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 智能系统学报  2020, Vol. 15 Issue (2): 271-280  DOI: 10.11992/tis.201809042 0

### 引用本文

WANG Keping, CAI Kaili, WANG Hongqi, et al. A global sparse rain removal model based on rain streaks main direction adaptation[J]. CAAI Transactions on Intelligent Systems, 2020, 15(2): 271-280. DOI: 10.11992/tis.201809042.

### 文章历史

A global sparse rain removal model based on rain streaks main direction adaptation
WANG Keping , CAI Kaili , WANG Hongqi , YANG Yi
College of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo 454150, China
Abstract: The existing single-image rain removal algorithms do not appropriately consider the influence of wind on the main direction of the rain streaks. When the rain streak deviates from the vertical direction, the existing methods do not take rotation or only rotate roughly, resulting in the phenomenon whereby rain streaks are residual or the background is blurred. Therefore, in this paper, we propose a global sparse rain removal model based on the rain streaks main direction adaptation. First, the image block with the smallest variance and the rain streaks image of the image library are matched according to the histogram of oriented gradients (HOG) feature, and the main direction of the rain streaks image with the highest matching degree is regarded as the main direction of the image block, which can determine the rotation angle of the global sparse model; then, the global sparse model with three sparse regular terms including rotation angles is used for rain removal. After removing the rain streaks from the global sparse model, the Y-channel image is enhanced by a color mask, and thus, some parts of the background are protected. Then, together with the original CbCr-channel images, the image after treatment is further reorganized, and the final image after rain removal is obtained. The results show that compared with three typical comparison algorithms, the peak signal-to-noise ratio and the structural similarity are improved, and the running time is shorter. The proposed method can retain the background details of the image as much as possible while effectively removing the rain streaks.
Key words: single-image rain removal    main directional of the rain streaks    the image block    HOG feature    global sparse model    sparse regularization term    color mask    reorganization by the channel image

1 单向全变分模型去噪

 ${ r} = { t} + { s}$ (1)

 ${ I}\left( {x,y} \right) = {{ I}_u}\left( {x,y} \right) + {{ I}_s}\left( {x,y} \right)$ (2)

 $\mathop {\min }\limits_{{{ I}_u}} { E}\left( {{{ I}_u}} \right) = {\rm T}{{\rm V}_y}\left( {{{ I}_u} - { I}} \right) + \tilde {\text{λ}} {\rm T}{{\rm V}_x}\left( {{{ I}_u}} \right)$ (3)

 $\mathop {\min }\limits_{{{ I}_u}} { E}\left( {{{ I}_u}} \right) = \int_\varOmega {\left( {\left| {\frac{{\partial \left( {{{ I}_u} - { I}} \right)}}{{\partial y}}} \right|} \right.} + \left. {\tilde {\text{λ}} \left| {\frac{{\partial {{ I}_u}}}{{\partial x}}} \right|} \right){\rm d}x{\rm d}y$ (4)

1)为了有助于条形噪声的去除，可将梯度信息分离为沿着垂直方向的保真项 $\int_\varOmega {\left| {\dfrac{{\partial \left( {{{ I}_u} - { I}} \right)}}{{\partial y}}} \right|} {\rm d}y$ 和沿着水平方向的正则化项 $\int_\varOmega {\left| {\dfrac{{\partial {{ I}_u}}}{{\partial x}}} \right|{\rm d}x}$

2)对保真项和正则化项应使用边缘保持范数，例如 ${\rm{TV}}$ 范数或 ${l_1}$ 范数等，避免背景图出现伪影现象。

2 基于雨线主方向自适应的全局稀疏去雨模型 2.1 自适应调节雨线主方向角度

 Download: 图 2 图像分块匹配确定雨线主方向 Fig. 2 Image block matching determines the main direction of the rain streaks

 $\theta = \left\{ \begin{array}{l} {90^ \circ } - \tilde \theta ,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{{{I}}_1},{{{I}}_2},{{{I}}_3},{{{I}}_4}\\ {0^ \circ },\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{{{I}}_5}\\ - \left| {\tilde \theta - {{90}^ \circ }} \right| = {90^ \circ } - \tilde \theta ,\;\;{{{I}}_6},{{{I}}_7},{{{I}}_8},{{{I}}_9} \end{array} \right.$ (5)

