Minmax-concave total variation denoising

Total variation (TV) denoising is a commonly used method for recovering 1-D signal or 2-D image from additive white Gaussian noise observation. In this paper, we define the Moreau enhanced function of L₁ norm as Φ_α(x) and introduce the minmax-concave TV (MCTV) in the form of Φ_α(Dx) , where D is the finite difference operator. We present that MCTV approaches ‖ Dx‖ ₀ if the non-convexity parameter α is chosen properly and apply it to denoising problem. MCTV can strongly induce the signal sparsity in gradient domain, and moreover, its form allows us to develop corresponding fast optimization algorithms. We also prove that although this regularization term is non-convex, the cost function can maintain convexity by specifying α in a proper range. Experimental results demonstrate the effectiveness of MCTV for both 1-D signal and 2-D image denoising.