Combined complex ridgelet shrinkage and total variation minimization

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.