 $\theta = \left\{ \begin{array}{l} \theta > {0^ \circ },\;\;\;\;\;\;{{{I}}_1},{{{I}}_2},{{{I}}_3},{{{I}}_4}\\ \theta = {0^ \circ },\;\;\;\;\;\;{{{I}}_5}\\ \theta < {0^ \circ },\;\;\;\;\;\;{{{I}}_6},{{{I}}_7},{{{I}}_8},{{{I}}_9}{\rm{ }} \end{array} \right.$ (6)

1) $\theta > {0^ \circ }$ ，需要逆时针旋转 $\theta$ ，旋转角的大小为 ${90^ \circ } - \tilde \theta$ ，从而使雨线近似沿着垂直方向降落；

2) $\theta = {0^ \circ }$ ，此时待处理雨图的雨线主方向近似垂直方向，(图像库中编号为 ${{{I}}_5}$ 的雨线图与方差最小图像块的HOG特征匹配度最高，即：待处理雨图的雨线主方向在 ${90^ \circ } \pm {5^ \circ }$ 的区间)，无需旋转待处理雨图，即旋转角 $\theta$ ${0^ \circ }$

3) $\theta < {0^ \circ }$ ，需要顺时针旋转 $\theta$ ，旋转角的大小为 $\left| {\tilde \theta - {{90}^ \circ }} \right|$ ，从而使雨线近似沿着垂直方向降落。

2.2 全局稀疏模型的确定

1)雨线的稀疏正则项： ${l_0}$ 范数可对雨线的稀疏性起到较好的稀疏约束作用，但 ${l_0}$ 范数的非凸性会使模型收敛到局部最优解； ${l_1}$ 范数既可以表示稀疏性又具有凸性，会使稀疏模型收敛到全局最优。故本文基于 ${l_1}$ 范数对雨线添加稀疏正则项：

 ${\rm{Re}}{{\rm{g}}^{\left( {\rm{1}} \right)}}\left( { s} \right) = {\left\| { s} \right\|_1}$ (7)

2)雨线主方向( $\hat y$ 轴方向)上的梯度稀疏性：在该方向上，雨线梯度沿降落方向具有稀疏平滑性。即可对 $\hat y$ 方向的雨线梯度添加稀疏正则项进行约束：

 ${\rm{Re}}{{\rm{g}}^{\left( {\rm{2}} \right)}}\left( { s} \right) = {\left\| {{\nabla _{\hat y}}{ s}} \right\|_1}$ (8)

${\nabla _{\hat y}}$ $\hat y$ 方向上的差分算子。该项可看作式(3)的第一个正则项的变换。

3)垂直于雨线主方向( $\hat x$ 轴)上的梯度稀疏性：受单向总变分(UTV)的启发，背景梯度在 $\hat x$ 方向受雨水影响最小，即：背景梯度具有稀疏性。在自然雨图像中，雨线图 ${ s}$ 具有非负性。线性叠加后，待处理含雨图 ${ r}$ 的强度值最大。因此，非负性约束条件为：

 ${ r} \geqslant { s} \geqslant 0$ (9)

 ${\rm{Re}}{{\rm{g}}^{\left( {\rm{3}} \right)}}\left( { s} \right) = {\left\| {{\nabla _{\hat x}}\left( {{ r} - { s}} \right)} \right\|_1}$ (10)

${D_\theta }$ 为旋转角为 $\theta$ 的旋转运算符，则带有 ${D_\theta }$ 的3个稀疏正则项分别为 ${\left\| {{D_\theta }{ s}} \right\|_1}$ ${\left\| {{\nabla _{\hat y}}\left( {{D_\theta }{ s}} \right)} \right\|_1}$ ${\left\| {{\nabla _{\hat x}}\left( {{D_\theta }\left( {{ r} - { s}} \right)} \right)} \right\|_1}$

