Associations among Image Assessments as Cost Functions in Linear Decomposition: MSE, SSIM, and Correlation Coefficient

The traditional methods of image assessment, such as mean squared error (MSE), signal-to-noise ratio (SNR), and Peak signal-to-noise ratio (PSNR), are all based on the absolute error of images. Pearson's inner-product correlation coefficient (PCC) is also usually used to measure the similarity between images. Structural similarity (SSIM) index is another important measurement which has been shown to be more effective in the human vision system (HVS). Although there are many essential differences among these image assessments, some important associations among them as cost functions in linear decomposition are discussed in this paper. Firstly, the selected bases from a basis set for a target vector are the same in the linear decomposition schemes with different cost functions MSE, SSIM, and PCC. Moreover, for a target vector, the ratio of the corresponding affine parameters in the MSE-based linear decomposition scheme and the SSIM-based scheme is a constant, which is just the value of PCC between the target vector and its estimated vector.