這個方法被應用於深度學習目標檢測的經典之作selective search方法中（Selective Search for Object Recognition），用於初始化分割區域。。論文題目：《Efficient Graph-Based Image Segmentation》

查閱了許多部落格，後來感覺，對於這個方法整體還是一知半解，於是花了一個下午閱讀了原始碼，做一個筆記，如有錯誤，希望大家指正

原始碼輸入5個引數，示意如下：

 sigma: Used to smooth the input image before segmenting it.
  k: Value for the
 
 threshold function.
  min: Minimum component size enforced by post-processing.
  input: Input image.
  output: Output image.

大意為，sigma：使用的平滑引數，k:給出的初始化閾值，min:最小分割塊大小（用於合併小塊），輸入圖片(input)和輸入圖片(output)（都要為pnm格式）

引數傳入segment.cpp中，如下

image<rgb> *seg = segment_image(input, sigma, k, min_size, &num_ccs); 
 

呼叫segment-image.h中的 image *segment_image，函式大概註釋如下：

/*
Copyright (C) 2006 Pedro Felzenszwalb

This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307 USA
*/
 


#ifndef SEGMENT_IMAGE
#define SEGMENT_IMAGE

#include <cstdlib>
#include <image.h>
#include <misc.h>
#include <filter.h>
#include <stdio.h>
#include "segment-graph.h"

// random color
rgb random_rgb(){ 
  rgb c;
  double r;

  c.r = (uchar)random();
  c.g = (uchar)random();
  c.b = (uchar)random();

  return c;
}

// dissimilarity measure between pixels
// 衡量不相似度，用rgb距離
static inline float diff(image<float> *r, image<float> *g, image<float> *b,
             int x1, int y1, int x2, int y2) {
  return sqrt(square(imRef(r, x1, y1)-imRef(r, x2, y2)) +
          square(imRef(g, x1, y1)-imRef(g, x2, y2)) +
          square(imRef(b, x1, y1)-imRef(b, x2, y2)));
}

/*
 * Segment an image
 *
 * Returns a color image representing the segmentation.
 *
 * im: image to segment.
 * sigma: to smooth the image.
 * c: constant for treshold function.
 * min_size: minimum component size (enforced by post-processing stage).
 * num_ccs: number of connected components in the segmentation.
 */
image<rgb> *segment_image(image<rgb> *im, float sigma, float c, int min_size,
              int *num_ccs) {
    //影象的寬和高
  int width = im->width();
  int height = im->height();
    // 初始化了三個通道的片
  image<float> *r = new image<float>(width, height);
  image<float> *g = new image<float>(width, height);
  image<float> *b = new image<float>(width, height);

  // smooth each color channel
    //平滑色彩通道
  for (int y = 0; y < height; y++) {
    for (int x = 0; x < width; x++) {
      imRef(r, x, y) = imRef(im, x, y).r;
      imRef(g, x, y) = imRef(im, x, y).g;
      imRef(b, x, y) = imRef(im, x, y).b;
    }
  }
  image<float> *smooth_r = smooth(r, sigma);
  image<float> *smooth_g = smooth(g, sigma);
  image<float> *smooth_b = smooth(b, sigma);
  delete r;
  delete g;
  delete b;

  // build graph
    //構建圖，邊的陣列是原來的4倍，就是四個方向,計算出了全部的點和點之間的相似性
    //這裡是初始化了width*height*4大小的陣列，其實用不了這麼多
  edge *edges = new edge[width*height*4];
  int num = 0;
  for (int y = 0; y < height; y++) {
    for (int x = 0; x < width; x++) {
        //對每個邊的左右點計算權重
      if (x < width-1) {
    edges[num].a = y * width + x;
    edges[num].b = y * width + (x+1);
    edges[num].w = diff(smooth_r, smooth_g, smooth_b, x, y, x+1, y);
    num++;
      }

      if (y < height-1) {
    edges[num].a = y * width + x;
    edges[num].b = (y+1) * width + x;
    edges[num].w = diff(smooth_r, smooth_g, smooth_b, x, y, x, y+1);
    num++;
      }

