Hybrid Multi-level Regularizations with Sparse Representation for Single Depth Map Super-Resolution

Abstract

Limited spatial resolution and varieties of degradations are the main restrictions of today’s captured depth map by active 3D sensing devices. Typical restrictions limit the direct use of the obtained depth maps in most of 3D applications. In this paper, we present a single depth map upsampling approach in contrast to the common work of using the corresponding combined color image to guide the upsampling process. The proposed approach employs a multi-level decomposition to convert the depth upsampling process to a classification-based problem via a multi-level classification-based learning algorithm. Hence, the lost high frequency details can be better preserved at different levels. The adopted multi-level decomposition algorithm utilizes $l_{1} ,$ and $l_{0}$ sparse regularization with total-variation regularization to keep structure- and edge-preserving smoothing with robustness to noisy degradations. In addition, the proposed classification-based learning algorithm supports the accuracy of discrimination by learning discriminative dictionaries that carry original features about each class and learning common shared dictionaries that represent the shared features between classes. The proposed algorithm has been validated via different experiments under variety of degradations using different datasets from different sensing devices. Results show superiority to the state of the art, especially in case of upsampling noisy low-resolution depth maps.

Appendices

Appendix 1

For solving the proposed learning model in Eq. 19 as

$$\arg \mathop {\hbox{min} }\limits_{{{\mathcal{A}}, S}} \left\| {{\mathcal{M}} - {\mathcal{A}}S} \right\|_{2}^{2} + \sum\limits_{i = 1}^{{{ \dot{C} }}} {\left\{ {\left\| {\dot{M}_{i}^{HL} - \dot{A}_{i}^{HL} S_{i}^{HL} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL} } \right\|_{2}^{2} + \mathop \sum \limits_{\varepsilon = 1,\varepsilon \ne i}^{{{{ \dot{C} }} + 1}} \left\| {S_{\varepsilon }^{HL} } \right\|_{2}^{2} + \sigma_{1} \mathop \sum \limits_{u = 1,u \ne i}^{{{{ \dot{C} }} + 1}} \left\| {\dot{A}_{i}^{{HL^{T} }} \dot{A}_{u}^{HL} } \right\|_{2}^{2} + \mathop \sum \limits_{j = 1}^{{ {C} _{i} }} \left( {\left\| {\dot{M}_{i,j}^{HL} - \dot{A}_{i,j}^{HL} S_{i,j}^{HL} - \dot{A}_{{i,{{Q}}}}^{HL} S_{{i,{{Q}}}}^{HL} } \right\|_{2}^{2} + \mathop \sum \limits_{\varUpsilon = 1,\varUpsilon \ne j}^{{ {C} _{i} + 1}} \left\| {S_{i,\varUpsilon }^{HL} } \right\|_{2}^{2} + \sigma_{2} \mathop \sum \limits_{j = 1}^{{ {C} _{i} + 1}} \mathop \sum \limits_{v = 1,v \ne j}^{{ {C} _{i} + 1}} \left\| {\dot{A}_{i,j}^{{HL^{T} }} \dot{A}_{i,v}^{HL} } \right\|_{2}^{2} } \right)} \right\} + \sigma_{3} \left\| S \right\|_{1} }$$

(19)

An alternative optimization process, by updating the targeted variable while fixing all other variables, is followed. In this procedure, we need to update the overall structured dictionary considering both the shared and main dictionaries besides updating the coding matrix $S$ considering both the shared and the main codes.

