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在X光图像融合算法中,首先要确定融合规则。古铜镜X光的源图像经过提升小波分解后可得到X光图像的低频分量(Al, Bl)和高频分量(Ah, Bh),需采用不同的规则分别对低频分量和高频分量进行融合(Fh, Fl)。目前,低频分量的融合方法主要有单像素融合,一般采用加权平均的方法进行融合。高频分量采用区域融合,包含有区域方差,模极大值等方法进行融合[18-21]。单像素点融合方法忽略了单点像素与周围像素的相关性,会使融合后的图像过渡不自然,但其融合速度快。加权平均的融合规则会降低图像前景与背景之间的差异,使得融合后图像的对比度降低。传统方法中高频分量大多使用区域特征来确定融合规则,增加了融合后图像的对比度浮动。融合框架如图 1所示。
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X光图像富有大量的能量信息,低频子带继承了X光图像的大部分能量。加权平均准则和取系数极大值等融合规则在融合过程中会使得融合图像的对比度减弱,使得图像的信息丢失。区域能量与区域方差相结合可以很好地避免这种情况,因为X光图像包含了大部分能量信息,低频分量包含了X光图像的能量信息,使用区域能量作为系数可以有效地保留X光图像的信息。区域方差反映了图像灰度的离散程度。极大地保留了图像目标特征。因此本文中提出使用区域方差与区域能量相结合,综合两者可更好地表达源图像的信息。
首先,计算低频子带以点(i, j)为中心的区域方差,区域方差定义如下:
$ {V_{1,M(N)}}(i,j) = \frac{1}{{S \times T}}\sum\limits_{m \in S} {\sum\limits_{n \in T} {\left[ {{C_{1,M(N)}}(i + m,j + n) - } \right.} } \\ \left.\bar{C}_{1, M(N)}(i, j)\right]^{2} $
(1) 式中,Vl, M(N)(i, j)为图像M, N低频子带, 以(i, j)为中心,区域大小为S×T的方差,本文中的S×T的取值为3×3;Cl, M(N)为图像第i行、第j列的低频系数;$ \bar{C}_{1, M(N)}$为该部分的系数平均值。
其次,图像以第i行、第j列为中心的低频子带区域能量定义如下:
$ {E_{1, M(N)}}(i, j) = \sum\limits_{m \in S} {\sum\limits_{n \in T} {{{\left[ {{C_{1, M}}(i + m, j + n)} \right]}^2}} } $
(2) 融合图像的低频子带系数以区域能量与区域方差来确定,当图像M的区域能量和区域方差都大于图像N时,选取图像M的低频子带系数作为融合图像系数;当图像M的区域能量和区域方差都小于图像N时,选取图像N的低频子带系数作为融合系数;当图像M的区域能量和区域方差都不大于或都不小于图像N时,本文中以区域能量的系数作为加权融合,图像M、N的区域能量占比值如下:
$ \left\{\begin{array}{l} {K_{M}=\frac{E_{1, M}(i, j)}{E_{1, M}(i, j)+E_{1, N}(i, j)}} \\ {K_{N}=\frac{E_{1, N}(i, j)}{E_{1, M}(i, j)+E_{1, N}(i, j)}} \end{array}\right. $
(3) 最后,低频子带融合后的系数可表示为:
$ C_{l, f}(i, j)=\left\{\begin{array}{l} {C_{1, M}(i, j), \left(V_{1, M}(i, j)>\right.} \\ {\left.V_{1, N}(i, j) \cap E_{1, u}(i, j)>E_{1, N}(i, j)\right)} \\ {C_{1, N}(i, j), \left(V_{1, M}(i, j) <V_{1, N}(i, j) \cap\right.} \\ {\left.E_{1, M}(i, j) <E_{1, N}(i, j)\right)} \\ {K_{M} C_{1, N}(i, j)+K_{N} C_{1, M}(i, j), (\text { other })} \end{array}\right. $
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图像边缘信息与图像纹理信息是高频子带融合的关键,图像的边缘及纹理含有大量的信息,X光图像边缘细节包含了纹饰区的大部分信息。X光图像的边缘细节是X光图像的重要特征。人类视觉特征主要以边缘及轮廓认识图像, 包含了图像的细节特性。目前大多数高频融合规则都是取图像特征求取理论最优解,而古铜镜X光根据镜缘区和纹饰区的材质与厚度已取得相应区域的最优解。因此使用空间频率在源图像上选取最优解,避免了融合时取特征权值后,降低或增加图像对比度和信息丢失的可能性。空间频率反映了图像的总体活动与清晰度。空间频率包含了行频率与列频率, 空间频率可定义为:
$ {f_{\rm{s}}} = \sqrt {f_{\rm{r}}^2 + f_{\rm{c}}^{\rm{2}}} $
(5) 式中,fr表示空间频率的行频率,fc表示空间频率的列频率,对于X×Y大小的图像,fr与fc的定义为:
$ f_{\mathrm{r}}=\sqrt{\frac{1}{X Y} \sum\limits_{i=1}^{X} \sum\limits_{j=2}^{Y}[f(i, j)-f(i, j-1)]^{2}} $
(6) $ f_{\mathrm{c}}=\sqrt{\frac{1}{X Y} \sum\limits_{i=2}^{X} \sum\limits_{j=1}^{Y}[f(i, j)-f(i-1, j)]^{2}} $
(7) 式中,X, Y表示总行数和总列数; f(i, j)表示位置(i, j)处的灰度值。
定义空间频率比$ \theta = {f_{{\rm{s}}, M, k}}\left( {{2^l}, i, j} \right)/{f_{{\rm{s}}, N, k}}\left( {{2^l}, i, j} \right)$。${f_{{\rm{s}}, M, k}}\left( {{2^l}, i, j} \right) $与$ {f_{{\rm{s}}, N, k}}\left( {{2^l}, i, j} \right)$分别表示在第k个融合子带、在分辨率为2l下, 点(i, j)的空间频率。按以下准则初步确定融合后的子带数据:
