, anisotropic conduction and viscosity, and cosmic rays (CRs) injected by supernovae (including anisotropic diffusion, streaming, adiabatic, hadronic and Coulomb losses).

^{[17]}Therefore, in this work we present a comparative study of ten filters for the elimination of noise in confocal microscopy: median, anisotropic diffusion, bilateral, propagated, improved propagated, Rudin-Osher-Fatemi (ROF), TVL1, non-local means, K-SVD, and Wavelet 'A trous' and Haar filters, with and without preprocessing images with the Wiener filter, taking as criteria the noise reduction and the conservation of edges.

^{[18]}They include combinations of CNN and "classical" algorithms like Non-Local Means, BM3D, Bilateral, Anisotropic diffusion, Total Variation (TV), etc.

^{[19]}We develop a new theoretical model called anisotropic diffusion through transient trapping in zones to explain CTCF dynamics.

^{[20]}Specifically, directional water diffusivity (anisotropic diffusion) within neural tissues can provide information about axoplasmic flow and through that derive the histological architecture and microstructure of nerves, tract course and integrity, the severity of peripheral neuropathies, and potentially serve as a marker for nerve regeneration.

^{[21]}This observed anisotropic diffusion is explained here by determining the orientation-dependent dif.

^{[22]}We model the contribution of the nearest young supernova remannt Vela to the local cosmic ray flux taking into account both the influence of the Local Superbubble and the effect of anisotropic diffusion.

^{[23]}In this paper, a local activity measurement of the clipped and normalized variance or standard deviation is proposed to drive anisotropic diffusion and relative total variation (RTV) to work better for structural preservation.

^{[24]}

## partial differential equation

The proposed model is a fusion of Maximum Likelihood Expectation Maximization (MLEM) with partial differential equation based anisotropic diffusion (AD) filter.^{[1]}Then the noise of the resultant acceleration (vertical direction) signal is removed by the smoothing process of the anisotropic diffusion based on the partial differential equation.

^{[2]}The innovation of this method lies in the introduction of anisotropic diffusion equation of partial differential equation.

^{[3]}The output of modified OSL undertakes in Fourth order partial differential equation (PDE) based Anisotropic Diffusion regularization approach to deal with an ill-posed.

^{[4]}In this paper, we propose an anisotropic diffusion partial differential equation (PDE) driven hybrid CNN framework, named PM-HCNN.

^{[5]}This paper proposes a despeckling approach named the Gabor-based anisotropic diffusion coupled with the lattice Boltzmann method (GAD-LBM), which uses the lattice Boltzmann method (LBM) to fast solve the partial differential equation of an anisotropic diffusion model embedded with the Gabor edge detector.

^{[6]}To address this problem, an improved partial differential equation method for neutron image denoising is proposed, in which the anisotropic diffusion properties are considered.

^{[7]}

## nonlinear second order

The considered nonlinear second-order anisotropic diffusion model is then particularized for an image restoration task.^{[1]}So, the second section describes the anisotropic diffusion models for image restoration based on nonlinear second-order parabolic and hyperbolic partial differential equations, proposed by us.

^{[2]}It is based on a nonlinear second-order anisotropic diffusion-based model that is adapted for the Poisson distribution.

^{[3]}The proposed restoration technique is based on a nonlinear second-order anisotropic diffusion model that removes effectively the additive and multiplicative noise, and also overcomes the unintended effects, while preserving the photo's details.

^{[4]}

## cylindrical representative volume

The goal of the proposed review paper is to comprise a collection of the knowledge of the last seven decades dealing with the following topics: • stress fields induced by edge and screw dislocation; a presentation of the most recent solution technique and discussion of previous solutions; • eigenstrains of carbon and hydrogen atoms placed in octahedral and tetrahedral sites; a selection of the values of reliable sources; • assembling of the interstitial atoms, acting as inclusions with a misfit eigenstrain state, in cylindrical representative volume elements; • eigenstrain states in cylindrical representative volume elements due to occupancy of interstitial atoms described by X1, X2 and X3; • eigenstress fields generated by eigenstrains in cylindrical representative volume elements; • formulation of the generalized chemical potential of interstitial atoms in the stressed bcc lattice; • anisotropic diffusion equations for interstitials in a stressed bcc lattice, accounting for elastodiffusion and the effect of occupancy of different types of interstitial sites.^{[1]}