For Updating Dictionaries

The overall composite dictionary ${\mathcal{A}}$ is expressed as ${\mathcal{A}} = \left[ {\dot{A}_{{B_{2} }}^{HL} , \dot{A}_{{B_{2} ,{{Q}}}}^{HL} ,\dot{A}_{{D_{1} }}^{HL} ,\dot{A}_{{D_{1} ,{{Q}}}}^{HL} ,\dot{A}_{{D_{2} }}^{HL} ,\dot{A}_{{D_{2} ,{{Q}}}}^{HL} ,\dot{A}_{{{Q}}}^{HL} } \right]$ = $\left[ {\dot{A}_{{B_{2} ,1}}^{HL} , \ldots ,\dot{A}_{{B_{2} ,C_{{B_{2} }} }}^{HL} ,\dot{A}_{{B_{2} ,{{Q}}}}^{HL} ,\dot{A}_{{D_{1} ,1}}^{HL} , \ldots ,\dot{A}_{{D_{1} ,C_{{D_{1} }} }}^{HL} ,\dot{A}_{{D_{1} ,{{Q}}}}^{HL} , \dot{A}_{{D_{2} ,1}}^{HL} , \ldots ,\dot{A}_{{D_{2} ,C_{{D_{2} }} }}^{HL} ,\dot{A}_{{D_{2} ,{{Q}}}}^{HL} , \dot{A}_{{{Q}}}^{HL} } \right]$. We have two types of dictionaries main particular dictionaries $\dot{A}_{i}^{HL}$ and common dictionaries $\dot{A}_{{i,{{Q}}}}^{HL}$ and $\dot{A}_{{{Q}}}^{HL} .$ Hence, in each time, we will update only one kind of dictionaries while fixing the others beside, of course, all coding matrices

2.1 For Updating $\dot{\varvec{A}}_{\varvec{i}}^{{\varvec{HL}}}$

$\dot{A}_{i}^{HL}$ can be updated subclass by subclass. Hence, when updating the subclass dictionary $\dot{A}_{i,j}^{HL}$, all others $\dot{A}_{a,b}^{HL}$,$a \ne i$ & $b \ne j$ are fixed. Hence, the energy minimization problem in Eq. 19 is turned into

$$\left\langle {\dot{A}_{i,j}^{HL} } \right\rangle = \arg \mathop {\hbox{min} }\limits_{{\dot{A}_{i,j}^{HL} }} \left\{ {\left\| {{\mathcal{M}} - {\mathcal{A}}S} \right\|_{2}^{2} + \left\| {\dot{M}_{i}^{HL} - \dot{A}_{i}^{HL} S_{i}^{HL} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL} } \right\|_{2}^{2} + \left\| {\dot{M}_{i,j}^{HL} - \dot{A}_{i,j}^{HL} S_{i,j}^{HL} - \dot{A}_{{i,{{Q}}}}^{HL} {\text{S}}_{{i,{{Q}}}}^{HL} } \right\|_{2}^{2} + \sigma_{2} \mathop \sum \limits_{v = 1, v \ne j}^{{C_{i} + 1}} \left\| {\dot{A}_{i,j}^{{HL^{T} }} \dot{A}_{i,v}^{HL} } \right\|_{2}^{2} } \right\}$$

(19.1)

where $\mathop \sum \nolimits_{v = 1, v \ne j}^{{C_{i} + 1}} \left\| {\dot{A}_{i,j}^{{HL^{T} }} \dot{A}_{i,v}^{HL} } \right\|_{2}^{2}$ is a general incoherence term between $\dot{A}_{i,j}^{HL}$ and all other sub-classes’ dictionaries. Then, Eq. (19.1) can be converted into Eq. (19.2) in terms of the targeted $\dot{A}_{i,j}^{HL}$ as

