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Hough Line Transform {#tutorial_hough_lines}
====================
Goal
----
In this tutorial you will learn how to:
- Use the OpenCV functions @ref cv::HoughLines and @ref cv::HoughLinesP to detect lines in an
image.
Theory
------
@note The explanation below belongs to the book **Learning OpenCV** by Bradski and Kaehler.
Hough Line Transform
--------------------
-# The Hough Line Transform is a transform used to detect straight lines.
-# To apply the Transform, first an edge detection pre-processing is desirable.
### How does it work?
-# As you know, a line in the image space can be expressed with two variables. For example:
-# In the **Cartesian coordinate system:** Parameters: \f$(m,b)\f$.
-# In the **Polar coordinate system:** Parameters: \f$(r,\theta)\f$
For Hough Transforms, we will express lines in the *Polar system*. Hence, a line equation can be
written as:
\f[y = \left ( -\dfrac{\cos \theta}{\sin \theta} \right ) x + \left ( \dfrac{r}{\sin \theta} \right )\f]
Arranging the terms: \f$r = x \cos \theta + y \sin \theta\f$
-# In general for each point \f$(x_{0}, y_{0})\f$, we can define the family of lines that goes through
that point as:
\f[r_{\theta} = x_{0} \cdot \cos \theta + y_{0} \cdot \sin \theta\f]
Meaning that each pair \f$(r_{\theta},\theta)\f$ represents each line that passes by
\f$(x_{0}, y_{0})\f$.
-# If for a given \f$(x_{0}, y_{0})\f$ we plot the family of lines that goes through it, we get a
sinusoid. For instance, for \f$x_{0} = 8\f$ and \f$y_{0} = 6\f$ we get the following plot (in a plane
\f$\theta\f$ - \f$r\f$):
We consider only points such that \f$r > 0\f$ and \f$0< \theta < 2 \pi\f$.
-# We can do the same operation above for all the points in an image. If the curves of two
different points intersect in the plane \f$\theta\f$ - \f$r\f$, that means that both points belong to a
same line. For instance, following with the example above and drawing the plot for two more
points: \f$x_{1} = 4\f$, \f$y_{1} = 9\f$ and \f$x_{2} = 12\f$, \f$y_{2} = 3\f$, we get:
The three plots intersect in one single point \f$(0.925, 9.6)\f$, these coordinates are the
parameters (\f$\theta, r\f$) or the line in which \f$(x_{0}, y_{0})\f$, \f$(x_{1}, y_{1})\f$ and
\f$(x_{2}, y_{2})\f$ lay.
-# What does all the stuff above mean? It means that in general, a line can be *detected* by
finding the number of intersections between curves.The more curves intersecting means that the
line represented by that intersection have more points. In general, we can define a *threshold*
of the minimum number of intersections needed to *detect* a line.
-# This is what the Hough Line Transform does. It keeps track of the intersection between curves of
every point in the image. If the number of intersections is above some *threshold*, then it
declares it as a line with the parameters \f$(\theta, r_{\theta})\f$ of the intersection point.
### Standard and Probabilistic Hough Line Transform
OpenCV implements two kind of Hough Line Transforms:
a. **The Standard Hough Transform**
- It consists in pretty much what we just explained in the previous section. It gives you as
result a vector of couples \f$(\theta, r_{\theta})\f$
- In OpenCV it is implemented with the function @ref cv::HoughLines
b. **The Probabilistic Hough Line Transform**
- A more efficient implementation of the Hough Line Transform. It gives as output the extremes
of the detected lines \f$(x_{0}, y_{0}, x_{1}, y_{1})\f$
- In OpenCV it is implemented with the function @ref cv::HoughLinesP
Code
----
-# **What does this program do?**
- Loads an image
- Applies either a *Standard Hough Line Transform* or a *Probabilistic Line Transform*.
- Display the original image and the detected line in two windows.