      if ((x < width-1) && (y < height-1)) {
    edges[num].a = y * width + x;
    edges[num].b = (y+1) * width + (x+1);
    edges[num].w = diff(smooth_r, smooth_g, smooth_b, x, y, x+1, y+1);
    num++;
      }

      if ((x < width-1) && (y > 0)) {
    edges[num].a = y * width + x;
    edges[num].b = (y-1) * width + (x+1);
    edges[num].w = diff(smooth_r, smooth_g, smooth_b, x, y, x+1, y-1);
    num++;
      }
    }
  }
//   printf("%d %d", width*height*4, num);
//    printf("width is:%d height is:%d", width, height);
  delete smooth_r;
  delete smooth_g;
  delete smooth_b;

  // segment
    //分割步驟，。。。。跳出
  universe *u = segment_graph(width*height, num, edges, c);

  // post process small components
    //合併最小的塊，如果邊兩邊的塊小於min_size就合併
  for (int i = 0; i < num; i++) {
    int a = u->find(edges[i].a);
    int b = u->find(edges[i].b);
    if ((a != b) && ((u->size(a) < min_size) || (u->size(b) < min_size)))
      u->join(a, b);
  }
  delete [] edges;
  *num_ccs = u->num_sets();
    //寫入新的pnm圖片
  image<rgb> *output = new image<rgb>(width, height);

  // pick random colors for each component
  rgb *colors = new rgb[width*height];
  for (int i = 0; i < width*height; i++)
    colors[i] = random_rgb();

  for (int y = 0; y < height; y++) {
    for (int x = 0; x < width; x++) {
      int comp = u->find(y * width + x);
      imRef(output, x, y) = colors[comp];
    }
  }  

  delete [] colors;  
  delete u;

  return output;
}

#endif

其中 segment_graph(width*height, num, edges, c); 這一步跳到 segment-graph.h中

/*
Copyright (C) 2006 Pedro Felzenszwalb

This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307 USA
*/

#ifndef SEGMENT_GRAPH
#define SEGMENT_GRAPH

#include <algorithm>
#include <cmath>
#include "disjoint-set.h"

// threshold function
#define THRESHOLD(size, c) (c/size)

typedef struct {
  float w;
  int a, b;
} edge;

bool operator<(const edge &a, const edge &b) {
    //按照不相似程度從小到大排序
  return a.w < b.w;
}

/*
 * Segment a graph
 *
 * Returns a disjoint-set forest representing the segmentation.
 *
 * num_vertices: number of vertices in graph.
 * num_edges: number of edges in graph
 * edges: array of edges.
 * c: constant for treshold function.
 */
universe *segment_graph(int num_vertices, int num_edges, edge *edges, 
            float c) { 
  // sort edges by weight
    // 將edge按照權重進行排序,權重越大越不相似，這裡權重是不相似程度
  std::sort(edges, edges + num_edges);

  // make a disjoint-set forest
    // num_vertices 是頂點的數量
  universe *u = new universe(num_vertices);

  // init thresholds
    //這裡閾值是為了防止剛開始的時候每個點之間全部都分開了
  float *threshold = new float[num_vertices];
  for (int i = 0; i < num_vertices; i++)
      //c/1
    threshold[i] = THRESHOLD(1,c);

  // for each edge, in non-decreasing weight order...
  for (int i = 0; i < num_edges; i++) {
      //遍歷所有的邊
    edge *pedge = &edges[i];

