基于小波变换的显微图像细胞识别
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摘要
计算机医学图像在临床诊断和治疗中的作用越来越显著。而显微图像的自动分析是医学图像处理和分析的重要研究课题，它不仅为医生赢得了诊断时间而且提高了诊断的精度。
由于小波变换在不同尺度层上具有“变焦”的功能，根据显微图像的特点利用小波变换的自适应时频窗性质，对细胞边缘和背景噪声采取不同的阈值，再利用模角可分离的性质有效地分离出细胞边缘。对此细胞边缘运用圆弧算法实现显微图像的细胞识别和定量分析。
多分辨分析和小波变换
要构造一个空间中的小波基，关键是要找到它的相应的多尺度分析生成元，通过实际检测，三阶B样条函数更适合显微图像的处理，
给出M+1尺度上的离散采样值数据，要计算处M尺度层上的数据，这就是分解算法的要求。计算分解系数的过程和公式具体地给出如下：
= +
+ +
相应的重构算法为：
＝
＋
＋
＋
在正交小波分解中，用复杂地计算来且定初始数据是不合算的，此处用了变通的方法，简单取用
＝，
即用细尺度层上的采样值作为初始数据。经实际应用和推理这种假设是合理可行的。
在不同尺度下的图象一般是经过光滑处理的，我们考虑梯度向量：
＝
因为在处的梯度向量的方向指明了图像沿这些方向的最大绝对值，图像多尺度边缘就是二进小波变换的局部极值点，边缘点实际是曲面的拐点。
在尺度下，定义梯度向量的振幅为
幅角为：
幅角等于梯度和水平方向的夹角，我们沿幅角方向求出梯度向量的模的极大值点，即找到了图像的边缘点。
显微图像的去噪
经过分析和大量的实验，基于显微图像的特点，本文提出了加权组合的阈值选取——对信号小波分解中的低频部分，先检索出与突变点相对应的各尺度层上的小波变换模值，先找到一个突变点的模值，按此点的四个方向——0，45，90，135
找出各点的模值，通过下述公式计算出此点新的模值：
其中为输出值，为四个方向上的模值，为第i个方向上的模值与此点模值的差。对于新产生的模值设定一个阈值，大于此阈值的模值点保留此点及其附近的小波变换值；小于此阈值点的模值置为零。对信号小波分解中细节部分仅保留与突变点位置相应地小波地变换值，其它小波变换值用零值代替。这种处理方法相当于不同尺度层使用不同阈值处理，能较好地表现信号地突变部分，消除噪声表现，能较好地回复各个尺度层的回复信号，回复信号的消噪效果是明显的。
显微图像边缘提取
由于边缘是图像的灰度级不连续点，具有奇异性，因此，可以利用检测小波系数模极大值的方法来检测图像的边缘。图像二进小波变换的模正比于梯度向量的模，而小波变换的幅角等于梯度向量与水平方向的夹角。图像的边缘就是其梯度向量模值的局部最大值点，用二进小波变换对图像边缘进行检测，也就是寻找小波变换的模值沿幅角方向的局部极大值点。但在光栅格离散的状态下，与一个像素点紧密相连的周围的8个像素点，分别对应方向0，，，，，，，，而的取值范围为，在此设表示沿方向灰度
变换强度，则
首先找出沿＝0，，，这4个方向的边缘强度变换的极大值点，然后在将4个方向的极大值点进行叠加，即可得到该尺度下图像的边缘。试验结果表明，该方法可以检测出大部分的边缘点。
显微图像的细胞识别和定量分析
本文对显微图像的特点和细胞的边缘特征进行了研究，运用圆弧算法实现了红白细胞的识别和定量分析，取得了很好的效果。
通过对B样条函数进行小波变换得出奇变点出的模值，记录出模值大于阈值的点（此处的阈值取为14），然后对所得的点确定圆弧的候选链表，按照链表生成条件查找以此链表头为出发点的候选圆弧链。在此基础上提取表头来对各个圆弧进行识别处理。
首先，确定该圆弧的起始点和终止点。
其次，对圆（弧）定位，确定圆（弧）心和半径。根据弧的走向和圆心角的大小，弧可被完整地识别出来。
最后，根据红白细胞半径的大小差异可对红白细胞进行定量分析。
增强结果：
加权阈值选取后增强和二值化效果图
识别结果：
显微图片细胞的识别结果
测试结果
本文实验仿真采用VC++6.0编程实现。经对736张医学显微图像的测试表明：
红细胞的识别率＞95%；
白细胞的识别率＞92%；
整张显微图像的识别率＞87%；
（以上识别率均以人工阅片数为基准。）
The computer medical image has become more and more powerful in clinic diagnosis and therapy. How to analyze the micro image automatically is a very important research field in medical image processing and analysis, which can win the diagnosis time for the doctor and increase the accuracy of the diagnosis.
