Perona-Malik Model Research Articles

An Extended Perona–Malik Model Based on Probabilistic Models

The Perona–Malik model has been very successful at restoring images from noisy input. In this paper, we reinterpret the Perona–Malik model in the language of Gaussian scale mixtures and derive some extensions of the model. Specifically, we show that the expectation–maximization (EM) algorithm applied to Gaussian scale mixtures leads to the lagged-diffusivity algorithm for computing stationary points of the Perona–Malik diffusion equations. Moreover, we show how mean field approximations to these Gaussian scale mixtures lead to a modification of the lagged-diffusivity algorithm that better captures the uncertainties in the restoration. Since this modification can be hard to compute in practice, we propose relaxations to the mean field objective to make the algorithm computationally feasible. Our numerical experiments show that this modified lagged-diffusivity algorithm often performs better at restoring textured areas and fuzzy edges than the unmodified algorithm. As a second application of the Gaussian scale mixture framework, we show how an efficient sampling procedure can be obtained for the probabilistic model, making the computation of the conditional mean and other expectations algorithmically feasible. Again, the resulting algorithm has a strong resemblance to the lagged-diffusivity algorithm. Finally, we show that a probabilistic version of the Mumford–Shah segmentation model can be obtained in the same framework with a discrete edge prior.

A splitting algorithm for a novel regularization of Perona-Malik and application to image restoration

In this paper, we focus on a numerical method of a problem called the Perona-Malik inequality which we use for image denoising. This model is obtained as the limit of the Perona-Malik model and the p-Laplacian operator with p→∞. In Atlas et al., (Nonlinear Anal. Real World Appl 18:57–68, 2014), the authors have proved the existence and uniqueness of the solution of the proposed model. However, in their work, they used the explicit numerical scheme for approximated problem which is strongly dependent to the parameter p. To overcome this, we use in this work an efficient algorithm which is a combination of the classical additive operator splitting and a nonlinear relaxation algorithm. At last, we have presented the experimental results in image filtering show, which demonstrate the efficiency and effectiveness of our algorithm and finally, we have compared it with the previous scheme presented in Atlas et al., (Nonlinear Anal. Real World Appl 18:57–68, 2014).

On Lipschitz solutions for some forward–backward parabolic equations

We investigate the existence and properties of Lipschitz solutions for some forward–backward parabolic equations in all dimensions. Our main approach to existence is motivated by reformulating such equations into partial differential inclusions and relies on a Baire's category method. In this way, the existence of infinitely many Lipschitz solutions to certain initial-boundary value problem of those equations is guaranteed under a pivotal density condition. Under this framework, we study two important cases of forward–backward anisotropic diffusion in which the density condition can be realized and therefore the existence results follow together with micro-oscillatory behavior of solutions. The first case is a generalization of the Perona–Malik model in image processing and the other that of Höllig's model related to the Clausius–Duhem inequality in the second law of thermodynamics.

Perona–Malik Model with a New Diffusion Coefficient for Image Denoising

This paper presents a new diffusion coefficient which is based on second order derivative and local entropy information for image denoising. In the proposed model, a second order derivative term is introduced, which reduces the staircasing effect and preserves edge in a processed image. The local entropy information can preserve texture. The Perona–Malik model with a new diffusion coefficient improves the denoised effects, and prevents edges from being over-smoothed. Comparative experiments show that the proposed model obtains more satisfied results than the other two existing models.

Improved anisotropic diffusion equation based on new non‐local information scheme for image denoising

This study presents a new Perona–Malik (PM) model which is based on new non‐local means (NLM) theory for an image denoising. In the proposed model, the intensity, space position and gradient information are introduced into NLM model, which weakens the staircasing effect and preserves edge in a processed image. The new PM model enhances the denoising capability, and decreases fine characteristics to be over‐smoothed. Comparative experiments show that the denoising capability of the proposed model is superior to the other three existing models.