    // components conected by this edge
      // 找出這個邊的左右頂點，這個頂點會隨著邊的合併而變動，為了保證閾值threshold[a]是最大的，就是這個域內的不相似程度最大
    int a = u->find(pedge->a);
    int b = u->find(pedge->b);
      // 如果 a==b 則說明在同個域裡面
    if (a != b) {
        //權值小於閾值，這裡a和b代表這個邊連結的兩個域的分別的最大不相似程度
      if ((pedge->w <= threshold[a]) &&
      (pedge->w <= threshold[b])) {
          //如果這個邊的長度（就是這個邊連結的兩個點的不相似程度）比兩邊的最大的不相似度都小，則將這兩個域連結起來
    u->join(a, b);
//          printf("size a is:%d ", u->size(a));
    a = u->find(a);
//          printf(" size  a is:%d \n", u->size(a));
          //這裡只更新了a的部分，這裡個人理解，不太確定，應該是因為在disjoint-set.h中程式碼find函式中總是找到下標最靠前的那個點，然後返回
    threshold[a] = pedge->w + THRESHOLD(u->size(a), c);
      }
    }
  }

  // free up
  delete threshold;
  return u;
}

#endif

下面是 universe *u = new universe(num_vertices); 的定義。。。在disjoint-set.h中

/*
Copyright (C) 2006 Pedro Felzenszwalb

This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307 USA
*/

#ifndef DISJOINT_SET
#define DISJOINT_SET

// disjoint-set forests using union-by-rank and path compression (sort of).

typedef struct {
    /*
     rank:等級，用來判定將要合併的塊合在左邊還是右邊
     p:用來找到這個塊最先的那個節點
     size:用來記錄現在這個塊有大
     */
  int rank;
  int p;
  int size;
} uni_elt;

class universe {
public:
  universe(int elements);
  ~universe();
  int find(int x);  
  void join(int x, int y);
  int size(int x) const { return elts[x].size; }
  int num_sets() const { return num; }

private:
  uni_elt *elts;
  int num;
};

universe::universe(int elements) {
  //初始化三個引數
  elts = new uni_elt[elements];
  num = elements;
  for (int i = 0; i < elements; i++) {
    elts[i].rank = 0;
    elts[i].size = 1;
    elts[i].p = i;
  }
}

universe::~universe() {
  delete [] elts;
}

int universe::find(int x) {
  int y = x;
  while (y != elts[y].p){
    // 個人理解，這裡不確定。這裡通過不斷的連結，查詢到最初的那個點，那個點作為這個域的標識
    y = elts[y].p;
  }
  elts[x].p = y;
  return y;
}

void universe::join(int x, int y) {
  //連結兩個塊，如果x的rank大於y就合併在x裡面，
  //反之則加在y裡面
  if (elts[x].rank > elts[y].rank) {
    elts[y].p = x;
    elts[x].size += elts[y].size;
  } else {
    elts[x].p = y;
    elts[y].size += elts[x].size;
    if (elts[x].rank == elts[y].rank)
      elts[y].rank++;
  }
  num--;
}

#endif

關於如何平滑（smooth）引數，在filter.h裡，就不貼了（裡面涉及一些數學公式，不看程式碼會看的很累）

個人覺得，大概思路是，先是每個點作為一個塊，對邊的權重進行排序，遍歷所有的邊，不斷結合塊，把邊遍歷過之後，也就合併所有的塊了，在把太小的塊融合起來。

最後，貼上一段原始論文中的描述，供大家參考

Algorithm 1 Segmentation algorithm.
The input is a graph G = (V, E), with n vertices and m edges. The output is a
segmentation of V into components S = (C 1 , . . . , C r ).
0. Sort E into π = (o 1 , . . . , o m ), by non-decreasing edge weight.
1. Start with a segmentation S 0 , where each vertex v i is in its own component.
2. Repeat step 3 for q = 1, . . . , m.
3. Construct S q given S q−1 as follows. Let v i and v j denote the vertices connected
by the q-th edge in the ordering, i.e., o q = (v i , v j ). If v i and v j are in disjoint
components of S q−1 and w(o q ) is small compared to the internal difference of
both those components, then merge the two components otherwise do nothing.
More formally, let C i q−1 be the component of S q−1 containing v i and C j q−1 the
component containing v j . If C i q−1 6 = C j q−1 and w(o q ) ≤ M Int(C i q−1 , C j q−1 ) then
S q is obtained from S q−1 by merging C i q−1 and C j q−1 . Otherwise S q = S q−1 .
4. Return S = S m .

最後的最後，程式碼和論文如下：這裡