 $\begin{gathered} \underset{{ s}}{\mathop{\min }}\,E\left( s \right)=\underset{s}{\mathop{\min }}\,\left\{ {{\textit{λ}}_{1}}{\rm T}{{{\rm V}}_{x}}\left( {{D}_{\theta }}\left( { r}-{ s} \right) \right)+{{\textit{λ}}_{2}}{{\left\| {{D}_{\theta }}{ s} \right\|}_{1}}+ \right. \\ \left. {\rm T}{{{\rm V}}_{y}}\left( {{D}_{\theta }}{ s} \right) \right\}=\underset{{ s}}{\mathop{\min }}\,\left\{ {{\textit{λ}}_{1}}{{\left\| {{\nabla }_{{\hat{x}}}}\left( {{D}_{\theta }}\left( { r}-{ s} \right) \right) \right\|}_{1}}+ \right. \\ \left. {{\textit{λ}}_{2}}{{\left\| {{D}_{\theta }}{ s} \right\|}_{1}}+{{\left\| {{\nabla }_{{\hat{y}}}}\left( {{D}_{\theta }}{ s} \right) \right\|}_{1}} \right\} \\ \end{gathered}$ (11)

 Download: 图 3 本文所提方法的流程 Fig. 3 The flow chart of the proposed method

 $\mathop {\min }\limits_{ s} \left\{ {{\lambda _1}{{\left\| { u} \right\|}_1} + {\lambda _2}{{\left\| { v} \right\|}_1} + {{\left\| { w} \right\|}_1}} \right\}$ (12)

 $\!\!\mathop {\min }\limits_{{ u},{ v},{ w},{ s}} \left. {\left\{ \begin{gathered} L\left( {{ u},{ v},{ w},{ s},{p_1},{p_2},{p_3}} \right) = \\ {{\textit{λ}}_1}{\left\| { u} \right\|_1} + {{\textit{λ}}_2}{\left\| { v} \right\|_1} + {\left\| { w} \right\|_1} + \left\langle {{p_1},{\nabla _{\hat x}}\left( {{r_{{D_\theta }}} - {{ s}_{{D_\theta }}}} \right) - { u}} \right\rangle + \\ \dfrac{{{\beta _1}}}{2}\left\| {{\nabla _{\hat x}}\left( {{{ r}_{{D_\theta }}} - {{ s}_{{D_\theta }}}} \right) - { u}} \right\|_2^2 + \\ \left\langle {{p_2},{{ s}_{{D_\theta }}} - { v}} \right\rangle + \dfrac{{{\beta _2}}}{2}\left\| {{{ s}_{{D_\theta }}} - { v}} \right\|_2^2 +\\ \left\langle {{p_3},{\nabla _{\hat y}}{{ s}_{{D_\theta }}} - { w}} \right\rangle + \dfrac{{{\beta _3}}}{2}\left\| {{\nabla _{\hat y}}{{ s}_{{D_\theta }}} - { w}} \right\|_2^2 \end{gathered} \right.} \right\}$ (13)

1) $u$ 的子问题：固定 $v$ $w$ ，最小化能量泛函：

 $\begin{gathered} \hat { u}{\rm{ = arg }}\mathop {{\rm{min}}}\limits_{ u} {{\textit{λ}}_{\rm{1}}}{\left\| { u} \right\|_1} + \left\langle {{p_1}{\rm{,}}{\nabla _{\hat x}}\left( {{{ r}_{{D_\theta }}} - {{ s}_{{D_\theta }}}} \right) - { u}} \right\rangle + \\ \dfrac{{{\beta _1}}}{2}\left\| {{\nabla _{\hat x}}\left( {{{ r}_{{D_\theta }}} - {{ s}_{{D_\theta }}}} \right) - { u}} \right\|_2^2 = \\ {\rm{arg }}\mathop {{\rm{min}}}\limits_{ u} {{\textit{λ}}_{\rm{1}}}{\left\| { u} \right\|_1} + \dfrac{{{\beta _1}}}{2}\left\| {{\nabla _{\hat x}}\left( {{{ r}_{{D_\theta }}} - {{ s}_{{D_\theta }}}} \right) - { u}{\rm{ + }}\dfrac{{{p_1}}}{{{\beta _1}}}} \right\|_2^2 \end{gathered}$ (14)

