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

MULTIRIDGELETS FOR TEXTURE ANALYSIS

Directional wavelets have orientation selectivity and thus are able to efficiently represent highly anisotropic elements such as line segments and edges. Ridgelet transform is a kind of directional multi-resolution transform and has been successful in many image processing and texture analysis applications. The objective of this research is to develop multi-ridgelet transform by applying multiwavelet transform to the Radon transform so as to attain attractive improvements. By adapting the cardinal orthogonal multiwavelets to the ridgelet transform, it is shown that the proposed cardinal multiridgelet transform (CMRT) possesses cardinality, approximate translation invariance, and approximate rotation invariance simultaneously, whereas no single ridgelet transform can hold all these properties at the same time. These properties are beneficial to image texture analysis. This is demonstrated in three studies of texture analysis applications. Firstly a texture database retrieval study taking a portion of the Brodatz texture album as an example has demonstrated that the CMRT-based texture representation for database retrieval performed better than other directional wavelet methods. Secondly the study of the LCD mura defect detection was based upon the classification of simulated abnormalities with a linear support vector machine classifier, the CMRT-based analysis of defects were shown to provide efficient features for superior detection performance than other competitive methods. Lastly and the most importantly, a study on the prostate cancer tissue image classification was conducted. With the CMRT-based texture extraction, Gaussian kernel support vector machines have been developed to discriminate prostate cancer Gleason grade 3 versus grade 4. Based on a limited database of prostate specimens, one classifier was trained to have remarkable test performance. This approach is unquestionably promising and is worthy to be fully developed