Infrared and Visible Light Image Fusion Based on Mahalanobis Distance and Guided Filter Weighting
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摘要: 为使红外与可见光融合图像获得更好的分辨率和清晰度，提出基于非下采样轮廓波变换（non-subsampled contourlet transform, NSCT）的马氏距离加权拉普拉斯能量和与引导滤波改进（frequency tuned, FT）结合的红外与可见光图像融合算法。首先，对可见光图像进行对比度受限的自适应直方图均衡（contrast limited adaptive histogram equalization, CLAHE），并将红外图像与CLAHE处理后可见光图像进行NSCT变换，分解为低频和高频; 其次，对FT算法使用引导滤波进行改进，利用改进的FT算法提取红外图像显著性图自适应加权融合低频图像，对高频图像使用基于马氏距离加权的拉普拉斯能量和取大融合; 最后，对融合的低频和高频图像进行NSCT逆变换获得融合图像。实验结果表明，该融合方法相较其他传统融合方法，在主观视觉上和客观指标上都有较好的表现。Abstract: To improve the definition of fusion images and obtain better target information during the fusion of infrared and visible light images using the characteristics of non-subsampled contourlet transform(NSCT) coefficients, an Manalanobis distance weighted Laplacian energy combined with guided filtering is proposed to improve the frequency tuned (FT) algorithm. First, the visible light image is subjected to contrast limited adaptive histogram equalization(CLAHE), and the infrared image and the CLAHE processed visible light image are decomposed into a low-frequency approximate image and a high-frequency detail image through a multi-scale and multi-directional NSCT transform. Second, the FT algorithm improved by guided filtering isused to extract the significance graph of infrared images, the adaptive weighted fusion rule based on the significance graph of infrared images is used for low-frequency images, and the fusion rule based on the Laplace energy and maximum weighted by the Manalanobis distance is used for high-frequency images. Finally, the fusion image is obtained by the NSCT inverse transformation of the fused low-frequency and high-frequency images. The experimental results show that this fusion method has better performance in terms of subjective vision and objective indexes than other traditional fusion methods.
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0. 引言
框架式稳定平台系统近年来发展迅速，广泛应用于飞行器上的目标探测系统、精确制导武器的导引系统等。三自由度框架式红外稳定平台实现惯性空间稳定和对目标跟踪的同时，还可以直接测量制导系统所需的视线角速度信息[1]。三自由度稳定平台在结构上由3个单轴运动框架复合而成，机械装配中产生的装调误差造成框架轴系偏差[2-3]、红外探测器位姿偏差以及陀螺敏感轴的交叉耦合[4-5]，使得基座角运动的耦合更加复杂[6]，对测量视线角速度带来不利的影响。
文献[7]研究了仅陀螺敏感轴交叉耦合情况下视线角速度的计算。所得结果是在框架轴系正交的假设下得到的。而实际系统中框架的装配必然会存在一定装调误差。本文系统研究了框架、陀螺和红外探测器均存在装调误差时，三自由度框架式红外视线角速度的计算方法，建立基于三轴稳定平台的轴系偏差的数学模型，分析了装调误差对视线角速度计算的影响，并进行仿真验证。
1. 稳定平台系统
三自由度框架式红外稳定平台系统的示意图如图 1。图中，O-XbYbZb表示载体坐标系。载体坐标系的原点取为稳定平台回转中心且坐标系和载体固连。