Adaptive cell-centered finite volume method for diffusion equations on a consistent quadtree grid

Models applied in image processing are often described by nonlinear PDEs in which a good approximation of gradient plays an important role especially in such cases where irregular finite volume grids are used. In image processing, such a situation can occur during a coarsening based on quadtree grids. We present a construction of a deformed quadtree grid in which the connection of representative points of two adjacent finite volumes is perpendicular to their common boundary enabling us to apply the classical finite volume methods. On the other hand, for such an adjusted grid, the intersection of representative points connection with a finite volume boundary is not a middle point of their common edge and standard methods cannot achieve a good accuracy. In this paper we present a new cell-centered finite volume method to evaluate solution gradients, which results into a solution of a simple linear algebraic system and we prove its unique solvability. Finally we present numerical experiments for the regularized Perona-Malik model in which we applied this new method.

Image Denoising Using Variations of Perona-Malik Model with Different Edge Stopping Functions

Abstract Anisotropic diffusion is used for both image enhancement and denoising. The Perona-Malik model makes use of anisotropic diffusion to filter out the noise. In Perona-Malik model the rate of diffusion is controlled by edge stopping function. The drawback of Perona-Malik model is that the sharp edges and fine details are not preserved well in the denoised image. But the sharp edges and fine details can be preserved well using appropriate edge stopping function. We have analysed the effect of different edge stopping functions in anisotropic diffusion in terms of how efficient they are in preserving edges. We have found that an edge stopping function which stops diffusion from low image gradient onwards well preserves the sharp edges and fine details. This property of an edge stopping function will also result in lower evolution in case of level set methods. But an edge stopping function which stops diffusion from high image gradient onwards will not preserve sharp edges and fine details, since they are blurred due to diffusion. We have also found that low values of gradient threshold parameter used in edge stopping function well preserves the sharp edges and fine details than high values of threshold parameter. By utilizing an edge stopping function which stops diffusion from low image gradient onwards or which has zero or insignificant value at low image gradient, we can well preserve the sharp edges and fine details in the denoised image.

Image smoothing model based on the combination of the gradient and curvature

In image processing, in order to keep the detailed information about image edge, we propose a curvature smoothing model based on the nature of diffusion coefficient and curvature. Considering the fact that the curvature will change significantly when the image is affected by noise pollution, in this article we will continue to take the level set curvature as a detection factor and substitute it into the model, then we present a new model which combines gradient and curvature. Analysis and simulation indicate that the new model can keep more image information than the Perona-Malik model, and it can strengthen the sharp edge of the image efficiently, and well keep the straight lines of image, and edges, corners, slopes and small-scale features of curve at the same time, so this model is an ideal model.

Image denoising model based on the improved Demons algorithm

According to image registration, we build a new Demons model of image denoising, in which the diffusion access is regarded as image registration. The experimental results indicate that the performance of the model is better than that of the Perona-Malik model: the ill-condition of the model is removed. It is not enough to describe local characteristics only by using the gradient information in the access of image denoising, so a level set curvature which is the driving force of image structure controlling is introduced into the denoising model. Therefore we propose a new model of image denoising based on two driving forces of gradient and curvature. The simulation results show that the two improved models can both suppress noise effectively, their definitions are enhanced obviously, the performance of image denoising model of two driving forces is more greatly improved.

Modified Perona-Malik Equation and Computer Simulation for Image Denoising

The edges and textures of a digital image may be destroyed by traditional denoising methods, which is a difficult problem in image denoising. In this paper, anisotropic diffusion algorithm based on Partial differential equation is studied. First, image denoising algorithms based on Perona-Malik model are studied. Second, a modified Perona-Malik model is proposed. In the proposed model, the gradient statistic and edge thresholds are embedded into the Perona-Malik equation. Finally, the effects of this model and some other models are compared and analyzed. The experimental results show that the proposed modified Perona-Malik model outperforms the original Perona-Malik model in removing Gaussian noise, and the edges and textures of the image are well preserved.

Adaptive Fourth Order Partial Differential Equation Filter from the Webers Total Variation for Image Restoration

By substituting an anisotropic diffusion operator for the isotropic Laplace operator in the directed diffusion equation, an improvd directed diffusion equation model is proposed. To overcome the staircasing effects and simultaneously avoid edge blurring, this paper proposed an adaptive fourth order partial differential equation from the Webers Total Variation for Image Restoration. This functional is not only to use Laplace operator but also to add the human psychology system, this paper show numerical evidence of the power of resolution of the model with respect to other known models as the Perona-Malik model. Compared results disctincly demonstrate the superiority of our proposed scheme , in terms of removing noise while sharply maintaining the edge features.

