Nonlinear Anisotropic Diffusion Research Articles

Perona-Malik Diffusion-Driven Regularization for Image Resolution Enhancement in Electrical Capacitance Tomography

Electrical Capacitance Tomography (ECT) is a non-invasive promising method for monitoring industrial processes, such as oil-gas flow in pipelines and solid-gas flow in pneumatic systems. Despite its potential benefits, ECT generates poor-quality images, often used only for qualitative analysis. A non-linear relationship between measured capacitances and permittivity distribution and the ill-posedness of the sensitivity matrix elements causes this limitation. This hinders the applicability of ECT in monitoring online industrial process applications. This work proposes a reconstruction method based on a nonlinear diffusion function to generate high-quality images from the measured capacitance data from the ECT system. The diffusion regularization functional helps to remove noise and preserve semantic features. Experimental results reveal that the proposed method generates high-quality and visually appealing images with a 15% reduction in distribution error and a 10% increase in correlation coefficient compared to state-of-the-art methods such as linear back projection and projected landweber. This allows further investigation into how nonlinear anisotropic diffusion can improve the applicability of ECT systems in industrial control and monitoring.

Scale-Aware Edge-Preserving Full Waveform Inversion with Diffusion Filter for Crosshole Sensor Arrays.

Full waveform inversion (FWI) is recognized as a leading data-fitting methodology, leveraging the detailed information contained in physical waveform data to construct accurate, high-resolution velocity models essential for crosshole surveys. Despite its effectiveness, FWI is often challenged by its sensitivity to data quality and inherent nonlinearity, which can lead to instability and the inadvertent incorporation of noise and extraneous data into inversion models. To address these challenges, we introduce the scale-aware edge-preserving FWI (SAEP-FWI) technique, which integrates a cutting-edge nonlinear anisotropic hybrid diffusion (NAHD) filter within the gradient computation process. This innovative filter effectively reduces noise while simultaneously enhancing critical small-scale structures and edges, significantly improving the fidelity and convergence of the FWI inversion results. The application of SAEP-FWI across a variety of experimental and authentic crosshole datasets clearly demonstrates its effectiveness in suppressing noise and preserving key scale-aware and edge-delineating features, ultimately leading to clear inversion outcomes. Comparative analyses with other FWI methods highlight the performance of our technique, showcasing its ability to produce images of notably higher quality. This improvement offers a robust solution that enhances the accuracy of subsurface imaging.

EWRD: Entropy‐Weighted Low‐Light Image Enhancement via Reverse Diffusion Model

Low‐light enhancement significantly aids vision tasks under poor illumination conditions. Existing methods primarily focus on enhancing severely degraded low‐light areas, improving illumination accuracy, or noise suppression, yet they often overlook the loss of additional details and color distortion during calculation. In this paper, we propose an innovative entropy‐weighted low‐light image enhancement method via the reverse diffusion model, aiming at addressing the limitations of the traditional Retinex decomposition model in preserving local pixel details and handling excessive smoothing issues. This method integrates an entropy‐weighting mechanism for improved image quality and entropy, along with a reverse diffusion model to address the detail loss in total variation regularization and refine the enhancement process. Furthermore, we utilize long short‐term memory networks for the learning reverse process and the simulation of image degradation, based on a thermodynamics‐based nonlinear anisotropic diffusion model. Comparative experiments reveal the superiority of our method over conventional Retinex‐based approaches in terms of detail preservation and visual quality. Extensive tests across diverse datasets demonstrate the exceptional performance of our method, evidencing its potential as a robust solution for low‐light image enhancement.