 ${{ u}^{k{\rm{ + 1}}}} = {\rm{Shrink}}\left( {{\nabla _{\hat x}}\left( {{{ r}_{{D_\theta }}} - { s}_{{D_\theta }}^k} \right) + \frac{{p_1^k}}{{{\beta _1}}},\frac{{{\lambda _{\rm{1}}}}}{{{\beta _1}}}} \right)$ (15)
 ${\rm{Shrink}}\left( {a{\rm{,}}b} \right) = {\rm{sign}}\left( a \right)\max \left( {\left| a \right| - b,0} \right)$ (16)

 $\text{sign}\left( a \right)=\left\{ \begin{matrix} \begin{matrix} 1, \\ 0, \\ -1, \\ \end{matrix} & \begin{matrix} a > 0 \\ a = 0 \\ a < 0 \\ \end{matrix} \\ \end{matrix} \right.$ (17)

2) $v$ 的子问题：固定 $u$ $w$ ，最小化能量泛函：

 $\begin{gathered} \hat { v}{\rm{ = arg }}\mathop {{\rm{min}}}\limits_{ v} {{\textit{λ}}_{\rm{2}}}{\left\| { v} \right\|_1} + \left\langle {{p_2}{\rm{,}}{{ s}_{{D_\theta }}} - { v}} \right\rangle + \dfrac{{{\beta _2}}}{2}\left\| {{{ s}_{{D_\theta }}} - { v}} \right\|_2^2 = \\ _{}^{}{\rm{arg }}\mathop {{\rm{min}}}\limits_{ v} {{\textit{λ}}_{\rm{2}}}{\left\| { v} \right\|_1} + \dfrac{{{\beta _2}}}{2}\left\| {{{ s}_{{D_\theta }}} - { v} + \dfrac{{{p_2}}}{{{\beta _2}}}} \right\|_2^2 \\ \end{gathered} \!\!\!\!\!\!\!\!$ (18)

 ${{ v}^{k + 1}} = {\rm{Shrink}}\left( {{ s}_{{D_\theta }}^k + \frac{{p_2^k}}{{{\beta _2}}},\frac{{{\lambda _{\rm{2}}}}}{{{\beta _2}}}} \right)$ (19)

3) $w$ 的子问题：固定 $u$ $v$ ，最小化能量泛函：

 $\begin{gathered} \hat { w}{\rm{ = arg }}\mathop {{\rm{min}}}\limits_{ w} {\left\| { w} \right\|_1} + \left\langle {{p_3}{\rm{,}}{\nabla _{\hat y}}{{{ s}}_{{D_\theta }}} - { w}} \right\rangle + \dfrac{{{\beta _3}}}{2}\left\| {{\nabla _{\hat y}}{{{ s}}_{{D_\theta }}} - { w}} \right\|_2^2 = \\ _{}^{} {\rm{arg }}\mathop {{\rm{min}}}\limits_{ w} {\left\| { w} \right\|_1} + \dfrac{{{\beta _3}}}{2}\left\| {{\nabla _{\hat y}}{{{ s}}_{{D_\theta }}} - { w} + \dfrac{{{p_3}}}{{{\beta _3}}}} \right\|_2^2 \\ \end{gathered} \!\!\!\!\!\!\!\!\!$ (20)

 ${w^{k + 1}} = {\rm{Shrink}}\left( {{\nabla _{\hat y}}{\rm{s}}_{{D_\theta }}^k + \frac{{p_3^k}}{{{\beta _3}}},\frac{1}{{{\beta _3}}}} \right)$ (21)