## skeleton strength map

Firstly, we propose advanced skeleton strength map (ASSM) algorithm to create the skeleton which computes the skeleton strength map by isotropic diffusion on the gradient vector field, selects critical points from the skeleton strength map and connects them by Kruskal's algorithm.^{[1]}

## Restricted Isotropic Diffusion

Amyloid-b-42 positively correlated with restricted isotropic diffusion in several white matter tracts.^{[1]}Contrary to the conventional wisdom that apparent diffusion coefficient (ADC) negatively correlates with increased tumor cellularity, we demonstrate disagreement between ADC and histologically confirmed tumor cellularity in glioblastoma specimens, whereas DBSI-derived restricted isotropic diffusion fraction positively correlated with tumor cellularity in the same specimens.

^{[2]}

## isotropic diffusion filter

First, to reduce noise, an improved curvature anisotropic diffusion filter is applied, which simultaneously stores edge information.^{[1]}The evaluated filters were Coherence-Enhancing Diffusion (CED), Speckle Reducing Anisotropic Diffusion (SRAD), Detail-Preserving Anisotropic Diffusion (DPAD), and Anisotropic Diffusion Filter with Memory Based on Speckle (ADMSS).

^{[2]}The proposed algorithm utilizes a powerful anisotropic diffusion filter to denoise the image.

^{[3]}It involves preprocessing of the acquired retinal SD-OCT image (B-scan) using the proposed automatic Anisotropic diffusion filter, followed with contrast stretching to suppress intrinsic speckle noise without blurring structural edges.

^{[4]}In this paper, we propose a non-local weighted operational anisotropic diffusion filter and evaluate its effect on magnetic resonance images and on kV/CBCT radiotherapy images.

^{[5]}Initially, the images are filtered using anisotropic diffusion filter.

^{[6]}An anisotropic diffusion filter is used to remove the noise and to smooth the digital images.

^{[7]}Anisotropic diffusion filters can be used to reduce noise and preserve the edges in the image.

^{[8]}An anisotropic diffusion filter was used to reduce speckle noise before implementation of the edge detection method, which consists of three characteristic steps.

^{[9]}The images obtained were filtered using a combination of linear filters and anisotropic diffusion filter.

^{[10]}Because of the isotropic diffusion filter applied in this model, USM enhanced the edge information and noise simultaneously.

^{[11]}OBJECTIVE To develop a novel pulmonary vessel segmentation approach by incorporating vessel enhancement filters and the anisotropic diffusion filter with the variational region growing.

^{[12]}Therefore, the spatial filter can be adapted according to the amount of information in each area, so that small features are reprojected in the Kernel PCA domain where image details are reconstructed efficiently, and large features are denoised by an anisotropic diffusion filter to recover the homogenous regions.

^{[13]}To effectively balance speckle smoothing and preservation of edges and radiation, a novel anisotropic diffusion filter was developed that uses a directional coherent coefficient.

^{[14]}Firstly, anisotropic diffusion filter is applied for removal of noise because input MRI images are mostly noisy and inhomogeneous contrast.

^{[15]}The region of interest (ROI) is enhanced using a proposed partial differential diffusion filter (PDDF) which is a modified form of anisotropic diffusion filter.

^{[16]}The Anisotropic Diffusion Filter is an image smoothing method often applied to improve segmentation and classification tasks.

^{[17]}In this paper we compare two different methods to filter prediction signals in video coding: The anisotropic diffusion filter modeled using a (modified) heat equation and the ADMM (Alternating Direction Method of Multipliers) filter using directional total variation.

^{[18]}In the preprocessing step, anisotropic diffusion filter used to filter unwanted distortions from MRI image.

^{[19]}In this study, we propose an improved adaptive anisotropic diffusion filter to effectively reduce noise and reinforce image contours.

^{[20]}Furthermore, we altered the speckle reducing anisotropic diffusion filter to improve the performance of disturbance suppression.

^{[21]}The images obtained by this technique are processed with an anisotropic diffusion filter to reduce the noise and preserve edges.