Non-local means theory based Perona–Malik model for image denosing

Among various kinds of image denoising methods, the Perona–Malik model is a representative Partial Differential Equation based (PDE-based) algorithm which effectively removes the noise as well as having edge enhancement simultaneously through anisotropic diffusion controlled by the diffusion coefficient. However, the unstable behavior of the Perona–Malik model introduces staircasing artifacts in the processed images. To realize less diffusion in the texture region and to get more smooth in flat region while implementing image denoising, we propose an improved Perona–Malik model based on non-local means theory, which assumes that the image contains an extensive amount of self-similarity and uses the similarity between the region around the center pixel and the region outside the center pixel to give a more reasonable description of the image. The improved algorithm is applied on numerical simulation and practical images, and the quantitative analyzing results prove that the modified anisotropic diffusion model can preserve textures effectively while ruling out the noise, meanwhile, the staircasing effects are decreased obviously.

Image denoising using modified Perona–Malik model based on directional Laplacian

The Perona–Malik (PM) model, or the anisotropic diffusion, shows good performance for image restoration. However, it suffers from the so-called staircasing effect if the contrast parameter is small and otherwise cannot preserve edges. This paper proposes a modified Perona–Malik (MPM) model based on directional Laplacian, which diffuses image along the edge direction of the original image. A slightly weighted Laplacian is also integrated to suppress noise. The proposed model can alleviate the staircasing effect, preserve sharp discontinuities, and remove noise simultaneously. Experimental results compared with several relevant methods demonstrate the good performance of the proposed algorithm.

HPM-Based Dynamic Wavelet Transform and Its Application in Image Denoising

Wavelet-based multiscale interpolation operator is often employed to construct the adaptive numerical method for PDEs, in which the computational complexity of the wavelet transform is one of the main factors affecting the algorithm efficiency. As the wavelet transform just acts as the detector of the characteristic points in the interpolation operator, the multiscale wavelet interpolation operator can be viewed as a nonlinear problem. Based on this assumption, we construct an approximate dynamic interpolation operator with the homotopy perturbation method (HPM), which decreases the computational complexity of the wavelet transform appearing in the wavelet interpolation operator fromO((1/3)42J−1)toO(4J), whereJis the amount of the wavelet scales. Then an adaptive algorithm solving the Perona-Malik model on image denoising is constructed with the HPM-based interpolation operator. Last, the quasi-Shannon wavelet is employed to design the experiments on the medical image and some artificial images denoising. The experiment results show that the simplified wavelet interpolation operator based on HPM possesses the adaptability and nonsensitivity to the time step, which is helpful to improve the algorithm efficiency. This illustrates that the HPM-based wavelet interpolation operator is an effective tool to solve the problems in image processing.

Nonlinear diffusion filtering of data on the Earth’s surface

The paper deals with data filtering on closed surfaces using linear and nonlinear diffusion equations. We define a surface finite-volume method to approximate numerically parabolic partial differential equations on closed surfaces, namely on a sphere, ellipsoid or the Earth’s surface. The closed surface as a computational domain is approximated by a polyhedral surface created by planar triangles and we construct a dual co-volume grid. On the co-volumes we define a weak formulation of the problem by applying Green’s theorem to the Laplace–Beltrami operator. Then the finite-volume method is applied to discretize the weak formulation. Weak forms of elliptic operators are expressed through surface gradients. In our numerical scheme we use a piece-wise linear approximation of a solution in space and the backward Euler time discretization. Furthermore, we extend a linear diffusion on surface to the regularized surface Perona–Malik model. It represents a nonlinear diffusion equation, which at the same time reduces noise and preserves main edges and other details important for a correct interpretation of the real data. We present four numerical experiments. The first one has an illustrative character showing how an additive noise is filtered out from an artificial function defined on a sphere. Other three examples deal with the real geodetic data on the Earth’s surface, namely (i) we reduce a stripping noise from the GOCE satellite only geopotential model up to degree 240, (ii) we filter noise from the real GOCE measurements (the component $$T_{zz})$$ , and (iii) we reduce a stripping noise from the satellite only mean dynamic topography at oceans. In all experiments we focus on a comparison of the results obtained by both the linear and nonlinear models presenting advantages of the nonlinear diffusion.