4) ${{ s}_{{D_\theta }}}$ 的子问题：

 $\begin{gathered} {{{\hat{ s}}}_{{{D}_{\theta }}}}=\text{arg}\underset{{{ s}_{{{D}_{\theta }}}}}{\mathop{\text{min}}}\,\left\langle {{p}_{1}},{{\nabla }_{{\hat{x}}}}\left( {{ r}_{{{D}_{\theta }}}}-{{ s}_{{{D}_{\theta }}}} \right)-{ u} \right\rangle +\left\langle {{p}_{2}},{{ s}_{{{D}_{\theta }}}}-{ v} \right\rangle \text{+} \\ \left\langle {{p}_{3}},{{\nabla }_{{\hat{y}}}}{{ s}_{{{D}_{\theta }}}}-{ w} \right\rangle +\dfrac{{{\beta }_{1}}}{2}\left\| {{\nabla }_{{\hat{x}}}}\left( {{ r}_{{{D}_{\theta }}}}-{{ s}_{{{D}_{\theta }}}} \right)-{ u} \right\|_{2}^{2}+ \\ \dfrac{{{\beta }_{2}}}{2}\left\| {{ s}_{{{D}_{\theta }}}}-{ v} \right\|_{2}^{2}+\dfrac{{{\beta }_{3}}}{2}\left\| {{\nabla }_{{\hat{y}}}}{{ s}_{{{D}_{\theta }}}}-{ w} \right\|_{2}^{2}=\text{arg}\underset{{{ s}_{{{D}_{\theta }}}}}{\mathop{\text{min}}}\,\dfrac{{{\beta }_{1}}}{2} \\ \left\| {{\nabla }_{{\hat{x}}}}\left( {{ r}_{{{D}_{\theta }}}}-{{ s}_{{{D}_{\theta }}}} \right)-{ u}\text{+}\dfrac{{{p}_{1}}}{{{\beta }_{1}}} \right\|_{2}^{2}+\dfrac{{{\beta }_{2}}}{2}\left\| {{ s}_{{{D}_{\theta }}}}-{ v}\text{+}\dfrac{{{p}_{2}}}{{{\beta }_{2}}} \right\|_{2}^{2}+ \\ \dfrac{{{\beta }_{3}}}{2}\left\| {{\nabla }_{{\hat{y}}}}{{ s}_{{{D}_{\theta }}}}-{ w}\text{+}\dfrac{{{p}_{3}}}{{{\beta }_{3}}} \right\|_{2}^{2} \end{gathered}$ (22)

 $\begin{gathered} \left( {{\beta _1}\nabla _{\hat x}^T{\nabla _{\hat x}}{\rm{ + }}{\beta _2}{{I + }}{\beta _3}\nabla _{\hat y}^T{\nabla _{\hat y}}} \right)\tilde { s}_{{D_\theta }}^{k{\rm{ + 1}}} = \nabla _{\hat y}^T\left( {{\beta _{3{w^{k{\rm{ + 1}}}}}} - \dfrac{{p_{\rm{3}}^k}}{{{\beta _3}}}} \right){\rm{ + }}\\ {\beta _2}{{ v}^{k{\rm{ + 1}}}} - p_{\rm{2}}^k + \nabla _{\hat x}^T\left( {{\beta _1}{\nabla _{\hat x}}{{ r}_{{D_\theta }}} - {\beta _1}{{ u}^{k{\rm{ + 1}}}} + p_{\rm{1}}^k} \right)\pi \end{gathered}$ (23)

 ${ s}_{{D_\theta }}^{k + 1} = {\rm{min}}\left( {{{ r}_{{D_\theta }}},{\rm{max}}\left( {\tilde { s}_{{D_\theta }}^{k{\rm{ + 1}}},0} \right)} \right)$ (24)

 $\begin{gathered} p_{\rm{1}}^{k + 1} = p_{\rm{1}}^k + {\beta _1}\left( {{\nabla _{\hat x}}\left( {{{ r}_{{D_\theta }}} - { s}_{{D_\theta }}^{k + 1}} \right) - {{ u}^{k{\rm{ + 1}}}}} \right) \\ p_{\rm{2}}^{k + 1} = p_{\rm{2}}^k + {\beta _2}\left( {{ s}_{{D_\theta }}^{k + 1} - {{ v}^{k{\rm{ + 1}}}}} \right) \\ p_{\rm{3}}^{k + 1} = p_{\rm{3}}^k + {\beta _3}\left( {{\nabla _{\hat y}}{ s}_{{D_\theta }}^{k + 1} - {{ w}^{k{\rm{ + 1}}}}} \right) \\ \end{gathered}$ (25)

