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A Globally Convergent Algorithm for a Constrained Non-Lipschitz Image Restoration Model

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Abstract

In this paper, we study a non-Lipschitz and box-constrained model for both piecewise constant and natural image restoration with Gaussian noise removal. It consists of non-Lipschitz isotropic first-order \(\ell _{p}\) (\(0<p<1\)) and second-order \(\ell _{1}\) as regularization terms to keep edges and overcome staircase effects in smooth regions simultaneously. The model thus combines the advantages of non-Lipschitz and high order regularization, as well as box constraints. Although this model is quite complicated to analyze, we establish a motivating theorem, which induces an iterative support shrinking algorithm with proximal linearization. This algorithm can be easily implemented and is globally convergent. In the convergence analysis, a key step is to prove a lower bound theory for the nonzero entries of the gradient of the iterative sequence. This theory also provides a theoretical guarantee for the edge preserving property of the algorithm. Extensive numerical experiments and comparisons indicate that our method is very effective for both piecewise constant and natural image restoration.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (NSFC) (Nos. 11871035, 11531013), Recruitment Program of Global Young Expert, Zhejiang Provincial Natural Science Foundation of China (No. LQ20A010007), and Grants RG(R)-RC/17-18/02-MATH, GRF-12300819, NSFC/RGC-N_HKBU214/19.

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Correspondence to Chunlin Wu.

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Appendix

Appendix

1.1 The Proof of Lemma 3.4

Firstly, we introduce more index sets to clearly give our proof.

Definition 7.1

For any \(u \in {\mathscr {U}}\), we define:

  1. (1)

    \(B_{1}(u)=\{i\in \Omega _1(u) \cap B(u): u_{i+1} \in B(u) \ \text {and} \ u_{i+n} \in B(u)\}\).

  2. (2)

    \(L_{1}(u)= \Omega _1(u) \setminus B_{1}(u).\)

  3. (3)

    \(B_{0}(u)=\Omega _0(u)\cap B(u)\).

  4. (4)

    \(I_{0}(u)= \Omega _0(u) \setminus B_{0}(u).\)

Here, the subset \(B_{1}(u)\) of \(\Omega _1(u)\) includes pixels equal to \( l_{1}\) or \( l_{2}\) with its left and right pixels being \( l_{1}\) or \( l_{2}\), while pixels in the subset \(L_{1}(u)\) are in \((l_{1}, l_{2})\). Similarly, the subset \(B_{0}(u)\) of \(\Omega _0(u)\) consists of pixels equal to \( l_{1}\) or \( l_{2}\), where its left and right pixels are the same as itself. In contrast, the subset \(I_{0}(u)\) of \(\Omega _0(u)\) consists of pixels in \((l_{1}, l_{2})\).

A simple example is shown in Fig. 7 to explain these index sets in Definitions 3.2 and 7.1, where we consider the periodic boundary condition. Here, \(l_{1}=0\), \(l_{2}=1\), u is

$$\begin{aligned} u=\left[ 0,1,\frac{1}{2},\frac{1}{2},1,\frac{1}{4},\frac{1}{2},\frac{1}{4},1,1,\frac{1}{4},\frac{1}{4},1,\frac{1}{2},\frac{1}{4},\frac{1}{4}\right] ^{T}, \end{aligned}$$

and \((\Vert d_{i} u\Vert )_{i \in J}\) is \([\sqrt{2}, \frac{\sqrt{13}}{4}, 0, \frac{\sqrt{13}}{4}, \frac{3}{4}, \frac{\sqrt{10}}{4}, \frac{\sqrt{2}}{4}, \frac{3}{4}, 0, \frac{\sqrt{13}}{4}, 0, \frac{3}{4}, \frac{\sqrt{5}}{2}, \frac{\sqrt{5}}{4},\frac{1}{4},\frac{\sqrt{10}}{4}]^{T}.\) Then,

  1. (I)

    \(I(u)=\{3,4,6,7,8,11,12,14,15,16\}, \ B(u)=\{1,2,5,9,10,13\};\)

  2. (II)

    \(\Omega _1(u)=\{1,2,4,5,6,7,8,10,12,13,14,15,16\}, \ B_{1}(u)=\{1\}\),

    \(L_{1}(u)=\{2,4,5,6,7,8,10,12,13,14,15,16\};\)

  3. (III)

    \(\Omega _0(u)=\{3,9,11\}, \ B_{0}(u)=\{9\}, \ I_{0}(u)=\{3,11\}.\)

Fig. 7
figure 7

A simple example. a The illustration of an image and its gradient. b Pixel values in black and norms of gradient in red (Color figure online)

We observe that if \(I(u)=\emptyset \) or \(L_{1}(u)=\emptyset \), then either \(\Vert d_{i} u\Vert =0\) or \(\Vert d_{i} u\Vert =l_{2}-l_{1}\) or \(\Vert d_{i} u\Vert =\sqrt{2}(l_{2}-l_{1})\), for \(\forall i\in J\). Thus, we assume that \(I(u)\ne \emptyset \) and \(L_{1}(u)\ne \emptyset \) in the rest of our paper.

