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Every pixel of the image is scanned, and a sliding window is used to compute the Hough transform of small regions of the image. Peaks of the Hough image (which correspond to line segments) are then extracted, and a rectangle is detected when four extracted peaks satisfy certain geometric conditions.
The Hough transform (HT) can be used to detect lines circles or • The Hough transform (HT) can be used to detect lines, circles or other parametric curves. It was introduced in 1962 (Hough 1962) and first used to find lines in images a decade later (Duda 1972). The goal is to find the location of lines in images.
If two edge points lay on the same line, their corresponding cosine curves will intersect each other on a specific (ρ, θ) pair. Thus, the Hough Transform algorithm detects lines by finding the (ρ, θ) pairs that have a number of intersections larger than a certain threshold.
The Hough Transform (HT) is a popular technique for detecting straight lines and curves on gray-scale images. It maps image data from image space to a parameter space, where curve detection becomes peak detection problem.
For those of you wondering about the paper, it's:
Rectangle Detection based on a Windowed Hough Transform by Cláudio Rosito Jung and Rodrigo Schramm.
Now according to the paper, the intersection points are expressed as polar coordinates, obviously you implementation may be different (the only way to tell is to show us your code).
Assuming you are being consistent with his notation, your peaks should be expressed as:
You must then perform peak paring given by equation (3) in section 4.3 or
where 
represents the angular threshold corresponding to parallel lines
and 
is the normalized threshold corresponding to lines of similar length.
The accuracy of the Hough space should be dependent on two main factors.
The accumulator maps onto Hough Space. To loop through the accumulator array requires that the accumulator divide the Hough Space into a discrete grid.
The second factor in accuracy in Linear Hough Space is the location of the origin in the original image. Look for a moment at what happens if you do a sweep of \theta for any given change in \rho. Near the origin, one of these sweeps will cover far less pixels than a sweep out near the edges of the image. This has the consequence that near the edges of the image you need a much higher \rho \theta resolution in your accumulator to achieve the same level of accuracy when transforming back to Cartesian.
The problem with increasing the resolution of course is that you will need more computational power and memory to increase it. Also If you uniformly increase the accumulator resolution you have wasted resolution near the origin where it is not needed.
Some ideas to help with this.
(x-a)^2 + (y-b)^2 = \rho^2
circle equation where
- x,y are the current pixel
- a,b are your chosen origin
- \rho is the radius
once the radius is known adjust your accumulator
resolution accordingly. You will have to keep
track of the center of each \rho \theta bin.
for transforming back to Cartesian
The link to the referenced paper does not work, but if you used the standard hough transform than the four intersection points will be expressed in cartesian coordinates. In fact, the four lines detected with the hough tranform will be expressed using the "normal parametrization":
rho = x cos(theta) + y sin(theta)
so you will have four pairs (rho_i, theta_i) that identifies your four lines. After checking for orthogonality (for example just by comparing the angles theta_i) you solve four equation system each of the form:
rho_j = x cos(theta_j) + y sin(theta_j)
rho_k = x cos(theta_k) + y sin(theta_k)
where x and y are the unknowns that represents the cartesian coordinates of the intersection point.
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