图像处理之高斯混合模型

图像处理之高斯混合模型

   一：概述

高斯混合模型(GMM)在图像分割、对象识别、视频分析等方面均有应用，对于任意给定的数据样本集合，根据其分布概率， 可以计算每个样本数据向量的概率分布，从而根据概率分布对其进行分类，但是这些概率分布是混合在一起的，要从中分离出单个样本的概率分布就实现了样本数据聚类，而概率分布描述我们可以使用高斯函数实现，这个就是高斯混合模型-GMM。

这种方法也称为D-EM即基于距离的期望最大化。

   三：算法步骤

    1.初始化变量定义-指定的聚类数目K与数据维度D

    2.初始化均值、协方差、先验概率分布

    3.迭代E-M步骤

         - E步计算期望

         - M步更新均值、协方差、先验概率分布

         -检测是否达到停止条件（最大迭代次数与最小误差满足），达到则退出迭代，否则继续E-M步骤

    4.打印最终分类结果

四：代码实现

package com.gloomyfish.image.gmm;import java.util.ArrayList;import java.util.Arrays;import java.util.List;/** *  * @author gloomy fish * */public class GMMProcessor {public final static double MIN_VAR = 1E-10;public static double[] samples = new double[]{10, 9, 4, 23, 13, 16, 5, 90, 100, 80, 55, 67, 8, 93, 47, 86, 3};private int dimNum;private int mixNum;private double[] weights;private double[][] m_means;private double[][] m_vars;private double[] m_minVars;/*** *  * @param m_dimNum - 每个样本数据的维度， 对于图像每个像素点来说是RGB三个向量 * @param m_mixNum - 需要分割为几个部分，即高斯混合模型中高斯模型的个数 */public GMMProcessor(int m_dimNum, int m_mixNum) {dimNum = m_dimNum;mixNum = m_mixNum;weights = new double[mixNum];m_means = new double[mixNum][dimNum];m_vars = new double[mixNum][dimNum];m_minVars = new double[dimNum];}/*** * data - 需要处理的数据 * @param data */public void process(double[] data) {int m_maxIterNum = 100;double err = 0.001;boolean loop = true;double iterNum = 0;double lastL = 0;double currL = 0;int unchanged = 0;initParameters(data);int size = data.length;double[] x = new double[dimNum];double[][] next_means = new double[mixNum][dimNum];double[] next_weights = new double[mixNum];double[][] next_vars = new double[mixNum][dimNum];List<DataNode> cList = new ArrayList<DataNode>();while(loop) {Arrays.fill(next_weights, 0);cList.clear();for(int i=0; i<mixNum; i++) {Arrays.fill(next_means[i], 0);Arrays.fill(next_vars[i], 0);}lastL = currL;currL = 0;for (int k = 0; k < size; k++){for(int j=0;j<dimNum;j++)x[j]=data[k*dimNum+j];double p = getProbability(x); // 总的概率密度分布DataNode dn = new DataNode(x);dn.index = k;cList.add(dn);double maxp = 0;for (int j = 0; j < mixNum; j++){double pj = getProbability(x, j) * weights[j] / p; // 每个分类的概率密度分布百分比if(maxp < pj) {maxp = pj;dn.cindex = j;}next_weights[j] += pj; // 得到后验概率for (int d = 0; d < dimNum; d++){next_means[j][d] += pj * x[d];next_vars[j][d] += pj* x[d] * x[d];}}currL += (p > 1E-20) ? Math.log10(p) : -20;}currL /= size;// Re-estimation: generate new weight, means and variances.for (int j = 0; j < mixNum; j++){weights[j] = next_weights[j] / size;if (weights[j] > 0){for (int d = 0; d < dimNum; d++){m_means[j][d] = next_means[j][d] / next_weights[j];m_vars[j][d] = next_vars[j][d] / next_weights[j] - m_means[j][d] * m_means[j][d];if (m_vars[j][d] < m_minVars[d]){m_vars[j][d] = m_minVars[d];}}}}// Terminal conditionsiterNum++;if (Math.abs(currL - lastL) < err * Math.abs(lastL)){unchanged++;}if (iterNum >= m_maxIterNum || unchanged >= 3){loop = false;}}// print resultSystem.out.println("=================最终结果=================");for(int i=0; i<mixNum; i++) {for(int k=0; k<dimNum; k++) {System.out.println("[" + i + "]: ");System.out.println("means : " + m_means[i][k]);System.out.println("var : " + m_vars[i][k]);System.out.println();}}// 获取分类for(int i=0; i<size; i++) {System.out.println("data[" + i + "]=" + data[i] + " cindex : " + cList.get(i).cindex);}}/** *  * @param data */private void initParameters(double[] data) {// 随机方法初始化均值int size = data.length;for (int i = 0; i < mixNum; i++){for (int d = 0; d < dimNum; d++){m_means[i][d] = data[(int)(Math.random()*size)];}}// 根据均值获取分类int[] types = new int[size];for (int k = 0; k < size; k++){double max = 0;for (int i = 0; i < mixNum; i++){double v = 0;for(int j=0;j<dimNum;j++) {v += Math.abs(data[k*dimNum+j] - m_means[i][j]);}if(v > max) {max = v;types[k] = i;}}}double[] counts = new double[mixNum];for(int i=0; i<types.length; i++) {counts[types[i]]++;}// 计算先验概率权重for (int i = 0; i < mixNum; i++){weights[i] = counts[i] / size;}// 计算每个分类的方差int label = -1;int[] Label = new int[size];double[] overMeans = new double[dimNum];double[] x = new double[dimNum];for (int i = 0; i < size; i++){for(int j=0;j<dimNum;j++)x[j]=data[i*dimNum+j];label=Label[i];// Count each Gaussiancounts[label]++;for (int d = 0; d < dimNum; d++){m_vars[label][d] += (x[d] - m_means[types[i]][d]) * (x[d] - m_means[types[i]][d]);}// Count the overall mean and variance.for (int d = 0; d < dimNum; d++){overMeans[d] += x[d];m_minVars[d] += x[d] * x[d];}}// Compute the overall variance (* 0.01) as the minimum variance.for (int d = 0; d < dimNum; d++){overMeans[d] /= size;m_minVars[d] = Math.max(MIN_VAR, 0.01 * (m_minVars[d] / size - overMeans[d] * overMeans[d]));}// Initialize each Gaussian.for (int i = 0; i < mixNum; i++){if (weights[i] > 0){for (int d = 0; d < dimNum; d++){m_vars[i][d] = m_vars[i][d] / counts[i];// A minimum variance for each dimension is required.if (m_vars[i][d] < m_minVars[d]){m_vars[i][d] = m_minVars[d];}}}}System.out.println("=================初始化=================");for(int i=0; i<mixNum; i++) {for(int k=0; k<dimNum; k++) {System.out.println("[" + i + "]: ");System.out.println("means : " + m_means[i][k]);System.out.println("var : " + m_vars[i][k]);System.out.println();}}}/*** *  * @param sample - 采样数据点 * @return 该点总概率密度分布可能性 */public double getProbability(double[] sample){double p = 0;for (int i = 0; i < mixNum; i++){p += weights[i] * getProbability(sample, i);}return p;}/** * Gaussian Model -> PDF * @param x - 表示采样数据点向量 * @param j - 表示对对应的第J个分类的概率密度分布 * @return - 返回概率密度分布可能性值 */public double getProbability(double[] x, int j){double p = 1;for (int d = 0; d < dimNum; d++){p *= 1 / Math.sqrt(2 * 3.14159 * m_vars[j][d]);p *= Math.exp(-0.5 * (x[d] - m_means[j][d]) * (x[d] - m_means[j][d]) / m_vars[j][d]);}return p;}public static void main(String[] args) {GMMProcessor filter = new GMMProcessor(1, 2);filter.process(samples);}}

结构类DataNode

package com.gloomyfish.image.gmm;public class DataNode {public int cindex; // clusterpublic int index;public double[] value;public DataNode(double[] v) {this.value = v;cindex = -1;index = -1;}}

五：结果

这里初始中心均值的方法我是通过随机数来实现，GMM算法运行结果跟初始化有很大关系，常见初始化中心点的方法是通过K-Means来计算出中心点。大家可以尝试修改代码基于K-Means初始化参数，我之所以选择随机参数初始，主要是为了省事！

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