$$\left\langle {{\text{A}}_{i,j}^{HL} } \right\rangle = \arg \mathop { \hbox{min} }\limits_{{\dot{A}_{i,j}^{HL} }} \left\{ {\begin{array}{*{20}l} {\left\| {{\mathcal{M}} - \dot{A}_{i,j}^{HL} S_{i,j}^{HL} - \mathop \sum \limits_{{\begin{array}{*{20}c} {a = 1,\forall a \ne i } \\ \end{array} }}^{{{\dot{C}}}} \mathop \sum \limits_{{\begin{array}{*{20}c} {b = 1,\forall b \ne j } \\ \end{array} }}^{{{{\dot{C}}}_{a} + 1}} \dot{A}_{a,b}^{HL} S_{a,b}^{HL} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL} } \right\|_{2}^{2} } \hfill \\ {\quad + \left\| {{\dot{\text{M}}}_{i}^{HL} - \dot{A}_{i,j}^{HL} S_{i,j}^{HL} - \mathop \sum \limits_{p = 1}^{{{{\dot{C}}}_{i} + 1}} \dot{A}_{i,p}^{HL} S_{i,p}^{HL} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL} } \right\|_{2}^{2} } \hfill \\ {\quad + \left\| {{\dot{\text{M}}}_{i,j}^{HL} - \dot{A}_{i,j}^{HL} S_{i,j}^{HL} - \dot{A}_{{i,{{Q}}}}^{HL} S_{{i,{{Q}}}}^{HL} } \right\|_{2}^{2} } \hfill \\ {\quad + \sigma_{2} \mathop \sum \limits_{v = 1, v \ne j}^{{C_{i} + 1}} \left\| {\dot{A}_{i,j}^{{HL^{T} }} \dot{A}_{i,v}^{HL} } \right\|_{2}^{2} } \hfill \\ \end{array} } \right\}$$

(19.2)

Then, to the following simplified formula as

$$\left\langle {\dot{A}_{i,j}^{HL} } \right\rangle = \arg \mathop {\hbox{min} }\limits_{{\dot{A}_{i,j}^{HL} }} \left\{ {\begin{array}{*{20}c} {\left\| {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathcal{M}} } - \dot{A}_{i,j}^{HL} S_{i,j}^{HL} } \right\|_{2}^{2} } \\ { + \left\| {\overset \smile{\dot {\text M}}_{i}^{HL} - \dot{A}_{i,j}^{HL} S_{i,j}^{HL} } \right\|_{2}^{2} + \left\| {\dot{M}_{i,j}^{HL} - \dot{A}_{i,j}^{HL} S_{i,j}^{HL} } \right\|_{2}^{2} + } \\ {\sigma_{2} \mathop \sum \limits_{v = 1, v \ne j}^{{C_{i} + 1}} \left\| {\dot{A}_{i,j}^{{HL^{T} }} \dot{A}_{i,v}^{HL} } \right\|_{2}^{2} } \\ \end{array} } \right\}$$

(19.3)

where ${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathcal{M}} } = {\mathcal{M}} - \mathop \sum \nolimits_{{\begin{array}{*{20}c} {a = 1,\forall a \ne i } \\ \end{array} }}^{{{\dot{C}}}} \mathop \sum \nolimits_{{\begin{array}{*{20}c} {b = 1,\forall b \ne j } \\ \end{array} }}^{{{{\dot{C}}}_{a} + 1}} \dot{A}_{a,b}^{HL} S_{a,b}^{HL} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL}$, and $\overset \smile{\dot {\text M}}_{i}^{HL} = \dot{M}_{i}^{HL} - \mathop \sum \nolimits_{p = 1}^{{{{\dot{C}}}_{i} + 1}} \dot{A}_{i,p}^{HL} S_{i,p}^{HL} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL}$. Equation (19.3) is a quadratic programming problem can be solved by updating $\dot{A}_{i,j}^{HL}$ atom by atom [39].