From the Definitions 3.2 and 7.1, we have a lemma similar to [12] as follows:

Lemma 7.2

[12] For a given vector \(u\in {\mathscr {U}}\), let \(I(u),B(u),\Omega _1(u),B_{1}(u),L_{1}(u),\Omega _0(u),B_{0}(u),I_{0}(u)\) be the index sets defined in Definitions 3.2 and 7.1. Then, the following statements hold:

  1. (a)

    \(L_{1}(u)\cap B_{1}(u)=\emptyset \) and \(L_{1}(u)\cup B_{1}(u)=\Omega _1(u)\);

  2. (b)

    \((d^{x}_{i})_{I(u)} = 0\) and \((d^{y}_{i})_{I(u)}=0\), for any \(i\in B_{1}(u)\);

  3. (c)

    \((d^{x}_{i})_{I(u)}\ne 0\) or \((d^{y}_{i})_{I(u)}\ne 0\), for any \(i\in L_{1}(u)\);

  4. (d)

    \(I_{0}(u)\cap B_{0}(u)=\emptyset \) and \(I_{0}(u)\cup B_{0}(u)=\Omega _0(u)\);

  5. (e)

    \((d^{x}_{i})_{I(u)}=0\) and \((d^{y}_{i})_{I(u)}=0\), for any \(i\in B_{0}(u)\);

  6. (f)

    \((d^{x}_{i})_{B(u)}=0\) and \((d^{y}_{i})_{B(u)}=0\), for any \(i\in I_{0}(u)\);

For simplifying symbols, let \(L^{*}_{1}=L_{1}(u^{*}),B^{*}_{1}=B_{1}(u^{*}), I^{*}_{0}=I_{0}(u^{*}),B^{*}_{0}=B_{0}(u^{*})\) and some descriptions in (5) be

$$\begin{aligned} g^{*}= & {} f- H_{B^{*}}u^{*}_{B^{*}},\\ t^{x,*}_{i}= & {} (d^{x}_{i})_{B^{*}}u^{*}_{B^{*}},\ t^{y,*}_{i} = (d^{y}_{i})_{B^{*}}u^{*}_{B^{*}}, \ \forall i\in J,\\ s^{*}_{i}= & {} (q_{i})_{B^{*}}u^{*}_{B^{*}}, \ \forall i\in {\widetilde{J}}. \end{aligned}$$

If \(i\in I_{0}^{*}\), then \(t^{x,*}_{i}=t^{y,*}_{i}=0\) by Lemma 7.2(f); if \(i\in B_{0}^{*}\), then \(t^{x,*}_{i} = (d^{x}_{i})_{I^{*}}u^{*}_{I^{*}}+(d^{x}_{i})_{B^{*}}u^{*}_{B^{*}}= (d^{x}_{i})u^{*}=0\) and \(t^{y,*}_{i} = (d^{y}_{i})_{I^{*}}u^{*}_{I^{*}}+(d^{y}_{i})_{B^{*}}u^{*}_{B^{*}}= (d^{y}_{i})u^{*}=0\) by Lemma 7.2(e). Since \(I_{0}^{*}\cup B_{0}^{*}=\Omega _0^{*}\) by Lemma 7.2(d), we have

$$\begin{aligned} t^{x,*}_{i}=t^{y,*}_{i}=0, \ \forall i\in \Omega _0^{*}. \end{aligned}$$
(30)

Since \((d_{i})_{I^{*}} z = 0\) for \(\forall i \in B_{0}^{*}\) by Lemma 7.2(e), \((d_{i})_{I^{*}} z = 0\) for all \(i \in \Omega _0^{*}\) when \(z \in \{z\in {\mathbb {R}}^{|I^{*}|}: (d_{i})_{I^{*}}z=0, \ \forall i\in I^{*}_0\}\).

Proof

When \(z^{*}=u^{*}_{I^{*}}\), \(R(z^{*})=F(u^{*})\). Next, we prove that \(z^{*}\) is a local minimizer of (5).

From \(u^{*}\) being a local minimizer of (3), we have \(F(u)> F(u^{*})\) for any \(u\in {\mathscr {U}}\) and \(\Vert u-u^{*}\Vert < \epsilon ^{*}\) with \(\epsilon ^{*}>0.\) If \(z^{*}=u^{*}_{I^{*}}\) is not a local minimizer, then there exists \({\widetilde{z}}\in \{{\widetilde{z}}: l_{1}< {\widetilde{z}}_{i} < l_{2}\}\) satisfying \(\Vert {\widetilde{z}}-z^{*}\Vert < \epsilon ^{*}\) and \((d_{i})_{I^{*}}{\widetilde{z}}=0\), for \(\forall i \in I^{*}_{0}(=\Omega _0^{*}\cap I^{*})\), such that

$$\begin{aligned} R({\widetilde{z}})<R(z^{*}). \end{aligned}$$
(31)

In particular, we define \({\widetilde{u}}=({\widetilde{z}};u^{*}_{B^{*}})\). Obviously, we have \({\widetilde{u}}\in {\mathscr {U}}\), \(\Vert {\widetilde{u}}-u^{*}\Vert < \epsilon ^{*}\) and thus \(F({\widetilde{u}})> F(u^{*})\). Moreover, the relationship between \(R({\widetilde{z}})\) and \(F({\widetilde{u}})\) is