Because the wavelet transform has different focus for different scale layer, people can set different threshold value for the cell edge and background noise by using the self adaptive time-frequency feature of the wavelet according to characters of the micro images. Then the cell edge can be effectively separated by separating the modulo angle. So the cell identification and quantitative analysis for the micro image is realized by performing circular arc algorithm to the cell edge.
1. Multiscale analysis and wavelet transform
The key to construct a wavelet base in a space is to find its corresponding generatorof the multiscale analysis. The third degree B-spline is more suitable for the micro image processing which has been proved in practice.
To use the analysis algorithm, the discrete sample value data for M+1 scale must be given and the data on the M scale layer must be computed. The process and formulas of computing the analysis coefficients are as follows:
= +
+ +
The corresponding reconstruction algorithm is as follows:
＝
＋
＋
＋
In the orthonormal wavelet analysis process, it is not convenient to find the original data at the price of complicated computation. So here we simplify the question by let
＝，
That is let the sample value of the fine scale layer be the original data. The hypothesis has been proved to be feasible and reasonable in the practice and theory.
The images for different scales usually have been soothed. We consider the gradient vector ：
＝
Because the direction of the gradient vector atindicates the maximum absolute value of the image in this direction, the multiscale edge of the image is the local extreme value of the 2-d wavelet transform and the edge points are the inflexions of the curve .
In scale, define the amplitude of the gradient vector as
and the angle as .
angle is equal to the angle of the gradient in the horizontal direction. We compute the extreme value of the gradient vector in the angle direction, which is the edge point of the image.
2. Noise removal of the micro image
This paper put forward a method to select the threshold value of the weighted combination. For the low frequency in the signal wavelet analysis, we first search for the corresponding wavelet transform modulo value in every scale layer of the mutant point, then after finding the modulo value of each mutant point, compute the modulo values in four directions: 0，45，90，135, by the formula
where is the output value，is the modulo value in four direction，is the difference between the modulo value in the ith direction and the modulo value of the point. We set a threshold value for the new modulo value. If the modulo point is greater than the threshold value, keep the transform value of this point and its neighborhood as its value, while the modulo point is zero if it is less than the threshold value. That is the detail part in the signal analysis which only detains the wavelet transform value corresponding with the position of the mutant points while the other wavelet transform values are substituted by zero. This method is different from the ones which use different threshold value for different scale layer, which lead to better representation of the mutant part in the signal, elimination of the
noise behavior and better feedback to the feedback signal for each scale layer and the effect of the noise removal is obvious.
3. Edge extraction of the micro image
Because the edge is the discontinuous point of the image gray grade which has singularity, image edge can be detected by searching for the maximum of the wavelet coefficient modulo. The modulo of the image 2-d wavelet transform is in dire
