Compressed-sensing (CS) MRI is frequently based on the minimization of an objective function consisting of the $$$\ell_1$$$-norm of a sparse image representation and the total variation (TV) of the image. Various definitions of the TV exist (isotropic/anisotropic; first-order/second-order/combined; and, particularly for complex-valued image data, several ways of dealing with real and imaginary parts of the image). The purpose of this abstract is to explain these variants, to provide implementation details, and to compare several variants with respect to CS reconstruction.
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