Morphing from the TV-Norm to the l -Norm

For piecewise constant objects, the images can be reconstructed with under-sampled measurements. The gradient image of a piecewise image is sparse. If a sparse solution is a desired solution, an $$ -norm minimization method is effective to solve an under-determined system. However, the $$ -norm is not differentiable, and it is not straightforward to minimize an $$ -norm. This paper suggests a function that is like the $$ -norm function, and we refer to this function as meta $$ -norm. The subdifferential of the meta $$ -norm has a simple explicit expression. Thus, it is straightforward to derive a gradient descent algorithm to enforce the sparseness in the solution. In fact, the proposed meta norm is a transition that varies between the TV-norm and the $$ -norm. As an application, this paper uses the proposed meta $$ -norm for few-view tomography. Computer simulation results indicate that the proposed meta $$ -norm effectively guides the image reconstruction algorithm to a piecewise constant solution. It is not clear whether the TV-norm or the $$ -norm is more effective in producing a sparse solution. Index Terms-Inverse problem, optimization, total-variation minimization, $$ -norm minimization, few-view tomography, iterative algorithm, image reconstruction.