由于小波变换在不同尺度层上具有“变焦”的功能，根据显微图像的特点利用小波变换的自适应时频窗性质，对细胞边缘和背景噪声采取不同的阈值，再利用模角可分离的性质有效地分离出细胞边缘。对此细胞边缘运用圆弧算法实现显微图像的细胞识别和定量分析。
多分辨分析和小波变换
要构造一个空间中的小波基，关键是要找到它的相应的多尺度分析生成元，通过实际检测，三阶B样条函数更适合显微图像的处理，
给出M+1尺度上的离散采样值数据，要计算处M尺度层上的数据，这就是分解算法的要求。计算分解系数的过程和公式具体地给出如下：
= +
+ +
相应的重构算法为：
＝
＋
＋
＋
在正交小波分解中，用复杂地计算来且定初始数据是不合算的，此处用了变通的方法，简单取用
＝，
即用细尺度层上的采样值作为初始数据。经实际应用和推理这种假设是合理可行的。
在不同尺度下的图象一般是经过光滑处理的，我们考虑梯度向量：
＝
因为在处的梯度向量的方向指明了图像沿这些方向的最大绝对值，图像多尺度边缘就是二进小波变换的局部极值点，边缘点实际是曲面的拐点。
在尺度下，定义梯度向量的振幅为
幅角为：
幅角等于梯度和水平方向的夹角，我们沿幅角方向求出梯度向量的模的极大值点，即找到了图像的边缘点。
显微图像的去噪
经过分析和大量的实验，基于显微图像的特点，本文提出了加权组合的阈值选取——对信号小波分解中的低频部分，先检索出与突变点相对应的各尺度层上的小波变换模值，先找到一个突变点的模值，按此点的四个方向——0，45，90，135
找出各点的模值，通过下述公式计算出此点新的模值：
其中为输出值，为四个方向上的模值，为第i个方向上的模值与此点模值的差。对于新产生的模值设定一个阈值，大于此阈值的模值点保留此点及其附近的小波变换值；小于此阈值点的模值置为零。对信号小波分解中细节部分仅保留与突变点位置相应地小波地变换值，其它小波变换值用零值代替。这种处理方法相当于不同尺度层使用不同阈值处理，能较好地表现信号地突变部分，消除噪声表现，能较好地回复各个尺度层的回复信号，回复信号的消噪效果是明显的。
显微图像边缘提取
由于边缘是图像的灰度级不连续点，具有奇异性，因此，可以利用检测小波系数模极大值的方法来检测图像的边缘。图像二进小波变换的模正比于梯度向量的模，而小波变换的幅角等于梯度向量与水平方向的夹角。图像的边缘就是其梯度向量模值的局部最大值点，用二进小波变换对图像边缘进行检测，也就是寻找小波变换的模值沿幅角方向的局部极大值点。但在光栅格离散的状态下，与一个像素点紧密相连的周围的8个像素点，分别对应方向0，，，，，，，，而的取值范围为，在此设表示沿方向灰度
变换强度，则
首先找出沿＝0，，，这4个方向的边缘强度变换的极大值点，然后在将4个方向的极大值点进行叠加，即可得到该尺度下图像的边缘。试验结果表明，该方法可以检测出大部分的边缘点。
显微图像的细胞识别和定量分析
本文对显微图像的特点和细胞的边缘特征进行了研究，运用圆弧算法实现了红白细胞的识别和定量分析，取得了很好的效果。
通过对B样条函数进行小波变换得出奇变点出的模值，记录出模值大于阈值的点（此处的阈值取为14），然后对所得的点确定圆弧的候选链表，按照链表生成条件查找以此链表头为出发点的候选圆弧链。在此基础上提取表头来对各个圆弧进行识别处理。
首先，确定该圆弧的起始点和终止点。
其次，对圆（弧）定位，确定圆（弧）心和半径。根据弧的走向和圆心角的大小，弧可被完整地识别出来。
最后，根据红白细胞半径的大小差异可对红白细胞进行定量分析。
增强结果：
加权阈值选取后增强和二值化效果图
识别结果：
显微图片细胞的识别结果
测试结果
本文实验仿真采用VC++6.0编程实现。经对736张医学显微图像的测试表明：
红细胞的识别率＞95%；
白细胞的识别率＞92%；
整张显微图像的识别率＞87%；
（以上识别率均以人工阅片数为基准。）
The computer medical image has become more and more powerful in clinic diagnosis and therapy. How to analyze the micro image automatically is a very important research field in medical image processing and analysis, which can win the diagnosis time for the doctor and increase the accuracy of the diagnosis.
Because the wavelet transform has different focus for different scale layer, people can set different threshold value for the cell edge and background noise by using the self adaptive time-frequency feature of the wavelet according to characters of the micro images. Then the cell edge can be effectively separated by separating the modulo angle. So the cell identification and quantitative analysis for the micro image is realized by performing circular arc algorithm to the cell edge.
1. Multiscale analysis and wavelet transform
The key to construct a wavelet base in a space is to find its corresponding generatorof the multiscale analysis. The third degree B-spline is more suitable for the micro image processing which has been proved in practice.
To use the analysis algorithm, the discrete sample value data for M+1 scale must be given and the data on the M scale layer must be computed. The process and formulas of computing the analysis coefficients are as follows:
= +
+ +
The corresponding reconstruction algorithm is as follows:
＝
＋
＋
＋
In the orthonormal wavelet analysis process, it is not convenient to find the original data at the price of complicated computation. So here we simplify the question by let
＝，
That is let the sample value of the fine scale layer be the original data. The hypothesis has been proved to be feasible and reasonable in the practice and theory.
The images for different scales usually have been soothed. We consider the gradient vector ：
＝
Because the direction of the gradient vector atindicates the maximum absolute value of the image in this direction, the multiscale edge of the image is the local extreme value of the 2-d wavelet transform and the edge points are the inflexions of the curve .
