Parametric ratio-based method for efficient contrast-preserving decolorization

Abstract

Decolorization is a fundamental technology indispensable for image processing, digital printing, and photograph rendering, while it suffers from information loss. This paper presents a novel parametric ratio-based method for efficient contrast-preserving decolorization (PrDecolor). For better preserving the details, features and visual distinctiveness of original color contrast, the proposed method simplifies and improves the current ratio-based decolorization model by incorporating the multivariate parametrical constraint. Moreover, the newly defined model is solved by efficient augmented Lagrangian and alternating direction method. The presented algorithm exhibits excellent iterative convergence and robustness with respect to the parameters. Extensive experiments under a variety of test images and a comprehensive evaluation against existing state-of-the-art methods consistently demonstrate the potential of the proposed algorithm.

Appendix

In earlier literatures [11] and [19], we found that the AL and ADM have been applied to solve the low-rank factorization based matrix separation problem and dictionary learning problem. For these non-convex problems they both provided a weak convergence result for their algorithm, i.e. under mild conditions any limit point of the iteration sequence generated by the algorithm is a KKT point. In our work, for this new non-convex problem we also give a similar result with regard to the convergence of the algorithm PrDecolor. It should be emphasized that although the following convergence result far from being satisfactory, it provides an assurance for the behavior of the algorithm.

Since the sub-gradient of the L(w, s) are as follows:

$$ {\displaystyle \begin{array}{l}\frac{\partial }{\partial w}L\left(w,s\right)=\frac{\partial }{\partial w}\left\{\frac{\gamma }{2}{\left\Vert \sum \limits_{I_l\in {Z}_2}{w}_l\nabla {I}_l-\left({s}_{x,y}-\lambda /\gamma \right)\right\Vert}_2^2\right\}\\ {}\kern7.5em =\gamma \sum \limits_{I_l\in {Z}_2}\nabla {I_l}^T\left[w\nabla {I}_l-\left({s}_{x,y}-\lambda /\gamma \right)\right]=0\end{array}} $$

(14)

$$ {\displaystyle \begin{array}{l}\frac{\partial }{\partial s}L\left(w,s\right)=\frac{\partial }{\partial s}\left\{-2K\sum \limits_{\left(x,y\right)\in P}\left|{\delta}_{x,y}\right|\frac{1}{2}{\left\Vert \sqrt{t}{s}_{x,y}\right\Vert}_2^2+\frac{\left(2+\gamma \right)}{2}{\left\Vert {s}_{x,y}-\frac{\gamma \sum \limits_{I_l\in {Z}_2}{w}_l\nabla {I}_l+\lambda }{2+\gamma}\right\Vert}_2^2\right\}\\ {}=\frac{\partial }{\partial s}\left\{-2K\sum \limits_{\left(x,y\right)\in P}\left|{\delta}_{x,y}\right|\frac{1}{2}{\left\Vert \sqrt{t}{s}_{x,y}\right\Vert}_2^2+\frac{\left(2+\gamma \right)}{2}{\left\Vert \frac{\left(2+\gamma \right){s}_{x,y}-\left(\gamma \sum \limits_{I_l\in {Z}_2}{w}_l\nabla {I}_l+\lambda \right)}{2+\gamma}\right\Vert}_2^2\right\}\\ {}=\frac{\partial }{\partial s}\left\{-2K\sum \limits_{\left(x,y\right)\in P}\left|{\delta}_{x,y}\right|\frac{1}{2}{\left\Vert \sqrt{t}{s}_{x,y}\right\Vert}_2^2+\frac{\left(2+\gamma \right)}{2}{\left\Vert \frac{\gamma \left({s}_{x,y}-\sum \limits_{I_l\in {Z}_2}{w}_l\nabla {I}_l\right)+2{s}_{x,y}-\lambda }{2+\gamma}\right\Vert}_2^2\right\}\\ {}=\left(2-2 Kt\sum \limits_{\left(x,y\right)\in P}\left|{\delta}_{x,y}\right|\right){s}_{x,y}-\lambda =0\end{array}} $$

(15)

Additionally, by considering the constraints ${s}_{x,y}=\sum_{I_l\in {Z}_2}{w}_l\nabla {I}_l$, it is straightforward to obtain the KKT conditions for (5) are as follows:

$$ {\displaystyle \begin{array}{l}\sum \limits_{I_l\in {Z}_2}\lambda \nabla {I_l}^T=0;\\ {}\left(2-2 Kt\sum \limits_{\left(x,y\right)\in P}\left|{\delta}_{x,y}\right|\right){s}_{x,y}-\lambda =0;\\ {}{s}_{x,y}=\sum \limits_{I_l\in {Z}_2}{w}_l\nabla {I}_l\end{array}} $$

