We develop algorithms for sparse X-ray computed tomography (CT) image reconstruction of objects with known contour, where the signal outside the contour is assumed to be zero. We first propose a constrained residual squared error minimization criterion that incorporates both the knowledge of the object's contour and signal sparsity in an appropriate transform domain. We then present convex relaxation and greedy approaches to approximately solving this minimization problem; our greedy mask iterative hard thresholding schemes guarantee monotonically non-increasing residual squared error. We also apply mask minimum norm (mask MN) and least squares (mask LS) methods that ignore signal sparsity and solve the residual squared error minimization problem that imposes only the object contour constraint. We compare the proposed schemes with existing large-scale sparse signal reconstruction methods via numerical simulations and demonstrate that, by exploiting both the object contour information in the underlying image and sparsity of its discrete wavelet transform (DWT) coefficients, we can reconstruct this image using a significantly smaller number of measurements than the existing methods. We apply the proposed methods to reconstruct images from simulated X-ray CT measurements and demonstrate their superior performance compared with the existing approaches.