2.2 For Updating the Common Featured Dictionary of Subclasses $\dot{\varvec{A}}_{{\varvec{i},\varvec{Q}}}^{{\varvec{HL}}}$

The energy minimization problem in Eq. 19 is turned into the following problem in terms of $\dot{\varvec{A}}_{{\varvec{i},{\mathbf{{Q}}}}}^{{\varvec{HL}}}$ as

$$\left\langle {\dot{A}_{{i,{{Q}}}}^{HL} } \right\rangle = \arg \mathop {\hbox{min} }\limits_{{\dot{A}_{{i,{{Q}}}}^{HL} }} \left\{ {\begin{array}{*{20}c} {\left\| {{\mathcal{M}} - {\mathcal{A}}S} \right\|_{2}^{2} + \left\| {\dot{M}_{i,j}^{HL} - \dot{A}_{i,j}^{HL} S_{i,j}^{HL} - \dot{A}_{{i,{{Q}}}}^{HL} S_{{i,{{Q}}}}^{HL} } \right\|_{2}^{2} } \\ {\sigma_{2} \mathop \sum \limits_{j = 1}^{{C_{i} }} \left\| {\dot{A}_{{i,{{Q}}}}^{{HL^{T} }} \dot{A}_{i,j}^{HL} } \right\|_{2}^{2} } \\ \end{array} } \right\}$$

(19.4)

which can be rewritten as

$$\arg \mathop {\hbox{min} }\limits_{{\dot{A}_{{i,{{Q}}}}^{HL} }} \left\{ {\begin{array}{*{20}c} {\left\| {{\mathcal{M}} - \dot{A}_{{i,{{Q}}}}^{HL} S_{{i,{{Q}}}}^{HL} - \mathop \sum \limits_{i = 1}^{{{\dot{C}}}} \mathop \sum \limits_{{\begin{array}{*{20}c} {j = 1 } \\ \end{array} }}^{{C_{i} }} \dot{A}_{i,j}^{HL} S_{i,j}^{HL} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL} } \right\|_{2}^{2} + } \\ {\left\| {\dot{M}_{i,j}^{HL} - \dot{A}_{i,j}^{HL} S_{i,j}^{HL} - \dot{A}_{{i,{{Q}}}}^{HL} S_{{i,{{Q}}}}^{HL} } \right\|_{2}^{2} + } \\ {\sigma_{2} \mathop \sum \limits_{j = 1}^{{C_{i} }} \left\| {\dot{A}_{{i,{{Q}}}}^{{HL^{T} }} \dot{A}_{i,j}^{HL} } \right\|_{2}^{2} } \\ \end{array} } \right\}$$

(19.5)

$$\arg \mathop {\hbox{min} }\limits_{{\dot{A}_{{i,{{Q}}}}^{HL} }} \left\{ {\begin{array}{*{20}c} {\left\| {\overline{\overline{{\mathcal{M}}}} - \dot{A}_{{i,{{Q}}}}^{HL} S_{{i,{{Q}}}}^{HL} } \right\|_{2}^{2} + \left\| {\overline{\overline{\dot{\text{M}}}}_{i,j}^{HL} - \dot{A}_{{i,{{Q}}}}^{HL} S_{{i,{{Q}}}}^{HL} } \right\|_{2}^{2} + } \\ {\sigma_{2} \mathop \sum \limits_{j = 1}^{{{{\dot{C}}}_{i} }} \left\| {\dot{A}_{{i,{{Q}}}}^{{HL^{T} }} \dot{A}_{i,j}^{HL} } \right\|_{2}^{2} } \\ \end{array} } \right\}$$

(19.6)

where $\overline{\overline{{\mathcal{M}}}} = {\mathcal{M}} - \mathop \sum \nolimits_{i = 1}^{{{C}}} \mathop \sum \nolimits_{{\begin{array}{*{20}c} {j = 1 } \\ \end{array} }}^{{C_{i} }} \dot{A}_{i,j}^{HL} S_{i,j}^{HL} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL}$, and $\overline{\overline{{{\dot{\text{M}}}}}}_{i,j}^{HL} = \dot{M}_{i,j}^{HL} - \dot{A}_{i,j}^{HL} S_{i,j}^{HL}$. Similar to Eq. (19.3), Eq. (19.6) is another quadratic problem can be solved by updating $\dot{A}_{{i,{{Q}}}}^{HL}$ atom by atom [39].

