分水岭算法（Watershed Algorithm）

　　分水岭分割方法（Watershed Segmentation），是一种基于拓扑理论的数学形态学的分割方法，其基本思想是把图像看作是测地学上的拓扑地貌，图像中每一点像素的灰度值表示该点的海拔高度，每一个局部极小值及其影响区域称为集水盆，而集水盆的边界则形成分水岭。分水岭的概念和形成可以通过模拟浸入过程来说明。在每一个局部极小值表面，刺穿一个小孔，然后把整个模型慢慢浸入水中，随着浸入的加深，每一个局部极小值的影响域慢慢向外扩展，在两个集水盆汇合处构筑大坝，即形成分水岭。
　　分水岭的计算过程是一个迭代标注过程。分水岭比较经典的计算方法是L. Vincent提出的。在该算法中，分水岭计算分两个步骤，一个是排序过程，一个是淹没过程。首先对每个像素的灰度级进行从低到高排序，然后在从低到高实现淹没过程中，对每一个局部极小值在h阶高度的影响域采用先进先出（FIFO）结构进行判断及标注。
　　分水岭变换得到的是输入图像的集水盆图像，集水盆之间的边界点，即为分水岭。显然，分水岭表示的是输入图像极大值点。因此，为得到图像的边缘信息，通常把梯度图像作为输入图像，即：
　　grad(f(x,y))=((f(x-1,y)-f(x+1,y))^2 + (f(x,y-1)-f(x,y+1))^2)^0.5
　　式中，f(x,y)表示原始图像，grad(.)表示梯度运算。
　　分水岭算法对微弱边缘具有良好的响应，图像中的噪声、物体表面细微的灰度变化，都会产生过度分割的现象。但同时应当看出，分水岭算法对微弱边缘具有良好的响应，是得到封闭连续边缘的保证的。另外，分水岭算法所得到的封闭的集水盆，为分析图像的区域特征提供了可能。
　　为消除分水岭算法产生的过度分割，通常可以采用两种处理方法，一是利用先验知识去除无关边缘信息。二是修改梯度函数使得集水盆只响应想要探测的目标。
　　为降低分水岭算法产生的过度分割，通常要对梯度函数进行修改，一个简单的方法是对梯度图像进行阈值处理，以消除灰度的微小变化产生的过度分割。即：
　　g(x,y)=max(grad(f(x,y)),gθ)
　　式中，gθ表示阈值。
　　程序可采用方法：用阈值限制梯度图像以达到消除灰度值的微小变化产生的过度分割，获得适量的区域，再对这些区域的边缘点的灰度级进行从低到高排序，然后在从低到高实现淹没的过程，梯度图像用Sobel算子计算获得。对梯度图像进行阈值处理时，选取合适的阈值对最终分割的图像有很大影响，因此阈值的选取是图像分割效果好坏的一个关键。缺点：实际图像中可能含有微弱的边缘，灰度变化的数值差别不是特别明显，选取阈值过大可能会消去这些微弱边缘。

　　下面用C++实现分水岭算法：

#define _USE_MATH_DEFINES#include <cstddef>#include <cstdlib>#include <cstring>#include <climits>#include <cfloat>#include <ctime>#include <cmath>#include <cassert>#include <vector>#include <stack>#include <queue>using namespace std;typedef voidGVVoid;typedef boolGVBoolean;typedef charGVChar;typedef unsigned charGVByte;typedef shortGVInt16;typedef unsigned shortGVUInt16;typedef intGVInt32;typedef unsigned intGVUInt32;typedef long longGVInt64;typedef unsigned long longGVUInt64;typedef floatGVFloat32;typedef doubleGVFloat64;const GVBoolean GV_TRUE= true;const GVBoolean GV_FALSE= false;const GVByteGV_BYTE_MAX = UCHAR_MAX;const GVInt32GV_INT32_MAX = INT_MAX;const GVInt32GV_INT32_MIX = INT_MIN;const GVInt64GV_INT64_MAX = LLONG_MAX;const GVInt64GV_INT64_MIN = LLONG_MIN;const GVFloat32 GV_FLOAT32_MAX= FLT_MAX;const GVFloat32 GV_FLOAT32_MIN= FLT_MIN;const GVFloat64 GV_FLOAT64_MAX= DBL_MAX;const GVFloat64 GV_FLOAT64_MIN= DBL_MIN;classGVPoint;class GVPoint {public:GVInt32x;GVInt32y;public:GVPoint() : x(0), y(0) { }GVPoint(const GVPoint &obj) : x(obj.x), y(obj.y) { }GVPoint(GVInt32 x, GVInt32 y) : x(x), y(y) { }public:GVBoolean operator ==(const GVPoint &right) const {return ((x == right.x) && (y == right.y));}GVBoolean operator !=(const GVPoint &right) const {return (!(x == right.x) || !