The ridgelet transform with non-linear threshold for seismic noise attenuation in marine carbonates

Wavelet transforms have been successfully used in seismic data processing with their ability for local time - frequency analysis. However, identification of directionality is limited because wavelet transform coefficients reveal only three spatial orientations. Whereas the ridgelet transform has a superior capability for direction detection and the ability to process signals with linearly changing characteristics. In this paper, we present the issue of low signal-to-noise ratio （SNR） seismic data processing based on the ridgelet transform. Actual seismic data with low SNR from south China has been processed using ridgelet transforms to improve the SNR and the continuity of seismic events. The results show that the ridgelet transform is better than the wavelet transform for these tasks.

Denoising of seismic data via multi-scale ridgelet transform

Noise has traditionally been suppressed or eliminated in seismic data sets by the use of Fourier filters and, to a lesser degree, nonlinear statistical filters. Although these methods are quite useful under specific conditions, they may produce undesirable effects for the low signal to noise ratio data. In this paper, a new method, multi-scale ridgelet transform, is used in the light of the theory of ridgelet transform. We employ wavelet transform to do sub-band decomposition for the signals and then use non-linear thresholding in ridgelet domain for every block. In other words, it is based on the idea of partition, at sufficiently fine scale, a curving singularity looks straight, and so ridgelet transform can work well in such cases. Applications on both synthetic data and actual seismic data from Sichuan basin, South China, show that the new method eliminates the noise portion of the signal more efficiently and retains a greater amount of geologic data than other methods, the quality and consecutiveness of seismic event are improved obviously as well as the quality of section is improved.

Application of multi-resolution analysis in sonar image denoising

Sonar images have complex background, low contrast, and deteriorative edges; these characteristics make it difficult for researchers to dispose the sonar objects. The multi-resolution analysis represents the signals in different scales efficiently, which is widely used in image processing. Wavelets are successful in disposing point discontinuities in one dimension, but not in two dimensions. The finite Ridgelet transform （FRIT） deals efficiently with the singularity in high dimension. It presents three improved denoising approaches, which are based on FRIT and used in the sonar image disposal technique. By experiment and comparison with traditional methods, these approaches not only suppress the artifacts, but also obtain good effect in edge keeping and SNR of the sonar image denoising.

SPREAD SPECTRUM WATERMARK DETECTION IN DRT-DOMAIN

The traditional correlation-based detector is optimal only for Gaussian data, but the Laplacian Probability Density Function (PDF) is more appropriate to model the coefficients in the Discrete Ridgelet Transform (DRT) domain. An additive maximum-likelihood detector based on the Laplacian PDF is analyzed and the theoretical result of its performance is given. The experiments show that the error of the Laplacian model for the DRT coefficients of many images is smaller than that of the Gaussian model. The experiments also prove that the Laplacian detector is superior to the tradi- tional correlation-based detector.

A 4-quadrant Curvelet Transform for Denoising Digital Images

The conventional discrete wavelet transform （DWT） introduces artifacts during denoising of images containing smooth curves. Finite ridgelet transform （FRIT） solved this problem by mapping the curves in terms of small curved ridges. However, blind application of FRIT all over an image is computationally heavy. Finite curvelet transform （FCT） selectively applies FRIT only to the tiles containing small portions of a curve. In this work, a novel curvelet transform named as 4-quadrant finite curvelet transform （4QFCT） based on a new concept of 4-quadrant finite ridgelet transform （4QFRIT） has been proposed. An image is band pass filtered and the high frequency bands are divided into small non-overlapping square tiles. The 4QFRIT is applied to the tiles containing at least one curve element. Unlike FRIT, the 4QFRIT takes 4 sets of radon projections in all the 4 quadrants and then averages them in time and frequency domains after denoising. The proposed algorithm is extensively tested and benchmarked for denoising of images with Gaussian noise using mean squared error （MSE） and peak signal to noise ratio （PSNR）. The results confirm that 4QFCT yields consistently better denoising performance quantitatively and visually.