2.3 For Updating the Common Dictionary $\dot{\varvec{A}}_{\varvec{Q}}^{{\varvec{HL}}}$

The energy minimization of Eq. 19 can be turned, in terms of $\dot{A}_{\varvec{Q}}^{{\varvec{HL}}}$, into

$$\left\langle {\dot{A}_{{{Q}}}^{HL} } \right\rangle = \arg \mathop {\hbox{min} }\limits_{{\dot{A}_{{{Q}}}^{HL} }} \left\| {{\mathcal{M}} - {\mathcal{A}}S} \right\|_{2}^{2} + \mathop \sum \limits_{i = 1}^{{{\dot{C}}}} \left( {\left\| {\dot{M}_{i}^{HL} - \dot{A}_{i}^{HL} S_{i}^{HL} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL} } \right\|_{2}^{2} + \left\| {\sigma_{1} \dot{A}_{{{Q}}}^{{HL^{T} }} \dot{A}_{i}^{HL} } \right\|_{2}^{2} } \right)$$

(19.7)

which can be turned into

$$\left\langle {\dot{A}_{{{Q}}}^{HL} } \right\rangle = \arg \mathop {\hbox{min} }\limits_{{\dot{A}_{{{Q}}}^{HL} }} \left\{ {\begin{array}{*{20}c} {\left\| {{\mathcal{M}} - \mathop \sum \limits_{i = 1}^{{{\dot{C}}}} \left( {\dot{A}_{i}^{HL} S_{i}^{HL} + \dot{A}_{{i,{{Q}}}}^{HL} S_{{i,{{Q}}}}^{HL} } \right) - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL} } \right\|_{2}^{2} } \\ { + \mathop \sum \limits_{i = 1}^{{{\dot{C}}}} \left( {\left\| {\dot{M}_{i}^{HL} - \dot{A}_{i}^{HL} S_{i}^{HL} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL} } \right\|_{2}^{2} + \sigma_{1} \left\| {\dot{A}_{{{Q}}}^{{HL^{T} }} \dot{A}_{i}^{HL} } \right\|_{2}^{2} } \right)} \\ \end{array} } \right\}$$

(19.8)

which can be simplified to

$$\left\langle {\dot{A}_{{{Q}}}^{HL} } \right\rangle = \arg \mathop {\hbox{min} }\limits_{{{\text{A}}_{{{Q}}}^{HL} }} \left\| {{\bar{\mathcal{M}}} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL} } \right\|_{2}^{2} + \mathop \sum \limits_{i = 1}^{{{ \dot{C} }}} \left( {\left\| {{{\bar{\dot{{\text{M}}}}}}_{i}^{HL} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL} } \right\|_{2}^{2} + \sigma_{1} \left\| {\dot{A}_{{{Q}}}^{{HL^{T} }} \dot{A}_{i}^{HL} } \right\|_{2}^{2} } \right)$$

(19.9)

where ${\bar{\mathcal{M}}} = {\mathcal{M}} - \mathop \sum \nolimits_{i = 1}^{{{ \dot{C} }}} \left( {\dot{A}_{i}^{HL} S_{i}^{HL} + \dot{A}_{{i,{{Q}}}}^{HL} S_{{i,{{Q}}}}^{HL} } \right)$, and ${{\bar{\dot{\text{{M}}}}}}_{i}^{HL} = \dot{M}_{i}^{HL} - \dot{A}_{i}^{HL} S_{i}^{HL}$. Equation (19.9) is again another quadratic problem can be solved by updating $\dot{A}_{{{Q}}}^{HL}$ atom by atom [39].

Updating the Coding Coefficients

In this step when updating any type of coding matrices, all dictionaries are kept fixed.

