Spectral computed tomography(CT)provides information on material characterization and quantification because of its ability to separate different basis materials.Dual-energy(DE)CT provides two sets of measurements at two different source energies.In principle,two materials can be accurately decomposed from DECT measurements.However,many clinical and industrial applications require three or more material images.For triple-material decomposition,a third constraint,such as volume conservation,mass conservation or both,is required to solve three sets of unknowns from two sets of measurements.The recently proposed flexible image-domain(ID)multi-material decomposition)method assumes each pixel contains at most three materials out of several possible materials and decomposes a mixture pixel by pixel.We propose a penalized-likelihood(PL)method with edge-preserving regularizers for each material to reconstruct multi-material images using a similar constraint from sinogram data.We develop an optimization transfer method with a series of pixel-wise separable quadratic surrogate(PWSQS)functions to monotonically decrease the complicated PL cost function.The PWSQS algorithm separates pixels to allow simultaneous update of all pixels,but keeps the basis materials coupled to allow faster convergence rate than our previous proposed material-and pixel-wise SQS algorithms.Comparing with the ID method using 2-D fan-beam simulations,the PL method greatly reduced noise,streak and cross-talk artifacts in the reconstructed basis component images,and achieved much smaller root mean square errors.