Ma, Jianwei; Fenn, Markus Combined complex ridgelet shrinkage and total variation minimization. (English) Zbl 1114.65160 SIAM J. Sci. Comput. 28, No. 3, 984-1000 (2006). Summary: A new algorithm for the characterization of engineering surface topographies with line singularities is proposed. It is based on thresholding complex ridgelet coefficients combined with total variation (TV) minimization. The discrete ridgelet transform is designed by first using a discrete Radon transform based on the nonequispaced fast Fourier transform (NFFT) and then applying a dual-tree complex wavelet transform (DT CWT). The NFFT-based approach of the Radon transform completely avoids linear interpolations of the Cartesian-to-polar grid and requires only \(O(n^{2} \log n)\) arithmetic operations for \(n\) by \(n\) arrays, while its inverse preserves the good reconstruction quality of the filtered backprojection. The DT CWT in the second step of the ridgelet transform provides approximate shift invariance on the projections of the Radon transform. After hard thresholding the ridgelet coefficients, they are restored using TV minimization to eliminate the pseudo-Gibbs artifacts near the discontinuities. Numerical experiments demonstrate the remarkable ability of the methodology to extract line scratches. Cited in 7 Documents MSC: 65T50 Numerical methods for discrete and fast Fourier transforms 65T60 Numerical methods for wavelets 68U10 Computing methodologies for image processing 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 65D18 Numerical aspects of computer graphics, image analysis, and computational geometry Keywords:nonequispaced fast Fourier transform; ridgelets; complex wavelets; shift invariance; total variation minimization; detection of line singularities; surface characterization; algorithm; engineering surface topographies; discrete ridgelet transform; discrete Radon transform; dual-tree complex wavelet transform; reconstruction; filtered backprojection; numerical experiments Software:DT-CWT; NFFT; NUFFT PDFBibTeX XMLCite \textit{J. Ma} and \textit{M. Fenn}, SIAM J. Sci. Comput. 28, No. 3, 984--1000 (2006; Zbl 1114.65160) Full Text: DOI