PDE-based Technique Research Articles

New Approach for Multimodal Medical Image Fusion using PDE-based Technique

Accurate diagnosis of diseases is essential for effective medical treatment, and medical image fusion plays a crucial role in achieving this goal. This article proposes a novel technique for multimodal medical image fusion based on fourth order PDE-based techniques. The proposed technique uses inputted CT and brain MRI images, and applies a PDE-based technique for noise reduction and edge preservation. Subsequently, a computed fusion rule using principal component analysis is applied to generate a fused image of the brain that presents clear and essential components for medical diagnosis and analysis. The article provides a comparative study of the proposed technique with other existing techniques, demonstrating its optimal performance. The proposed technique can significantly enhance medical image diagnosis and analysis for accurate disease detection and treatment.

Efficient exposure computation by risk factor decomposition

The focus of this paper is the efficient computation of counterparty credit risk exposure on portfolio level. Here, the large number of risk factors rules out traditional PDE-based techniques and allows only a relatively small number of paths for nested Monte Carlo simulations, resulting in large variances of estimators in practice. We propose a novel approach based on Kolmogorov forward and backward PDEs, where we counter the high dimensionality by a generalization of anchored-ANOVA decompositions. By computing only the most significant terms in the decomposition, the dimensionality is reduced effectively, such that a significant computational speed-up arises from the high accuracy of PDE schemes in low dimensions compared to Monte Carlo estimation. Moreover, we show how this truncated decomposition can be used as control variate for the full high-dimensional model, such that any approximation errors can be corrected while a substantial variance reduction is achieved compared to the standard simulation approach. We investigate the accuracy for a realistic portfolio of exchange options, interest rate and cross-currency swaps under a fully calibrated 10-factor model.

A mesh-free algorithm for ROF model

The total variation (TV) denoising method is a PDE-based technique that preserves the edges well but has undesirable staircase effect in some cases, namely, the translation of smooth regions (ramps) into piecewise constant regions (stairs). This paper introduces a novel mesh-free approach using TV (ROF model) regularization and radial basis function (RBF) for the numerical approximation of TV-based model to remove the additive noise from the measurements. This approach is structured on local collocation and multiquadric radial basis function. These features enable this strategy not only to eliminate noise from images and preserve the edges but also has the advantage to minimize the staircase effect substantially from real and artificial images which cause the image to look blocky. Experimental results demonstrate that the proposed mesh-free approach is robust and performs well in visual improvement as well as peak signal-to-noise ratio compared with the recent partial differential equation (PDE)-based traditional methods.

Lyapunov exponents and adaptive mesh refinement for high-speed flows using a discontinuous Galerkin scheme

This paper proposes two important improvements to shock-capturing strategies using a discontinuous Galerkin scheme, namely, accurate shock identification via finite-time Lyapunov exponent (FTLE) operators and efficient shock treatment through a point-implicit discretization of a PDE-based artificial viscosity technique. The advocated approach is based on the FTLE operator, originally developed in the context of dynamical systems theory to identify certain types of coherent structures in a flow. We propose the application of FTLEs in the detection of shock waves and demonstrate the operator's ability to identify strong and weak shocks equally well. The detection algorithm is coupled with a mesh refinement procedure and applied to transonic and supersonic flows. While the proposed strategy can be used potentially with any numerical method, a high-order discontinuous Galerkin solver is used in this study. In this context, two artificial viscosity approaches are employed to regularize the solution near shocks: an element-wise constant viscosity technique and a PDE-based smooth viscosity model. As the latter approach is more sophisticated and preferable for complex problems, a point-implicit discretization in time is proposed to reduce the extra stiffness introduced by the PDE-based technique, making it more competitive in terms of computational cost.

Improved Seamless Cloning based Image In painting

Image in-painting is a restoration of image parts which are missing either accidentally or intentionally, and also image recovery is a research hot spot in the computer graphics and computer vision in the current. Two major inpainting techniques are introduced, those are PDE-based and Texture based algorithms. In PDE-based technique a conventional image matching techniques may be classified as either area based or feature based methods. In this paper, taking an input image and apply Pre-processing, After we get gray image and contour extraction algorithm is used to extract the contour features of damaged area, which determine the contour feature location. After that Matching algorithms play a key role in deciding correspondences between two image scenes. Area-based matching algorithm is used to search alternative matching area .Contour-based Similarity Distance function (CSD) is used to determine the most similar matching area. According to the damaged area and the matching area, the source cloning domains and target cloning domains of Seamless Cloning algorithm is improved. Using mean value coordinates to achieve regional pixel cloning from the source cloning domain to the target cloning domain. Finally, the improved SC algorithm is used to repair the image. Experimental results confirm the effectiveness of the proposed algorithm, and that the method can not only repair small-scale scratches, but also repair a large area damaged areas.