^{[22]}The technique presented takes in different image processing techniques for vessel extraction, viz image enhancement and smoothening with contrast-limited adaptive histogram equalization (CLAHE) and anisotropic diffusion filter, respectively, segmentation by Kirsch’s template and morphological cleaning to get the final segmented image.

^{[23]}Another technique used for characterizing spatial information on the images at different scales is based on computing profiles relying on edge-preserving filters such as anisotropic diffusion filters.

^{[24]}This is emphasized in the implementation of the anisotropic diffusion filter and illumination correction method.

^{[25]}This non-invasive and less expensive methodology is composed of a contrast enhancement algorithm, an anisotropic diffusion filter (denoising) and an edge detector.

^{[26]}

## isotropic diffusion equation

We formulate the system in terms of a high-dimensional, time-dependent, anisotropic diffusion equation and employ matched asymptotic analysis to derive the effective trapping rate.^{[1]}NCSF is especially useful when coupled with an anisotropic diffusion equation in noisy configurations.

^{[2]}This paper presents a new tailored finite point method (TFPM) for the strongly anisotropic diffusion equation on Cartesian coordinates.

^{[3]}The innovation of this method lies in the introduction of anisotropic diffusion equation of partial differential equation.

^{[4]}We consider a coupled system for image denoising, which consists of an anisotropic diffusion equation and a global sparse gradient model (GSG).

^{[5]}This paper investigates an anisotropic diffusion equation with degeneracy on the boundary.

^{[6]}We achieved this smoothing by solving an anisotropic diffusion equation with a finite-difference method.

^{[7]}The goal of the proposed review paper is to comprise a collection of the knowledge of the last seven decades dealing with the following topics: • stress fields induced by edge and screw dislocation; a presentation of the most recent solution technique and discussion of previous solutions; • eigenstrains of carbon and hydrogen atoms placed in octahedral and tetrahedral sites; a selection of the values of reliable sources; • assembling of the interstitial atoms, acting as inclusions with a misfit eigenstrain state, in cylindrical representative volume elements; • eigenstrain states in cylindrical representative volume elements due to occupancy of interstitial atoms described by X1, X2 and X3; • eigenstress fields generated by eigenstrains in cylindrical representative volume elements; • formulation of the generalized chemical potential of interstitial atoms in the stressed bcc lattice; • anisotropic diffusion equations for interstitials in a stressed bcc lattice, accounting for elastodiffusion and the effect of occupancy of different types of interstitial sites.

^{[8]}Based on the concept of the nonlinear anisotropic diffusion equation, we develop a computer code which uses noise-driven parameterization to optimize the image denoising process.

^{[9]}Direct generalization of the Random Walk on Spheres method to anisotropic diffusion equations is not possible, therefore, we have derived approximations of the probability densities for the first passage time and the exit point on a small sphere.

^{[10]}

## isotropic diffusion filtering

This paper demonstrates a method to segment liver lesions automatically in CT abdomen images by using Anisotropic Diffusion filtering, Chan-Vese algorithm, K-means clustering, and morphological binary operations.^{[1]}, nonlocal operators, anisotropic diffusion filtering).

^{[2]}MyoSAT combines several unconventional approaches including advanced background leveling, Perona-Malik anisotropic diffusion filtering, and Steger’s line detection algorithm to aid in pre-processing and enhancement of the muscle image.

^{[3]}By avoiding common problems of linear filtering such as blurring and localization, anisotropic diffusion filtering has been applied as preprocessing operation to enhance the edge region/boundaries between adjacent yarns of the fabric images.

^{[4]}Anisotropic diffusion filtering (ADF) has been recently applied to restore THG images of normal brain, but ADF is hard‐to‐code, time‐consuming and only reconstructs salient edges.

^{[5]}Whereas for the detail coefficient, Soft-Thresholding will be carried out and followed by the Anisotropic Diffusion Filtering process.

^{[6]}In the pre-processing, the images will be filtered through anisotropic diffusion filtering algorithm so as to remove the noises and avoid the indistinctness.

^{[7]}The speckle noise in the images is smoothened using anisotropic diffusion filtering technique, and segmentation of Retinal Nerve Fiber layer (RNFL) along with Ganglion Cell Layer (GCL) and Inner Plexiform Layer (IPL) complex is performed using the axial gradient canny edge detection combined with a level set method.

