﻿ 基于六维前馈神经网络模型的图像增强算法
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 山东大学学报(工学版)  2018, Vol. 48 Issue (4): 10-19  DOI: 10.6040/j.issn.1672-3961.0.2018.063 0

### 引用本文

ZHANG Xianhong, ZHANG Chunrui. Image enhancement algorithm based on six dimensional feedforward neural network model[J]. Journal of Shandong University (Engineering Science), 2018, 48(4): 10-19. DOI: 10.6040/j.issn.1672-3961.0.2018.063.

### 文章历史

1. 黑龙江工程学院计算机科学与技术学院，黑龙江 哈尔滨 150001;
2. 东北林业大学数学系，黑龙江 哈尔滨 150001

Image enhancement algorithm based on six dimensional feedforward neural network model
ZHANG Xianhong1, ZHANG Chunrui2
1. College of Computer Science and Technology, Heilongjiang Institue of Technogly, Harbin 150001, Heilongjiang, China;
2. Department of Mathematics, Northeast Forestry University, Harbin 150001, Heilongjiang, China
Abstract: Aiming at the problems of weakening the edges caused by filtering denoising, partially indistinct images and low contrast, an image enhancement algorithm based on the six dimension feedforward neural network model was proposed on the basis of fully analyzing the dynamic properties of the model. The experiment showed that the image enhancement algorithm based on the six dimensional feedforward neural network model could better achieve a very good enhancement effect. Compared with other enhancement algorithms, the enhancement effect was clearer and the algorithm was better.
Key words: image processing    image enhancement    neural network    six dimensional feed-forward neural network model    dynamic system
0 引言

1 前馈神经网络模型的动力系统性质分析

2004年，文献[7]首先提出一种带有离散时滞和分布时滞的二维神经网络模型

 $\left\{ \begin{array}{l} {{\dot x}_1} = - \mu {x_1}\left( t \right) + {a_{11}}{f_{11}}\left( {\int_{ - \infty }^t {F\left( {t - s} \right){x_1}\left( {s - \tau } \right){\rm{d}}s} } \right) + {a_{12}}{f_{12}}\left( {{x_2}\left( {t - \tau } \right)} \right)\\ {{\dot x}_2} = - \mu {x_2}\left( t \right) + {a_{22}}{f_{22}}\left( {\int_{ - \infty }^t {F\left( {t - s} \right){x_2}\left( {s - \tau } \right){\rm{d}}s} } \right) + {a_{21}}{f_{21}}\left( {{x_1}\left( {t - \tau } \right)} \right) \end{array} \right.,$

 $\left\{ \begin{array}{l} {{\dot x}_1} = - {x_1}\left( t \right) + af\left( {{x_2}\left( t \right)} \right) + bf\left( {{x_3}\left( {t - \tau } \right)} \right)\\ {{\dot x}_2} = - \gamma {x_2}\left( t \right) + {x_1}\left( {t - \tau } \right)\\ {{\dot x}_3} = - {x_3}\left( t \right) + af\left( {{x_4}\left( t \right)} \right) + bf\left( {{x_1}\left( {t - \tau } \right)} \right)\\ {{\dot x}_4} = - \gamma {x_4}\left( t \right) + {x_3}\left( {t - \tau } \right) \end{array} \right.。$

 $\left\{ \begin{array}{l} {{\dot x}_1} = - {x_1}\left( t \right) + af\left( {{x_2}\left( t \right)} \right) + bf\left( {{x_1}\left( {t - \tau } \right)} \right)\\ {{\dot x}_2} = - \gamma {x_2}\left( t \right) + {x_1}\left( {t - \tau } \right)\\ {{\dot x}_3} = - {x_3}\left( t \right) + af\left( {{x_4}\left( t \right)} \right) + bf\left( {{x_1}\left( {t - \tau } \right)} \right)\\ {{\dot x}_4} = - \gamma {x_4}\left( t \right) + {x_3}\left( {t - \tau } \right)\\ {{\dot x}_5} = - {x_5}\left( t \right) + af\left( {{x_6}\left( t \right)} \right) + bf\left( {{x_3}\left( {t - \tau } \right)} \right)\\ {{\dot x}_6} = - \gamma {x_6}\left( t \right) + {x_5}\left( {t - \tau } \right) \end{array} \right.,$