$$\begin{aligned} F({\widetilde{u}})&= R({\widetilde{z}}) + \sum _{i \in \Omega _0^{*}} \varphi _1(\sqrt{(d^{x}_{i})_{I^{*}}{\widetilde{z}}+t^{x,*}_{i})^2+ ((d^{y}_{i})_{I^{*}}{\widetilde{z}}+t^{y,*}_{i})^2})\\ [\hbox { by } 30\,]&= R({\widetilde{z}}) + \sum _{i \in \Omega _0^{*}}\varphi _1(\Vert (d_{i})_{I^{*}}{\widetilde{z}}\Vert )\\&= R({\widetilde{z}}) + \sum _{i \in I^{*}_{0}} \varphi _1(\Vert (d_{i})_{I^{*}}{\widetilde{z}}\Vert ) + \sum _{i \in B^{*}_{0}} \varphi _1(\Vert (d_{i})_{I^{*}}{\widetilde{z}}\Vert )\\ [\hbox { by Lemma}~7.2\hbox {(e) }]&= R({\widetilde{z}}) + \sum _{i \in I^{*}_{0}} \varphi _1(\Vert (d_{i})_{I^{*}}{\widetilde{z}}\Vert ) \\ [\hbox { by } (d_{i})_{I^{*}}{\widetilde{z}}=0\, ]&= R({\widetilde{z}}). \end{aligned}$$

We observe that \(R({\widetilde{z}})=F({\widetilde{u}})> F(u^{*})=R(z^{*})\), which contradicts (31). Thus, \(z^{*}\) is a local minimizer of the optimization problem (5). \(\square \)

1.2 The Proof of Theorem 3.5

1.2.1 More Definitions and Properties of \(u^{*}\)

The proof uses some important tools and technologies from [39] briefly reviewed here. We first introduce some notations of the matrix repesentation \(U^{*} \in {\mathbb {R}}^{n\times n}\) (\(m=n^2\)) corresponding to \(u^{*}\in {\mathbb {R}}^m\) and then present some definitions and existing results.

Denote \(\mathbf{I }^{*}=\{(i_{x},i_{y}):l_{2}<U^{*}_{i_{x},i_{y}}<l_{1}\}\), \(\mathbf{B }^{*}=\mathbf{J }\setminus \mathbf{I }^{*}\), \(\mathbf{B }_{l_{1}}^{*}=\{(i_{x},i_{y}):U^{*}_{i_{x},i_{y}}=l_{1}\}\), \(\mathbf{B }_{l_{2}}^{*}=\{(i_{x},i_{y}):U^{*}_{i_{x},i_{y}}=l_{2}\}\), \({\mathbf {I}}^{*}_{0}=\{(i_{x},i_{y})\in {\mathbf {I}}^{*}: \sqrt{(D_{x}^{+}U^{*})^{2}_{i_{x},i_{y}}+(D_{y}^{+}U^{*})^{2}_{i_{x},i_{y}}}= 0\}\) and \(\mathbf {\Omega }_1^{K} = \left\{ (i_{x},i_{y})\in {\mathbf {J}}: \sqrt{(D_{x}^{+}U^{*})^{2}_{i_{x},i_{y}}+(D_{y}^{+}U^{*})^{2}_{i_{x},i_{y}}} \ne 0\right\} .\) Similar to the one-to-one correspondence between J and \({\mathbf {J}}\), i.e., \(J \rightarrow {\mathbf {J}}\): \(i\mapsto (i_x,i_y)\), we have one-to-one correspondences for \(I^{*} \rightarrow {\mathbf {I}}^{*}\), \(B^{*} \rightarrow {\mathbf {B}}^{*}\), \(I_{0}^{*} \rightarrow {\mathbf {I}}_{0}^{*}\) and \(\Omega _1^{K} \rightarrow \mathbf {\Omega }_1^{K}\).

For a pixel \((i_{x},i_{y})\), let \({\mathcal {N}}_{1}(i_{x},i_{y})=\{(i_{x},i_{y}),(i_{x},i_{y}+1),(i_{x}+1,i_{y})\}\) represent its 1-neighborhood, and two edges in \({\mathcal {N}}_{1}(i_{x},i_{y})\) are given by \( e^{x}_{i_{x},i_{y}}=\big ((i_{x},i_{y}),(i_{x},i_{y}+1)\big ),\ e^{y}_{i_{x},i_{y}}=\big ((i_{x},i_{y}),(i_{x}+1,i_{y})\big ). \) We also define \(e(i_{x},i_{y})=\{e^{x}_{i_{x},i_{y}}, e^{y}_{i_{x},i_{y}}\}\), and \(e(\hat{\mathbf{J }})=\bigcup _{(i_{x},i_{y})\in \hat{\mathbf{J }}} e(i_{x},i_{y})\), where \(\hat{\mathbf{J }}\) is a subset of \(\mathbf{J }\). Moreover, the 1-neighborhood of a set \(\hat{\mathbf{J }}\) is \( {\mathcal {N}}_{1}(\hat{\mathbf{J }})= \{(i_{x},i_{y})\in \mathbf{J }: (i_{x},i_{y})\in {\mathcal {N}}_{1}({\hat{i}}_{x},{\hat{i}}_{y}) \ \text {for some} \ ({\hat{i}}_{x},{\hat{i}}_{y})\in \hat{\mathbf{J }}\}. \)

For two pixels \(((i_{s+1})_{x},(i_{s+1})_{y})\) and \(((i_{s+t})_{x},(i_{s+t})_{y})\), if there exist a sequence of pixels

$$\begin{aligned} ((i_{s+2})_{x},(i_{s+2})_{y}),((i_{s+3})_{x},(i_{s+3})_{y}),\cdots , ((i_{s+t-1})_{x},(i_{s+t-1})_{y}) \end{aligned}$$

such that \(((i_{s+l})_{x},(i_{s+l})_{y})\) and \(((i_{s+l+1})_{x},(i_{s+l+1})_{y})\) being adjacent pixels for \(1 \le l \le t-1\), then these pixels form a path connecting \(((i_{s+1})_{x},(i_{s+1})_{y})\) and \(((i_{s+t})_{x},(i_{s+t})_{y})\).