框架式红外稳定平台系统一般将探测成像系统和速率陀螺安装在稳定平台上，稳定平台固定在内环框架上，成为内环框架的负载。内环框架和稳定平台组成内环本体组合，通过内环框架转轴固定在中环框架上，成为中环框架的负载。中环框架和内环本体组合通过中环框架转轴固定在外环框架上，成为外环框架的负载。外环框架转轴架固定在红外稳定平台的载体上。外环框架相对载体可以做滚转运动；外环框架处于零位时，中环框架相对载体可以做偏航运动；外环框架和中环框架处于零位时，内环框架相对载体可以做俯仰运动。通过内环、中环、外环3个框架的运动合成，可以实现稳定平台在惯性空间中绕回转中心转动。
2. 轴系偏差建模
针对三自由度红外稳定平台的结构特点，除了前面定义的载体坐标系O-XbYbZb，再建立外环坐标系O-XoYoZo、中环坐标系O-XmYmZm、平台坐标系O-XpYpZp和探测坐标系O-XdYdZd。这4个坐标系原点均为稳定平台回转中心，其中，外环坐标系X轴和外环框架转轴固连；中环坐标系Y轴与中环框架转轴固连；平台坐标系Z轴和内环框架转轴固连；探测坐标系和探测器光敏面固连，其X轴对应光敏面的中垂线（即探测成像系统光轴），Y轴和Z轴分别对应探测器光敏面的行和列。
在设计的理想状态下，探测成像系统光轴与内环框架转轴、内环框架转轴与中环框架转轴、中环框架转轴与外环框架转轴应分别正交，而外环框架转轴和载体纵轴完全重合。探测坐标系和平台坐标系重合且各框架处于零位时，4个坐标系和载体坐标系重合。记外环框架角为φw，中环框架角为φz，内环框架角为φn，角度正负按右手规则确定，那么各坐标系相互间的变换关系如图 2所示。
实际装配时，框架轴系不可能做到零误差。本文描述轴系装调误差的参数为α1、β1、α2、β2、α3、β3、α4、β4和γ4。其中，α1为外环框架转轴在载体坐标系XOZ面的投影与载体系X轴的夹角；β1为外环框架转轴与载体系XOZ面的夹角；α2为中环框架转轴在外环坐标系YOZ面的投影与外环系Y轴的夹角；β2为中环框架转轴与载体系YOZ面的夹角；α3为内环框架转轴在中环坐标系XOZ面的投影与中环系Z轴的夹角；β3为内环框架转轴与中环系β3面的夹角；α4为探测器光敏面中垂线在平台系XOZ面的投影与平台系X轴的夹角；β4为光敏面中垂线与平台系XOZ面的夹角；γ4和α4、β4一起构成一组平台系到探测系的欧拉角。角度正负号按右手规则确定。当这些装调误差存在时，各坐标系相互间的变换关系如图 3所示。
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图 3 框架轴系偏差时各坐标系之间的变换关系Figure 3. Transformation of coordinate systems with axis system deviation3. 视线角速度的计算
当稳定跟踪目标时，探测成像系统光轴和视线重合，那么视线角速度$\dot{q}$近似为光轴在惯性空间中转动的角速度在探测系YOZ面的投影。由于探测成像系统是固连在稳定平台上的，所以光轴在惯性空间中转动的角速度也是平台转动的角速度$\tilde{\omega }$。为了简化分析，本文假设载体不动，并定义如下矩阵函数：
$$ {\mathit{\boldsymbol{T}}_x}(\phi ) = \left[ {\begin{array}{*{20}{c}} 1&0&0\\ 0&{\cos \phi }&{ - \sin \phi }\\ 0&{\sin \phi }&{\cos \phi } \end{array}} \right], $$ $$ {\mathit{\boldsymbol{T}}_y}(\phi ) = \left[ {\begin{array}{*{20}{c}} {\cos \phi }&0&{\sin \phi }\\ 0&1&0\\ { - \sin \phi }&0&{\cos \phi } \end{array}} \right], $$ $$ {\mathit{\boldsymbol{T}}_z}(\phi ) = \left[ {\begin{array}{*{20}{c}} {\cos \phi }&{ - \sin \phi }&0\\ {\sin \phi }&{\cos \phi }&0\\ 0&0&1 \end{array}} \right]。 $$ 当存在轴系偏差时，各坐标系之间按图 3的方式进行变换。此时在惯性空间中，平台转动的角速度$\tilde \omega $在探测系中的投影为：
$$ \begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} {{{\tilde \omega }_{{\rm{d}}x}}}\\ {{{\tilde \omega }_{{\rm{d}}y}}}\\ {{{\tilde \omega }_{{\rm{d}}z}}} \end{array}} \right] = \\T_x^{ - 1}({\gamma _4})T_z^{ - 1}({\beta _4})T_y^{ - 1}({\alpha _4})\left( {\left[ {\begin{array}{*{20}{c}} 0\\ 0\\ {{\omega _{\rm{n}}}} \end{array}} \right] + T_z^{ - 1}({\phi _{\rm{n}}})T_x^{ - 1}({\beta _3})T_y^{ - 1}({\alpha _3})\left( {\left[ {\begin{array}{*{20}{c}} 0\\ {{\omega _z}}\\ 0 \end{array}} \right] + T_y^{ - 1}({\phi _z})T_z^{ - 1}({\beta _2})T_x^{ - 1}({\alpha _2})\left[ {\begin{array}{*{20}{c}} {{\omega _w}}\\ 0\\ 0 \end{array}} \right]} \right)} \right)}\\ { = {A_{\tilde \omega }}({\phi _z}, \, {\phi _{\rm{n}}}, \, {\alpha _2}, \, {\beta _2}, \, {\alpha _3}, \, {\beta _3}, \, {\alpha _4}, \, {\beta _4}, {\gamma _4})\left[ {\begin{array}{*{20}{c}} {{\omega _{\rm{w}}}}\\ {{\omega _{\rm{z}}}}\\ {{\omega _{\rm{n}}}} \end{array}} \right]} \end{array} $$ (1) 式中：φn为内环框架角；φz为中环框架角；ωn是位标器内环框架转动的角速度，其方向沿平台坐标系的Z轴；ωz是位标器中环框架转动的角速度，其方向沿中环坐标系的Y轴。