3.1 Updating the Main Coding Coefficients $\varvec{S}_{\varvec{i}}^{{\varvec{HL}}}$

The main coding coefficient matrix will be updated through the coding coefficient of subclasses $S_{i,j}^{HL}$ as

$$\begin{aligned} \left\langle {S_{i,j}^{HL} } \right\rangle & = \arg \mathop {\hbox{min} }\limits_{{S_{i,j}^{HL} }} \left\| {{\mathcal{M}} - {\mathcal{A}}S} \right\|_{2}^{2} + \left\| {\dot{M}_{i}^{HL} - \dot{A}_{i}^{HL} S_{i}^{HL} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL} } \right\|_{2}^{2} \\ & \quad + \left\| {\dot{M}_{i,j}^{HL} - \dot{A}_{i,j}^{HL} S_{i,j}^{HL} - \dot{A}_{{i,{{Q}}}}^{HL} S_{{i,{{Q}}}}^{HL} } \right\|_{2}^{2} + \sigma_{3} \left\| {S_{i,j}^{HL} } \right\|_{1} \\ \end{aligned}$$

(19.10)

which can turned into

$$\begin{aligned} \left\langle {S_{i,j}^{HL} } \right\rangle & = \arg \mathop {\hbox{min} }\limits_{{S_{i,j}^{HL} }} \left\| {{\mathcal{M}} - \dot{A}_{i,j}^{HL} S_{i,j}^{HL} - \mathop \sum \limits_{{\begin{array}{*{20}c} {a = 1,\forall a \ne i } \\ \end{array} }}^{{\dot{C}} + 1} \mathop \sum \limits_{{\begin{array}{*{20}c} {b = 1,\forall b \ne j } \\ \end{array} }}^{{C_{a} + 1}} \dot{A}_{a,b}^{HL} S_{a,b}^{HL} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL} } \right\|_{2}^{2} \\ & \quad + \left\| {\dot{M}_{i}^{HL} - \dot{A}_{i,j}^{HL} S_{i,j}^{HL} - \mathop \sum \limits_{p = 1}^{{C_{i} + 1}} \dot{A}_{i,p}^{HL} S_{i,p}^{HL} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL} } \right\|_{2}^{2} + \left\| {\dot{M}_{i,j}^{HL} - \dot{A}_{i,j}^{HL} S_{i,j}^{HL} - \dot{A}_{{i,{{Q}}}}^{HL} S_{{i,{{Q}}}}^{HL} } \right\|_{2}^{2} + \sigma_{3} \left\| {S_{i,j}^{HL} } \right\|_{1} \\ \end{aligned}$$

(19.11)

which can be simplified to

$$\left\langle {S_{i,j}^{HL} } \right\rangle = \arg \mathop {\hbox{min} }\limits_{{S_{i,j}^{HL} }} \left\{ \begin{array}{*{20}c} {\left\| {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{M} - \dot{A}_{i,j}^{HL} S_{i,j}^{HL} } \right\|_{2}^{2} } \\ + \left\| {\mathop{{\dot{M}}}\limits^{\smile}}_{i}^{HL} - \dot{A}_{i,j}^{HL} S_{i,j}^{HL} \right\|_{2}^{2} + \left\| {\mathop{{\dot{M}}}\limits^{\smile}}_{i,j}^{HL} - \dot{A}_{i,j}^{HL} S_{i,j}^{HL} \right\|_{2}^{2} + \\ {\sigma_{3} \left\| {S_{i,j}^{HL} } \right\|_{1} } \\ \end{array} \right\}$$

(19.12)

which is a LASSO problem can be solved using the feature sign algorithm [43].