Definition 7.3

For a subset \(\hat{{\mathbf {J}}}\subseteq {\mathbf {J}}\), if for any two pixels \(((i_{s+1})_{x},(i_{s+1})_{y})\) and \(((i_{s+t})_{x},(i_{s+t})_{y})\) of \({\mathcal {N}}_{1}(\hat{{\mathbf {J}}})\), there exists a path \(P=\{((i_{s+1})_{x},(i_{s+1})_{y}),\cdots ,(i_{s+l})_{x},(i_{s+l})_{y}),\cdots ,((i_{s+t})_{x},(i_{s+t})_{y})\}\) such that

  1. (1)

    \(((i_{s+l})_{x},(i_{s+l})_{y}) \in {\mathcal {N}}_{1}(\hat{{\mathbf {J}}}), \ \text {for} \ 1 \le l \le t\),

  2. (2)

    \(\big (((i_{s+l})_{x},(i_{s+l})_{y}), ((i_{s+l+1})_{x},(i_{s+l+1})_{y})\big ) \in e(\hat{{\mathbf {J}}}), \ \text {for}\ 1 \le l \le t-1\),

connecting them, then we call \(\hat{{\mathbf {J}}}\) is a connected set and P is a path of \(\hat{{\mathbf {J}}}\).

Clearly, for any two connected subsets \(\hat{\mathbf{J }}_{a}\) and \(\hat{\mathbf{J }}_{b}\), if \(\hat{\mathbf{J }}_{a}\cup \hat{\mathbf{J }}_{b}\) is not connected, then \({\mathcal {N}}_{1}(\hat{\mathbf{J }}_{a})\cap {\mathcal {N}}_{1}(\hat{\mathbf{J }}_{a})=\emptyset \). Moreover, a subset \(\hat{\mathbf{J }}\) can be divided into several connected subsets \(\hat{\mathbf{J }}_{1}, \cdots , \hat{\mathbf{J }}_{k}\) satisfying \({\mathcal {N}}_{1}(\hat{\mathbf{J }}_{i})\cap {\mathcal {N}}_{1}(\hat{\mathbf{J }}_{j})=\emptyset \), when \(i\ne j\), and every subset \(\hat{\mathbf{J }}_{i}\) is called a connected component of \(\hat{\mathbf{J }}\).

Lemma 7.4

[39] For a connected set \(\hat{{\mathbf {J}}} \in {\mathbf {J}}\) and any \((i_{x},i_{y}) \in \hat{{\mathbf {J}}}\), suppose that \(|(D_{x}^{+}U)_{i_{x},i_{y}}| \le C\) and \(|(D_{y}^{+}U)_{i_{x},i_{y}}| \le C\) with C being a given constant, and \(U_{(i_{s})_{x},(i_{s})_{y}}=0\) for a given pixel \(((i_{s})_{x},(i_{s})_{y}) \in {\mathcal {N}}_{1}(\hat{{\mathbf {J}}})\). Then, we have

$$\begin{aligned} |U_{i_{x},i_{y}}|&< m C, \quad \forall (i_{x},i_{y}) \in {\mathcal {N}}_{1}(\hat{{\mathbf {J}}});\\ \sum _{(i_{x},i_{y}) \in {\mathcal {N}}_{1}(\hat{{\mathbf {J}}})} |U_{i_{x},i_{y}}|^2&< \frac{m(m+1)(2m+1)}{6}C^2. \end{aligned}$$

Definition 7.5

[39] For an edge \(\big (((i_{s})_{x},(i_{s})_{y}),((i_{t})_{x},(i_{t})_{y})\big )\), if \(|U^{*}_{(i_{s})_{x},(i_{s})_{y}}-U^{*}_{(i_{t})_{x},(i_{t})_{y}}|> \frac{|l_{2}-l_{1}|}{m}\), then \(\big (((i_{s})_{x},(i_{s})_{y}),((i_{t})_{x},(i_{t})_{y})\big )\) is called a leaping edge. Furthermore, if \((i_{s})_{x}+(i_{s})_{y}<(i_{t})_{x}+(i_{t})_{y}\), then \(((i_{s})_{x},(i_{s})_{y})\) is called a leaping pixel; if else, \(((i_{t})_{x},(i_{t})_{y})\) is called a leaping pixel.

This definition of leaping pixel is important in the proofs of Theorems 3.5 and 4.3.

Define \({\mathbf {L}}^{*}=\{(i_{x},i_{y})\in {\mathbf {J}}: (i_{x},i_{y}) \ \text {is a leaping pixel}\}\) as a set consisting all leaping pixels. We have the following lemma for the connected subset \({\mathbf {J}}^{*}_{a}\) of \(({\mathbf {I}}^{*}_{0} \cup \mathbf {\Omega }_1^{K})\setminus {\mathbf {L}}^{*}\).

