Find intersection of two lines in polar coordinates

Question

I'm using an image processing library (aForge) to process images. In this case, I'm trying to pick out a playing card from an image. aForge calculates Hough Lines, for any edges that it detects. Each Hough Line is produced in a polar coordinate from the center of the image. Here's an example, along with angle (Theta) and radius (Rad).

I want to extract the card, in order run it through an image recognition to identify suit and rank.

To extract the card, I need to calculate the intersection of the lines, then cut out the rectangular(ish) image.

Here's a sample, the Hough Lines appear in pink, the selected 1st Hough Line appears in bold.

Answer 1

The intersection point $(x,y)$ of two Hough Lines

$$ \begin{aligned} x \cos \theta_1 + y \sin \theta_1 &= r_1 \\ x \cos \theta_2 + y \sin \theta_2 &= r_2 \\ \end{aligned}$$

is given by

$$\begin{aligned} x & = \frac{r_2 \sin \theta_1 - r_1 \sin \theta_2}{\sin(\theta_1-\theta_2)} \\ y & = \frac{r_1 \cos \theta_2 - r_2 \cos \theta_1}{\sin(\theta_1-\theta_2)} \\ \end{aligned}$$

Answer 2

The Hough line equation is $$ x\,\cos\theta + y\,\sin\theta = r $$ see, for example, this article.

In order to find the intersection, you have to find the solution of the following system of equations: $$ x\,\cos\theta_1 + y\,\sin\theta_1 = r_1 \\ x\,\cos\theta_2 + y\,\sin\theta_2 = r_2 $$ which can also be written as $$ \begin{pmatrix} \cos \theta_1 & \sin\theta_1 \\ \cos \theta_2 & \sin\theta_2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} r_1 \\ r_2 \end{pmatrix} $$

Answer 3

Here's a sample implementing John's answer.