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Replacing the lscr2 data fidelity term of the standard total variation (TV) functional with an lscr1 data fidelity term has been found to offer a number of theoretical and practical benefits. Efficient algorithms for minimizing this lscr1-TV functional have only recently begun to be developed, the fastest of which exploit graph representations, and are restricted to the denoising problem. We describe an alternative approach that minimizes a generalized TV functional, including both lscr2-TV and lscr1-TV as special cases, and is capable of solving more general inverse problems than denoising (e.g., deconvolution). This algorithm is competitive with the graph-based methods in the denoising case, and is the fastest algorithm of which we are aware for general inverse problems involving a nontrivial forward linear operator.