An Efficient Algorithm for Computing the Derivative of Mean Structural Similarity Index Measure

Abstract

Many inverse problems in imaging can be addressed using energy minimization. The Euclidean distance is traditionally used in data fidelity term of energy functionals, even though it is not an optimal measure of visual quality. Recently the use of Mean Structural Similarity Index Measure (MSSIM) in data fidelity expressions has been examined. Solving such problems requires derivative of MSSIM. We propose an efficient algorithm for computing this derivative using convolutions. We indicate how the computational cost will be reduced to ${\mathcal {O}}(N \log {} N)$ from the cost of traditional scheme ${\mathcal {O}}(m~ N \log {} N)$ where N is the size of an input image and m is the window size. The proposed algorithm can be used for any inverse problem that traditionally applies L2 norm as a data fidelity measure. We apply the proposed numerical scheme to the inverse problem of image denoising.