ωw是位标器外环框架转动的角速度，其方向沿外环坐标系的X轴。于是按本文中对视线角速度$\dot q$的近似，其在探测系中为：
$$ \left[ {\begin{array}{*{20}{c}} {{{\dot q}_{{\rm{d}}x}}}\\ {{{\dot q}_{{\rm{d}}y}}}\\ {{{\dot q}_{{\rm{d}}z}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0&0&0\\ 0&1&0\\ 0&0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{\tilde \omega }_{{\rm{d}}x}}}\\ {{{\tilde \omega }_{{\rm{d}}y}}}\\ {{{\tilde \omega }_{{\rm{d}}z}}} \end{array}} \right] = {\mathit{\boldsymbol{A}}_{\dot q}}({\phi _{\rm{z}}}, \, {\phi _{\rm{n}}}, \, {\alpha _2}, \, {\beta _2}, \, {\alpha _3}, \, {\beta _3}, \, {\alpha _4}, \, {\beta _4}, {\gamma _4})\left[ {\begin{array}{*{20}{c}} {{\omega _{\rm{w}}}}\\ {{\omega _{\rm{z}}}}\\ {{\omega _{\rm{n}}}} \end{array}} \right] $$ (2) 式中：${\mathit{\boldsymbol{A}}_{\dot q}} = \left[ {\begin{array}{*{20}{c}} 0&0&0\\ 0&1&0\\ 0&0&1 \end{array}} \right]{\mathit{\boldsymbol{A}}_{\bar \omega }}$。
对于三自由度框架式红外稳定平台系统，稳定平台上正交安装了偏航/俯仰陀螺分别测量平台相对惯性空间的偏航/俯仰角速度；外环框架上安装有外环陀螺，可以测量外框架相对惯性空间的滚转角速度。理想情况下，稳定平台偏航/俯仰陀螺的敏感轴分别平行于平台坐标系的Y轴和Z轴，外环陀螺敏感轴与外环坐标系X轴平行。这里仍用第2章轴系偏差建模的方法描述陀螺的装配误差，记误差参数为α5、β5、α6、β6、α7、β7。其中，α5为外环陀螺敏感轴在外环系XOZ面的投影与外环系X轴的夹角；β5为外环陀螺敏感轴与外环系XOZ面的夹角；α6为中环陀螺敏感轴在平台系YOZ面的投影与平台系Y轴的夹角；β6为中环陀螺敏感轴与平台系YOZ面的夹角；α7为内环陀螺敏感轴在平台系XOZ面的投影与平台系Z轴的夹角；β7为内环陀螺敏感轴与平台系XOZ面的夹角。角度正负号按右手规则确定。那么在考虑轴系偏差情形下，陀螺的输出和外、中、内环框架的角速度满足下式：
$$ \begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} {{{\hat \omega }_{\rm{w}}}}\\ {{{\hat \omega }_{\rm{z}}}}\\ {{{\hat \omega }_{\rm{n}}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\cos {\alpha _5}\cos {\beta _5}}&{\sin {\beta _5}}&{ - \sin {\alpha _5}\cos {\beta _5}}\\ 0&0&0\\ 0&0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\omega _{\rm{w}}}}\\ 0\\ 0 \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0&0&0\\ { - \sin {\beta _6}}&{\cos {\alpha _6}\cos {\beta _6}}&{\sin {\alpha _6}\cos {\beta _6}}\\ {\sin {\alpha _7}\cos {\beta _7}}&{ - \sin {\beta _7}}&{\cos {\alpha _7}\cos {\beta _7}} \end{array}} \right]}\\ {\quad \quad \quad \cdot \left( {\left[ {\begin{array}{*{20}{c}} 0\\ 0\\ {{\omega _{\rm{n}}}} \end{array}} \right] + T_z^{ - 1}({\phi _n})T_x^{ - 1}({\beta _3})T_y^{ - 1}({\alpha _3})\left( {\left[ {\begin{array}{*{20}{c}} 0\\ {{\omega _z}}\\ 0 \end{array}} \right] + T_y^{ - 