In scale, define the amplitude of the gradient vector as
and the angle as .
angle is equal to the angle of the gradient in the horizontal direction. We compute the extreme value of the gradient vector in the angle direction, which is the edge point of the image.
2. Noise removal of the micro image
This paper put forward a method to select the threshold value of the weighted combination. For the low frequency in the signal wavelet analysis, we first search for the corresponding wavelet transform modulo value in every scale layer of the mutant point, then after finding the modulo value of each mutant point, compute the modulo values in four directions: 0，45，90，135, by the formula
where is the output value，is the modulo value in four direction，is the difference between the modulo value in the ith direction and the modulo value of the point. We set a threshold value for the new modulo value. If the modulo point is greater than the threshold value, keep the transform value of this point and its neighborhood as its value, while the modulo point is zero if it is less than the threshold value. That is the detail part in the signal analysis which only detains the wavelet transform value corresponding with the position of the mutant points while the other wavelet transform values are substituted by zero. This method is different from the ones which use different threshold value for different scale layer, which lead to better representation of the mutant part in the signal, elimination of the
noise behavior and better feedback to the feedback signal for each scale layer and the effect of the noise removal is obvious.
3. Edge extraction of the micro image
Because the edge is the discontinuous point of the image gray grade which has singularity, image edge can be detected by searching for the maximum of the wavelet coefficient modulo. The modulo of the image 2-d wavelet transform is in dire
引文
Bevlkin , G. , Coifman ,R ， and Rokhlin , V. , Fast wavelet transforms and numerical algorithms I , Comm. Pure and Applied Math. ，1990，
Chui , C. K. and J. Z. Wang , An analysis of cardinal spline-wavelets, CAT Report #231, Texas A&M University ,1991.
Jaffard ,S. , and Laurencot , P., Orthonormal wavelets, analysis of operations and applications to numerical analysis, 第三次中法小波会议论文集，1992.
Meyer , Y. , wavelet and applications, 1990年京都国际数学家会议报告.
Simoncelli , E. P. ,Freeman , W. T. ,Adelon, E. H. ,and Heeger ,D.J. Shiftabale multiscale transforms, IEEE Trans. On Information Theory, Vol.38,1992.
Unser ,M. and , Aldroubi , A. , and Edem , M. , On the asymptopic convergence of B-spline wavelet to Gabor function , IEEE Trans. On Information Theory, Vol.38,1992.
Wilson ,R. , Calay , A.D. , and Pearson , E.R.S，A generalized wavelet transforms for Fourier analysis :The Multiresolution Fourier transform and
its application to image and audio signal analysis, IEEE Trans. On Information Theory, Vol.38,1992.
Yu , X.M. , Nonlinear wavelet approximation and the space . Theory & its appl. 1991
Zhong , S. , Edges representation from wavelet transform maxima, ph, New York Univ. 1990
秦前清，杨宗凯。 实用小波分析，西安电子科技大学出版社，1994。
徐长发，李国宽。 实用小波方法， 华中科技大学出版社，2004。
Chui , C. K. and J. Z. Wang , An analysis of cardinal spline-wavelets, CAT Report #231, Texas A&M University ,1991.
Jaffard ,S. , and Laurencot , P., Orthonormal wavelets, analysis of operations and applications to numerical analysis, 第三次中法小波会议论文集，1992.
Meyer , Y. , wavelet and applications, 1990年京都国际数学家会议报告.
Simoncelli , E. P. ,Freeman , W. T. ,Adelon, E. H. ,and Heeger ,D.J. Shiftabale multiscale transforms, IEEE Trans. On Information Theory, Vol.38,1992.
Unser ,M. and , Aldroubi , A. , and Edem , M. , On the asymptopic convergence of B-spline wavelet to Gabor function , IEEE Trans. On Information Theory, Vol.38,1992.
Wilson ,R. , Calay , A.D. , and Pearson , E.R.S，A generalized wavelet transforms for Fourier analysis :The Multiresolution Fourier transform and
its application to image and audio signal analysis, IEEE Trans. On Information Theory, Vol.38,1992.
Yu , X.M. , Nonlinear wavelet approximation and the space . Theory & its appl. 1991
Zhong , S. , Edges representation from wavelet transform maxima, ph, New York Univ. 1990
秦前清，杨宗凯。 实用小波分析，西安电子科技大学出版社，1994。
徐长发，李国宽。 实用小波方法， 华中科技大学出版社，2004。