k均值聚类
原文地址：http://blog.csdn.NET/hjimce/article/details/45200985
作者：hjimce
高斯混合模型和k均值聚类是聚类算法中的两种比较常用而简单的算法，这里先介绍k均值聚类算法。
一、K-means算法理论简介
K-means算法是硬聚类算法，是典型的基于原型的目标函数聚类方法的代表，它是数据点到原型的某种距离作为优化的目标函数，利用函数求极值的方法得到迭代运算的调整规则。K-means算法以欧式距离作为相似度测度，它是求对应某一初始聚类中心向量V最优分类，使得评价指标J最小。算法采用误差平方和准则函数作为聚类准则函数，总之就是要使得下面的公式最小化：
算法过程如下：
1）从N个文档随机选取K个文档作为质心
2）对剩余的每个文档测量其到每个质心的距离，并把它归到最近的质心的类
3）重新计算已经得到的各个类的质心
4）迭代2～3步直至新的质心与原质心相等或小于指定阈值，算法结束
二、K-means算法实现
K均值聚类算法实现分为四个步骤，设数据集为二维data，用matlab把数据绘制出来，如下所示：
绘制代码：
绘制结果：
现在假设要把该数据集分为4类，算法步骤如下：
1) 初始化聚类中心
由于聚类个数选择4，因此我们用matlab随机生成4个不重复的整数，且大小不超过数据点的个数，得到初始聚类中心A,B,C,D。
2) 计算每个点分别到4个聚类中心的距离，找出最小的那个。假设点p为数据集中的点，求出A、B、C、D中距离p点最近的那个点，设B距离P最近，那么就把p点聚类为B类。
3) 更新聚类中心。根据步骤2可得数据集的聚类结果，根据聚类结果，计算每个类的重心位置，作为更新的聚类中心。然后返回步骤2，重新进行聚类，如此循环步骤2与步骤3，直到迭代收敛。
最后贴一下代码：
- close all;
- clear;
- clc;
- % %生成高斯分布随机数1
- % mu = [2 3];
- % SIGMA = [1 0; 0 2];
- % r1 = mvnrnd(mu,SIGMA,100);
- % plot(r1(:,1),r1(:,2),'r+');
- % hold on;
- % %生成高斯分布随机数1
- % mu = [7 8];
- % SIGMA = [ 1 0; 0 2];
- % r2 = mvnrnd(mu,SIGMA,100);
- % plot(r2(:,1),r2(:,2),'*')
- %
- % data=[r1;r2]
- data=importdata('data.txt');
- % 算法流程
- figure(1);
- plot(data(:,1),data(:,2),'*');
- figure(2);
- [m n]=size(data);
- %聚类个数为4，则4个不重复的整数
- p=randperm(m);
- kn=4;
- d=p(1:kn);
- center(1:kn,:)=data(d(:),:);
- flag=zeros(1,m);
- %计算每个点到4个质心点的最近的那个点
- it=1;
- while(it<30 for="" ii="1:m;" flag="" ii="" -1="" mindist="inf;" for="" jj="1:kn;" dst="norm(data(ii,:)-center(jj,:));" if="" mindist="">dst;
- mindist=dst;
- flag(ii)=jj;
- end
- end
- end
- %更新聚类中心
- center=zeros(size(center));
- countflag=zeros(1,kn);
- for ii=1:m
- for jj=1:kn
- if(flag(ii)==jj)
- center(jj,:)=center(jj,:)+data(ii,:);
- countflag(jj)=countflag(jj)+1;
- end
- end
- end
- for jj=1:kn
- center(jj,:)=center(jj,:)./countflag(jj);
- end
- it=it+1;
- end
- hold on;
- for i=1:m;
- if(flag(i)==1);
- plot(data(i,1),data(i,2),'.y');
- elseif flag(i)==2
- plot(data(i,1),data(i,2),'.b');
- elseif (flag(i)==3)
- plot(data(i,1),data(i,2),'.k');
- elseif (flag(i)==4)
- plot(data(i,1),data(i,2),'.r');
- end
- end</30>
聚类结果如下：
opencv版kmeans：
声明：
- enum
- {
- KMEANS_RANDOM_CENTERS=0, // Chooses random centers for k-Means initialization
- KMEANS_PP_CENTERS=2, // Uses k-Means++ algorithm for initialization
- KMEANS_USE_INITIAL_LABELS=1 // Uses the user-provided labels for K-Means initialization
- };
- //! clusters the input data using k-Means algorithm
- CV_EXPORTS_W double kmeans( InputArray data, int K, CV_OUT InputOutputArray bestLabels,
- TermCriteria criteria, int attempts,
- int flags, OutputArray centers=noArray() );
实现函数：
- double kmeans( const Mat& data, int K, Mat& best_labels,
- TermCriteria criteria, int attempts,
- int flags, Mat* _centers )
- {
- const int SPP_TRIALS = 3;
- int N = data.rows > 1 ? data.rows : data.cols;
- int dims = (data.rows > 1 ? data.cols : 1)*data.channels();
- int type = data.depth();
- bool simd = checkHardwareSupport(CV_CPU_SSE);
- attempts = std::max(attempts, 1);
- CV_Assert( type == CV_32F && K > 0 );
- Mat _labels;
- if( flags & CV_KMEANS_USE_INITIAL_LABELS )
- {
- CV_Assert( (best_labels.cols == 1 || best_labels.rows == 1) &&
- best_labels.cols*best_labels.rows == N &&
- best_labels.type() == CV_32S &&
- best_labels.isContinuous());
- best_labels.copyTo(_labels);
- }
- else
- {
- if( !((best_labels.cols == 1 || best_labels.rows == 1) &&
- best_labels.cols*best_labels.rows == N &&
- best_labels.type() == CV_32S &&
- best_labels.isContinuous()))
- best_labels.create(N, 1, CV_32S);
- _labels.create(best_labels.size(), best_labels.type());
- }
- int* labels = _labels.ptr<int>();
- Mat centers(K, dims, type), old_centers(K, dims, type);
- vector<int> counters(K);