An accurate and robust strain field smoothing method based on polynomial fitting and anisotropic diffusion in digital image correlation

Digital Image Correlation (DIC) is a popular optical measurement method. However, the accuracy of strain values obtained by DIC may be adversely affected by random noise. This paper proposes an Anisotropic Nonlinear Diffusion Filtering (ADF) method, combining Pointwise Least Square (PLS) with the Perona-Malik (PM) model, extracts accurate and smooth strain fields from noisy displacement fields. The strain field is extracted point by point using PLS and then smoothed by the PM model. The ADF method was tested on simulated and real speckle deformation experiments. The results indicate that the ADF method significantly reduces the negative impact of inappropriate calculation parameters, such as the strain window size, compared to the PLS method. Furthermore, the PM model was found to be better suited for smoothing strain fields compared to other image denoising models. The ADF method demonstrates excellent noise suppression performance and accurate strain identification in both simulated and real experiments.

Pattern formation in photo-controlled bioconvection

The model eukaryotic microalgae Chlamydomonas reinhardtii is well known for its ability to generate bioconvection flows that are associated to intricate concentration patterns. Recently, it was demonstrated that the propensity of these algae to move toward a light source – a phenomenon termed phototaxis – can be exploited to locally concentrate micro-organisms and induce (photo)-bioconvection in algal suspensions by inducing a localised excess of density. In the present study we show experimentally that a cell population in a thin liquid layer self-organises in the presence of a heterogeneous light field and displays remarkable symmetry-breaking instabilities that are ruled by both the width of the light beam and the photo-bioconvection Rayleigh number. Beside circular stable states, fingers, dendrites and wave instabilities are reported, quantified and classified in a general phase diagram. Next, we use lubrication theory to develop an asymptotic model for bioconvection in a thin liquid layer, that includes the influences of both gyrotaxis and phototaxis. We obtain a single nonlinear anisotropic diffusion–drift equation describing the spatiotemporal evolution of the depth-averaged algal population. Analytical and numerical solutions are presented and show a very good agreement with the experimental results. In particular, we show that the dendrite instability arises as a result of a subtle coupling between the nonlinearity of the phototactic response, the gyrotactic effect and the self-induced bioconvective flows. Such complex flow fields might find applications in photo-bioreactors, through the efficient stirring of the harvested biomass.

A coupled nonlinear finite element scheme for anisotropic diffusion equation with nonlinear capacity term

A new nonlinear finite element method is studied to solve multi-dimensional anisotropic nonlinear diffusion equation with nonlinear capacity term, especially to secure a vivid simulation in the case of problems with transient physical quantities. In the scheme design, fully implicit two-layer coupled discretization is applied to assure high time accuracy, and finite element discretization is applied to assure high space accuracy. By introducing Ritz projection and new inductive reasoning methods, we develop the discrete functional analysis technique from that for finite difference schemes, and establish a new general analysis framework for nonlinear finite element schemes. Consequently, we overcome the difficulties arising from the strong coupling of the nonlinear finite element discretizations for the capacity term and diffusion operator, and prove the existence and boundedness of the nonlinear finite element solution, as well as its second-order time and optimal order space convergence. Numerical experiments and comparisons confirm the theoretical analysis results and demonstrate its high performance.

Nodal discrete duality numerical scheme for nonlinear diffusion problems on general meshes

Abstract Discrete duality finite volume (DDFV) schemes are known for their ability to approximate nonlinear and linear anisotropic diffusion operators on general meshes, but they possess several drawbacks. The most important drawback of DDFV is the simultaneous use of the cell and the node unknowns. We propose a discretization approach that incorporates DDFV ideas and the associated analysis techniques, but allows for a rapid elimination of the cell unknowns. Further, unlike the DDFV scheme, the new ‘Nodal Discrete Duality’ (NDD) scheme does not require specific adaptation in presence of discontinuities of the diffusion tensor along cell boundaries. We describe in detail the 2D NDD framework and its two 3D variants, focusing on the consistency properties of the discrete gradient and discrete divergence operators and on the key structural property of discrete duality. For the 2D scheme, convergence analysis is carried out and a series of numerical tests are provided for a large family of nonlinear anisotropic elliptic problems with zero Dirichlet boundary condition.

Separable solutions to nonlinear anisotropic diffusion equation in elliptic coordinates.

