基于小波图像阈值分割技术.doc

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.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L2（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jZ（Z整数集合），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jZ，是一组在实数轴上的等间隔采样点集合，如果ij,则表示高分辨率（较细）的采样点。假定f表示为一幅图像，g是图像f中最大灰度，则直方图表示为 （4-1）Therefore, for any j Z, the collection is a group of the real axis interval sampling point if i j, said high-resolutionrate (finer) sampling points. Assuming f expressed as an image, g is the maximum gradation of the image f is expressed as the histogram式中表示计数操作，是离散函数。令，离散函数表示成连续函数,看作是由几个分段常数函数组成。对于jZ，按采样点采样，则表示在j分辨率下的直方图。进一步可以用Haar尺度函数的平移与伸缩表示，即 （4-2） （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多层表达曲线可以表示如下 （4-4）对直方图进行小波分解，利用小波系数，按式（4-4）重建直方图，从近似直方图中选择阈值，完成分割阈值。步骤如下：Wavelet decomposition of the histogram, wavelet coefficients, according to the formula (4-4) to rebuild the histogram, choose from approximate histogram threshold, complete the segmentation threshold. The steps are as follows:步骤1：预设分割区域为M，分解级数，L为图像最大灰度值；步骤2：小波分解曲线，得到，令j=0,;步骤3：，将大于j分解层次的系数设置为0，用式（4-4）重建，在重建直方图中，找出满足和条件的标号l（灰度），并且统计标号l的个数n;步骤4：如果nM,则j=j+1,当jJ时，转向步骤:3；步骤5：从重建直方图中，找到阈值；步骤6：像素值与阈值比较，标出所在区域。Step 1: The default divided region M, decomposition level, L is the image maximum gradation value;Step 2: The wavelet decomposition curve obtained, j = 0;Step 3:, will be greater than the j the coefficients of the decomposition level is set to 0, the formula (4-4) Reconstruction meet and the conditions in the reconstruction in the histogram to identify the numeral l (gray scale), and the statistical reference numeral l of a number n;Step 4: If n M, then j = j +1, when j J, steering steps: 3;Step 5: from the reconstruction histogram, find the threshold;Step 6: the pixel value and the threshold value comparison, marked Area.4.3 阈值选取以及实验分析本论文所采用波谷点确定为图像分割的阈值点，两阈值平均点作为后一阈值和前一阈值之间区间灰度的代表值。This thesis uses the trough point threshold point is determined as the image segmentation, two threshold average point as a representative value of the interval between grayscale after a threshold value and a previous threshold.4.3.1 直方图分辨率的小波表示设图像的灰度范围为0，1，2，N-1，灰度值x(0xN-1)对应的像素为n0，则一幅图像总像素为M：Set the gradation of an image range of 0, 1, 2, ., N-1, the gray value x (0 x N-1) of pixels corresponding to N0, then a total image pixels as M: （4-5）灰度值x出现的概率为：The probability of occurrence of the gray-scale value x as: （4-6）由上式可以建立该图像的直方图，它反映了该图H(x)=P x，x=O，1，N-1)上灰度分布的统计特性，是基于像素灰度的图像分割方法的基础。Statistical properties, can create a histogram of the image by the above formula, which reflects the FIG H (x) = P-X, X = O, 1, ., N-1) on the gray level distribution is based on the pixel graythe basis of image segmentation methods.为了建立小波变换的多分辨率分解表示，引入尺度函数(x)，其傅立叶变换满足条件：In order to establish the multi-resolution wavelet transform decomposition, said the introduction of the scales function (x), its Fourier transform to meet the conditions （4-7）可见，(x)相当于低通滤波器，这样图像直方图H(x)的低通分量为：Visible, (x) is equivalent to a low pass filter, so that an image histogram H (x) of the low-pass component: （4-8）设原始图像直方图信号各尺度之间的各阶小波变换。可以证明：信号在在尺度为时被平滑掉的高频成分，可以用尺度的小波变换来恢复，我们称集合为图像直方图信号的多分辨率小波分解表示。直方图信号多分辨小波分解由一个最低分辨率下的近似信号和一组分辨率的细节信号所组成。这是一种介于频域和时域的表示。为图像分析提供了一个由粗到细的分层框架。Wavelet transform of the order set the original image histogram signal scales between . Can prove: a signal in the scale time-out of the high-frequency component is smoothed, can be used to scale wavelet transform to recover, we call collection is a multi-resolution wavelet decomposition of the image histogram signal. Histogram signal multiresolution wavelet decomposition composed by a minimum resolution approximation signal and a set of the details of the resolution signal. This is a cross between the frequency-domain and time-domain representation. Image analysis provides a layered framework from coarse to fine.4.3.2 多分辨率阈值选取基于直方图和小波变换的图像分割技术由以下几个步骤组成：首先由粗分辨率下的图像直方图细节信息确定分割区域类数；其次，在相邻峰之间自动确定最优阈值；最后用求出的最优阈值分割原图像。Histogram and wavelet transform-based image segmentation technology consists of the following steps: First class number of the divided region is determined by the coarse resolution image histogram details; automatically determine the optimal threshold between adjacent peaks; finalobtained optimal threshold segmentation of the original image.由于图像的原始直方图一般不够平滑或含有一定的噪声，因此，有必要对原始直方图进行平滑处理，以利于分割目标。其方法为：在空间域中采用保护边缘平滑方法平滑直方图，它既能保留原直方图基本变化特性，又能消除小峰的跳动。或者选取大尺度下的小波变换系数对直方图进行处理，也可以减小噪声的影响。As the raw histogram of the image is generally not smooth enough, or contains a certain amount of noise, therefore, it is necessary to smoothing process to the original histogram, to facilitate the segmented target. Its methods: to smooth histogram with protection edge smoothing method in the spatial domain, it not only to preserve the the original histogram basic change characteristics, but also eliminate the small peak of beating. Or select the large scale wavelet transform coefficient histogram processing, but also can reduce the impact of noise.在分辨率为2j时，由小波分解后的直方图近似信号的极大值确定初始区域类数，即确定峰的数目。对于灰度级数不多的原始影像，一个区域类通常对应直方图中的一个峰，然而，对于一幅复杂图像，经小波分解后平滑直方图中的每个峰则不一定都对应一个区域类，它也可能从属于邻近的一个峰，因而有必要通过检查认定哪些峰对应于分割区域类。At a resolution of 2j, the maximum value to determine the class number of the initial region of the histogram approximation signal by the wavelet decomposition, i.e. to determine the number of peaks.峰的独立性判断是为了消除不能成为一类的那些峰，独立峰应满足三个条件：Peak independence of judgment in order to eliminate the not become a class of those peaks, independent peaks should satisfy three c