On the discontinuity of images recovered by noncovex nonsmooth regularized isotropic models with box constraints

Abstract

Nonconvex nonsmooth regularizations have exhibited the ability of restoring images with neat edges in many applications, which has been provided a mathematical explanation by analyzing the discontinuity of the local minimizers of the variational models. Since in many applications the pixel intensity values in digital images are restricted in a certain given range, box constraints are adopted to improve the restorations. A similar property of nonconvex nonsmooth regularization for box-constrained models has been proved in the literature. While many theoretical results are available for anisotropic models, we investigate the isotropic case. We establish similar theoretical results for isotropic nonconvex nonsmooth models with box constraints. Numerical experiments are presented to validate our theoretical results.

Appendix The: calculation of the quadratic forms

For any function $\mathcal {F}(v) \in \mathcal {C}^{2}(\mathbb {R}^{m_{1}})$, our convention of the operator ∇ is $\nabla \mathcal {F} =(\frac {\partial }{\partial v_{1}} \mathcal {F},\ldots ,\frac {\partial }{\partial v_{m_{1}}} \mathcal {F})^{T}$. Then, the Hessian matrix of F is $\nabla ^{2} \mathcal {F} =\nabla \nabla ^{T} \mathcal {F}$.

The first and second differentials of $\mathcal {F}(v)$ at v^∗ are well defined and read

$$\begin{array}{@{}rcl@{}} \nabla \mathcal{F}(v^{*}) & =&\lambda {A_{1}^{T}}(A_{1}v^{*}-f^{\circ})+ \sum\limits_{i\in I_{1} \cup \partial B}\nabla \varphi(\|G_{i} u^{*}\|), \\ \nabla^{2} \mathcal{F}(v^{*}) & =&\lambda {A_{1}^{T}}A_{1} + \sum\limits_{i \in I_{1} \cup \partial B}\nabla^{2} \varphi(\|G_{i} u^{*}\|), \end{array} $$

where

$$\begin{array}{@{}rcl@{}} \nabla \varphi(\|G_{i} u^{*}\|) & =&\varphi^{\prime}(\|G_{i} u^{*}\|)\nabla(\|G_{i} u^{*}\|), \\ \nabla^{2} \varphi(\|G_{i} u^{*}\|) & =& \nabla (\nabla^{T} \varphi(\|G_{i} u^{*}\|)) = \nabla (\varphi^{\prime}(\|G_{i} u^{*}\|)\nabla^{T}(\|G_{i} u^{*}\|)). \end{array} $$

Notice that u^∗ = χ(v^∗) and ${G_{i}^{d}} \chi (v) = {g_{i}^{d}} v + {{\Delta }_{i}^{d}}$. We have

$$\begin{array}{@{}rcl@{}} 2\|G_{i} u^{*}\|\nabla(\|G_{i} u^{*}\|) &=& \nabla(\|G_{i} u^{*}\|^{2}) \\ & =& \nabla(({G_{i}^{x}} u^{*})^{2}+({G_{i}^{y}} u^{*})^{2}) \\ & =& 2({{g_{i}^{x}}}^{T} {G_{i}^{x}}u^{*}+{{g_{i}^{y}}}^{T} {G_{i}^{y}}u^{*}) \end{array} $$

It follows that

$$\nabla(\|G_{i} u^{*}\|)=\frac{{{g_{i}^{x}}}^{T} {G_{i}^{x}}u^{*}+{{g_{i}^{y}}}^{T} {G_{i}^{y}}u^{*}}{\|G_{i} u^{*}\|} =\frac{{g_{i}^{T}}G_{i}u^{*}}{\|G_{i} u^{*}\|}. $$

We have

$$\begin{array}{@{}rcl@{}} \nabla^{2} \varphi(\|G_{i} u^{*}\|) & =& \nabla \left( \varphi^{\prime}(\|G_{i} u^{*}\|)\frac{(G_{i}u^{*})^{T}g_{i}}{\|G_{i} u^{*}\|}\right) \\ & =& \nabla \varphi^{\prime}(\|G_{i} u^{*}\|)\frac{(G_{i}u^{*})^{T}g_{i}}{\|G_{i} u^{*}\|} + \varphi^{\prime}(\|G_{i} u^{*}\|)\nabla\left( \frac{(G_{i}u^{*})^{T}g_{i}}{\|G_{i} u^{*}\|}\right), \end{array} $$

