An accurate and efficient multiscale finite-difference frequency-domain method for the scalar Helmholtz equation

Frequency-domain numerical modeling of the seismic wave equation can readily describe frequency-dependent seismic wave behaviors, yet is computationally challenging to perform in finely discretized or large-scale geological models. Conventional finite-difference frequency-domain (FDFD) methods for solving the Helmholtz equation usually lead to large linear systems that are difficult to solve with a direct or iterative solver. Parallel strategies and hybrid solvers can partially alleviate the computational burden by improving the performance of the linear system solver. We develop a novel multiscale FDFD method to eventually construct a dimension-reduced linear system from the scalar Helmholtz equation based on the general framework of heterogeneous multiscale method (HMM). The methodology associated with multiscale basis functions in the multiscale finite-element method (MsFEM) is applied to the local microscale problems of this multiscale FDFD method. Solved from frequency- and medium-dependent local Helmholtz problems, these multiscale basis functions capture fine-scale medium heterogeneities and are finally incorporated into the dimension-reduced linear system by a coupling of scalar Helmholtz problem solutions at two scales. We use several highly heterogeneous models to verify the performance in terms of the accuracy, efficiency, and memory cost of our multiscale method. The results show that our new method can solve the scalar Helmholtz equation in complicated models with high accuracy and quite low time and memory costs compared with the conventional FDFD methods.

Comparative studies of multiscale edge detection using different edge detectors for MRI thigh

Edge detection plays an important role in computer vision to extract object boundary. Multiscale edge detection method provides a variety of image features by different resolution at multiscale of edges. The method extracts coarse and fine structure edges simultaneously in an image. Due to this, the multiscale method enables more reliable edges are detected. Most of the multiscale methods are not translation invariant due to the decimated process. They mostly depend on the corresponding transform coefficients. These methods need more computation and a larger storage space. This study proposes a multiscale method that uses an average filter to smooth image at three different scales. Three different classical edge detectors namely Prewitt, Sobel and Laplacian were used to extract the edges from the smooth images. The edges extracted from the different scales of smooth images were then combined to form the multiscale edge detection. The performances of the multiscale images extracted from the three classical edge detectors were then compared and discussed.