Lemma 7.6

[39] For a connected subset \({\mathbf {J}}^{*}_{a}\) of \(({\mathbf {I}}^{*}_{0} \cup \mathbf {\Omega }_1^{K})\setminus {\mathbf {L}}^{*}\), we have \({\mathcal {N}}_{1}({\mathbf {J}}^{*}_{a})\cap {\mathbf {B}}^{*} = \emptyset \), or \({\mathcal {N}}_{1}({\mathbf {J}}^{*}_{a})\cap {\mathbf {B}}^{*} \subseteq \mathbf{B }_{l_{1}}^{*}\), or \({\mathcal {N}}_{1}({\mathbf {J}}^{*}_{a})\cap {\mathbf {B}}^{*} \subseteq \mathbf{B }_{l_{2}}^{*}\).

Note that, we similarly have one-to-one correspondences for \(L^{*} \rightarrow {\mathbf {L}}^{*}\) and \({\mathcal {N}}_{1}(J^{*}_{a}) \rightarrow {\mathcal {N}}_{1}({\mathbf {J}}^{*}_{a})\).

1.2.2 The proof

Proof

By the first-order necessary condition for the local minimizer \(z^{*}\) of (5), we have

$$\begin{aligned} \left\langle \partial R(z^*), {\widehat{z}} \right\rangle = 0, \quad \forall \ {\widehat{z}} \in {\mathcal {K}}(I^{*}_0), \end{aligned}$$

where

$$\begin{aligned} {\mathcal {K}}(I^{*}_0)=\{z\in {\mathbb {R}}^{|I^{*}|}: (d_{i})_{I^{*}}z=0, \ \forall i\in I^{*}_0\}. \end{aligned}$$
(32)

In detail, we compute \(\left\langle \partial R(z^*), {\widehat{z}} \right\rangle \) as

$$\begin{aligned} \left\langle \partial R(z^*), {\widehat{z}} \right\rangle&= \left\langle \sum _{i \in \Omega _1^{*}} \varphi _1'(\Vert d_{i}u^{*}\Vert )\frac{d_{i}u^{*}}{\Vert d_{i}u^{*}\Vert }(d_{i})^{T}_{I^{*}} + \alpha _{1}\sum _{i \in {\widetilde{J}}} \partial \varphi _2(q_{i}u^{*})(q_{i})_{I^{*}}^{T} + \alpha _{2} H_{I^{*}}^{T}(H_{I^{*}}z^{*}-g^{*}), {\widehat{z}} \right\rangle \\&= \sum _{i \in \Omega _1^{*}} \varphi _1'(\Vert d_{i}u^{*}\Vert )\left\langle \frac{d_{i}u^{*}}{\Vert d_{i}u^{*}\Vert }, (d_{i})_{I^{*}}{\widehat{z}} \right\rangle + \alpha _{1}\sum _{i \in {\widetilde{J}}} \partial \varphi _2(q_{i}u^{*}) (q_{i})_{I^{*}}{\widehat{z}} + \alpha _{2} \left\langle H_{I^{*}}z^{*}-g^{*}, H_{I^{*}}{\widehat{z}} \right\rangle . \end{aligned}$$

That is,

$$\begin{aligned} \sum _{i \in \Omega _1^{*}} \varphi _1'(\Vert d_{i}u^{*}\Vert )\left\langle \frac{d_{i}u^{*}}{\Vert d_{i}u^{*}\Vert }, (d_{i})_{I^{*}}{\widehat{z}} \right\rangle&= -\alpha _{1}\sum _{i \in {\widetilde{J}}} \partial \varphi _2(q_{i}u^{*}) (q_{i})_{I^{*}}{\widehat{z}} -\alpha _{2} \left\langle H_{I^{*}}z^{*}-g^{*}, H_{I^{*}}{\widehat{z}} \right\rangle \\ [\hbox { by} |\partial \varphi _2(q_{i}u^{*})|\le 1\, ] \le&\alpha _{1}\sum _{i \in {\widetilde{J}}}\Vert q_{i}\Vert \Vert {\widehat{z}}\Vert +\alpha _{2}\sqrt{F(u^*)}\Vert H_{I^{*}}\Vert \Vert {\widehat{z}}\Vert \\&= (4\sqrt{6}m\alpha _{1}+\alpha _{2}\sqrt{F(u^*)}\Vert H_{I^{*}}\Vert )\Vert {\widehat{z}}\Vert :< {\widehat{\delta }}\Vert {\widehat{z}}\Vert , \end{aligned}$$

where \({\widehat{\delta }}>0\) is a constant independent on \(u^{*}\). The reason is that, for the given \({\widehat{u}}\), we can choose its neighbourhood including \(u^*\) such that \(4\sqrt{6}m\alpha _{1}+\alpha _{2}\sqrt{F(u^*)}\Vert H_{I^{*}}\Vert <{\widehat{\delta }}\).