where

$$\begin{array}{@{}rcl@{}} \nabla \varphi^{\prime}(\|G_{i} u^{*}\|) & =& \varphi^{\prime\prime}(\|G_{i} u^{*}\|)\nabla(\|G_{i} u^{*}\|)=\varphi^{\prime\prime}(\|G_{i} u^{*}\|)\frac{{g_{i}^{T}}G_{i}u^{*}}{\|G_{i} u^{*}\|}, \\ \nabla \left( \frac{(G_{i}u^{*})^{T}g_{i}}{\|G_{i} u^{*}\|}\right) & =& \nabla \left( \frac{1}{\|G_{i} u^{*}\|}\right)(G_{i}u^{*})^{T}g_{i}+\frac{1}{\|G_{i} u^{*}\|}\nabla ((G_{i}u^{*})^{T}g_{i}), \end{array} $$

and

$$\begin{array}{@{}rcl@{}} \nabla \left( \frac{1}{\|G_{i} u^{*}\|}\right) & =&\nabla (\|G_{i} u^{*}\|^{-1})=\frac{-1}{\|G_{i} u^{*}\|^{2}}\nabla (\|G_{i} u^{*}\|)=-\frac{{g_{i}^{T}}G_{i}u^{*}}{\|G_{i} u^{*}\|^{3}}, \\ \nabla ((G_{i}u^{*})^{T}g_{i}) & =& \nabla((G_{i}u^{*})^{T}) g_{i}, \end{array} $$

where $\nabla ((G_{i}u^{*})^{T}) = (\nabla ({G_{i}^{x}} u^{*}),\nabla ({G_{i}^{y}} u^{*}))=({{g_{i}^{x}}}^{T},{{g_{i}^{y}}}^{T})={g_{i}^{T}}$. Thus,

$$\nabla ((G_{i}u^{*})^{T}G_{i})= {g_{i}^{T}} g_{i}. $$

At last, we have

$$\begin{array}{@{}rcl@{}} \nabla^{2} \varphi(\|G_{i} u^{*}\|) & = & \frac{\varphi^{\prime\prime}(\|G_{i} u^{*}\|)}{\|G_{i} u^{*}\|^{2}}{g_{i}^{T}}\!G_{i}u^{*}({g_{i}^{T}}G_{i}u^{*})^{T} - \frac{\varphi^{\prime}(\|G_{i} u^{*}\|)}{\|G_{i} u^{*}\|^{3}}{g_{i}^{T}}\!G_{i}u^{*}({g_{i}^{T}}G_{i}u^{*})^{T} \\ && + \frac{\varphi^{\prime}(\|G_{i} u^{*}\|)}{\|G_{i} u^{*}\|}{g_{i}^{T}}g_{i}. \end{array} $$

Therefore,

$$\begin{array}{lll} \nabla^{2} \mathcal{F}(v^{*}) =& \lambda {A_{1}^{T}}A_{1} + \sum\limits_{i\in I_{1} \cup \partial B}\frac{\varphi^{\prime\prime}(\|G_{i} u^{*}\|)}{\|G_{i} u^{*}\|^{2}}{g_{i}^{T}}G_{i}u^{*}({g_{i}^{T}}G_{i}u^{*})^{T} \\ & -\sum\limits_{i\in I_{1} \cup \partial B}\frac{\varphi^{\prime}(\|G_{i} u^{*}\|)}{\|G_{i} u^{*}\|^{3}}{g_{i}^{T}}G_{i}u^{*}({g_{i}^{T}}G_{i}u^{*})^{T} \\ & + \sum\limits_{i\in I_{1} \cup \partial B}\frac{\varphi^{\prime}(\|G_{i} u^{*}\|)}{\|G_{i} u^{*}\|}{g_{i}^{T}}g_{i}. \end{array} $$

For any v ∈ K(I₀), any i ∈ I₁ ∪ ∂B, we have

$$\begin{array}{@{}rcl@{}} ({g_{i}^{T}}G_{i}u^{*})^{T}v & = \langle G_{i}u^{*},g_{i}v \rangle, \\ v^{T} {g_{i}^{T}} g_{i} v & = \|g_{i} v\|^{2}, \\ v^{T} {G_{i}^{T}} G_{i} v & = \|G_{i} v\|^{2}. \end{array} $$

It follows that

$$\begin{array}{@{}rcl@{}} v^{T} \nabla^{2} \mathcal{F}(v^{*})v &= & \lambda \|A_{1}v\|^{2}+ \sum\limits_{i \in I_{1} \cup \partial B} \frac{\varphi^{\prime\prime}(\|G_{i} u^{*}\|)}{\|G_{i} u^{*}\|^{2}} \langle G_{i} u^{*}, g_{i} v\rangle^{2} \\ && +\sum\limits_{i \in I_{1} \cup \partial B}\frac{\varphi^{\prime}(\|G_{i} u^{*}\|)}{\|G_{i} u^{*}\|^{3}}\left( \|G_{i} u^{*}\|^{2} \|g_{i}v\|^{2} - \langle G_{i} u^{*}, g_{i} v\rangle^{2} \right). \end{array} $$