Two- and three-dimensional exact solutions of the nonlinear diffusion equation are proved to exist in elliptic coordinates subject to an arbitrary piecewise constant azimuthal anisotropy. Degrees of freedom traditionally used to satisfy boundary conditions are instead employed to ensure continuity and conservation of mass across contiguity surfaces between subdomains of distinct diffusivities. Not all degrees of freedom are exhausted thereby, and conditions are given for the inclusion of higher harmonics. Degrees of freedom associated with one isotropic subdomain are always available to satisfy boundary conditions. The second harmonic is pivotal in the solution construction as well as the identification of partial symmetries in the domain partition. The anisotropy gives rise to an unconventional mixed type critical point that combines saddle and node-like characteristics. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

A nonlinear anisotropic diffusion model with non-standard growth for image segmentation

The anisotropic diffusion equation with non-standard growth embodies the physical characteristics of “point-by-point anisotropy” as well as has important potential value in computer vision. In this paper, a general anisotropic diffusion framework of the level set function is proposed for image segmentation in scalar-value and vector-value images. Specifically, we develop a new regularization term that uses a diffusion coefficient with non-standard growth conditions and diffusion tensors. The existence and uniqueness of the model are obtained by the Galerkin method. We establish the numerical algorithms for obtaining the texture feature and evolving the level set function of images. Some numerical tests on medical and natural images confirm the accuracy of the proposed method and the improvement in segmenting small features.

Bohemian sandstone for restoration of cultural heritage sites: 3D microstructure and mass transport properties

We characterised the microstructure of sandstone from the Msene locality (Czech Republic) by combining X-ray computed micro-tomography, back-scattered electron imaging, chemical composition analysis and textural analysis and gas permeation. Both, 2D and 3D images were commonly processed by linking an anisotropic non-linear diffusion filter and a segmentation method based on power watershed. This approach guaranteed binarised outputs that were almost the same in wide ranges of spatial-filter and power-watershed parameters, i.e., subjective choices of the parameters played the negligible role. The rock pore structure was found to be statistically homogeneous and almost isotropic with perfectly connected pore space. We also partitioned the void and solid phases into either grains or pores and throats, which enabled us to study characteristic sizes and connectivity of partitioned regions. By comparing pore and throat sizes, we demonstrated the significant convergent-divergent nature of the pore space because the throat size covered at most one half of the total surface area per pore (cavity). In addition, we calculated coordination numbers for all pores (cavities) to verify almost perfect connectivity of internal pores with those at the external surface. Effective (macroscopic) transport properties of the reconstructed pore space were simulated and the results were validated by experimentally observing steady state flow of inert gas. These findings appear to be a favorable starting point for future investigation of consolidation procedures. The well-connected pore structure with minimum occurrence of dead-end pores suggests that the consolidation agent is very likely to flood the whole pore space.

A Nonlinear Hybrid Diffusion Model for Image Denoising

AbstractIn this paper, a denoising technique is proposed based on a nonlinear anisotropic diffusion model with two diffusivities parameters, i.e., Charbonnier and total variation. These diffusivities are dependent on the diffusivity function for balancing. Furthermore, proof of the existence and uniqueness theorem of the model are presented. The convergent iterative scheme is proposed for the diffusion model. To discretize the diffusion model, the finite difference method with forward‐backward diffusivities is used. The numerical results are given in terms of peak signal‐to‐noise ratio (PSNR) as a metric.

Thermal Biometric Features for Drunk Person Identification Using Multi-Frame Imagery

In this work, multi-frame thermal imagery of the face of a person is employed for drunk identification. Regions with almost constant temperature on the face of sober and drunk persons are thoroughly examined for their capability to discriminate intoxication. Novel image processing approaches as well as feature extraction techniques are developed to support the drunk identification procedure. These techniques constitute novel ideas in the theory of image analysis and algorithm development. Nonlinear anisotropic diffusion is employed for a light smoothing on the images before feature extraction. Feature vector extraction is based on morphological operations performed on the isothermal regions on the face. The classifier chosen to verify the drunk person discrimination capabilities of the procedure is a Support Vector Machine (SVM). Obviously, the isothermal regions on the face change their shape and size with alcohol consumption. Consequently, intoxication identification can be carried out based only on the thermal signatures of the drunk person, while the signature of the corresponding sober person is not needed. A sample of 41 participants who drank in a controlled alcohol consumption procedure was employed for creating the database, which contains 4100 thermal images. The proposed method for intoxication identification achieves a success rate of over 86% and constitutes a fast non-invasive test that can replace existing breathalyzer check up.

