Four-Directional Total Variation Denoising Using Fast Fourier Transform and ADMM

Noise removal is a fundamental problem in image processing. Among many approaches, the total variation has attracted great attention because of its nice mathematical interpretation. Traditional total variation explores the gradient information of the vertical and the horizontal directions. Thus, the number of directions can be increased to further improve denoising performance. The resulting challenge is higher computation since multiple constraints are introduced in denoising model. This work first transforms the quaternion total variation constraints problem in the spatial domain into a problem in the frequency domain by using the fast Fourier transform and the convolution theorem. Then, it incorporates the alternating direction method of multipliers (ADMM) to enable fast image denoising. This fast computation is verified by the comparisons with other total variation based methods including state-of-the-art methods.