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Design of Linear Equalizers Optimized for the Structural Similarity Index

### Authors:

Sumohana S. Channappayya,
Alan C. Bovik,
Constantine Caramanis,
and
Robert W. Heath Jr.

### Reference:

Submitted to the IEEE Trans. on Image Processing, July 2007.

### Abstract:

We propose an algorithm for designing linear equalizers that maximize the Structural SIMilarity (SSIM) index between the reference and restored signals.
The SSIM index has enjoyed considerable application in the {\it evaluation} of image processing algorithms. Algorithms however, have not been designed yet to explicitly optimize for this measure. The design of such an algorithm is non-trivial due to the non-convex nature of the distortion measure. In this paper, we reformulate the non-convex problem as a quasi-convex optimization problem, which admits a tractable solution. We compute the optimal solution in near closed-form, with complexity of the resulting algorithm comparable to complexity of the linear minimum mean squared error (MMSE) solution, independent of the number of filter taps. To demonstrate the usefulness of the proposed algorithm, it is applied to restore images that have been blurred and corrupted with additive white gaussian noise. As a special case, we consider blur-free image denoising. In each case, its performance is compared to a locally adaptive linear MSE-optimal filter. We show that the images denoised and restored using the SSIM-optimal filter have higher SSIM index, and superior perceptual quality than those restored using the MSE-optimal adaptive linear filter. Through these results, we demonstrate that (a) designing image processing algorithms, and in particular denoising and restoration-type algorithms, can yield significant gains over existing (in particular linear MMSE-based) algorithms, by optimizing them for perceptual distortion measures, and (b) these gains may be obtained without significant increase in the computational complexity of the algorithm.