基于小波图像阈值分割技术
小波变换是近年来得到广泛应用的数学工具,与傅里叶变换、窗口傅里叶变换相比,小波变换是空间(时间) 和频率的局域变换，能有效地从信号中提取信息。 In recent years the wavelet transform widely used mathematical tools, and Fourier transform window compared to the Fourier transform, wavelet
transform is a spatial (time) and the frequency of the local transformation can be effectively extracted from the signal information.
4.1 基于小波阈值分割技术简述
本论文利用小波变换对含噪图像的直方图进行多尺度分解,先在较大的尺度下找出图像分割阈值的粗略值,然后逐渐减小尺度,精确定位分割阈值，算法采用MATLAB 编程仿真。
In this study, the wavelet transform of the noisy image histogram multiscale decomposition, to identify rough image segmentation threshold value in a larger scale, then gradually reduce the scale precise positioning segmentation threshold simulation algorithm using MATLAB programming.
基于小波变换的阈值法图像分割技术则能够有效地避免噪声的影响。Thresholding method based on wavelet transform image segmentation technology can effectively avoid the impact of noise.
该方法的基本思想是首先由二进制小波变换将图像的直方图分解为不同层次的小波系数, 然后依据给定的分割准则和小波系数选择阈值门限, 最后利用阈值标出图像分割的区域。The basic idea of the method is the first by the binary wavelet transform of the histogram of the image is decomposed into wavelet coefficients of the different levels to select the threshold value threshold, and segmentation based on a given criteria and wavelet coefficients, and finally using the threshold marked region of the image segmentation.
2
整个分割过程是从粗到细, 由尺度变化来控制, 即起始分割由粗略的L（R）
子空间上投影的直方图来实现, 如果分割不理想, 则利用直方图在精细的子空间上的小波系数逐步细化图像分割。Throughout the segmentation process from coarse to fine scale changes in control, that is the starting split roughly L2 (R) sub-space projection histogram, if the split is not ideal, the use of the
histogram in fine subspacewavelet coefficients on the gradual refinement of image segmentation. 4.2 小波分析
基于小波变换的阈值法图像分割技术能有效地弥补传统的图像阈值法分割技术的不足,具有较强的抗噪声性能,同时,对于直方图为多峰值的情况,可以利用小波的多分辨率分解,对灰度阈值进行合理地选择,实现对图像的分割处理。
Multi-resolution image segmentation techniques based on wavelet transform threshold method can effectively compensate for the deficiencies of the traditional image thresholding segmentation technology, has strong anti-noise performance, at the same time, for the histogram for multi-peak, you can take advantage of wavelet decompositiongray threshold reasonable selection, to achieve the image segmentation processing. 4.2.1小波变换
由于图像的直方图可以看作是一维信号，而直方图上的突变点（波峰点和波谷点），往往可以代表图像灰度变化的特征。
Since the histogram of the image can be seen as a one-dimensional signal, and the histogram on the mutation point (peak point and the valley point), can often be representative of the features of the image gray-scale variation.
因此Jean-Christophe Olivo提出了用小波变换对直方图进行处理的方法实现自动阈值提取。Olivo通过检测直方图小波变换的奇异点和区域极值点给出直方图峰值点的特性。而小波变换的波峰和波谷点可以代表图像中灰度代表值和阈值点。利用小波变换多尺度特性实现对图像的阈值分割。又由于小波变换具有多分辨率的特性，因此可以通过对医学图像直方图的小波变换，实现由粗到细的多层次结构的阈值分割。
Therefore, Jean-Christophe Olivo the histogram processing method using wavelet transform automatic threshold extraction. The Olivo by the singular point detection histogram wavelet transform and regional extreme point given the characteristics of the histogram peak point. And the peaks and troughs of the wavelet transform point can represent the gradation representative value
of the image and the threshold point. Multi-scale features using wavelet transform threshold of image segmentation. The characteristics of multi-resolution wavelet transform has, therefore by the wavelet transform for medical image histogram threshold segmentation from coarse to fine hierarchy.
首先在最低分辨率一层进行，然后逐渐向高层推进。小波变换W2j H?x?的零交叉点表示了在分辨率2j时低通信号的局部跳变点。当尺度2j减小时，信号的局部微小细节逐渐增多，因此，能够检测出各微小细节的灰度突变点；当尺度2j增大时，信号的局部细节逐渐消失，而结构较大的轮廓却能清晰地反映出来，因而能检测出该结构较大的灰度突变点。
First, a layer in the lowest resolution, then gradually advancing to senior. The wavelet transform of the zero-cross point indicates a partial transition point of the low resolution 2j communication number. When the scale 2j reduced, partial tiny details of the signal gradually increased, and therefore can detect the mutation point of each minute detail of the gradation; When the scale 2j increasing, a fragmentary detail of the signal gradually disappeared, and the structure is a larger contour able clearly reflected in the structure larger gradation mutation point, and thus can be detected.
因此，可以选择小波为光滑函数??x?的二阶导数，对图像的一维直方图信号进行小波变换，检测出直方图信号的突变点，由此搜索出两峰之问的谷点作为分割阈值点。这就是小波变换用于图像分割的基本原理。
Therefore, it is possible to select the wavelet is a smooth function of the second
derivative,
the
wavelet
transformation
is
performed
on
one-dimensional histogram of the image signal, and detect the mutation point of the histogram signal to thereby search out the valley point of the two peaks Q as the segmentation threshold point. This is the basic principle of the wavelet transform for image segmentation.
对图像的直方图来说，它的各层的小波分解系数表示不同分辨率下的细节信号，它与小波近似信号联合构成直方图的多分辨率小波分解表示。给定直方图，考虑其多分辨率小波分解表示的零交叉点和极值点来确定直方图的峰值点和谷点。
For the histogram of the image, its layers of wavelet coefficients showing the details of the different resolution signal and wavelet approximation signal jointly constitute a histogram of the multi-resolution wavelet decomposition expressed. Given histogram, consider the multi-resolution wavelet decomposition represented by the zero-cross point and the extreme point to determine a histogram of the peak points and valley points. 4.2.2 小波分割算法及步骤
分割算法的计算量与图像尺寸大小呈线性变化，本论文介绍直方图的多分辨率分析。对于每个整数j∈Z（Z整数集合），dj?k/2j;k?Z表示在j分辨率下的二进制有理数。Calculate the amount of the segmentation algorithm and the image size changes linearly, this paper introduces the multi-resolution histogram analysis. For each integer j ∈ Z (Z set of integers), expressed in the resolution of j binary rationals.
因此，对于任何j∈Z，dj是一组在实数轴上的等间隔采样点集合，如果i
??hf?k????x,y?:f?x,y??k?;k??0,g? （4-1）
Therefore, for any j ∈ Z, the collection is a group of the real axis interval sampling point if i
?(x)???1,?0n?Z0?x?1,其他 （4-2）
hjf(x)??hf(2?jn)?(2?jx?n) （4-3）
Wherein said counting operation is a discrete function. Order, discrete function expressed as a continuous function, seen as composed by several piecewise constant function. For j ∈ Z, sampling according to the sampling point, the histogram represents the j resolution. Further pan with telescopic Haar scaling function
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