^{[8]}Several image evaluation metrics were measured to compare the image quality of the proposed method versus the other denoising techniques, including the total variation minimization scheme, anisotropic diffusion filtering, nonlocal means, and Wiener filtering.

^{[9]}The anisotropic diffusion filtering algorithm has excellent smoothing performance for medical images, but the normal diffusion filtering algorithm will blur the edges and details.

^{[10]}

## isotropic diffusion problem

The finite volume method considered here has been developed recently in [3] to approximate heterogeneous and anisotropic diffusion problems using a general class of nonconforming meshes.^{[1]}Finite difference schemes and iterative methods of solving anisotropic diffusion problems governing multidimensional elliptic PDE with mixed derivatives are considered.

^{[2]}SUSHI (Scheme Using Stabilization and Hybrid Interfaces) is a finite volume method has been developed at the first time to approximate heterogeneous and anisotropic diffusion problems.

^{[3]}A nonlinear finite volume element scheme for anisotropic diffusion problems on general triangular meshes is proposed.

^{[4]}The advantages of the CRS are demonstrated by solving an anisotropic diffusion problem in a shrinking domain.

^{[5]}Finally, some numerical examples are presented to verify that it is second order accurate on 2D polygonal meshes for anisotropic diffusion problems.

^{[6]}In this paper, we present a high order finite difference solver for anisotropic diffusion problems based on the first-order hyperbolic system method.

^{[7]}

## isotropic diffusion tensor

We present a new method to study position-dependent, anisotropic diffusion tensors inside spherically confined systems—a geometry that is common to many chemical nanoreactors.^{[1]}The Smoluchowski equation for the many-particle probability distribution then acquires an anisotropic diffusion tensor.

^{[2]}First, anisotropic diffusion tensor (ADT) is utilized as high frequency edge information.

^{[3]}A complete convergence analysis is presented on the coupled model, using only weak regularity assumptions on the solution (which are satisfied in practical applications), and not relying on $$L^\infty $$L∞ bounds (which are impossible to ensure at the discrete level given the anisotropic diffusion tensors and the general grids used in applications).

^{[4]}Because of its additional freedom degrees, the MRT collision operator provides accurate approximations to space-fractional advection-diffusion equations, even in the cases which the BGK fails to represent because of anisotropic diffusion tensor or of flow rate destabilizing the BGK LBM scheme.

^{[5]}Diffusion basis spectrum imaging (DBSI) combines discrete anisotropic diffusion tensors and the spectrum of isotropic diffusion tensors to model the underlying multiple sclerosis (MS) pathologies.

^{[6]}

## isotropic diffusion model

The considered nonlinear second-order anisotropic diffusion model is then particularized for an image restoration task.^{[1]}To characterize the progression of misfolded proteins across the brain, we combine the classical Fisher–Kolmogorov equation for population dynamics with an anisotropic diffusion model and simulate misfolding across a sagittal section and across the entire brain.

^{[2]}So, the second section describes the anisotropic diffusion models for image restoration based on nonlinear second-order parabolic and hyperbolic partial differential equations, proposed by us.

^{[3]}An anisotropic diffusion model based on a log-normal distribution of a local gray-level is used to propose a way to denoise the panchromatic images.

^{[4]}A diffusion function based on mixed gradient and variable exponent is built by combining image characteristics in wavelet transform and spatial domains to solve the edge blurring problem of the traditional anisotropic diffusion model in image filtering and improve image filtering performance.

^{[5]}The proposed restoration technique is based on a nonlinear second-order anisotropic diffusion model that removes effectively the additive and multiplicative noise, and also overcomes the unintended effects, while preserving the photo's details.

^{[6]}

## isotropic diffusion algorithm

In this AI architecture, ultrasound images were segmented by the reaction diffusion level set model combined with the Gabor-based anisotropic diffusion algorithm.^{[1]}Finally, the reconstructed cone beam CT image is repaired by an anisotropic diffusion algorithm, which can suppress artifacts.

^{[2]}Dense images are computed using an anisotropic diffusion algorithm taking both channels into account.

^{[3]}We also propose the guided anisotropic diffusion algorithm, which improves semantic segmentation results using the input images as guides to perform edge preserving filtering, and is used in conjunction with the iterative training method to improve results.

