OpenCV 之 霍夫变换

Hough 变换，对图像中直线的残缺部分、噪声、以及其它的共存结构不敏感，因此，具有很强的鲁棒性。

它常用来检测 直线和曲线 (圆形)，识别图像中的几何形状，甚至可用来分割重叠或有部分遮挡的物体。

1  平面坐标和极坐标

1)  平面坐标的点 <=> 极坐标(平面化)的曲线

所谓极坐标平面化是指, 将 ρ-θ 的关系像 x-y 那样在平面内展开。

公式推导: x-y坐标中的点 (x₀, y₀), 代入极坐标 ρ-θ 中得

$\quad \begin{align*}\rho = x_{0}\: cos\theta + y_{0}\: sin\theta = \sqrt{x_{0}^{2}+y_{0}^{2}}\:\left (\frac{x_{0}}{\sqrt{x_{0}^{2}+y_{0}^{2}}}\: cos\theta  +  \frac{y_{0}}{\sqrt{x_{0}^{2}+y_{0}^{2}}}\:sin\theta\right )
= \sqrt{x_{0}^{2}+y_{0}^{2}} \; sin(\varphi_{0} +\theta ) \end{align*} $

其中, $sin\varphi_{0} = \frac{x_{0}}{\sqrt{x_{0}^{2}+y_{0}^{2}}}$

由上述公式可以明显的看出ρ与θ之间的函数关系。

2)  极坐标的点 <=> 平面坐标的直线

    公式推导: ρ-θ 极坐标中的点 (ρ₀, θ₀), 代入平面坐标 x-y 中得

$\quad y = ( -\tfrac{cos\theta_{0} }{sin\theta_{0}} )\:x + \tfrac{\rho_{0} }{sin\theta_{0}}$

由此可知, 极坐标中的一个定点(ρ₀, θ₀), 对应于平面坐标内的一条直线。若极坐标中有N条曲线相交于一点(如下左图), 则说明在平面坐标中, 该点对应的直线包含N条曲线对应的N个点, 也即这N个点组成了一条直线。

              

上面左图中, 是平面坐标中的三个点 (8,6), (4, 9), (12,3) 分别对应的三条曲线。

因此, 霍夫变换的目的是将平面坐标内的直线检测, 转换为极坐标内的交点检测, 显然寻找一个交点要比寻找直线(实际上是寻找许多点)容易得多了。

2  OpenCV中的函数

1)  HoughLines

void cv::HoughLines(
　　InputArray image,  // 输入图像
　　OutputArray lines, // 输出位直线数组
　　double rho,        // 用像素个数表示的长度分辨率
　　double theta,      // 用像素个数表示的角度分辨率
　　int threshold,     // 累加器阈值 (只有大于该阈值的才有足够的投票)
　　double srn = ,    // 若 srn = stn = 0，则使用的是经典霍夫变换
　　double stn =,     
　　double min_theta = ,     // 取值为 0 ~ max_theta
　　double max_theta = CV_PI  // 取值为 min_theta ~ CV_PI
)；

未完待续-源代码留待以后剖析：

/*
Here image is an input raster;
step is it's step; size characterizes it's ROI;
rho and theta are discretization steps (in pixels and radians correspondingly).
threshold is the minimum number of pixels in the feature for it
to be a candidate for line. lines is the output
array of (rho, theta) pairs. linesMax is the buffer size (number of pairs).
Functions return the actual number of found lines.
*/
static void
HoughLinesStandard( const Mat& img, float rho, float theta,
                    int threshold, std::vector<Vec2f>& lines, int linesMax,
                    double min_theta, double max_theta )
{
    int i, j;
    float irho =  / rho;
 
    CV_Assert( img.type() == CV_8UC1 );
 
    const uchar* image = img.ptr();
    int step = (int)img.step;
    int width = img.cols;
    int height = img.rows;
 
    if (max_theta < min_theta ) {
        CV_Error( CV_StsBadArg, "max_theta must be greater than min_theta" );
    }
    int numangle = cvRound((max_theta - min_theta) / theta);
    int numrho = cvRound(((width + height) *  + ) / rho);
 
