Perona-Malik Model Research Articles

Destriping method for infrared images based on wavelet domain unidirectional variational and improved Perona Malik model

Infrared imagery is widely used in military, industry, and medicine. Stripe noise frequently affects the quality of infrared images. This study introduces a method that combines wavelet domain unidirectional variation (UV) and an enhanced Perona Malik (PM) model to remove stripes. The method involves wavelet decomposition to isolate components, UV optimization for the approximate part, and the improved PM model for the vertical part. By merging the processed components back, stripe-free infrared images are obtained, as confirmed by experimental validation.

Low illumination fog noise image denoising method based on ACE-GPM.

The Perona-Malik (P-M) model exhibits deficiencies such as noise amplification, new noise introduction, and significant gradient effects when processing noisy images. To address these issues, this paper proposes an image-denoising algorithm, ACE-GPM, which integrates an Automatic Color Equalization (ACE) algorithm with a gradient-adjusted P-M model. Initially, the ACE algorithm is employed to enhance the contrast of low-light images obscured by fog and noise. Subsequently, the Otsu method, a technique to find the optimal threshold based on between-class variance, is applied for precise segmentation, enabling more accurate identification of different regions within the image. After that, distinct gradients enhance the image's foreground and background via an enhancement function that accentuates edge and detailed information. The denoising process is finalized by applying the gradient P-M model, employing a gradient descent approach to further emphasize image edges and details. Experimental evidence indicates that the proposed ACE-GPM algorithm not only elevates image contrast and eliminates noise more effectively than other denoising methods but also preserves image details and texture information, evidenced by an average increase of 0.42 in the information entropy value. Moreover, the proposed solution achieves these outcomes with reduced computational resource expenditures while maintaining high image quality.

Enhancing image denoising: A novel non-local anisotropic diffusion framework based on Caputo derivatives and Gaussian convolution for the Perona–Malik model

In the field of image restoration, denoising is considered one of the most important techniques. It is a pre-processing approach aims to refine image clarity and enhance its overall quality by effectively reducing noise present in the image. The aim is to obtain good-quality images from a version degraded by additive noise or convolutional noise that introduces blur. As a result, more advanced treatments can be performed on the resulting image. In order to remove Gaussian noise from input images, we propose the following methodology using a fractional differential equation in time-space based on Gaussian convolution, where the integer and fractional order derivatives of Caputo can be discretized using finite difference and L1−approximations. Once the equation is solved numerically, the scheme is applied to grayscale digital images using the presented algorithm. The parameters must be optimized and adjusted. As a result of testing with natural images, we are able to successfully suppress the noise present in the images. Aside from that. Our model demonstrates strong visual quality, as verified by the calculation of indexes such as Peak Signal-To-Noise Ratio (PSNR) and Structural Similarity Index Measure (SSIM). Our denoising technique demonstrates its effectiveness in mitigating noise present in both MRI and X-ray images, resulting in improved image quality and diagnostic accuracy for these vital medical imaging modalities.

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.

The use of physics-informed neural network approach to image restoration via nonlinear PDE tools

In this work, we propose a physics-informed solution for a blending PDE model with application in image denoising and edge-preserving. The blending model contains isotropic diffusion (ID) model and the modified Perona-Malik model introduced by Catte et al. Then we solve this model applying a physics-informed approach. Solving the new blending model is done using DeepXDE, which is a deep learning library for solving differential equations, by considering the image as input. Error analysis for neural network approximations of the new PDE model is shown. Applying PSNR criteria for some images, it can be seen that the proposed method can compete with solving denoising PDE models using finite difference method (FDM).

Natural digital image mixed noise removal using regularization Perona–Malik model and pulse coupled neural networks

Natural digital image mixed noise removal using regularization Perona–Malik model and pulse coupled neural networks

Regularized Perona & Malik model involving Caputo time-fractional derivative with application to image denoising

This work tackles a novel partial differential equation based on a time-fractional order derivative for image denoising and restoration. We ameliorate the classical Perona–Malik model by considering Caputo time-fractional order derivative with a regularized diffusion. We start by investigating the existence of the solution to the proposed model. As a first result, we use the Faedo–Galerkin approach to prove the existence and uniqueness of a weak solution to an auxiliary fractional problem. Secondly, we establish the existence and uniqueness of a weak solution to our model via Schauder fixed point method. To validate the efficiency of our model, we present some numerical experiments on images with different features and various noise levels. We begin by introducing an adaptive discrete scheme to the proposed model. We illustrate the sensitivity of the time-fractional order derivative with respect to some well-known criteria. Finally, the obtained numerical results show a great efficiency and robustness against noise compared to the competitive models.

Adaptive Perona–Malik model based on dynamical threshold for image multi–noise removal with details preservation

Adaptive Perona–Malik model based on dynamical threshold for image multi–noise removal with details preservation

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.