Moreover, we have

$$\begin{aligned} \begin{aligned} {\widehat{\delta }}\Vert {\widehat{z}}\Vert > \sum _{i \in \Omega _1^{*}} \varphi _1'(\Vert d_{i}u^{*}\Vert )\left\langle \frac{d_{i}u^{*}}{\Vert d_{i}u^{*}\Vert }, (d_{i})_{I^{*}}{\widehat{z}} \right\rangle&= \left( \sum _{i \in L_1^{*}}+\sum _{i \in B_1^{*}}\right) \varphi _1'(\Vert d_{i}u^{*}\Vert )\left\langle \frac{d_{i}u^{*}}{\Vert d_{i}u^{*}\Vert }, (d_{i})_{I^{*}}{\widehat{z}}\right\rangle \\ [\hbox { by Lemma}~7.2\hbox {(b) }] =&\sum _{i \in L_1^{*}} \varphi _1'(\Vert d_{i}u^{*}\Vert )\left\langle \frac{d_{i}u^{*}}{\Vert d_{i}u^{*}\Vert }, (d_{i})_{I^{*}}{\widehat{z}}\right\rangle . \end{aligned} \end{aligned}$$
(33)

By analyzing (33), we prove \(\Omega _{1}^{*}=\Omega _{1}(u^*) \subseteq \Omega _1({\widehat{u}})\).

According to the definitions and properties of \(u^{*}\) and its matrix representation \(U^{*}\) in  7.2.1, we know that \(\Vert d_{i}u^{*}\Vert > \frac{l_{2}-l_{1}}{m}\) for \(\forall i \in L^{*}\). Accordingly, since \(u^*\) is very near to \({\widehat{u}}\), we can find a constant \(\vartheta >0\) such that \(\Vert d_{i}{\widehat{u}}\Vert >\vartheta \) for \(\forall i \in L^{*}\). Clearly, \(L^{*} \subseteq \Omega _1({\widehat{u}})\).

Next, we verify \((\Omega _{1}^{*}\setminus L^{*})\subseteq \Omega _1({\widehat{u}})\) by induction. From \(B_{1}^{*} \subseteq L^{*}\), we obtain \(\Omega _{1}^{*}\setminus L^{*}=L_1^{*}\setminus L^{*}\). Then, we just show \(L_1^{*}\setminus L^{*}\subseteq \Omega _1({\widehat{u}})\) by induction.

For all \(i \in L_1^{*}\setminus L^{*}\), we sort the values of \(\Vert d_{i}u^{*}\Vert \) as

$$\begin{aligned} \Theta ^{*}=\{\varpi ^{*}_{1}, \varpi ^{*}_{2},\cdots ,\varpi ^{*}_{r}\}, \end{aligned}$$

where \(\varpi ^{*}_{1}>\varpi ^{*}_{2}>\cdots >\varpi ^{*}_{r}\) and \(r\le \#(L_1^{*}\setminus L^{*})<m\). Define \(E^{*}_{j}=\{i\in L_1^{*}\setminus L^{*}: \Vert d_{i}u^{*}\Vert =\varpi ^{*}_{j}\}\).

First, we select a \({z}^{1}\) such that

$$\begin{aligned} (d_{i})_{I^{*}}{z}^{1}=\frac{1}{\varpi ^{*}_{1}}d_{i}u^{*},\quad \forall i\in (I^{*}_{0}\cup \Omega _{1}^{*})\setminus L^{*}. \end{aligned}$$
(34)

To easily find a solution of (34), we construct the following problem

$$\begin{aligned} \left\{ \begin{aligned}&d_{i}{u}^{1}=\frac{1}{\varpi ^{*}_{1}}d_{i}u^{*},&\forall i\in (I^{*}_{0}\cup \Omega _{1}^{*})\setminus L^{*},\\&{u}^{1}_{i}=0,&\forall i\in {\mathcal {N}}_{1}((I^{*}_{0}\cup \Omega _{1}^{*})\setminus L^{*})\cap B^{*}. \end{aligned} \right. \end{aligned}$$
(35)

Once (35) is solved, (34) will be solved by setting \({z}^{1}=({u}^{1})_{I^{*}}\). Suppose all connected subsets of \((I^{*}_{0}\cup \Omega _{1}^{*})\setminus L^{*}\) are \(J_{1},J_{2},\cdots ,J_{l}\). The problem (35) is further equivalent to

$$\begin{aligned} \left\{ \begin{aligned}&d_{i}{u}^{1}=\frac{1}{\varpi ^{*}_{1}}d_{i}u^{*},&\forall i\in J_{j},\\&{u}^{1}_{i}=0,&\forall i\in {\mathcal {N}}_{1}(J_{j})\cap B^{*}, \end{aligned} \right. \end{aligned}$$
(36)

where \(1\le j \le l\).

Similar to [39], since either \({\mathcal {N}}_{1}(J_{j})\cap B^{*}=\emptyset \) or \({\mathcal {N}}_{1}(J_{j})\cap B^{*}\subseteq B_{l_{1}}^{*}= \{i\in J: u_{i} = l_{1}\}\) or \({\mathcal {N}}_{1}(J_{j})\cap B^{*}\subseteq B_{l_{2}}^{*}= \{i\in J: u_{i} = l_{2}\}\) from Lemma 7.6, a solution of (36) is given by

$$\begin{aligned} {u}^{1}_{i}= {\left\{ \begin{array}{ll} \frac{{u}^{*}_{i}-c_{j}}{\varpi ^{*}_{1}}, &{} \text {if} \ i\in {\mathcal {N}}_{1}(J_{j}) \ \text {with} \ {\mathcal {N}}_{1}(J_{j})\cap B^{*}=\emptyset ,\\ \frac{{u}^{*}_{i}-l_{1}}{\varpi ^{*}_{1}}, &{}\text {if} \ i\in {\mathcal {N}}_{1}(J_{j}) \ \text {with} \ {\mathcal {N}}_{1}(J_{j})\cap B^{*}\subseteq B_{l_{1}}^{*},\\ \frac{{u}^{*}_{i}-l_{2}}{\varpi ^{*}_{1}}, &{}\text {if} \ i\in {\mathcal {N}}_{1}(J_{j}) \ \text {with} \ {\mathcal {N}}_{1}(J_{j})\cap B^{*}\subseteq B_{l_{2}}^{*},\\ 0, &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$
(37)

where \(c_{j}={u}^{*}_{m_{j}}\) with \(m_{j}=\mathrm {min}({\mathcal {N}}_{1}(J_{j}))\). Clearly, (36) is solvable and so is (35).