On the use of nonlinear anisotropic diffusion filters for seismic imaging using the full waveform

Nonlinear anisotropic diffusion filters have been introduced in the field of image processing for image denoising and image restoration. They are based on the solution of partial differential equations involving a nonlinear anisotropic diffusion operator. From a mathematical point of view, these filters enjoy attractive properties, such as minimum–maximum principle, and an inherent decomposition of the images in different scales. We investigate in this study how these filters can be applied to help solving data-fitting inverse problems. We focus on seismic imaging using the full waveform, a well known nonlinear instance of such inverse problems. In this context, we show how the filters can be applied directly to the solution space, to enhance the structural coherence of the parameters representing the subsurface mechanical properties and accelerate the convergence. We also show how they can be applied to the seismic data itself. In the latter case, the method results in an original low-frequency data enhancement technique making it possible to stabilize the inversion process when started from an initial model away from the basin of attraction of the global minimizer. Numerical results on a 2D realistic synthetic full waveform inversion case study illustrate the interesting properties of both approaches.

A time-splitting local meshfree approach for time-fractional anisotropic diffusion equation: application in image denoising

Image denoising approaches based on partial differential modeling have attracted a lot of attention in image processing due to their high performance. The nonlinear anisotropic diffusion equations, specially Perona–Malik model, are powerful tools that improve the quality of the image by removing noise while preserving details and edges. In this paper, we propose a powerful and accurate local meshless algorithm to solve the time-fractional Perona–Malik model which has an adjustable fractional derivative making the control of the diffusion process more convenient than the classical one. In order to overcome the complexities of the problem, a suitable combination of the compactly supported radial basis function method and operator splitting technique is proposed to convert a complex time-fractional partial differential equation into sparse linear algebraic systems that standard solvers can solve. The numerical results of classical and fractional models are explored in different metrics to demonstrate the proposed scheme’s effectiveness. The numerical experiments confirm that the method is suitable to denoise digital images and show that the fractional derivative increases the model’s ability to remove noise in images.

Mean curvature motion facilitates the segmentation and surface visualization of electron tomograms

Three-dimensional visualization of biological samples is essential for understanding their architecture and function. However, it is often challenging due to the macromolecular crowdedness of the samples and low signal-to-noise ratio of the cryo-electron tomograms. Denoising and segmentation techniques address this challenge by increasing the signal-to-noise ratio and by simplifying the data in images. Here, mean curvature motion is presented as a method that can be applied to segmentation results, created either manually or automatically, to significantly improve both the visual quality and downstream computational handling. Mean curvature motion is a process based on nonlinear anisotropic diffusion that smooths along edges and causes high-curvature features, such as noise, to disappear. In combination with level-set methods for image erosion and dilation, the application of mean curvature motion to electron tomograms and segmentations removes sharp edges or spikes in the visualized surfaces, produces an improved surface quality, and improves overall visualization and interpretation of the three-dimensional images.

An anisotropic diffusion system with nonlinear time-delay structure tensor for image enhancement and segmentation

Anisotropic diffusion model has been established as one of the most significant image processing techniques, because it not only allows for space-variant filtering, but also realizes real direction-specific anisotropic smoothing of image structures. In this paper, an anisotropic nonlinear diffusion system is proposed for image enhancement and segmentation by taking into account the structure tensor constructed from the time-delay regularization and a nonlinear isotropic diffusion equation. Meanwhile, the diffusion system makes full use of the source term which increases the contrast and makes the image be less susceptible to noise, even in high level noise images. The source term also causes the restored image to tend to the binary image, which has proven to be an effective strategy for image segmentation. We analyze the proposed model in the view of the existence and uniqueness of the weak solution by using Galerkin's method. Some other theoretical analysis such as the maximum property is also discussed in the paper. With the appropriate and adaptive parameters, the anisotropic diffusion system achieves a better trade-off between computation time and image processing task. Furthermore, various image experiments illustrate that the proposed algorithm achieves much better efficiency and universality.

