Linear scan cone-beam computed tomography (LSCBCT) is a technically simple computed tomography (CT) configuration and is powerful enough to inspect long objects. As one kind of limited-angle problem, the image reconstruction of LSCBCT is ill posed. However, the total variation minimization (TVM)-projection on convex sets (POCS) reconstruction algorithm, which is based on the TVM and POCS, has been proven effective for solving the limited-angle problem. While applying the TVM-POCS algorithm to the LSCBCT reconstruction, the reconstructed image is distorted near the edges of the object. To solve this problem, an improved iterative reconstruction algorithm was developed. The improved method integrated simultaneous algebraic reconstruction technique, TVM, and C-V model. The C-V model can detect objects whose boundaries are not necessarily defined by gradient. The developed algorithm can reduce artifacts by piecewise constant modification and get a more accurate image. Numerical simulations are presented to illustrate the effectiveness of the algorithm. Moreover, the developed algorithm can be applied to other x-ray CT reconstruction problems.