A Semi-Lagrangian approximation for the AMSS model of image processing

Among the various nonlinear image filtering techniques, the Affine Morphological Scale Space (AMSS) model is characterized by especially appealing theoretical properties. In this paper we propose a Semi-Lagrangian scheme to approximate the AMSS model, prove convergence of its monotone version and apply it to some typical problem of image filtering. We also carry out a comparison with the Level Lines Affine Shortening implementation of the AMSS model, as well as with other nonlinear filtering techniques, namely the Mean Curvature equation and the Perona–Malik model.

Adaptive Perona–Malik Model Based on the Variable Exponent for Image Denoising

This paper introduces a class of adaptive Perona-Malik (PM) diffusion, which combines the PM equation with the heat equation. The PM equation provides a potential algorithm for image segmentation, noise removal, edge detection, and image enhancement. However, the defect of traditional PM model is tending to cause the staircase effect and create new features in the processed image. Utilizing the edge indicator as a variable exponent, we can adaptively control the diffusion mode, which alternates between PM diffusion and Gaussian smoothing in accordance with the image feature. Computer experiments indicate that the present algorithm is very efficient for edge detection and noise removal.

Well-posedness for a class of fourth order diffusions for image processing

A number of image denoising models based on higher order parabolic partial differential equations (PDEs) have been proposed in an effort to overcome some of the problems attendant to second order methods such as the famous Perona–Malik model. However, there is little analysis of these equations to be found in the literature. In this paper, methods of maximal regularity are used to prove the existence of unique local solutions to a class of fourth order PDEs for noise removal. The proof is laid out explicitly for two newly proposed fourth order models, and an outline is given for how to apply the techniques to other proposed models.

3D early embryogenesis image filtering by nonlinear partial differential equations

We present nonlinear diffusion equations, numerical schemes to solve them and their application for filtering 3D images obtained from laser scanning microscopy (LSM) of living zebrafish embryos, with a goal to identify the optimal filtering method and its parameters. In the large scale applications dealing with analysis of 3D+time embryogenesis images, an important objective is a correct detection of the number and position of cell nuclei yielding the spatio-temporal cell lineage tree of embryogenesis. The filtering is the first and necessary step of the image analysis chain and must lead to correct results, removing the noise, sharpening the nuclei edges and correcting the acquisition errors related to spuriously connected subregions. In this paper we study such properties for the regularized Perona-Malik model and for the generalized mean curvature flow equations in the level-set formulation. A comparison with other nonlinear diffusion filters, like tensor anisotropic diffusion and Beltrami flow, is also included. All numerical schemes are based on the same discretization principles, i.e. finite volume method in space and semi-implicit scheme in time, for solving nonlinear partial differential equations. These numerical schemes are unconditionally stable, fast and naturally parallelizable. The filtering results are evaluated and compared first using the Mean Hausdorff distance between a gold standard and different isosurfaces of original and filtered data. Then, the number of isosurface connected components in a region of interest (ROI) detected in original and after the filtering is compared with the corresponding correct number of nuclei in the gold standard. Such analysis proves the robustness and reliability of the edge preserving nonlinear diffusion filtering for this type of data and lead to finding the optimal filtering parameters for the studied models and numerical schemes. Further comparisons consist in ability of splitting the very close objects which are artificially connected due to acquisition error intrinsically linked to physics of LSM. In all studied aspects it turned out that the nonlinear diffusion filter which is called geodesic mean curvature flow (GMCF) has the best performance.

Ramp preserving Perona–Malik model

The Perona–Malik (PM) model, a classical anisotropic diffusion, can preserve edges while removing the noise. However, the defect of traditional PM model is tending to contain the staircase effect in the processed image. For this reason, a novel ramp preserving Perona–Malik (RPPM) model based on a new edge indicator is presented. In the RPPM model, the diffusion coefficient of the PM model is adaptively determined to introduce following property: isotropic diffusion in flat and ramp regions to prevent the staircase effect and anisotropic diffusion in edge regions to preserve edges. Comparative results on synthetic and real image denoising demonstrate that our model can preserve important structures, such as edges and ramps, while avoiding the speckles and the complex numerical implementation arising from high-order partial differential equations (PDEs).