Comparison of PDE based and other techniques for speckle reduction from digitally reconstructed holographic images

In this paper, the partial differential equation (PDE) based homomorphic filtering technique is proposed for speckle reduction from digitally reconstructed holographic images based on the concepts of complex diffusion processes. For digital implementations, the proposed scheme was discretized using finite differences scheme. Further, the performance of the proposed PDE-based technique is compared with other speckle reduction techniques such as homomorphic anisotropic diffusion filter based on extended concept of Perona and Malik (1990) [2], homomorphic Weiner filter, Lee filter, Frost filter, Kuan filter, speckle reducing anisotropic diffusion (SRAD) filter and hybrid filter in the context of digital holography. For the comparison of various speckle reduction techniques, the performance is evaluated quantitatively in terms of all possible parameters that justify the applicability of a scheme for a specific application. The chosen parameters are mean-square-error (MSE), normalized mean-square-error (NMSE), peak signal-to-noise ratio (PSNR), speckle index, average signal-to-noise ratio (SNR), effective number of looks (ENL), correlation parameter (CP), mean structure similarity index map (MSSIM) and execution time in seconds. For experimentation and computer simulation MATLAB 7.0 has been used and the performance is evaluated and tested for various sample holographic images for varying amount of speckle variance. The results obtained justify the applicability of proposed schemes.

A Wavelet-Laplace Variational Technique for Image Deconvolution and Inpainting

We construct a new variational method for blind deconvolution of images and inpainting, motivated by recent PDE-based techniques involving the Ginzburg-Landau functional, but using more localized wavelet-based methods. We present results for both binary and grayscale images. Comparable speeds are achieved with better sharpness of edges in the reconstruction.

Dynamic PDE-based surface design using geometric and physical constraints

PDE surfaces, which are defined as solutions of partial differential equations (PDEs), offer many modeling advantages in surface blending, free-form surface modeling, and specifying surface’s aesthetic or functional requirements. Despite the earlier advances of PDE surfaces, previous PDE-based techniques exhibit certain difficulties such as lack of interactive sculpting capabilities and restrained topological structure of modeled objects. This paper presents an integrated approach that can incorporate PDE surfaces into the powerful physics-based modeling framework, to realize the full potential of PDE methodology. We have developed a prototype system that allows interactive design of flexible topological surfaces as PDE surfaces and displacements using generalized boundary conditions as well as a variety of geometric and physical constraints, hence supporting various interactive techniques beyond the conventional boundary control. The system offers a set of sculpting toolkits that allow users to interactively modify arbitrary points, curve spans, and/or regions of interest across the entire PDE surfaces and displacements in an intuitive and physically meaningful way. To achieve real-time performance, we employ several simple, yet efficient numerical techniques, including the finite-difference discretization, the multigrid-like subdivision, and the mass-spring approximation of elastic PDE surfaces and displacements. In addition, we present the standard bivariant B-spline finite element approximations of dynamic PDEs, which can subsequently be sculpted and deformed directly in real-time subject to the intrinsic PDE constraints. Our experiments demonstrate many attractive advantages of the physics-based PDE formulation such as intuitive control, real-time feedback, and usability to both professional and common users.

Singular Integrals, Image Smoothness, and the Recovery of Texture in Image Deblurring

Total variation (TV) image deblurring is a PDE-based technique that preserves edges, but often eliminates vital small-scale information, or {\em texture}. This phenomenon reflects the fact that most natural images are not of bounded variation. The present paper reconsiders the image deblurring problem in Lipschitz spaces $\Lambda(\alpha, p, q)$, wherein a wide class of nonsmooth images can be accommodated. A new and fast FFT-based deblurring method is developed that can recover texture in cases where TV deblurring fails completely. Singular integrals, such as the Poisson kernel, are used to create an effective new image analysis tool that can calibrate the lack of smoothness in an image. It is found that a rich class of images $\in \Lambda(\alpha, 1, \infty) \cap \Lambda(\beta, 2, \infty)$, with $0.2 < \alpha, \beta < 0.7$. The Poisson kernel is then used to regularize the deblurring problem by appropriately constraining its solutions in $\Lambda(\alpha, 2, \infty)$ spaces, leading to new L2 error bounds that substantially improve on the Tikhonov--Miller method. This so-called Poisson Singular Integral or PSI method is only one of an infinite variety of singular integral deblurring methods that can be constructed. The method is found to be well-behaved in both the L1 and L2 norms, producing results closely matching those obtained in the theoretically optimal, but practically unrealizable, case of true Wiener filtering. Deblurring experiments on synthetically defocused images illustrate the PSI method's very significant improvements over both the total variation and Tikhonov--Miller methods. In addition, successful reconstructions with inexact prior Lipschitz space information, highlight the robustness and practicality of the PSI method.

High-Order Total Variation-Based Image Restoration

The total variation (TV) denoising method is a PDE-based technique that preserves edges well but has the sometimes undesirable staircase effect, namely, the transformation of smooth regions ( ramps) into piecewise constant regions ( stairs). In this paper we present an improved model, constructed by adding a nonlinear fourth order diffusive term to the Euler--Lagrange equations of the variational TV model. Our technique substantially reduces the staircase effect, while preserving sharp jump discontinuities (edges). We show numerical evidence of the power of resolution of this novel model with respect to the TV model in some 1D and 2D numerical examples.

An efficient algorithm for a visibility-based surveillance-evasion game

Abstract : We present an algorithm which computes the value function and optimal paths for a two-player static game, where the goal of one player is to maintain visibility of an adversarial player for as long as possible, and that of the adversarial player is to minimize this time. In a static game both players choose their controls at initial time and run in open-loop for t 0 until the end-game condition is met. Compared to closed-loop (feedback strategy) games which require solving a PDE in high dimensions, we demonstrate that the static game can be solved directly in the state space by the proposed PDE-based technique. This results in significant savings in both memory and computational cost, at the expense of a simpler information pattern that is more conservative towards one player. In addition, we describe how this algorithm can be easily generalized to games with multiple evaders. Applications to target tracking and an extension to a feedback control game are also presented.