Meaning that each pair :math:`(r_{\theta},\theta)` represents each line that passes by :math:`(x_{0}, y_{0})`.
#. If for a given :math:`(x_{0}, y_{0})` we plot the family of lines that goes through it, we get a sinusoid. For instance, for :math:`x_{0} = 8` and :math:`y_{0} = 6` we get the following plot (in a plane :math:`\theta` - :math:`r`):
We consider only points such that :math:`r > 0` and :math:`0< \theta < 2 \pi`.
#. 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 :math:`\theta` - :math:`r`, 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: :math:`x_{1} = 9`, :math:`y_{1} = 4` and :math:`x_{2} = 12`, :math:`y_{2} = 3`, we get:
The three plots intersect in one single point :math:`(0.925, 9.6)`, these coordinates are the parameters (:math:`\theta, r`) or the line in which :math:`(x_{0}, y_{0})`, :math:`(x_{1}, y_{1})` and :math:`(x_{2}, y_{2})` 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 :math:`(\theta, r_{\theta})` 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 :math:`(\theta, r_{\theta})`
* In OpenCV it is implemented with the function :hough_lines:`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 :math:`(x_{0}, y_{0}, x_{1}, y_{1})`
* In OpenCV it is implemented with the function :hough_lines_p:`HoughLinesP <>`
#. The sample code that we will explain can be downloaded from |TutorialHoughLinesSimpleDownload|_. A slightly fancier version (which shows both Hough standard and probabilistic with trackbars for changing the threshold values) can be found |TutorialHoughLinesFancyDownload|_.
#. Display the original image and the detected lines:
..code-block:: cpp
imshow("source", src);
imshow("detected lines", cdst);
#. Wait until the user exits the program
..code-block:: cpp
waitKey();
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.
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).