$ D_{k}\left(2^{l}, i, j\right)=\left\{\begin{array}{l} {D_{M, k}\left(2^{l}, i, j\right), (\theta>1)} \\ {D_{N, k}\left(2^{l}, i, j\right), (\text { else })} \end{array}\right. $
(8) 式中,$ {D_k}\left( {{2^l}, i, j} \right)$表示在第k个融合子带、在分辨率为2l下, 点(i, j)的融合值,${{D_{M, k}}\left( {{2^l}, i, j} \right)} $与$ {{D_{N, k}}\left( {{2^l}, i, j} \right)}$分别表示源图像M、源图像N在第k个融合子带,点(i, j)位置的融合值。
邻域选取S×T(本文中取3×3)大小的验证窗口,中心像素点的系数使用邻域8个像素点的系数来源窗口来验证并修正。修正的方式如下:目标像素点融合后的图像来源于图像M,且邻域8个像素超过1/2来源于N时,将中心像素替换为N在该位置的系数; 中心像素来源于图像N,且周围像素超过1/2来源于M时,将该中心像素位置的系数修改为M。
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本实验中采用3组6幅古铜镜X光图像作为实验数据,采用的图像数据为陕西省文物保护研究院采集的古铜镜X光图像。实验中采用MATLAB仿真,平台为Intel i5-2400 3.1GHz四核处理器、4G内存。采集到的3组6幅古铜镜图像如图 2所示。图 2中,每组古铜镜X光图像都是采用边缘区和纹饰区的最佳投射强度采集的X光图像。
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为了能更加客观地评价融合图像的性能,实验对比了除本文中算法以外其它3种不同的算法,分别为加权平均+模极大值、加权平均+邻域方差、传统小波变换以及本文中的方法。实验结果如图 3、图 4和图 5所示。
融合得到的效果图一般采用图像信息熵、平均梯度与标准差来对融合图像进行客观评价比较。信息熵U反映了图像中平均信息量的多少,图像的信息熵越大,说明图像的保留的信息就越丰富,其定义为[7]:
$ U = - \sum\limits_{i = 0}^{L - 1} {{P_i}} {\log _2}\left( {{P_i}} \right) $
(9) 式中,Pi表示某一灰度值在图像中出现的概率,L为图像总灰度级数。
平均梯度V表现了图像的边界与影线两侧的灰度情况变化差异,也是图像的清晰度,图像的平均梯度越大,表明图像越清晰。平均梯度V的定义为:
$ \begin{aligned} V &=\frac{1}{(X-1)(Y-1)} \sum\limits_{m-1}^{X-1} \sum\limits_{n=1}^{Y-1} \frac{1}{4} \times \\ & \sqrt{\left[\frac{\delta g(m, n)}{\delta m}\right]^{2}+\left[\frac{\delta g(m, n)}{\delta n}\right]^{2}} \end{aligned} $
(10) 式中, g(m, n)表示图像在第m行、第n列的灰度值。
标准差W可以反映图像的灰度均值的状况,标准差越大,表现出的视觉效果越好,标准差的定义为:
$ W=\sqrt{\frac{\sum\limits_{m=1}^{X} \sum\limits_{n=1}^{Y}[G(m, n)-\bar{G}]^{2}}{X \times Y}} $
(11) 式中,G(m, n)表示点(m, n)的灰度值,${\bar G} $表示点(m, n)的平均灰度。
Table 1. Performance comparison of different fusion algorithms for ancient bronze mirror Ⅰ
information entropy average gradient standard deviation algorithm 1 6.0742 0.0220 30.4525 algorithm 2 6.0789 0.0225 30.2534 algorithm 3 6.0648 0.0243 30.6245 method of this paper 6.3854 0.0342 33.9648 Table 2. Performance comparison of different fusion algorithms for ancient bronze mirror Ⅱ
information entropy average gradient standard deviation algorithm 1 2.8542 0.0273 37.7843 algorithm 2 2.8631 0.0280 38.3145 algorithm 3 2.8653 0.0278 37.5247 method of this paper 2.9775 0.0332 39.6012 Table 3. Performance comparison of different fusion algorithms for ancient bronze mirror Ⅲ
information entropy average gradient standard deviation algorithm 1 3.5739 0.0343 36.7548 algorithm 2 3.6032 0.0335 35.6574 algorithm 3 3.5468 0.0344 35.8962 method of this paper 3.8625 0.0398 38.6214 从表中可以看出,本文中的算法在图像的信息熵、平均梯度和标准差上都略高于其它3种算法。分析表中数据可知,在3组铜镜实验中, 相对于其它3种算法,本文中的算法信息熵平均提升了5.76%,平均梯度平均提升了28.70%,标准差平均提升了7.70%,由此可知,本文中算法的融合图像在信息量清晰度视觉效果方面都优于其它算法。综合分析和对照数据可知:本文中提出的算法有效地保留了源图像的信息,对于边缘的传递效果更优秀。
基于提升小波的古铜镜X光图像融合方法研究
Study on fusion method of X-ray images of ancient bronze mirrors based on lifting wavelet