3.2 Updating the Shared Subclasses Coding Matrix $\varvec{S}_{{\varvec{i},\varvec{Q}}}^{{\varvec{HL}}}$

For updating $S_{{i,{{Q}}}}^{HL}$, the energy minimization problem of Eq. 19 is converted to

$$\arg \mathop {\hbox{min} }\limits_{{S_{{i,{{Q}}}}^{HL} }} \left\{ {\begin{array}{*{20}c} {\left\| {{\mathcal{M}} - \dot{A}_{{i,{{Q}}}}^{HL} S_{{i,{{Q}}}}^{HL} - \mathop \sum \limits_{i = 1}^{{{\dot{C}}}} \mathop \sum \limits_{{\begin{array}{*{20}c} {j = 1 } \\ \end{array} }}^{{{{\dot{C}}}_{i} }} \dot{A}_{i,j}^{HL} S_{i,j}^{HL} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL} } \right\|_{2}^{2} + } \\ {\left\| {\dot{M}_{i,j}^{HL} - \dot{A}_{i,j}^{HL} S_{i,j}^{HL} - \dot{A}_{{i,{{Q}}}}^{HL} S_{{i,{{Q}}}}^{HL} } \right\|_{2}^{2} + } \\ { + \sigma_{3} \left\| {S_{{i,{{Q}}}}^{HL} } \right\|_{1} } \\ \end{array} } \right\}$$

(19.13)

which can be turned into

$$\arg \mathop {\hbox{min} }\limits_{{\dot{A}_{{i,{{Q}}}}^{HL} }} \left\{ {\begin{array}{*{20}c} {\left\| {\overline{\overline{{\mathcal{M}}}} - \dot{A}_{{i,{{Q}}}}^{HL} S_{{i,{{Q}}}}^{HL} } \right\|_{2}^{2} + \left\| {\overline{\overline{{\dot{M}}}}_{i,j}^{HL} - \dot{A}_{{i,{{Q}}}}^{HL} S_{{i,{{Q}}}}^{HL} } \right\|_{2}^{2} } \\ { + \sigma_{3} \left\| {S_{{i,{{Q}}}}^{HL} } \right\|_{1} } \\ \end{array} } \right\}$$

(19.14)

which is another LASSO problem can be solved using the feature sign algorithm [43]

3.3 Updating the Classes Shared Coding Matrix $\varvec{S}_{{\mathbf{{Q}}}}^{{\varvec{HL}}}$

For updating $S_{{{Q}}}^{HL}$, the energy minimization problem of Eq. 19 is converted to

$$\left\langle {S_{{{Q}}}^{HL} } \right\rangle = \arg \mathop {\hbox{min} }\limits_{{S_{{{Q}}}^{HL} }} \left\{ {\begin{array}{*{20}c} {\left\| {{\mathcal{M}} - \mathop \sum \limits_{i = 1}^{{{ \dot{C} }}} \left( {\dot{A}_{i}^{HL} S_{i}^{HL} + \dot{A}_{{i,{{Q}}}}^{HL} S_{{i,{{Q}}}}^{HL} } \right) - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL} } \right\|_{2}^{2} + } \\ {\mathop \sum \limits_{i = 1}^{{{ \dot{C} }}} \left\| {\dot{M}_{i}^{HL} - \dot{A}_{i}^{HL} S_{i}^{HL} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL} } \right\|_{2}^{2} + \sigma_{3} \left\| {S_{{{Q}}}^{HL} } \right\|_{1} } \\ \end{array} } \right\}$$

(19.15)

which can be simplified as

$$\left\langle {S_{{{Q}}}^{HL} } \right\rangle = \arg \mathop {\hbox{min} }\limits_{{S_{{{Q}}}^{HL} }} \left\| {{\bar{\mathcal{M}}} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL} } \right\|_{2}^{2} + \mathop \sum \limits_{i = 1}^{{{ \dot{C} }}} \left\| {\bar{\dot{M}}_{i}^{HL} - \dot{A}_{{{Q}}}^{HL} S_{{{Q}}}^{HL} } \right\|_{2}^{2} + \sigma_{3} \left\| {S_{{{Q}}}^{HL} } \right\|_{1}$$

(19.16)

which is another LASSO problem can be solved using the feature sign algorithm [43].