We develop a sparse image reconstruction method for Poisson distributed polychromatic X-ray computed tomography (CT) measurements under the blind scenario where the material of the inspected object and the incident energy spectrum are unknown. We employ our mass-attenuation spectrum parameterization of the noiseless measurements for single-material objects and express the mass-attenuation spectrum as a linear combination of B-spline basis functions of order one. A block coordinate descent algorithm is developed for constrained minimization of a penalized Poisson negative log-likelihood (NLL) cost function, where constraints and penalty terms ensure nonnegativity of the spline coefficients and nonnegativity and sparsity of the density-map image; the image sparsity is imposed using a convex total-variation (TV) norm penalty term. This algorithm alternates between a Nesterov’s proximal-gradient (NPG) step for estimating the density-map image and a limited-memory Broyden-Fletcher-Goldfarb-Shanno with box constraints (LBFGS- B) step for estimating the incident-spectrum parameters. We establish conditions for biconvexity of the penalized NLL objective function, which, if satisfied, ensures monotonicity of the NPG-BFGS iteration. We also show that the penalized NLL objective satisfies the Kurdyka-Łojasiewicz property, which is important for establishing local convergence of block-coordinate descent schemes in biconvex optimization problems. Simulation examples demonstrate the performance of the proposed scheme.