- vector<Vec2f> _box(dims);
- Vec2f* box = &_box[0];
- double best_compactness = DBL_MAX, compactness = 0;
- RNG& rng = theRNG();
- int a, iter, i, j, k;
- if( criteria.type & TermCriteria::EPS )
- criteria.epsilon = std::max(criteria.epsilon, 0.);
- else
- criteria.epsilon = FLT_EPSILON;
- criteria.epsilon *= criteria.epsilon;
- if( criteria.type & TermCriteria::COUNT )
- criteria.maxCount = std::min(std::max(criteria.maxCount, 2), 100);
- else
- criteria.maxCount = 100;
- if( K == 1 )
- {
- attempts = 1;
- criteria.maxCount = 2;
- }
- const float* sample = data.ptr<float>(0);
- for( j = 0; j < dims; j++ )
- box[j] = Vec2f(sample[j], sample[j]);
- for( i = 1; i < N; i++ )
- {
- sample = data.ptr<float>(i);
- for( j = 0; j < dims; j++ )
- {
- float v = sample[j];
- box[j][0] = std::min(box[j][0], v);
- box[j][1] = std::max(box[j][1], v);
- }
- }
- for( a = 0; a < attempts; a++ )
- {
- double max_center_shift = DBL_MAX;
- for( iter = 0; iter < criteria.maxCount && max_center_shift > criteria.epsilon; iter++ )
- {
- swap(centers, old_centers);
- if( iter == 0 && (a > 0 || !(flags & KMEANS_USE_INITIAL_LABELS)) )
- {
- if( flags & KMEANS_PP_CENTERS )
- generateCentersPP(data, centers, K, rng, SPP_TRIALS);
- else
- {
- for( k = 0; k < K; k++ )
- generateRandomCenter(_box, centers.ptr<float>(k), rng);
- }
- }
- else
- {
- if( iter == 0 && a == 0 && (flags & KMEANS_USE_INITIAL_LABELS) )
- {
- for( i = 0; i < N; i++ )
- CV_Assert( (unsigned)labels[i] < (unsigned)K );
- }
- // compute centers
- centers = Scalar(0);
- for( k = 0; k < K; k++ )
- counters[k] = 0;
- for( i = 0; i < N; i++ )
- {
- sample = data.ptr<float>(i);
- k = labels[i];
- float* center = centers.ptr<float>(k);
- for( j = 0; j <= dims - 4; j += 4 )
- {
- float t0 = center[j] + sample[j];
- float t1 = center[j+1] + sample[j+1];
- center[j] = t0;
- center[j+1] = t1;
- t0 = center[j+2] + sample[j+2];
- t1 = center[j+3] + sample[j+3];
- center[j+2] = t0;
- center[j+3] = t1;
- }
- for( ; j < dims; j++ )
- center[j] += sample[j];
- counters[k]++;
- }
- if( iter > 0 )
- max_center_shift = 0;
- for( k = 0; k < K; k++ )
- {
- float* center = centers.ptr<float>(k);
- if( counters[k] != 0 )
- {
- float scale = 1.f/counters[k];
- for( j = 0; j < dims; j++ )
- center[j] *= scale;
- }
- else
- generateRandomCenter(_box, center, rng);
- if( iter > 0 )
- {
- double dist = 0;
- const float* old_center = old_centers.ptr<float>(k);
- for( j = 0; j < dims; j++ )
- {
- double t = center[j] - old_center[j];
- dist += t*t;
- }
- max_center_shift = std::max(max_center_shift, dist);
- }
- }
- }
- // assign labels
- compactness = 0;
- for( i = 0; i < N; i++ )
- {
- sample = data.ptr<float>(i);
- int k_best = 0;
- double min_dist = DBL_MAX;
- for( k = 0; k < K; k++ )
- {
- const float* center = centers.ptr<float>(k);
- double dist = distance(sample, center, dims, simd);
- if( min_dist > dist )
- {
- min_dist = dist;
- k_best = k;
- }
- }
- compactness += min_dist;
- labels[i] = k_best;
- }
- }
- if( compactness < best_compactness )
- {
- best_compactness = compactness;
- if( _centers )
- centers.copyTo(*_centers);
- _labels.copyTo(best_labels);
- }
- }
- return best_compactness;
- }
- }
调用方法：
- const int kMeansItCount = 10; //迭代次数
- const int kMeansType = cv::KMEANS_PP_CENTERS; //Use kmeans++ center initialization by Arthur and Vassilvitskii
- cv::Mat bgdLabels, fgdLabels; //记录背景和前景的像素样本集中每个像素对应GMM的哪个高斯模型，论文中的kn
- //kmeans中参数_bgdSamples为：每行一个样本
- //kmeans的输出为bgdLabels，里面保存的是输入样本集中每一个样本对应的类标签（样本聚为componentsCount类后）
- kmeans( _fgdSamples, GMM::componentsCount, fgdLabels,
- cv::TermCriteria( CV_TERMCRIT_ITER, kMeansItCount, 0.0), 0, kMeansType );
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