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A new theory named compressed sensing (CS) for simultaneous sampling and compression of signals indicates novel mechanism and design for measurement instrumentation. In this paper, we concern the recovery methods for the CS measurements. First, we investigate how the iterative curvelet thresholding (ICT) can be improved for sparse reconstruction of CS undetermined linear inverse problem, by considering several accelerated strategies, including the following: 1) Bioucas-Dias and Figueiredo's two-step iteration; 2) Beck and Teboulle's fast method; 3) and Osher linearized Bregman iteration. Secondly, we propose a two-stage active-set anisotropic-total-variation-(ATV) minimization-based ICT. In the first stage, a curvelet thresholding is applied to obtain a rough approximation of objects, and the index of remained significant coefficients is labeled as an active set. A Barzilai-Borwein-Dai-Yuan (BBDY) step size is used to accelerate the gradient line search. In the second stage, an active-set-constrained ATV minimization is applied, in which only insignificant coefficients beyond the active set are changed into small values, subjecting to ATV minimization of reconstructed objects. Numerical experiments show good performance of the improved ICT methods for single-pixel imaging and Fourier-domain CS imaging in remote sensing and medical engineering.