(y == right.y));}};/* * <Parameter> * <image> image data; * <width> image width; * <height> image height; * <label out> image of labeled watersheds. */GVVoid gvWatershed(const GVByte *image,GVInt32 width,GVInt32 height,GVInt32 *label){// Local constantsconst GVInt32 WSHD = 0;const GVInt32 INIT = -1;const GVInt32 MASK = -2;const GVPoint FICT_PIXEL = GVPoint(~0, ~0);// Image statistics and sortingGVInt32 size = width * height;GVInt32 *image_stat = new GVInt32[GV_BYTE_MAX + 1];GVInt32 *image_space = new GVInt32[GV_BYTE_MAX + 1];GVPoint *image_sort = new GVPoint[size];::memset(image_stat, 0, sizeof (GVInt32) * (GV_BYTE_MAX + 1));::memset(image_space, 0, sizeof (GVInt32) * (GV_BYTE_MAX + 1));::memset(image_sort, 0, sizeof (GVPoint) * size);for (GVInt32 i = 0; !(i == size); ++i) {image_stat[image[i]]++;}for (GVInt32 i = 0; !(i == GV_BYTE_MAX); ++i) {image_stat[i + 1] += image_stat[i];}for (GVInt32 i = 0; !(i == height); ++i) {for (GVInt32 j = 0; !(j == width); ++j) {GVByte space = image[i * width + j];GVInt32 index = image_stat[space] - (++image_space[space]);image_sort[index].x = j;image_sort[index].y = i;}}for (GVInt32 i = GV_BYTE_MAX; !(i == 0); --i) {image_stat[i] -= image_stat[i - 1];}// Watershed algorithmGVPoint *head = image_sort;GVInt32 space = 0;GVInt32 *dist = new GVInt32[size];GVInt32 dist_cnt = 0;GVInt32 label_cnt = 0;std::queue<GVPoint> queue;::memset(dist, 0, sizeof (GVInt32) * size);::memset(label, ~0, sizeof (GVInt32) * size);for (GVInt32 h = 0; !(h == (GV_BYTE_MAX + 1)); ++h) {head += space;space = image_stat[h];for (GVInt32 i = 0; !(i == space); ++i) {GVInt32 index = head[i].y * width + head[i].x;GVInt32 index_l = ((head[i].x - 1) < 0) ? -1 : ((head[i].x - 1) + head[i].y * width);GVInt32 index_r = !((head[i].x + 1) > width) ? -1 : ((head[i].x + 1) + head[i].y * width);GVInt32 index_t = ((head[i].y - 1) < 0) ? -1 : (head[i].x + (head[i].y - 1) * width);GVInt32 index_b = !((head[i].y + 1) > height) ? -1 : (head[i].x + (head[i].y + 1) * width);label[index] = MASK;if ( (!(index_l < 0) && !(label[index_l] < WSHD))|| (!(index_r < 0) && !(label[index_r] < WSHD))|| (!(index_t < 0) && !(label[index_t] < WSHD))|| (!(index_b < 0) && !(label[index_b] < WSHD))) {dist[index] = 1;queue.push(head[i]);}}dist_cnt = 1;queue.push(FICT_PIXEL);while (GV_TRUE) {GVPoint top = queue.front();GVInt32 index = top.y * width + top.x;GVInt32 index_l = ((top.x - 1) < 0) ? -1 : ((top.x - 1) + top.y * width);GVInt32 index_r = !((top.x + 1) > width) ? -1 : ((top.x + 1) + top.y * width);GVInt32 index_t = ((top.y - 1) < 0) ? -1 : (top.x + (top.y - 1) * width);GVInt32 index_b = !((top.y + 1) > height) ? -1 : (top.x + (top.y + 1) * width);queue.pop();if (top == FICT_PIXEL) {if (queue.empty()) break;else {++dist_cnt;queue.push(FICT_PIXEL);top = queue.front();queue.pop();}}if (!(index_l < 0)) {if ((dist[index_l] < dist_cnt) && !(label[index_l] < WSHD)) {if (label[index_l] > WSHD) {if ((label[index] == MASK) || (label[index] = WSHD)) label[index] = label[index_l];else if (!