1Hough Line Transform {#tutorial_hough_lines} 2==================== 3 4Goal 5---- 6 7In this tutorial you will learn how to: 8 9- Use the OpenCV functions @ref cv::HoughLines and @ref cv::HoughLinesP to detect lines in an 10 image. 11 12Theory 13------ 14 15@note The explanation below belongs to the book **Learning OpenCV** by Bradski and Kaehler. 16 17Hough Line Transform 18-------------------- 19 20-# The Hough Line Transform is a transform used to detect straight lines. 21-# To apply the Transform, first an edge detection pre-processing is desirable. 22 23### How does it work? 24 25-# As you know, a line in the image space can be expressed with two variables. For example: 26 27 -# In the **Cartesian coordinate system:** Parameters: \f$(m,b)\f$. 28 -# In the **Polar coordinate system:** Parameters: \f$(r,\theta)\f$ 29 30  31 32 For Hough Transforms, we will express lines in the *Polar system*. Hence, a line equation can be 33 written as: 34 35 \f[y = \left ( -\dfrac{\cos \theta}{\sin \theta} \right ) x + \left ( \dfrac{r}{\sin \theta} \right )\f] 36 37Arranging the terms: \f$r = x \cos \theta + y \sin \theta\f$ 38 39-# In general for each point \f$(x_{0}, y_{0})\f$, we can define the family of lines that goes through 40 that point as: 41 42 \f[r_{\theta} = x_{0} \cdot \cos \theta + y_{0} \cdot \sin \theta\f] 43 44 Meaning that each pair \f$(r_{\theta},\theta)\f$ represents each line that passes by 45 \f$(x_{0}, y_{0})\f$. 46 47-# If for a given \f$(x_{0}, y_{0})\f$ we plot the family of lines that goes through it, we get a 48 sinusoid. For instance, for \f$x_{0} = 8\f$ and \f$y_{0} = 6\f$ we get the following plot (in a plane 49 \f$\theta\f$ - \f$r\f$): 50 51  52 53 We consider only points such that \f$r > 0\f$ and \f$0< \theta < 2 \pi\f$. 54 55-# We can do the same operation above for all the points in an image. If the curves of two 56 different points intersect in the plane \f$\theta\f$ - \f$r\f$, that means that both points belong to a 57 same line. For instance, following with the example above and drawing the plot for two more 58 points: \f$x_{1} = 4\f$, \f$y_{1} = 9\f$ and \f$x_{2} = 12\f$, \f$y_{2} = 3\f$, we get: 59 60  61 62 The three plots intersect in one single point \f$(0.925, 9.6)\f$, these coordinates are the 63 parameters (\f$\theta, r\f$) or the line in which \f$(x_{0}, y_{0})\f$, \f$(x_{1}, y_{1})\f$ and 64 \f$(x_{2}, y_{2})\f$ lay. 65 66-# What does all the stuff above mean? It means that in general, a line can be *detected* by 67 finding the number of intersections between curves.The more curves intersecting means that the 68 line represented by that intersection have more points. In general, we can define a *threshold* 69 of the minimum number of intersections needed to *detect* a line. 70-# This is what the Hough Line Transform does. It keeps track of the intersection between curves of 71 every point in the image. If the number of intersections is above some *threshold*, then it 72 declares it as a line with the parameters \f$(\theta, r_{\theta})\f$ of the intersection point. 73 74### Standard and Probabilistic Hough Line Transform 75 76OpenCV implements two kind of Hough Line Transforms: 77 78a. **The Standard Hough Transform** 79 80- It consists in pretty much what we just explained in the previous section. It gives you as 81 result a vector of couples \f$(\theta, r_{\theta})\f$ 82- In OpenCV it is implemented with the function @ref cv::HoughLines 83 84b. **The Probabilistic Hough Line Transform** 85 86- A more efficient implementation of the Hough Line Transform. It gives as output the extremes 87 of the detected lines \f$(x_{0}, y_{0}, x_{1}, y_{1})\f$ 88- In OpenCV it is implemented with the function @ref cv::HoughLinesP 89 90Code 91---- 92 93-# **What does this program do?** 94 - Loads an image 95 - Applies either a *Standard Hough Line Transform* or a *Probabilistic Line Transform*. 96 - Display the original image and the detected line in two windows. 