1}({\phi _z})T_z^{ - 1}({\beta _2})T_x^{ - 1}({\alpha _2})\left[ {\begin{array}{*{20}{c}} {{\omega _{\rm{w}}}}\\ 0\\ 0 \end{array}} \right]} \right)} \right)}\\ { = {\mathit{\boldsymbol{A}}_g}({\phi _z}, \, {\phi _n}, \, \, {\alpha _2}, \, {\beta _2}, \, {\alpha _3}, \, {\beta _3}, \, {\alpha _5}, \, {\beta _5}, \, {\alpha _6}, \, {\beta _6}, \, {\alpha _7}, \, {\beta _7})\left[ {\begin{array}{*{20}{c}} {{\omega _{\rm{w}}}}\\ {{\omega _{\rm{z}}}}\\ {{\omega _{\rm{n}}}} \end{array}} \right]} \end{array} $$ (3) 式中：${\hat \omega _{\rm{w}}}$是外环陀螺的输出；${\hat \omega _{\rm{z}}}$是偏航陀螺的输出；${\hat \omega _{\rm{n}}}$是内环陀螺的输出。
将式(3)代入式(2)，得到视线角速度在探测坐标系中的测量计算公式为：
$$ \left[ {\begin{array}{*{20}{c}} {{{\dot q}_{{\rm{d}}x}}}\\ {{{\dot q}_{{\rm{d}}y}}}\\ {{{\dot q}_{{\rm{d}}z}}} \end{array}} \right] = \mathit{\boldsymbol{T}}({\phi _z}, \, {\phi _n}, \, {\alpha _2}, \, {\beta _2}, \, {\alpha _3}, \, {\beta _3}, \, {\alpha _4}, \, {\beta _4}, {\gamma _4}\, {\alpha _5}, \, {\beta _5}, \, {\alpha _6}, \, {\beta _6}, \, {\alpha _7}, \, {\beta _7})\left[ {\begin{array}{*{20}{c}} {{{\hat \omega }_{\rm{w}}}}\\ {{{\hat \omega }_{\rm{z}}}}\\ {{{\hat \omega }_{\rm{n}}}} \end{array}} \right] $$ (4) 式中：$\mathit{\boldsymbol{T}} = {A_{\dot q}}A_g^{ - 1}$。
最后将其按图 3的坐标变换关系可得视线角速度在载体系中的测量计算公式为：
$$ \left[ {\begin{array}{*{20}{c}} {{{\dot q}_{bx}}}\\ {{{\dot q}_{by}}}\\ {{{\dot q}_{bz}}} \end{array}} \right] = {\mathit{\boldsymbol{T}}_y}({\alpha _1}){\mathit{\boldsymbol{T}}_z}({\beta _1}){\mathit{\boldsymbol{T}}_x}({\phi _w}){\mathit{\boldsymbol{T}}_x}({\alpha _2}){\mathit{\boldsymbol{T}}_z}({\beta _2}){\mathit{\boldsymbol{T}}_y}({\phi _z}){\mathit{\boldsymbol{T}}_y}({\alpha _3}){\mathit{\boldsymbol{T}}_x}({\beta _3}){\mathit{\boldsymbol{T}}_z}({\phi _n}){\mathit{\boldsymbol{T}}_y}({\alpha _4}){\mathit{\boldsymbol{T}}_z}({\beta _4}){\mathit{\boldsymbol{T}}_x}({\gamma _4})\mathit{\boldsymbol{T}}\left[ {\begin{array}{*{20}{c}} {{{\hat \omega }_{\rm{w}}}}\\ {{{\hat \omega }_{\rm{z}}}}\\ {{{\hat \omega }_{\rm{n}}}} \end{array}} \right] $$ (5) 当各误差参数都取零时，式(4)即蜕化成：
$$ \left[ {\begin{array}{*{20}{c}} {{{\dot q}_{{\rm{d}}x}}}\\ {{{\dot q}_{{\rm{d}}y}}}\\ {{{\dot q}_{{\rm{d}}z}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0&0&0\\ 0&1&0\\ 0&0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{\hat \omega }_{\rm{w}}}}\\ {{{\hat \omega }_{\rm{z}}}}\\ {{{\hat \omega }_{\rm{n}}}} \end{array}} \right] $$ (6) 将其坐标变换到载体系可得：