Due to \(\frac{|d^{x}_{i}u^{*}|}{\varpi ^{*}_{1}} \le 1\) and \(\frac{|d^{y}_{i}u^{*}|}{\varpi ^{*}_{1}} \le 1\), we can obtain \(|{u}^{1}_{i}| < m\) and \(\Vert {u}^{1}\Vert ^{2}<\frac{m(m+1)(2m+1)}{6}\) by Lemma 7.4. Denote \(\Gamma =\frac{m(m+1)(2m+1)}{6}\). Accordingly, we have \(\Vert {z}^{1}\Vert ^{2}<\Gamma \ \text {and} \ |{z}^{1}_{i}| < m, \forall i \in I^{*}.\) Moreover, \(|(d^{x}_{i})_{I^{*}}{z}^{1}|<2m\) and \(|(d^{y}_{i})_{I^{*}}{z}^{1}|<2m\) for \(\forall i \in J\). Thus, \( \Vert (d_{i})_{I^{*}}{z}^{1}\Vert ^2=|(d^{x}_{i})_{I^{*}}{z}^{1}|^2+|(d^{y}_{i})_{I^{*}}{z}^{1}|^2<8m^2,\ \forall i \in J. \) By (34), it yields

$$\begin{aligned} \left\langle d_{i}u^{*}, (d_{i})_{I^{*}}{z}^{1}\right\rangle =\frac{\Vert d_{i}u^{*}\Vert ^2}{\varpi ^{*}_{1}}, \ \forall i \in L_1^{*}\setminus L^{*}. \end{aligned}$$

When choosing \({\widehat{z}}={z}^{1}\) in (33), we obtain

$$\begin{aligned} \begin{aligned} {\widehat{\delta }}\sqrt{\Gamma }> {\widehat{\delta }}\Vert {z}^{1}\Vert&\ge \sum _{i \in L_1^{*}\setminus L^{*}} \varphi _1'(\Vert d_{i}u^{*}\Vert )\frac{\Vert d_{i}u^{*}\Vert }{\varpi ^{*}_{1}}+ \sum _{i \in L^{*}\cap L_1^{*}}\varphi _1'(\Vert d_{i}u^{*}\Vert )\left\langle \frac{d_{i}u^{*}}{\Vert d_{i}u^{*}\Vert }, (d_{i})_{I^{*}}{z}^{1}\right\rangle \\ [\hbox { by Remark}~3.1\hbox {(a) }]&\ge \sum _{i \in E^{*}_1} \varphi _1'(\Vert d_{i}u^{*}\Vert )\frac{\Vert d_{i}u^{*}\Vert }{\varpi ^{*}_{1}}- \sum _{i \in L^{*}\cap L_1^{*}}\varphi _1'(\Vert d_{i}u^{*}\Vert )\frac{\Vert d_{i}u^{*}\Vert }{\Vert d_{i}u^{*}\Vert }\Vert (d_{i})_{I^{*}}{z}^{1}\Vert \\&> \sum _{i \in E^{*}_1} \varphi _1'(\Vert d_{i}u^{*}\Vert )-8m^2\sum _{i \in L^{*}\cap L_1^{*}}\varphi _1'(\Vert d_{i}u^{*}\Vert ). \end{aligned} \end{aligned}$$

Then,

$$\begin{aligned} \begin{aligned} \sum _{i \in E^{*}_1} \varphi _1'(\Vert d_{i}u^{*}\Vert )< {\widehat{\delta }}\sqrt{\Gamma } + 8m^2\sum _{i \in L^{*}\cap L_1^{*}}\varphi _1'(\Vert d_{i}u^{*}\Vert ) < {\widehat{\delta }}\sqrt{\Gamma } + 8m^2\sum _{i \in L^{*}\cap L_1^{*}}\varphi _1'\left( \frac{l_{2}-l_{1}}{m}\right) \end{aligned}\nonumber \\ \end{aligned}$$
(38)

Based on (38), we prove \(E^{*}_{1} \subseteq \Omega _1({\widehat{u}})\) by contradiction. If there exists \(j\in E^{*}_{1}\) such that \(\Vert d_{j}{\widehat{u}}\Vert = 0\), then \(\Vert d_{j}u^*\Vert \) can be very close to \(\Vert d_{j}{\widehat{u}}\Vert =0\) because of \(u^*\) being very near to \({\widehat{u}}\). From \(\varphi _1'(t)|_{0+}=+\infty \) in Remark 3.1(a), we know that the Eq. (38) is impossible to be true. Thus, \(\Vert d_{j}{\widehat{u}}\Vert \ne 0\), i.e., \(j\in \Omega _1({\widehat{u}})\), which implies \(E^{*}_{1} \subseteq \Omega _1({\widehat{u}})\).

