MCMC-Net: accelerating Markov Chain Monte Carlo with neural networks for inverse problems
Dr Sudeb Majee, Anuj Abhishek, Thilo Strauss, and Taufiquar Khan
Source Title: Inverse Problems, Quartile: Q1
View abstract ⏷
In many computational problems, using the Markov Chain Monte Carlo (MCMC) can be prohibitively time-consuming. We propose MCMC-Net, a simple yet efficient way to accelerate MCMC via neural networks. The key idea of our approach is to substitute the true likelihood function of the MCMC method with a neural operator based surrogate. We extensively evaluate the accuracy and speedup of our method on three different partial differential equation-based inverse problems where likelihood computations are computationally expensive, namely electrical impedance tomography, diffuse optical tomography, and quantitative photoacoustic tomography. MCMC-Net performs similar to the classical likelihood counterpart but with a significant speedup. We conjecture that the method can be applied to any problem with a sufficiently expensive likelihood function. We also analyze MCMC-Net in a theoretical setting for the different use cases. We prove a universal approximation theorem-type result to show that the proposed network can approximate the mapping resulting from forward model evaluations to a desired accuracy. Furthermore, we establish convergence of the surrogate posterior to the true posterior under Hellinger distance.
Well-posedness of a variable-exponent telegraph equation applied to image despeckling
Source Title: Evolution Equations and Control Theory, Quartile: Q1
View abstract ⏷
In this paper, we present a telegraph diffusion model with variable exponents for image despeckling. Moving beyond the traditional assumption of a constant exponent in the telegraph diffusion framework, we explore three distinct variable exponents for edge detection. All of these depend on the gray level of the image or its gradient. We rigorously prove the existence and uniqueness of weak solutions of our model in a functional setting and perform numerical experiments to assess how well it can despeckle noisy gray-level images. We consider both a range of natural images contaminated by varying degrees of artificial speckle noise and synthetic aperture radar (SAR) images. We finally compare our method with the nonlocal speckle removal technique and find that our model outperforms the latter at speckle elimination and edge preservation.
On the Existence and Uniqueness of Weak Solutions of a Coupled Diffusion System Related to Image Restoration
Dr Sudeb Majee, Subit K. Jain and Rajendra K. Ray
Source Title: Inverse Problems and Imaging, Quartile: Q2
View abstract ⏷
In this study, the existence and uniqueness of the weak solution of a coupled diffusion system is presented. Since the considered problem is coupled and nonlinear, first we consider a corresponding linearized problem and then use a weak convergence method with Schauder fixed-point theorem to prove the existence of a weak solution of the underlying problem in an appropriate Hilbert space. Moreover, the computational experiments show that the considered model could be applied to image restoration problems.
CDM: A Coupled Deformable Model for Image Segmentation with Speckle Noise and Severe Intensity Inhomogeneity
Dr Sudeb Majee, Ankit Kumar and Subit K. Jain
Source Title: Chaos, Solitons & Fractals, Quartile: Q1
A New Non-Linear Hyperbolic-Parabolic Coupled PDE Model for Image Despeckling
Dr Sudeb Majee, Rajendra K. Ray, and Ananta K. Majee
Source Title: IEEE Transactions on Image Processing, Quartile: Q1
View abstract ⏷
In this work, we propose a non-linear hyperbolic-parabolic coupled Partial Differential Equation (PDE) based model for image despeckling. Here, a separate equation is used to calculate the edge variable, which improves the quality of edge information in the despeckled images. The existence of the weak solution of the present system is achieved via Schauder fixed point theorem. We used a generalized weighted average finite-difference scheme and the Gauss-Seidel iterative technique to solve the coupled system. Numerical studies are reported to show the effectiveness of the proposed approach with respect to standard PDE-based and nonlocal methods available in the literature. Numerical experiments are performed over gray-level images degraded by artificial speckle noise. Additionally, we investigate the noise removal efficiency of the proposed algorithm when applied to real synthetic aperture radar (SAR) and Ultrasound images. Overall, our study confirms that in most cases, the present model performs better than the other PDE-based models and shows competitive performance with the nonlocal technique. To the best of our knowledge, the proposed despeckling approach is the first work that utilizes the advantage of the non-linear coupled hyperbolic-parabolic PDEs for image despeckling.
A Fuzzy Edge Detector Driven Telegraph Total Variation Model for Image Despeckling
Dr Sudeb Majee, Subit K. Jain, Rajendra K. Ray, and Ananta K. Majee
Source Title: Inverse Problems and Imaging, Quartile: Q2
View abstract ⏷
Speckle noise suppression is a challenging and crucial pre-processing stage for higher-level image analysis. In this work, a new attempt has been made using telegraph total variation equation and fuzzy set theory for image despeckling. The intuitionistic fuzzy divergence function has been used to distinguish between edges and noise. To the best of the authors' knowledge, most of the studies on the multiplicative speckle noise removal process focus only on diffusion-based filters, and little attention has been paid to the study of fuzzy set theory. The proposed approach enjoys the benefits of both telegraph total variation equation and fuzzy edge detector, which is robust to noise and preserves image structural details. Moreover, we establish the existence and uniqueness of weak solutions of a regularized version of the present system using the Schauder fixed point theorem. With the proposed technique, despeckling is carried out on natural, real synthetic aperture radar, and real ultrasound images. The experimental results computed by the suggested method are reported, which are found better in terms of noise elimination and detail/edge preservation, concerning the existing approaches.
On the Development of a Coupled Nonlinear Telegraph-Diffusion Model for Image Restoration
Dr Sudeb Majee, Subit K. Jain, Rajendra K. Ray, and Ananta K. Majee
Source Title: Computers & Mathematics with Applications, Quartile: Q1
A Gray Level Indicator-Based Regularized Telegraph Diffusion Model: Application to Image Despeckling
Dr Sudeb Majee, Rajendra K. Ray, and Ananta K. Majee
Source Title: SIAM Journal on Imaging Sciences, Quartile: Q1
View abstract ⏷
In this work, a gray level indicator-based nonlinear telegraph diffusion model is presented for image despeckling. Most of the researchers focus only on diffusion equation-based filter for multiplicative noise removal process. The proposed technique uses the benefit of the combined effect of diffusion equation as well as the wave equation. The wave nature of the system preserves the high oscillatory and texture patterns in an image. In this model, the diffusion coefficient depends not only on the image gradient but also on the gray level of the image, which controls the diffusion process better than only gradient-based diffusion approaches. Moreover, we prove the well-posedness of the present system using the Schauder fixed point theorem. Furthermore, we show the superiority of the proposed method over three recently developed methods on a set of gray level test images corrupted by speckle noise and check the noise removal capability of the present technique over some real SAR images corrupted by speckle noise with different noise levels.