97 98-# The sample code that we will explain can be downloaded from [here](https://github.com/Itseez/opencv/tree/master/samples/cpp/houghlines.cpp). A slightly fancier version 99 (which shows both Hough standard and probabilistic with trackbars for changing the threshold 100 values) can be found [here](https://github.com/Itseez/opencv/tree/master/samples/cpp/tutorial_code/ImgTrans/HoughLines_Demo.cpp). 101 @include samples/cpp/houghlines.cpp 102 103Explanation 104----------- 105 106-# Load an image 107 @code{.cpp} 108 Mat src = imread(filename, 0); 109 if(src.empty()) 110 { 111 help(); 112 cout << "can not open " << filename << endl; 113 return -1; 114 } 115 @endcode 116-# Detect the edges of the image by using a Canny detector 117 @code{.cpp} 118 Canny(src, dst, 50, 200, 3); 119 @endcode 120 Now we will apply the Hough Line Transform. We will explain how to use both OpenCV functions 121 available for this purpose: 122 123-# **Standard Hough Line Transform** 124 -# First, you apply the Transform: 125 @code{.cpp} 126 vector<Vec2f> lines; 127 HoughLines(dst, lines, 1, CV_PI/180, 100, 0, 0 ); 128 @endcode 129 with the following arguments: 130 131 - *dst*: Output of the edge detector. It should be a grayscale image (although in fact it 132 is a binary one) 133 - *lines*: A vector that will store the parameters \f$(r,\theta)\f$ of the detected lines 134 - *rho* : The resolution of the parameter \f$r\f$ in pixels. We use **1** pixel. 135 - *theta*: The resolution of the parameter \f$\theta\f$ in radians. We use **1 degree** 136 (CV_PI/180) 137 - *threshold*: The minimum number of intersections to "*detect*" a line 138 - *srn* and *stn*: Default parameters to zero. Check OpenCV reference for more info. 139 140 -# And then you display the result by drawing the lines. 141 @code{.cpp} 142 for( size_t i = 0; i < lines.size(); i++ ) 143 { 144 float rho = lines[i][0], theta = lines[i][1]; 145 Point pt1, pt2; 146 double a = cos(theta), b = sin(theta); 147 double x0 = a*rho, y0 = b*rho; 148 pt1.x = cvRound(x0 + 1000*(-b)); 149 pt1.y = cvRound(y0 + 1000*(a)); 150 pt2.x = cvRound(x0 - 1000*(-b)); 151 pt2.y = cvRound(y0 - 1000*(a)); 152 line( cdst, pt1, pt2, Scalar(0,0,255), 3, LINE_AA); 153 } 154 @endcode 155-# **Probabilistic Hough Line Transform** 156 -# First you apply the transform: 157 @code{.cpp} 158 vector<Vec4i> lines; 159 HoughLinesP(dst, lines, 1, CV_PI/180, 50, 50, 10 ); 160 @endcode 161 with the arguments: 162 163 - *dst*: Output of the edge detector. It should be a grayscale image (although in fact it 164 is a binary one) 165 - *lines*: A vector that will store the parameters 166 \f$(x_{start}, y_{start}, x_{end}, y_{end})\f$ of the detected lines 167 - *rho* : The resolution of the parameter \f$r\f$ in pixels. We use **1** pixel. 168 - *theta*: The resolution of the parameter \f$\theta\f$ in radians. We use **1 degree** 169 (CV_PI/180) 170 - *threshold*: The minimum number of intersections to "*detect*" a line 171 - *minLinLength*: The minimum number of points that can form a line. Lines with less than 172 this number of points are disregarded. 173 - *maxLineGap*: The maximum gap between two points to be considered in the same line. 174 175 -# And then you display the result by drawing the lines. 176 @code{.cpp} 177 for( size_t i = 0; i < lines.size(); i++ ) 178 { 179 Vec4i l = lines[i]; 180 line( cdst, Point(l[0], l[1]), Point(l[2], l[3]), Scalar(0,0,255), 3, LINE_AA); 181 } 182 @endcode 183-# Display the original image and the detected lines: 184 @code{.cpp} 185 imshow("source", src); 186 imshow("detected lines", cdst); 187 @endcode 188-# Wait until the user exits the program 189 @code{.cpp} 190 waitKey(); 191 @endcode 192 193Result 194------ 195 196@note 197 The results below are obtained using the slightly fancier version we mentioned in the *Code* 198 section. It still implements the same stuff as above, only adding the Trackbar for the 199 Threshold. 200 201Using an input image such as: 202 203 204 205We get the following result by using the Probabilistic Hough Line Transform: 206 207 208 209You may observe that the number of lines detected vary while you change the *threshold*. The 210explanation is sort of evident: If you establish a higher threshold, fewer lines will be detected 211(since you will need more points to declare a line detected). 212