Now, we assume that \(E^{*}_{j}\subseteq \Omega _1({\widehat{u}})\) for any \(j\in \{1,2,\ldots , s-1\}\) with \(1<s<r\). Next, we prove that \(E^{*}_{s}\subseteq \Omega _1({\widehat{u}})\).

Let \(E^{*}=(\cup _{j=1}^{s-1}E^{*}_{j})\cup L^{*}\). Similarly, we want to select a \({z}^{s}\) such that

$$\begin{aligned} (d_{i})_{I^{*}}{z}^{s}=\frac{1}{\varpi ^{*}_{s}}d_{i}u^{*},\quad \forall i\in (I^{*}_{0}\cup \Omega _{1}^{*})\setminus E^{*}. \end{aligned}$$
(39)

As the discussion of \({z}^{1}\), (39) has a solution \({z}^{s}\) satisfying

$$\begin{aligned} \begin{aligned}&\Vert {z}^{s}\Vert ^{2}<\Gamma ; \ |{z}^{s}_{i}|< m, \quad \forall i \in I^{*}; \\&\left\langle d_{i}u^{*}, (d_{i})_{I^{*}}{z}^{s}\right\rangle =\frac{\Vert d_{i}u^{*}\Vert ^2}{\varpi ^{*}_{s}}, \quad \forall i \in L_1^{*}\setminus E^{*}; \\&\Vert (d_{i})_{I^{*}}{z}^{s}\Vert ^2=|(d^{x}_{i})_{I^{*}}{z}^{s}|^2+|(d^{y}_{i})_{I^{*}}{z}^{s}|^2<8m^2, \quad \forall i \in J. \end{aligned} \end{aligned}$$

Choosing \({\widehat{z}}={z}^{s}\) in (33), we have

$$\begin{aligned} \begin{aligned} {\widehat{\delta }}\sqrt{\Gamma }> {\widehat{\delta }}\Vert {z}^{s}\Vert&\ge \sum _{i \in L_1^{*}\setminus E^{*}} \varphi _1'(\Vert d_{i}u^{*}\Vert )\frac{\Vert d_{i}u^{*}\Vert }{\varpi ^{*}_{s}}+ \sum _{i \in E^{*}\cap L_1^{*}}\varphi _1'(\Vert d_{i}u^{*}\Vert ) \left\langle \frac{d_{i}u^{*}}{\Vert d_{i}u^{*}\Vert }, (d_{i})_{I^{*}}{z}^{s}\right\rangle \\ [\hbox { by Remark}~3.1\hbox {(a) }]&\ge \sum _{i \in E^{*}_s} \varphi _1'(\Vert d_{i}u^{*}\Vert )\frac{\Vert d_{i}u^{*}\Vert }{\varpi ^{*}_{s}}- \sum _{i \in E^{*}\cap L_1^{*}}\varphi _1'(\Vert d_{i}u^{*}\Vert )\frac{\Vert d_{i}u^{*}\Vert }{\Vert d_{i}u^{*}\Vert }\Vert (d_{i})_{I^{*}}{z}^{s}\Vert \\&> \sum _{i \in E^{*}_s} \varphi _1'(\Vert d_{i}u^{*}\Vert )-8m^2\sum _{i \in E^{*}\cap L_1^{*}}\varphi _1'(\Vert d_{i}u^{*}\Vert ). \end{aligned} \end{aligned}$$

From the inductive hypothesis of \(E^{*}_{j}\subseteq \Omega _1({\widehat{u}})\) for any \(j\in \{1,\cdots , s-1\}\) and \(\Vert d_{i}u^{*}\Vert \) being close to \(\Vert d_{i}{\hat{u}}\Vert \), \(\Vert d_{i}u^{*}\Vert \) is bounded for \(\forall i \in E^{*}_{j}\). Thus, \(\sum _{i \in E^{*}\cap L_1^{*}}\varphi _1'(\Vert d_{i}u^{*}\Vert )\) is also bouned, and there exists a constant \(\Lambda \) such that \(\sum _{i \in E^{*}\cap L_1^{*}}\varphi _1'(\Vert d_{i}u^{*}\Vert )<\Lambda \). Then, the inequality above is respresented as

$$\begin{aligned} \begin{aligned} \sum _{i \in E^{*}_s} \varphi _1'(\Vert d_{i}u^{*}\Vert ) < {\widehat{\delta }}\sqrt{\Gamma } + 8m^2\Lambda . \end{aligned} \end{aligned}$$
(40)

From (40), we also use contradiction to prove \(E^{*}_{s} \subseteq \Omega _1({\widehat{u}})\) as proving \(E^{*}_{1} \subseteq \Omega _1({\widehat{u}})\).

To sum up, from all \(E^{*}_{j} \subseteq \Omega _1({\widehat{u}})\) for \(\forall j \in \{1,2,\cdots ,r\}\), then \(L_1^{*}\setminus L^{*}=\Omega _1^{*}\setminus L^{*} \subseteq \Omega _1({\widehat{u}})\). In addition, we have shown \(L^{*} \subseteq \Omega _1({\widehat{u}})\). Thus, \(\Omega _1^{*}=\Omega _1(u^{*}) \subseteq \Omega _1({\widehat{u}})\), which completes the whole proof. \(\square \)

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Wang, W., Wu, C. & Tai, XC. A Globally Convergent Algorithm for a Constrained Non-Lipschitz Image Restoration Model. J Sci Comput 83, 14 (2020). https://doi.org/10.1007/s10915-020-01190-4

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