A non‐convex ternary variational decomposition and its application for image denoising

Abstract

A non‐convex ternary variational decomposition model is proposed in this study, which decomposes the image into three components including structure, texture and noise. In the model, a non‐convex total variation (NTV) regulariser is utilised to model the structure component, and the weaker G and E spaces are used to model the texture and noise components, respectively. The proposed model provides a very sparse representation of the structure in total variation (TV) transform domain due to the use of non‐convex regularisation and cleanly separates the texture and noise since two different weaker spaces are used to model these two components, respectively. In image denoising application, the proposed model can successfully remove noise while effectively preserving image edges and constructing textures. An alternating direction iteration algorithm combining with iteratively reweighted l1 algorithm, projection algorithm and wavelet soft threshold algorithm is introduced to effectively solve the proposed model. Numerical results validate the model and the algorithm for both synthetic and real images. Furthermore, compared with several state‐of‐the‐art image variational restoration models, the proposed model yields the best performance in terms of the peak signal to noise ratio (PSNR) and the mean structural similarity index (SSIM).

Ternary image decomposition with automatic parameter selection via auto- and cross-correlation

“…A similar effect can be observed in Fig. 10 where our decomposition approach is compared with the results shown in [23]. The image b has been corrupted by AWGN with σ = 20.…”

“…This means that the solution ĉ(µ) of the quadratic minimization problem (23) has the same mean value of the vector u, independently of the regularization parameter µ, and that the associated residual t(µ) = u − ĉ(µ) has mean zero. The 2D discrete Fourier transformed cartoon component ĉ(µ) in (30) can be rewritten in more compact form as follows…”

“…The unconstrained quadratic optimization problem in ( 43)-( 44) has exactly the same form of problem (23), with q (k) in place of q and γ in place of µ. Hence, according to the cross-correlation principle outlined in Section 5.2, we can preliminarily determine the value γ (k+1) -hence, the value µ (k+1) = β γ (k+1)which minimizes the normalized cross-correlation between the obtained cartoon component c (k+1) and the associated texture component t (k+1) = u * − c (k+1) .…”

“…In this last example we present the qualitative performance of the proposed decomposition approach with respect to other variational models which aim to separate an image into three components (structure, texture, and noise). Specifically, the original model in [2], based on total variation, G-norm, and Besovnorm (which amounts to a wavelet shrinkage), the adaptive model in [5], where differently weighted Gnorms are used to separate noise from texture using a space-variant regularisation parameter, and, finally, the ternary variational decomposition introduced in [23], where a non-convex penalty function replaces the TV term, while G-norm, and Besov-norm are used to model the texture and noise components, respectively. To avoid introducing artifacts due to naive reproduction of the various algorithms and associated parameter setting within, the images compared and here reported in Fig.…”

“…The image b has been corrupted by AWGN with σ = 20. The difficulty in separating texture from noise by [23] assuming any knowledge of noise kind and variance, but taking advantage, instead, of an automatic parameter selection to achieve a sub-optimal texture -cartoon separation.…”