^{[4]}

## isotropic diffusion along

More precisely, the method performs a PDE-based anisotropic diffusion along directions defined by local structures in the 4D ray space.^{[1]}The transport processes considered here are either anisotropic diffusion along axonal bundles or discrete Laplacian transport along a network.

^{[2]}The transport processes considered here are either anisotropic diffusion along axonal bundles or discrete Laplacian transport along a network.

^{[3]}

## isotropic diffusion property

Many biological tissues inside the human body exhibit highly anisotropic diffusion properties; for example, tissues of the nervous system and white matter in the brain.^{[1]}Owing to the systematic alignment and ordering of fatty acid and ceramide chains, lipid layers in biological membranes have strongly anisotropic diffusion properties.

^{[2]}To address this problem, an improved partial differential equation method for neutron image denoising is proposed, in which the anisotropic diffusion properties are considered.

^{[3]}

## isotropic diffusion method

We illustrate our technique on synthetic and real-world 3D data and compare our method with the median method, weighted least squares, L0 volume gradient minimization, and edge aware anisotropic diffusion method.^{[1]}In order to study the processing method of speckle noise reduction, this paper takes the OCT fundus image as the research object, and uses the median filtering, wavelet threshold method, Wiener filtering and anisotropic diffusion method to denoise the OCT fundus image.

^{[2]}Condensed anisotropic diffusion method is employed as a preprocessing tool for despeckling, and the resultant image is processed by multiscale approach in the Laplacian pyramid (LP) domain.

^{[3]}

## isotropic diffusion coefficient

The results indicated two mobile "free" states of the N-terminal domain undergoing global diffusive motions, with isotropic diffusion coefficients of 0.^{[1]}all coefficients anisotropic, is compared with both the case possessing isotropic diffusion coefficients and the case of partial anisotropy, i.

^{[2]}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.

^{[3]}

## isotropic diffusion behavior

This PDE is based on anisotropic diffusion behavior by adding some constraints to the Weickert coherence enhancing diffusion tensor in order to control the diffusion process near edges and avoids tiny corners destruction.^{[1]}The ab initio molecular dynamics simulations exhibit the anisotropic diffusion behaviors of Na ions on both o-ScC2 and o-ScN2 monolayers with low energy barriers of 0.

^{[2]}In order to realistically model the diffusion process that drives degradation, an experimental/numerical study is conducted on the anisotropic diffusion behavior of laminated composites.

^{[3]}

## isotropic diffusion encoding

Using a single‐shot isotropic diffusion encoding (IDE) preparation, we eliminate signal confounds caused by anisotropic diffusion, including microscopic anisotropy, and acquire in vivo diffusion‐weighted images (DWIs) over a wide range of diffusion sensitizations.^{[1]}Here we combine isotropic diffusion encoding with ultra-strong diffusion gradients (240 mT/m) to achieve high diffusion-weightings with high signal to noise ratio, while suppressing signal arising from anisotropic water compartments with significant mobility along at least one axis (e.

^{[2]}Isotropic diffusion encoding efficiently encodes additional microstructural information in combination with conventional linear diffusion encoding.

^{[3]}

## isotropic diffusion technique

Therefore, in this work, a novel method of optimal anisotropic diffusion technique along with contrast limited adaptive histogram equalization (CLAHE), Otsu's optimal thresholding and morphological thinning operation are applied over the fractographs for edge enhancement, overcoming inhomogeneous illumination, edge segmentation and thinning, respectively, to detect voids, automatically.^{[1]}Wavelet-based filters and anisotropic diffusion techniques have proven their worth in reducing both Gaussian additive noise and speckle noise.

^{[2]}

## isotropic diffusion scheme

It is based on a multi-scale anisotropic diffusion scheme.^{[1]}It is based on a multi-scale anisotropic diffusion scheme.

^{[2]}

## isotropic diffusion function

Meanwhile, the diffusion filtering method with adaptive image structure is introduced into image registration, and the anisotropic diffusion function with the ability of feature preservation and consistency enhancement is defined as the regularization term of the model.^{[1]}The median filter as a nonlinear operator utilized for an oriented anisotropic diffusion function to reduce noise and preserve the edges, is performed on medical images.

^{[2]}