#if defined HAVE_IPP && !defined(HAVE_IPP_ICV_ONLY) && IPP_VERSION_X100 >= 810 && IPP_DISABLE_BLOCK
    CV_IPP_CHECK()
    {
        IppiSize srcSize = { width, height };
        IppPointPolar delta = { rho, theta };
        IppPointPolar dstRoi[] = {{(Ipp32f) -(width + height), (Ipp32f) min_theta},{(Ipp32f) (width + height), (Ipp32f) max_theta}};
        int bufferSize;
        int nz = countNonZero(img);
        int ipp_linesMax = std::min(linesMax, nz*numangle/threshold);
        int linesCount = ;
        lines.resize(ipp_linesMax);
        IppStatus ok = ippiHoughLineGetSize_8u_C1R(srcSize, delta, ipp_linesMax, &bufferSize);
        Ipp8u* buffer = ippsMalloc_8u(bufferSize);
        if (ok >= ) ok = ippiHoughLine_Region_8u32f_C1R(image, step, srcSize, (IppPointPolar*) &lines[], dstRoi, ipp_linesMax, &linesCount, delta, threshold, buffer);
        ippsFree(buffer);
        if (ok >= )
        {
            lines.resize(linesCount);
            CV_IMPL_ADD(CV_IMPL_IPP);
            return;
        }
        lines.clear();
        setIppErrorStatus();
    }
#endif
 
    AutoBuffer<int> _accum((numangle+) * (numrho+));
    std::vector<int> _sort_buf;
    AutoBuffer<float> _tabSin(numangle);
    AutoBuffer<float> _tabCos(numangle);
    int *accum = _accum;
    float *tabSin = _tabSin, *tabCos = _tabCos;
 
    memset( accum, , sizeof(accum[]) * (numangle+) * (numrho+) );
 
    float ang = static_cast<float>(min_theta);
    for(int n = ; n < numangle; ang += theta, n++ )
    {
        tabSin[n] = (float)(sin((double)ang) * irho);
        tabCos[n] = (float)(cos((double)ang) * irho);
    }
 
    // stage 1. fill accumulator
    for( i = ; i < height; i++ )
        for( j = ; j < width; j++ )
        {
            if( image[i * step + j] !=  )
                for(int n = ; n < numangle; n++ )
                {
                    int r = cvRound( j * tabCos[n] + i * tabSin[n] );
                    r += (numrho - ) / ;
                    accum[(n+) * (numrho+) + r+]++;
                }
        }
 
    // stage 2. find local maximums
    for(int r = ; r < numrho; r++ )
        for(int n = ; n < numangle; n++ )
        {
            int base = (n+) * (numrho+) + r+;
            if( accum[base] > threshold &&
                accum[base] > accum[base - ] && accum[base] >= accum[base + ] &&
                accum[base] > accum[base - numrho - ] && accum[base] >= accum[base + numrho + ] )
                _sort_buf.push_back(base);
        }
 
    // stage 3. sort the detected lines by accumulator value
    std::sort(_sort_buf.begin(), _sort_buf.end(), hough_cmp_gt(accum));
 
    // stage 4. store the first min(total,linesMax) lines to the output buffer
    linesMax = std::min(linesMax, (int)_sort_buf.size());
    double scale = ./(numrho+);
    for( i = ; i < linesMax; i++ )
    {
        LinePolar line;
        int idx = _sort_buf[i];
        int n = cvFloor(idx*scale) - ;
        int r = idx - (n+)*(numrho+) - ;
        line.rho = (r - (numrho - )*0.5f) * rho;
        line.angle = static_cast<float>(min_theta) + n * theta;
        lines.push_back(Vec2f(line.rho, line.angle));
    }
}

2)  HoughLinesP

void cv::HoughLinesP ( InputArray image, OutputArray lines,
                        double rho,  double theta, int threshold,
                        double minLineLength=,  double maxLineGap= )
/*
1) image
    8-bit, single-channel binary source image. The image may be modified by the function.
2) lines
    Output vector of lines. Each line is represented by a 4-element vector x1, y1, x2,y2), where (x1, y1) and (x2, y2) are the ending points of each detected line segment.
3) rho
    Distance resolution of the accumulator in pixels.
4) theta
    Angle resolution of the accumulator in radians.
5) threshold
    Accumulator threshold parameter. Only those lines are returned that get enough votes(>threshold).
6) minLineLength
    Minimum line length. Line segments shorter than that are rejected.
7) MaxLineGap
    Maximum allowed gap between points on the same line to link them. */

Progressive Probabilistic Hough Transform Algorithm Outline:

/* 摘自文献 J. Matas, Robust Detection of Lines Using the Progressive Probabilistic Hough Transform
1. Check the input image; if it is empty then finish.
2. Update the accumulator with a single pixel randomly selected from the input image.
3. Remove the selected pixel from input image.
4. Check if the highest peak in the accumulator that was modified by the new pixel is higher than threshold thr(N). If not then goto 1.
5. Look along a corridor specified by the peak in the accumulator, and find the longest segment that either is continuous
   or exhibits a gap not exceeding a given threshold.
6. Remove the pixels in the segment from input image.
7. “Unvote” from the accumulator all the pixels from the line that have previously voted.
8. If the line segment is longer than the minimum length add it into the output list.
9. Goto 1. */