Rolling Bearing Fault Vibration Signal Denoising Based on Adaptive Morphological Wavelet Perona–Malik Filter Algorithm

This paper proposes an adaptive Perona–Malik filtering algorithm based on the morphological Haar wavelet, which is used for vibration signal denoising in rolling bearing fault diagnosis with strong noise. First, the morphological Haar wavelet operator is utilized to presmooth the noisy signal, and the gradient of the presmooth signal is estimated. Second, considering the uncertainty of gradient at the strong noise point, a strong noise point recognition operator is constructed to adaptively identify the strong noise point. Third, the two-step gradient average value of the strong noise point in the same direction is used to substitute, and the second derivative is introduced into the diffusion coefficient. Finally, diffusion filtering is performed based on the improved Perona–Malik model. The simulation experiment result indicated that not only the algorithm can denoise effectively, but also the average gradient and second derivative in the same direction can effectively suppress the back diffusion of strong noise points to improve the denoising signal-to-noise ratio. The experimental results of rolling bearing show that the algorithm can adaptively filter out strong noise points and keep the information of peak in the signal well, which can improve the accuracy of rolling bearing fault diagnosis.

Comparative Analysis of Multiplicative and Additive Noise Based Automated Regularizations in Non-Linear Diffusion Image Reconstruction

Multiplicative and additive noises are often introduced in image signals during the image acquisition process and result into degradation of image features. The work done by Perona and Malik in 1990 and its modified versions revolutionized the way through which noises or speckles are removed. The Perona-Malik model requires tuning of the regularization parameter to control and prevent staircase artifacts in restored images. The current manual tuning is a challenging and time consuming practice when a long queue of images is registered for processing. Attempt to automate the regularization parameter appeared in Perona-Malik model with self-adjusting shape-defining constant. Although both multiplicative and additive noise based automated regularizations were presented, the paper stayed silent on matters concerning the automation method that fits with speckle reduction. This paper therefore, presents a comparative analysis of additive and multiplicative noise based automated regularizations. Simulation results and paired samples T-tests reveal that the multiplicative noise based automation outperforms the additive noise based automation for small speckle variances. However, the two automation methods do not significantly differ when large speckle variances are assumed.

Perona-Malik Model with Diffusion Coefficient Depending on Fractional Gradient via Caputo-Fabrizio Derivative

The Perona-Malik (PM) model is used successfully in image processing to eliminate noise while preserving edges; however, this model has a major drawback: it tends to make the image look blocky. This work proposes to modify the PM model by introducing the Caputo-Fabrizio fractional gradient inside the diffusivity function. Experiments with natural images show that our model can suppress efficiently the blocky effect. Also, our model has good performance in visual quality, high peak signal-to-noise ratio (PSNR), and lower value of mean absolute error (MAE) and mean square error (MSE).

Shannon-Cosine Wavelet Precise Integration Method for Locust Slice Image Mixed Denoising

A novel denoising method for removing mixed noise from locust slice images is proposed by means of Shannon-cosine wavelet and the nonlinear variational model for the image processing. This method includes two parts that are the sparse representation of the slice images and the novel numerical algorithm for solving the variation model on image denoising based on the sparse representation. In the first part, a parametric Shannon-cosine wavelet function is introduced to construct the multiscale wavelet transform matrix, which is applied to represent the slice images sparsely by adjusting the parameters adaptively based on the texture of the locust slice images. By multiplying the matrix with the signal, the multiscale wavelet transform coefficients of the signal can be obtained at one time, which can be used to identify the salt-and-pepper noises in the slice images. This ensures that the salt-and-pepper noise points are kept away from the sparse representation of the slice images. In the second part, a semianalytical method on solving the system of the nonlinear differential equations is constructed based on the sparse representation of the slice images, which is named the sparse wavelet precise integration method (SWPIM). Substituting the sparse representation of the slice images into the Perona–Malik model which is the famous edge-preserving smoothing model for removing the Gaussian noises of the biomedical images, a system of nonlinear differential equations is obtained, whose scale is far smaller than the one obtained by the difference method. The numerical experiments show that both the values of SSIM and PSNR of the denoised locust slice images are better than the classical methods besides the algorithm efficiency.

Image Denoising with Nonlinear Hybrid Diffusion-Modified Perona-Malik Model (NHD-MPM) and Parameter Optimization by Artificial Bee Colony (ABC) Algorithm

Image Denoising with Nonlinear Hybrid Diffusion-Modified Perona-Malik Model (NHD-MPM) and Parameter Optimization by Artificial Bee Colony (ABC) Algorithm

Mixed Finite Element Approximation for Bivariate Perona–Malik Model Arising in 2D and 3D Image Denoising

In this paper, we present a new model of nonlinear diffusion arising in image processing field based on the Perona–Malik equation which we call the bivariate Perona–Malik model. The aim of this model is to remove image noise while preserving edges, boundaries, and textures. To solve this model we use a new algorithm based on mixed finite element method. In this context we prove mathematically the existence and uniqueness of the solution of the proposed model in a well chosen space. At last, we present the experimental results in 2D and 3D images filtering, which demonstrate the efficiency and effectiveness of our algorithm and finally, we compare it with other well known methods such as the finite difference method presented by Perona et al. (IEEE Trans Pattern Anal Mach Intell 12:629–639, 1990), the finite element method and the finite volume method studied by Handlovicova et al. (J Vis Commun Image Represent 13:217–237, 2002).