 图 1 六维前馈神经网络模型 Figure 1 Six dimensional feedforward neural network

(1) 余维一简单或双零的分支分析

 $\left\{ \begin{array}{l} {{\dot x}_1} = - {x_1}\left( t \right) + a{x_2}\left( t \right) + b{x_1}\left( {t - \tau } \right)\\ {{\dot x}_2} = - \gamma {x_2}\left( t \right) + {x_1}\left( {t - \tau } \right)\\ {{\dot x}_3} = - {x_3}\left( t \right) + a{x_4}\left( t \right) + b{x_1}\left( {t - \tau } \right)\\ {{\dot x}_4} = - \gamma {x_4}\left( t \right) + {x_3}\left( {t - \tau } \right)\\ {{\dot x}_5} = - {x_5}\left( t \right) + a{x_6}\left( t \right) + b{x_3}\left( {t - \tau } \right)\\ {{\dot x}_6} = - \gamma {x_6}\left( t \right) + {x_5}\left( {t - \tau } \right) \end{array} \right.,$

 $\Delta = {\Delta _1}\Delta _2^2 = 0,$

(a) 如果条件1成立，则Δ=0的根λ=0重数为1，在原点线性化方程在平凡解处经历余维一的简单零分支。

(b) 如果条件2成立，则Δ=0的根λ=0重数为2。在原点线性化方程在平凡解处经历余维一的双零Pitchfork分支。

 $\frac{{{\rm{d}}{\Delta _1}\left( \lambda \right)}}{{{\rm{d}}\lambda }}\left| {_{\lambda = 0}} \right. = 1 + \left( {a + b} \right)\tau ,\frac{{{\rm{d}}{\Delta _2}\left( \lambda \right)}}{{{\rm{d}}\lambda }}\left| {_{\lambda = 0}} \right. = 1 + a\tau 。$

(2) 1:1共振Hopf分支分析

 ${\lambda _{1,2}} = \frac{{ - \left( {1 - b + \gamma } \right) \pm \sqrt {{{\left( {1 - b + \gamma } \right)}^2} - 4\left( {\gamma \left( {1 - b} \right) - a} \right)} }}{2}$

 $- {\omega ^2} + {\rm{i}}\omega \left( {1 + \gamma } \right) + \gamma - {\rm{i}}\omega b{{\rm{e}}^{ - {\rm{i}}\mu \tau }} - \left( {\gamma b + a} \right){{\rm{e}}^{ - {\rm{i}}\mu \tau }} = 0,$

 $- {\omega ^2} + {\rm{i}}\omega \left( {1 + \gamma } \right) + \gamma - a{{\rm{e}}^{ - {\rm{i}}\mu \tau }} = 0,$

 $\begin{array}{*{20}{c}} {\tau = \tau _j^ \pm \left( {j = 0,1, \cdots } \right):{{\left( {\frac{{{\rm{d}}{\mathop{\rm Re}\nolimits} \lambda \left( \tau \right)}}{{{\rm{d}}\tau }}} \right)}^{ - 1}}\left| {\tau = {\tau _j} = \left( {2\omega _ + ^4 - 2\omega \left( { - {\omega ^2} + {b^2} - {a^2}} \right)} \right)} \right./}\\ {{{\left( {4{\omega ^6} + \left( {{\omega ^2} - {\omega ^2}\left( {{\omega ^2} + {b^2} - {a^2}} \right)} \right)} \right)}^2} > 0,} \end{array}$

 $\left\{ \begin{array}{l} {{\dot x}_1} = - \tau {x_1}\left( t \right) + \tau a\tanh {x_2}\left( t \right) + \tau b\tanh \left[ {{x_3}\left( {t - 1} \right)} \right]\\ {{\dot x}_2} = - \tau \gamma {x_2}\left( t \right) + \tau {x_1}\left( {t - 1} \right)\\ {{\dot x}_3} = - \tau {x_3}\left( t \right) + \tau a\tanh {x_4}\left( t \right) + \tau b\tanh \left[ {{x_1}\left( {t - 1} \right)} \right]\\ {{\dot x}_4} = - \tau \gamma {x_4}\left( t \right) + \tau {x_3}\left( {t - 1} \right)\\ {{\dot x}_5} = - \tau {x_5}\left( t \right) + \tau a\tanh {x_6}\left( t \right) + \tau b\tanh \left[ {{x_3}\left( {t - 1} \right)} \right]\\ {{\dot x}_6} = - \tau \gamma {x_6}\left( t \right) + \tau {x_5}\left( {t - 1} \right) \end{array} \right.,$

 ${L_{0\emptyset }} = \int\limits_{ - 1}^0 {{\rm{d}}\eta \left( \theta \right)\emptyset \left( \theta \right)} ,$

 $\eta \left( \theta \right) = \tau \mathit{\boldsymbol{A}}\delta \left( \theta \right) - \tau \mathit{\boldsymbol{B}}\delta \left( {\theta + 1} \right),$