3)  HoughCircles

void cv::HoughCircles ( InputArray image, OutputArray circles,
                        int method,  double dp,  double minDist,
                        double param1 = ,  double param2 = ,
                        int minRadius = ,  int maxRadius = )
/*
1) image
    8-bit, single-channel, grayscale input image.
2) circles
    Output vector of found circles. Each vector is encoded as a 3-element floating-point vector (x, y, radius).
3) method
    Detection method, see cv::HoughModes. Currently, the only implemented method is HOUGH_GRADIENT
4) dp
    Inverse ratio of the accumulator resolution to the image resolution.
    For example, if dp=1 , the accumulator has the same resolution as the input image.
    If dp=2 , the accumulator has half as big width and height.
5) minDist
    Minimum distance between the centers of the detected circles.
    If the parameter is too small, multiple neighbor circles may be falsely detected in addition to a true one.
    If it is too large, some circles may be missed.
6) param1
    First method-specific parameter.
    In case of CV_HOUGH_GRADIENT , it is the higher threshold of the two passed to the Canny edge detector (the lower one is twice smaller).
7) param2
    Second method-specific parameter.
    In case of CV_HOUGH_GRADIENT , it is the accumulator threshold for the circle centers at the detection stage.
    The smaller it is, the more false circles may be detected. Circles, corresponding to the larger accumulator values, will be returned first.
8) minRadius
    Minimum circle radius.
9) maxRadius
    Maximum circle radius. */

3  代码示例

在进行霍夫变换之前，首先要对图像进行"边缘检测"的预处理。

3.1  检测直线 (带滑动条)

#include "opencv2/imgcodecs.hpp"
#include "opencv2/highgui.hpp"
#include "opencv2/imgproc.hpp"
 
using namespace cv;
using namespace std;
 
// 全局变量
Mat src, src_gray, edges;
Mat standard_hough, probabilistic_hough;
int min_threshold = ;
int max_trackbar = ;
 
const char* probabilistic_name = "Probabilistic Hough Lines";
 
int p_trackbar = max_trackbar;
 
void Probabilistic_Hough( int, void* );  // 函数声明

int main( int, char** argv )
{
   // 读图
   src = imread("line_demo.png");
   if(src.empty())
       return -;
 
   // 灰度图
   cvtColor(src, src_gray, COLOR_RGB2GRAY);
 
   // cany 边缘检测
   Canny(src_gray, edges, , , );
 
   // 阈值滑动条
   char thresh_label[];
   sprintf( thresh_label, "Thres: %d + input", min_threshold );
 
   namedWindow(probabilistic_name, WINDOW_AUTOSIZE );
   createTrackbar(thresh_label, probabilistic_name, &p_trackbar, max_trackbar, Probabilistic_Hough);
 
   // 回调函数
   Probabilistic_Hough(, );
 
   waitKey();
}
 
void Probabilistic_Hough(int, void*)
{
  vector<Vec4i> p_lines;
  // gray -> rgb
  cvtColor(edges, probabilistic_hough, COLOR_GRAY2BGR);
  // Probabilistic Hough Transform
  HoughLinesP(edges, p_lines, , CV_PI/, min_threshold + p_trackbar, ,  );
  // 显示
  for( size_t i = ; i < p_lines.size(); i++ ) {
      Vec4i l = p_lines[i];
      line(probabilistic_hough, Point(l[], l[]), Point(l[], l[]), Scalar(,,), , LINE_AA);
  }
   imshow( probabilistic_name, probabilistic_hough );
}

3.2  检测圆形 (不带滑动条)

#include <opencv2/imgproc.hpp>
#include <opencv2/highgui.hpp>
 
using namespace cv;
using namespace std;
 
int main(int argc, char** argv)
{
    // 读图
    Mat src, src_gray;
    src = imread("circle_demo.png");
    if(src.empty())
        return -;
 
    // 灰度图 + 高斯滤波
    cvtColor(src, src_gray, COLOR_BGR2GRAY);
    GaussianBlur(src_gray, src_gray, Size(, ), ,  );
 
    // 画圆
    vector<Vec3f> circles;
    HoughCircles(src_gray, circles, HOUGH_GRADIENT, , src_gray.rows/, ,  );
    for(size_t i=; i<circles.size(); ++i)
    {
         Point center(cvRound(circles[i][]), cvRound(circles[i][]));
         int radius = cvRound(circles[i][]);
         circle(src, center, , Scalar(,,), -, ,  ); // 圆心
         circle(src, center, radius, Scalar(,,), , ,  ); // 圆形
    }
 
    // 窗体
    namedWindow("circles", WINDOW_AUTOSIZE);
    imshow("circles", src);
 
    waitKey();
}

参考资料：

<图像处理、分析与机器视觉> 第4版， 6.2.6 Hough 变换

OpenCV Tutorials / Image Processing (imgproc module) / Hough Line Transform

OpenCV Tutorials / Image Processing (imgproc module) / Hough Circle Transform