(16)

Proposition 1

Let X ≜ (w, s) and ${\left\{{X}^k\right\}}_{k=1}^{\infty }$ be generated by our algorithm PrDecolor, Assume that ${\left\{{X}^k\right\}}_{k=1}^{\infty }$ is bounded and $\underset{k\to \infty }{\lim}\left({X}^{k+1}-{X}^k\right)=0$. Then any accumulation of ${\left\{{X}^k\right\}}_{k=1}^{\infty }$ satisfies the KKT conditions (16). In particular, whenever ${\left\{{X}^k\right\}}_{k=1}^{\infty }$ converges, it converges to a KKT point of (5).

Proof

Firstly, since ${w}^{k+1}=\frac{\sum_{I_l\in {Z}_2}\nabla {I_l}^T\left({s}_i^{k+1}-{\lambda}^{k+1}/\gamma \right)}{\sum_{I_l\in {Z}_2}\nabla {I_l}^T\nabla {I}_l}$, it follows that

$$ {\displaystyle \begin{array}{l}{w}^{k+1}-{w}^k=\frac{\sum \limits_{I_l\in {Z}_2}\nabla {I_l}^T\left({s}_i^{k+1}-{\lambda}^{k+1}/\gamma \right)}{\sum \limits_{I_l\in {Z}_2}\nabla {I_l}^T\nabla {I}_l}-{w}^k\\ {}\kern7em =\frac{\sum \limits_{I_l\in {Z}_2}\nabla {I_l}^T\left({s}_i^{k+1}-\sum \limits_{I_l\in {Z}_2}\nabla {I}_l{w}^k-{\lambda}^{k+1}/\gamma \right)}{\sum \limits_{I_l\in {Z}_2}\nabla {I_l}^T\nabla {I}_l}\end{array}} $$

(17)

Secondly,

$$ {\displaystyle \begin{array}{l}{s}_{x,y}^{k+1}-{s}_{x,y}^k=\frac{\gamma \sum \limits_{I_l\in {Z}_2}{w}_l^{k+1}\nabla {I}_l+{\lambda}^k}{2+\gamma -2 Kt\sum \limits_{\left(x,y\right)\in P}\left|{\delta}_{x,y}\right|}-{s}_{x,y}^k\\ {}\kern6.5em =\frac{\gamma \sum \limits_{I_l\in {Z}_2}{w}_l^{k+1}\nabla {I}_l+{\lambda}^k-{s}_{x,y}^k\left(2+\gamma -2 Kt\sum \limits_{\left(x,y\right)\in P}\left|{\delta}_{x,y}\right|\right)}{2+\gamma -2 Kt\sum \limits_{\left(x,y\right)\in P}\left|{\delta}_{x,y}\right|}\\ {}\kern6.5em =\frac{\gamma \left(\sum \limits_{I_l\in {Z}_2}{w}_l^{k+1}\nabla {I}_l\_{s}_{x,y}^k\right)+{\lambda}^k-{s}_{x,y}^k\left(2-2 Kt\sum \limits_{\left(x,y\right)\in P}\left|{\delta}_{x,y}\right|\right)}{2+\gamma -2 Kt\sum \limits_{\left(x,y\right)\in P}\left|{\delta}_{x,y}\right|}\end{array}} $$

(18)

Finally, it follows that

$$ {\lambda}^{k+1}={\lambda}^k+\gamma \left(\sum_{I_l\in {Z}_2}{w}_l^{k+1}\nabla {I}_l-{s}^{k+1}\right) $$

(19)

Hence $\underset{k\to \infty }{\lim}\left({X}^{k+1}-{X}^k\right)=0$ implies that both sides of (17)(18)(19) all tend to zero as k goes to infinity. Consequently,

$$ {\displaystyle \begin{array}{l}s-\sum \limits_{I_l\in {Z}_2}{w}_l\nabla {I}_l\to 0;\\ {}\sum \limits_{I_l\in {Z}_2}\lambda \nabla {I_l}^T\to 0;\\ {}\left(2-2 Kt\sum \limits_{\left(x,y\right)\in P}\left|{\delta}_{x,y}\right|\right){s}_{x,y}-\lambda \to 0;\end{array}} $$

(20)

where the first limit in (20) is used to derive other limits. That is, the sequence ${\left\{{X}^k\right\}}_{k=1}^{\infty }$ asymptotically satisfies the KKT conditions (16), from which the conclusions of the proposition follow readily. This completes the proof.