Perona–Malik model with self-adjusting shape-defining constant

For decades, the Perona–Malik (PM) diffusion model has been receiving a considerable attention of scholars for its ability to restore detailed scenes. The model, despite its promising results, demands manual tuning of the shape-defining constant—a process that consumes time, prompts for human intervention, and limits flexibility of the model in real-time systems. Most works have tried to address other weaknesses of the PM model (non-convexity and non-monotonicity, which produce chances for instability and multiple solutions), but automating PM remains an open-ended question. In this work, we have introduced a new implementation approach that fully automates the PM model. In particular, the tuning parameters have been conditioned to ensure that the model guarantees convergence and is entirely convex over the scale-space domain. Experiments show that our implementation strategy is flexible, automatic, and achieves convincing results.

An approach to extract edge maps in infrared based breast images using inverse Perona-Malik diffusion filter

Infrared thermography is the adjunctive tool for early diagnosis of breast cancer. The infrared breast images are of low Signal to Noise Ratio (SNR) and amorphous in nature which makes analysis a challenging task. In this work, an attempt is made to extract the edge map from infrared breast images using inverse Perona-Malik (PM) model. This non-linear filter varies the diffusion near the interferences and edges using inverse gradient and new nearest-neighbour scheme. The edge maps are extracted for various gradient thresholds. The statistical features such as average gradient, contrast, entropy and variance are extracted from the edge map to find the optimal gradient threshold. The diffused image of optimal gradient threshold value is validated using SNR. Results show that the statistical features are found to have maximum value for the gradient threshold of 5. It is also observed that SNR obtained from the diffused image is improved by 30 dB compared to raw image. Inverse PM model is found to reduce the noise and enhance the edges. The integration of denoising and edge map extraction would result in accurate segmentation and aid for early diagnosis.

An approach to extract edge maps in infrared based breast images using inverse Perona-Malik diffusion filter

Infrared thermography is the adjunctive tool for early diagnosis of breast cancer. The infrared breast images are of low Signal to Noise Ratio (SNR) and amorphous in nature which makes analysis a challenging task. In this work, an attempt is made to extract the edge map from infrared breast images using inverse Perona-Malik (PM) model. This non-linear filter varies the diffusion near the interferences and edges using inverse gradient and new nearest-neighbour scheme. The edge maps are extracted for various gradient thresholds. The statistical features such as average gradient, contrast, entropy and variance are extracted from the edge map to find the optimal gradient threshold. The diffused image of optimal gradient threshold value is validated using SNR. Results show that the statistical features are found to have maximum value for the gradient threshold of 5. It is also observed that SNR obtained from the diffused image is improved by 30 dB compared to raw image. Inverse PM model is found to reduce the noise and enhance the edges. The integration of denoising and edge map extraction would result in accurate segmentation and aid for early diagnosis.

A Hybrid Model for Image Denoising Combining Modified Isotropic Diffusion Model and Modified Perona-Malik Model

In this paper, a hybrid image denoising algorithm based on directional diffusion is proposed. Specifically, we developed a new noise-removal model by combining the modified isotropic diffusion model and the modified Perona-Malik (PM) model. The novel hybrid model can adapt the diffusion process along the tangential direction of edges in the original image via a new control function based on the patch similarity modulus. In addition, the patch similarity modulus is used as the new structure indicator for the modified Perona-Malik model. The feature of second-order directional derivative of edge's tangential direction allows the proposed model to reduce the aliasing and the noise around edge during edge preserving smoothing. The proposed method is thus able to efficiently preserve the edges, textures, thin lines, weak edges, and fine details, meanwhile preventing the staircase effects. Computer experiments on synthetic image and nature images demonstrate that the proposed model achieves a better performance than the conventional partial differential equations models and some recent advanced models.

An adaptive fourth-order partial differential equation for image denoising

To overcome the weakness of second order methods such as Perona–Malik model for image denoising, various high order models have been proposed and studied. However, there is not too much analysis of these equations to be found in the literature. In this paper, we propose an adaptive fourth-order partial differential equation, which joints a fourth-order term and a second-order term. The model takes advantage of the fourth-order model’s better image avoiding staircase effect and the second-order model’s better edge preserving effect. By introducing a functional framework and k-bounded partial variation (BPVk) space, we prove the existence of a weak solution of the proposed model. Experimental results show that the proposed model can alleviate the staircase effect and preserve edges accurately.