Simultaneous denoising and enhancement of signals by a fractal conservation law

In this paper, a new filtering method is presented for simultaneous noise reduction and enhancement of signals using a fractal scalar conservation law which is simply the forward heat equation modified by a fractional anti-diffusive term of lower order. This kind of equation has been first introduced by physicists to describe morphodynamics of sand dunes. To evaluate the performance of this new filter, we perform a number of numerical tests on various signals. Numerical simulations are based on finite difference schemes or Fast and Fourier Transform. We used two well-known measuring metrics in signal processing for the comparison. The results indicate that the proposed method outperforms the well-known Savitzky-Golay filter in signal denoising. Interesting multi-scale properties w.r.t. signal frequencies are exhibited allowing to control both denoising and contrast enhancement

Efficient Denoising Of High Resolution Color Digital Images Utilizing Krylov Subspace Spectral Methods

The solution to a parabolic nonlinear diffusion equation using a Krylov Subspace Spectral method is applied to high resolution color digital images with parallel processing for efficient denoising. The evolution of digital image technology, processing power, and numerical methods must evolve to increase efficiency in order to meet current usage requirements. Much work has been done to perfect the edge detector in Perona-Malik equation variants, while minimizing the effects of artifacts. It is demonstrated that this implementation of a regularized partial differential equation model controls backward diffusion, achieves strong denoising, and minimizes blurring and other ancillary effects. By adaptively tuning edge detector parameters so as to not require human interaction, we propose to automatically adapt the parameters to specific images. It is anticipated that with KSS methods, in conjunction with efficient block processing, we will set new standards for efficiency and automation

Bilevel optimization, deep learning and fractional Laplacian regularization with applications in tomography

The article of record as published may be located at https://doi.org/10.1088/1361-6420/ab80d7Funded by Naval Postgraduate SchoolIn this work we consider a generalized bilevel optimization framework for solv- ing inverse problems. We introduce fractional Laplacian as a regularizer to improve the reconstruction quality, and compare it with the total variation regularization. We emphasize that the key advantage of using fractional Laplacian as a regularizer is that it leads to a linear operator, as opposed to the total varia- tion regularization which results in a nonlinear degenerate operator. Inspired by residual neural networks, to learn the optimal strength of regularization and the exponent of fractional Laplacian, we develop a dedicated bilevel opti- mization neural network with a variable depth for a general regularized inverse problem. We illustrate how to incorporate various regularizer choices into our proposed network. As an example, we consider tomographic reconstruction as a model problem and show an improvement in reconstruction quality, especially for limited data, via fractional Laplacian regularization. We successfully learn the regularization strength and the fractional exponent via our proposed bilevel optimization neural network. We observe that the fractional Laplacian regular- ization outperforms total variation regularization. This is specially encouraging, and important, in the case of limited and noisy data.The first and third authors are partially supported by NSF grants DMS-1818772, DMS-1913004, the Air Force Office of Scientific Research under Award No.: FA9550-19-1-0036, and the Department of Navy, Naval PostGraduate School under Award No.: N00244-20-1-0005. The third author is also partially supported by a Provost award at George Mason University under the Industrial Immersion Program. The second author is partially supported by DOE Office of Science under Contract No. DE-AC02-06CH11357.The first and third authors are partially supported by NSF grants DMS-1818772, DMS-1913004, the Air Force Office of Scientific Research under Award No.: FA9550-19-1-0036, and the Department of Navy, Naval PostGraduate School under Award No.: N00244-20-1-0005. The third author is also partially supported by a Provost award at George Mason University under the Industrial Immersion Program. The second author is partially supported by DOE Office of Science under Contract No. DE-AC02-06CH11357

Tomographic inversion using ℓ1\ell_1-norm regularization of wavelet coefficients

We propose the use of $\ell_1$ regularization in a wavelet basis for the solution of linearized seismic tomography problems $Am=d$, allowing for the possibility of sharp discontinuities superimposed on a smoothly varying background. An iterative method is used to find a sparse solution $m$ that contains no more fine-scale structure than is necessary to fit the data $d$ to within its assigned errors.Comment: 19 pages, 14 figures. Submitted to GJI July 2006. This preprint does not use GJI style files (which gives wrong received/accepted dates). Corrected typ

Geometric Approaches for 3D Shape Denoising and Retrieval

A key issue in developing an accurate 3D shape recognition system is to design an efficient shape descriptor for which an index can be built, and similarity queries can be answered efficiently. While the overwhelming majority of prior work on 3D shape analysis has concentrated primarily on rigid shape retrieval, many real objects such as articulated motions of humans are nonrigid and hence can exhibit a variety of poses and deformations. Motivated by the recent surge of interest in content-based analysis of 3D objects in computeraided design and multimedia computing, we develop in this thesis a unified theoretical and computational framework for 3D shape denoising and retrieval by incorporating insights gained from algebraic graph theory and spectral geometry. We first present a regularized kernel diffusion for 3D shape denoising by solving partial differential equations in the weighted graph-theoretic framework. Then, we introduce a computationally fast approach for surface denoising using the vertexcentered finite volume method coupled with the mesh covariance fractional anisotropy. Additionally, we propose a spectral-geometric shape skeleton for 3D object recognition based on the second eigenfunction of the Laplace-Beltrami operator in a bid to capture the global and local geometry of 3D shapes. To further enhance the 3D shape retrieval accuracy, we introduce a graph matching approach by assigning geometric features to each endpoint of the shape skeleton. Extensive experiments are carried out on two 3D shape benchmarks to assess the performance of the proposed shape retrieval framework in comparison with state-of-the-art methods. The experimental results show that the proposed shape descriptor delivers best-in-class shape retrieval performance

スペクトルの線形性を考慮したハイパースペクトラル画像のノイズ除去とアンミキシングに関する研究

This study aims to generalize color line to M-dimensional spectral line feature (M>3) and introduce methods for denoising and unmixing of hyperspectral images based on the spectral linearity.For denoising, we propose a local spectral component decomposition method based on the spectral line. We first calculate the spectral line of an M-channel image, then using the line, we decompose the image into three components: a single M-channel image and two gray-scale images. By virtue of the decomposition, the noise is concentrated on the two images, thus the algorithm needs to denoise only two grayscale images, regardless of the number of channels. For unmixing, we propose an algorithm that exploits the low-rank local abundance by applying the unclear norm to the abundance matrix for local regions of spatial and abundance domains. In optimization problem, the local abundance regularizer is collaborated with the L2, 1 norm and the total variation.北九州市立大