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摘要: 为了将古铜镜X光图像信息综合到同一图像中, 采用提升小波的方式将源图像进行分解, 并分别对低频、高频采用不同的融合规则进行了图像融合。低频采用区域能量与区域方差相结合的方法, 高频采用空间频率加邻域像素点规范中间像素点的方法, 最后经提升小波逆变换得到目标图像; 同时进行了理论分析和实验验证, 取得了融合图像的信息熵、平均梯度和标准差数据。结果表明, 在3组实验中, 相对于其它几种算法, 本文中算法的信息熵平均提升了5.76%, 平均梯度平均提升了28.70%, 标准差平均提升了7.70%, 算法有效地保留了源图像的信息, 对于边缘的传递效果更优秀。这一结果对古铜镜X光图像的融合是有帮助的。Abstract: In order to synthesize X-ray image information of bronze mirrors into the same image, source image was decomposed by lifting wavelet. Different fusion rules were used for image fusion at low and high frequencies respectively. The method of combining regional energy with regional variance was used in low frequency. Spatial frequency combined with the method of neighborhood pixels to standardize intermediate pixels was used in high frequency. Finally, the target image was obtained by lifting inverse wavelet transform. Theoretical analysis and experimental verification were carried out. Information entropy, average gradient and standard deviation of the fused image were obtained. The results show that, in three groups of experiments, compared with the other three algorithms, information entropy of the proposed algorithm in this paper is increased by 5.76% on average. Average gradient is increased by 28.70%. Standard deviation is increased by 7.70% on average. The algorithm effectively preserves the information of the source image. Edge transmission effect is better. This result is helpful for the fusion of X-ray images of bronze mirrors.
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Key words:
- image processing /
- image fusion /
- lifting wavelet /
- X-ray image /
- fusion rule
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Figure 2. X-ray images of ancient bronze mirrors
a—fusion source image a of ancient bronze mirrorⅠ b—fusion source image b of ancient bronze mirrorⅠ c—fusion source image a of ancient bronze mirrorⅡ d—fusion source image b of ancient bronze mirrorⅡ e—fusion source image a of ancient bronze mirrorⅢ f—fusion source image b of ancient bronze mirrorⅢ
Table 1. Performance comparison of different fusion algorithms for ancient bronze mirror Ⅰ
information entropy average gradient standard deviation algorithm 1 6.0742 0.0220 30.4525 algorithm 2 6.0789 0.0225 30.2534 algorithm 3 6.0648 0.0243 30.6245 method of this paper 6.3854 0.0342 33.9648 Table 2. Performance comparison of different fusion algorithms for ancient bronze mirror Ⅱ
information entropy average gradient standard deviation algorithm 1 2.8542 0.0273 37.7843 algorithm 2 2.8631 0.0280 38.3145 algorithm 3 2.8653 0.0278 37.5247 method of this paper 2.9775 0.0332 39.6012 Table 3. Performance comparison of different fusion algorithms for ancient bronze mirror Ⅲ
information entropy average gradient standard deviation algorithm 1 3.5739 0.0343 36.7548 algorithm 2 3.6032 0.0335 35.6574 algorithm 3 3.5468 0.0344 35.8962 method of this paper 3.8625 0.0398 38.6214 -
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