The amalgamated projection method for convex feasibility and optimization problems has recently been proposed and the stable convergence under summable perturbations has been derived. As an application in computerized tomography (CT), the accuracy and the rate of convergence of the cyclic projection method and Cimmino algorithm incorporated with total variation minimization under certain conditions are significantly improved based on the theory of compressed sensing. In this paper, a varying block cyclic projection method and a block Cimmino's algorithm in the compressed sensing framework are proposed and their convergence are derived with an application of the convergence theorem of the amalgamated projection methods. An example is given to illustrate the convergence behavior of new algorithms.