Image Texture Analysis and Edge Detection Algorithm Based on Anisotropic Diffusion Equation

This paper uses partial differential equation image processing techniques to establish image texture analysis models based on nonlinear anisotropic diffusion equations for image denoising, image segmentation, and image decomposition. This paper proposes a class of denoising models based on the hybrid anisotropic diffusion equation from the characteristics of different noise types. The model exhibits anisotropic diffusion near the image boundary, which can protect the boundary well, and isotropic diffusion inside the image; so, it can remove the noise effectively. We use the immovable point theory to prove the uniqueness of the model solution and further discuss other properties such as asymptotics of the solution. We propose a class of image texture analysis algorithms based on anisotropic diffusion equations and discrete gray level sets. First, a class of nonconvex generalized functions is proposed to remove the noise from the original image to obtain a smooth image while sharpening the edges. Then, an energy generalization function based on the gray level set is proposed, and the existence of the global minimum of this energy generalization function is discussed. Finally, an equivalent form of this energy generalization is given in the discrete case, and an image texture analysis algorithm is designed based on the equivalent form. The algorithm is improved by initial position optimization, dynamic adjustment of parameters, and adaptive selection of thresholds so that the ants can search along the real edges. Experiments show that the improved algorithm for image edge detection can obtain more complete edges and better detection results. The energy generalization function is calculated directly on the discrete gray level set instead of solving the corresponding partial differential equation, which can avoid the selection of the initial level set and the reinitialization of the level set, thus greatly improving the segmentation efficiency. The new algorithm has a high improvement in segmentation efficiency and can efficiently handle large size complex images.

A Nonlinear Anisotropic Diffusion Equation for Image Restoration with Forward-backward Diffusivities

Background: In the image processing area, deblurring and denoising are the most challenging hurdles. The deblurring image by a spatially invariant kernel is a frequent problem in the field of image processing. Methods: For deblurring and denoising, the total variation (TV norm) and nonlinear anisotropic diffusion models are powerful tools. In this paper, nonlinear anisotropic diffusion models for image denoising and deblurring are proposed. The models are developed in the following manner: first multiplying the magnitude of the gradient in the anisotropic diffusion model, and then apply priori smoothness on the solution image by Gaussian smoothing kernel. Results: The finite difference method is used to discretize anisotropic diffusion models with forward- backward diffusivities. Conclusion: The results of the proposed model are given in terms of the improvement.

Theoretical and numerical analysis for a hybrid tumor model with diffusion depending on vasculature

In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes an anisotropic nonlinear diffusion term with a diffusion velocity increasing with respect to vasculature. First, we prove the existence of global in time weak-strong solutions using a regularization technique via an artificial diffusion in the ODE-system and a fixed point argument. In addition, stability results of the critical points are given under some constraints on parameters. Finally, we design a fully discrete finite element scheme for the model which preserves the pointwise and energy estimates of the continuous problem.

Existence theory for the EED inpainting problem

An existence theory is developed for an elliptic boundary value problem in image analysis known as edge-enhancing diffusion (EED) inpainting. The EED inpainting problem aims at restoration of missing data in an image as the steady state of a nonlinear anisotropic diffusion process where the known data provide Dirichlet boundary conditions. The existence of a weak solution is established by applying the Leray–Schauder fixed point theorem, and it is shown that the set of all possible weak solutions is bounded. Moreover, it is demonstrated that under certain conditions the sequences resulting from iterative application of the operator from the existence theory contain convergent subsequences.