 $\delta \left( \theta \right) = \left\{ \begin{array}{l} 0,\theta \ne 0\\ 1,\theta = 0 \end{array} \right.,$
 $\mathit{\boldsymbol{A}} = \left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} { - 1}&a&0\\ 0&{ - \gamma }&0\\ 0&0&{ - 1} \end{array}}&{\begin{array}{*{20}{c}} 0&0&0\\ 0&0&0\\ a&0&0 \end{array}}\\ {\begin{array}{*{20}{c}} 0&0&0\\ 0&0&0\\ 0&0&0 \end{array}}&{\begin{array}{*{20}{c}} { - \gamma }&0&0\\ 0&{ - 1}&a\\ 0&0&{ - \gamma } \end{array}} \end{array}} \right),$
 $\mathit{\boldsymbol{B = }}\left( {\begin{array}{*{20}{c}} b&0&0&0&0&0\\ 1&0&0&0&0&0\\ b&0&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&b&0&0&0\\ 0&0&0&0&1&0 \end{array}} \right)。$

 $\dot X\left( t \right) = {L_0}{X_t},$
 ${X_t} = X\left( {t + \theta } \right),$

 $\left\langle {\psi ,\varphi } \right\rangle = \psi \left( 0 \right)\varphi \left( 0 \right) - \int_{ - 1}^0 {\int_{\xi = 0}^\theta {\psi \left( {\xi - \theta } \right){\rm{d}}\eta \left( \theta \right)\varphi \left( \xi \right){\rm{d}}\xi } } ,$

${\dot x}$1=-τx1(t)+τatanhx2(t)+τbtanh[x3(t-1)]和${\dot x}$2=-τγx2(t)+τx1(t-1)的局部解趋近于(x1, x2)=(0, 0)。

τ=τj(j=0, 1, …)，由于x1x2趋近于0，因此，方程：

${\dot x}$3=-τx3(t)+τatanhx4(t)+τbtanh[x1(t-1)]和${\dot x}$4=-τγx4(t)+τx3(t-1)经历标准Hopf分支。

 图 2 当γ=1.3, a=-1.6, b=0.25, τ=1.5时各振子波形图 Figure 2 Waveform of each oscillator when γ=1.3, a=-1.6, b=0.25, τ=1.5
 图 3 当γ=1.3, a=-1.6, b=0.25, τ=4.5时各振子波形图 Figure 3 Waveform of each oscillator when γ=1.3, a=-1.6, b=0.25, τ=4.5
2 前馈神经网络模型在图象增强中的应用

 图 4 相位差图 Figure 4 Phase difference diagram
 图 5 振子5作为输出的模型增强效果对比 Figure 5 Comparison of the model enhancement effect of oscillator 5 as output
3 试验分析

3.1 灰度图像增强方法仿真效果比较

 图 6 各增强方法效果对比 Figure 6 Contrast of the enhancement methods

 ${\rm{mean}} = \frac{1}{m}\sum\limits_i {ip\left( i \right)} .$

 $s = \sqrt {\frac{1}{{n - 1}}\sum\limits_{i = 1}^n {{{\left( {{\mathit{\boldsymbol{x}}_i} - \mathit{\boldsymbol{x}}} \right)}^2}} } ,$

 $I = - \sum\limits_{i = 0}^{255} {p\left( i \right) * \ln p\left( i \right)} ,$

3.2 彩色图像增强效果仿真

 $\left\{ \begin{array}{l} V = \left( {R + G + B} \right)/3\\ S = 1 - \frac{3}{{\left( {R + G + B} \right)}}\left[ {\min \left( {R,G,B} \right)} \right]\\ H = {\cos ^{ - 1}}\left( {\frac{{\left[ {\left( {R - G} \right) + \left( {R - B} \right)} \right]/2}}{{{{\left( {R - G} \right)}^2}\left( {R - B} \right){{\left( {R - G} \right)}^{\frac{1}{2}}}}}} \right)/360 \end{array} \right.。$

 图 7 几种增强方法效果对比 Figure 7 Contrast of the enhancement methods

 $\left\{ \begin{array}{l} {\mu _i} = \frac{1}{N}\sum\limits_{j = 1}^N {{I_{ij}}} \\ {\sigma _i} = {\left[ {\frac{1}{N}\sum\limits_{j = 1}^N {{{\left( {{I_{ij}} - \mu } \right)}^2}} } \right]^{\frac{1}{2}}}\\ {\zeta _i} = {\left[ {\frac{1}{N}\sum\limits_{j = 1}^N {{{\left( {{I_{ij}} - \mu } \right)}^3}} } \right]^{\frac{1}{3}}} \end{array} \right.,$

4 结论

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