2.3 颜色掩膜保护背景

 ${m_i} = \left\{ {\begin{array}{*{20}{c}} {1,\;\;{w_i} > T} \\ {0,\;\;{w_i} < T} \end{array}} \right.$ (26)

1)输入　雨图 $r$ ，通过式(5)确定旋转角 $\theta$

2)循环　 $k = k + 1$

 $\begin{gathered} \left\| {{{ r}_{{D_\theta }}} - { s}_{{D_\theta }}^k} \right\|/\left\| {{{ r}_{{D_\theta }}} - { s}_{{D_\theta }}^{k - 1}} \right\| > {\rm tol} = {{\rm e}^{ - 3}} \\ {\rm and}\;k < {M_{{\rm ite}{ r}}} = {{2}}00 \\ \end{gathered}$

3)通过式(15)、(19)、(21)对应求解 ${{ u}^{k + 1}}{\text{、}}$ ${{ v}^{k + 1}}{\text{、}}{{ w}^{k + 1}}$ ；通过式(24)求解 ${ s}_{{D_\theta }}^{k + 1}$ ，通过式(25)求解 $p_1^{k + 1}{\text{、}}p_2^{k + 1}{\text{、}}p_3^{k + 1}$ ，即：

 ${{ u}^{k{\rm{ + 1}}}} = {\rm{Shrink}}\left( {{\nabla _{\hat x}}\left( {{{ r}_{{D_\theta }}} - { s}_{{D_\theta }}^k} \right) + \frac{{p_1^k}}{{{\beta _1}}},\frac{{{{\textit{λ}} _{\rm{1}}}}}{{{\beta _1}}}} \right)$
 ${{ v}^{k{\rm{ + 1}}}} = {\rm{Shrink}}\left( {{ s}_{{D_\theta }}^k + \frac{{p_2^k}}{{{\beta _2}}},\frac{{{{\textit{λ}} _{\rm{2}}}}}{{{\beta _2}}}} \right)$
 ${{ w}^{k{\rm{ + 1}}}} = {\rm{Shrink}}\left( {{\nabla _{\hat y}}{{s}}_{{D_\theta }}^k + \frac{{p_3^k}}{{{\beta _3}}},\frac{1}{{{\beta _3}}}} \right)$
 ${ s}_{{D_\theta }}^{k{\rm{ + 1}}} = {\rm{min}}\left( {{{ r}_{{D_\theta }}},{\rm{max}}\left( {\tilde { s}_{{D_\theta }}^{k{\rm{ + 1}}},0} \right)} \right)$
 $\begin{gathered} p_{\rm{1}}^{k + 1} = p_{\rm{1}}^k + {\beta _1}\left( {{\nabla _{\hat x}}\left( {{{ r}_{{D_\theta }}} - { s}_{{D_\theta }}^{k + 1}} \right) - {{ u}^{k{\rm{ + 1}}}}} \right) \\ p_{\rm{2}}^{k + 1} = p_{\rm{2}}^k + {\beta _2}\left( {{ s}_{{D_\theta }}^{k + 1} - {{ v}^{k{\rm{ + 1}}}}} \right) \\ p_{\rm{3}}^{k + 1} = p_{\rm{3}}^k + {\beta _3}\left( {{\nabla _{\hat y}}{ s}_{{D_\theta }}^{k + 1} - {{ w}^{k{\rm{ + 1}}}}} \right) \\ \end{gathered}$

4)当有下式成立时

 $\begin{gathered} \left\| {{{ r}_{{D_\theta }}} - { s}_{{D_\theta }}^k} \right\|/\left\| {{{ r}_{{D_\theta }}} - { s}_{{D_\theta }}^{k - 1}} \right\| > {\rm tol} = {{\rm e}^{ - 3}} \\ {\rm{or }}\;k < {M_{\rm iter}} = {\rm{2}}00 \\ \end{gathered}$

 ${{ t}_{{D_\theta }}} = {{ r}_{{D_\theta }}} - {{ s}_{{D_\theta }}}$

3 实验结果与分析 3.1 合成数据定性分析

3.2 合成数据定量分析

3.3 真实数据分析

 Download: 图 6 真实图的去雨效果对比 Fig. 6 Comparison of rain streaks removal on real images

4 结束语

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