$$ \left[ {\begin{array}{*{20}{c}} {{{\dot q}_{bx}}}\\ {{{\dot q}_{by}}}\\ {{{\dot q}_{bz}}} \end{array}} \right] = {\mathit{\boldsymbol{T}}_x}({\phi _w}){\mathit{\boldsymbol{T}}_y}({\phi _z}){\mathit{\boldsymbol{T}}_z}({\phi _{\rm{n}}})\left[ {\begin{array}{*{20}{c}} 0\\ {{{\hat \omega }_{\rm{z}}}}\\ {{{\hat \omega }_{\rm{n}}}} \end{array}} \right] $$ (7) 也即理想情况下，三自由度框架式红外稳定平台系统稳定跟踪目标时，稳定平台上正交安装的偏航/俯仰陀螺可以直接测量出视线角速度。特别当框架轴系误差参数取零时，式(4)即蜕化成文献[7]中的结果。因此，式(4)也可以认为是对装调误差进行补偿，而且较文献[7]中的结果更具有一般性。
4. 仿真分析
对三自由度框架式红外稳定平台系统进行视线角速度测量试验。试验中载体静止，目标转台做30°/s匀速运动。记录稳定跟踪目标时的陀螺输出和框架角输出，如图 4和图 5所示。
用实测数据按第3章的计算公式进行离线仿真。仿真时，设置4种条件：忽略所有误差、忽略陀螺安装误差、忽略框架轴系误差和综合考虑各装调误差，具体误差参数如表 1所示。
表 1 装调误差参数设置Table 1. Parameters setting of installation errorsAxis system deviation /° Alignment error of gyros/° (α1, β1) (α2, β2) (α3, β3) (α4, β4, γ4) (α5, β5) (α6, β6) (α7, β7) 1 (0, 0) (0, 0) (0, 0) (0, 0, 0) (0, 0) (0, 0) (0, 0) 2 (0, －0.02) (0.02, 0.1) (－0.03, 0) (0.03, 0.08, 2.5) (0, 0) (0, 0) (0, 0) 3 (0, 0) (0, 0) (0, 0) (0, 0, 0) (－0.05, 0.04) (0.03, －0.4) (－0.02, 0.1) 4 (0, －0.02) (0.02, 0.1) (－0.03, 0) (0.03, 0.08, 2.5) (－0.05, 0.04) (0.03, －0.4) (－0.02, 0.1) 图 6给出了4种情况下的视线角速度曲线，其中实线表示忽略所有误差测量得到的结果，点划线是忽略陀螺安装误差的结果，长虚线是忽略框架轴系误差的结果，带“+”实线是综合考虑各装调误差得到的结果。图 7是图 6的局部放大。
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图 6 仿真结果比较图Figure 6. Comparison with simulation results (solid line shows the result of neglecting all errors; dash dot line shows the result of neglecting alignment error of gyros; dash line shows the result of neglecting axis system deviation; solid line with "+" shows the result of considering all errors)仿真试验结果表明，对装调误差进行补偿，可以提高视线角速度测量的精度。忽略装调误差时，测量计算的视线角速度较理论值最大偏差为4.08°/s；仅对框架轴系误差补偿时，视线角速度最大偏差减小到2.53°/s；仅对陀螺安装误差时，视线角速度最大偏差减小到1.49°/s；综合考虑各装调误差进行补偿，视线角速度最大偏差进一步减小到1.18°/s。总体来看，陀螺敏感轴交叉耦合对视线角速度精度的影响较框架轴系误差更显著。
5. 结论
本文系统研究了框架和陀螺均存在装调误差时，三自由度框架式红外视线角速度的计算方法，并进行仿真分析。结果表明，在计算视线角速度时如果对误差进行补偿，可以提高视线角速度的测量精度。在提高线角速度测量精度方面，补偿陀螺敏感轴交叉耦合的效果比补偿框架轴系偏差更显著。所以陀螺敏感轴交叉耦合对视线角速度的影响在各装调误差中最大。此结果对新型框架式稳定平台系统总体设计时的误差指标分配有重要的参考价值。
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表 1 融合图像客观评价结果
Table 1 Objective evaluation results of fusion image
Image name Fusion method EI SD AG SF Ship DWT 4.9016 10.4666 1.4100 3.1531 NSCT 4.9139 10.4807 1.3980 3.1546 NSCT-FT 5.9540 21.1184 1.6376 3.9024 NSCT-M 6.5735 25.8154 4.7976 10.1821 Man DWT 6.5266 31.5238 2.9829 5.5125 NSCT 6.5491 31.7851 3.2272 6.3206 NSCT-FT 7.1864 61.6516 3.4935 7.1168 NSCT-M 7.6698 58.7864 8.8359 15.5185 Street DWT 5.9299 20.6524 3.1668 7.7725 NSCT 5.9442 21.8888 3.7054 12.7396 NSCT-FT 5.5269 33.4513 4.0396 13.8090 NSCT-M 6.8136 41.2933 8.4553 20.3821 
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