(label[index] == label[index_l])) label[index] = WSHD;} else if (label[index] == MASK) {label[index] = WSHD;}} else if ((label[index_l] == MASK) && (dist[index_l] == 0)) {dist[index_l] = dist_cnt + 1;queue.push(GVPoint(top.x - 1, top.y));}}if (!(index_r < 0)) {if ((dist[index_r] < dist_cnt) && !(label[index_r] < WSHD)) {if (label[index_r] > WSHD) {if ((label[index] == MASK) || (label[index] = WSHD)) label[index] = label[index_r];else if (!(label[index] == label[index_r])) label[index] = WSHD;} else if (label[index] == MASK) {label[index] = WSHD;}} else if ((label[index_r] == MASK) && (dist[index_r] == 0)) {dist[index_r] = dist_cnt + 1;queue.push(GVPoint(top.x + 1, top.y));}}if (!(index_t < 0)) {if ((dist[index_t] < dist_cnt) && !(label[index_t] < WSHD)) {if (label[index_t] > WSHD) {if ((label[index] == MASK) || (label[index] = WSHD)) label[index] = label[index_t];else if (!(label[index] == label[index_t])) label[index] = WSHD;} else if (label[index] == MASK) {label[index] = WSHD;}} else if ((label[index_t] == MASK) && (dist[index_t] == 0)) {dist[index_t] = dist_cnt + 1;queue.push(GVPoint(top.x, top.y - 1));}}if (!(index_b < 0)) {if ((dist[index_b] < dist_cnt) && !(label[index_b] < WSHD)) {if (label[index_b] > WSHD) {if ((label[index] == MASK) || (label[index] = WSHD)) label[index] = label[index_b];else if (!(label[index] == label[index_b])) label[index] = WSHD;} else if (label[index] == MASK) {label[index] = WSHD;}} else if ((label[index_b] == MASK) && (dist[index_b] == 0)) {dist[index_b] = dist_cnt + 1;queue.push(GVPoint(top.x, top.y + 1));}}}for (GVInt32 i = 0; !(i == space); ++i) {GVInt32 index = head[i].y * width + head[i].x;dist[index] = 0;if (label[index] == MASK) {label_cnt++;label[index] = label_cnt;queue.push(head[i]);while (!queue.empty()) {GVPoint top = queue.front();GVInt32 index_l = ((top.x - 1) < 0) ? -1 : ((top.x - 1) + top.y * width);GVInt32 index_r = !((top.x + 1) > width) ? -1 : ((top.x + 1) + top.y * width);GVInt32 index_t = ((top.y - 1) < 0) ? -1 : (top.x + (top.y - 1) * width);GVInt32 index_b = !((top.y + 1) > height) ? -1 : (top.x + (top.y + 1) * width);queue.pop();if (!(index_l < 0) && (label[index_l] == MASK)) {queue.push(GVPoint(top.x - 1, top.y));label[index_l] = label_cnt;}if (!(index_r < 0) && (label[index_r] == MASK)) {queue.push(GVPoint(top.x + 1, top.y));label[index_r] = label_cnt;}if (!(index_t < 0) && (label[index_t] == MASK)) {queue.push(GVPoint(top.x, top.y - 1));label[index_t] = label_cnt;}if (!(index_b < 0) && (label[index_b] == MASK)) {queue.push(GVPoint(top.x, top.y + 1));label[index_b] = label_cnt;}}}}}// Release resourcesdelete[] image_stat;delete[] image_space;delete[] image_sort;delete[] dist;}