-# The sample code that we will explain can be downloaded from [here](https://github.com/opencv/opencv/tree/master/samples/cpp/houghlines.cpp). A slightly fancier version
(which shows both Hough standard and probabilistic with trackbars for changing the threshold
values) can be found [here](https://github.com/opencv/opencv/tree/master/samples/cpp/tutorial_code/ImgTrans/HoughLines_Demo.cpp).
@include samples/cpp/houghlines.cpp
Explanation
-----------
-# Load an image
@code{.cpp}
Mat src = imread(filename, 0);
if(src.empty())
{
help();
cout << "can not open " << filename << endl;
return -1;
}
@endcode
-# Detect the edges of the image by using a Canny detector
@code{.cpp}
Canny(src, dst, 50, 200, 3);
@endcode
Now we will apply the Hough Line Transform. We will explain how to use both OpenCV functions
available for this purpose:
-# **Standard Hough Line Transform**
-# First, you apply the Transform:
@code{.cpp}
vector<Vec2f> lines;
HoughLines(dst, lines, 1, CV_PI/180, 100, 0, 0 );
@endcode
with the following arguments:
- *dst*: Output of the edge detector. It should be a grayscale image (although in fact it
is a binary one)
- *lines*: A vector that will store the parameters \f$(r,\theta)\f$ of the detected lines
- *rho* : The resolution of the parameter \f$r\f$ in pixels. We use **1** pixel.
- *theta*: The resolution of the parameter \f$\theta\f$ in radians. We use **1 degree**
(CV_PI/180)
- *threshold*: The minimum number of intersections to "*detect*" a line
- *srn* and *stn*: Default parameters to zero. Check OpenCV reference for more info.
-# And then you display the result by drawing the lines.
@code{.cpp}
for( size_t i = 0; i < lines.size(); i++ )
{
float rho = lines[i][0], theta = lines[i][1];
Point pt1, pt2;
double a = cos(theta), b = sin(theta);
double x0 = a*rho, y0 = b*rho;
pt1.x = cvRound(x0 + 1000*(-b));
pt1.y = cvRound(y0 + 1000*(a));
pt2.x = cvRound(x0 - 1000*(-b));
pt2.y = cvRound(y0 - 1000*(a));
line( cdst, pt1, pt2, Scalar(0,0,255), 3, LINE_AA);
}
@endcode
-# **Probabilistic Hough Line Transform**
-# First you apply the transform:
@code{.cpp}
vector<Vec4i> lines;
HoughLinesP(dst, lines, 1, CV_PI/180, 50, 50, 10 );
@endcode
with the arguments:
- *dst*: Output of the edge detector. It should be a grayscale image (although in fact it
is a binary one)
- *lines*: A vector that will store the parameters
\f$(x_{start}, y_{start}, x_{end}, y_{end})\f$ of the detected lines
- *rho* : The resolution of the parameter \f$r\f$ in pixels. We use **1** pixel.
- *theta*: The resolution of the parameter \f$\theta\f$ in radians. We use **1 degree**
(CV_PI/180)
- *threshold*: The minimum number of intersections to "*detect*" a line
- *minLinLength*: The minimum number of points that can form a line. Lines with less than
this number of points are disregarded.
- *maxLineGap*: The maximum gap between two points to be considered in the same line.
-# And then you display the result by drawing the lines.
@code{.cpp}
for( size_t i = 0; i < lines.size(); i++ )
{
Vec4i l = lines[i];
line( cdst, Point(l[0], l[1]), Point(l[2], l[3]), Scalar(0,0,255), 3, LINE_AA);
}
@endcode
-# Display the original image and the detected lines:
@code{.cpp}
imshow("source", src);
imshow("detected lines", cdst);
@endcode
-# Wait until the user exits the program
@code{.cpp}
waitKey();
@endcode
Result
------
@note
The results below are obtained using the slightly fancier version we mentioned in the *Code*
section. It still implements the same stuff as above, only adding the Trackbar for the
Threshold.
Using an input image such as:
We get the following result by using the Probabilistic Hough Line Transform:
You may observe that the number of lines detected vary while you change the *threshold*. The
explanation is sort of evident: If you establish a higher threshold, fewer lines will be detected
(since you will need more points to declare a line detected).
|
