Examples of parametric models are: lines / circles/ arbitrary shapes/templates.
As an easy example: Line.
Line model has three requirements: 1. how many lines are there? 2. which points combine which line? 3. what is the line?
Parametric models use voting systems to fill in occlusions and cancel noisy lines, by eliminating edges of features that received fewer votes compared to the rest of the edges of features.
Hough(parametric) Transform as a voting system:
Is just one of the techniques that does this voting system:
Hough (parameter) Space is in a representation of -m and b.
Hough(parameter) space is re-arranged, b is on the vertical instead of y and -m is on the horizontal.
A point in Hough Space is (-m,b)
A line in Hough Space is b = -mx + y, but because vertical lines in algebra have an infinite m(slope) it’s very hard to calculate and so the lines will be represented in polar representation(xsin(θ) – ycos(θ) = d ) as a sinusoid, but the concept is the same.
Image/Regular Space is in a representation of y and x.
A point in Image Space is (x,y)
y = -mx + b .
There is a duality between Hough and Image Spaces as follows:
1: Every point in Image Space creates a line in Hough Space and vice versa.
2: Every point in Hough Space creates a line in Image Space and vice versa.
Hough Transofrm works as followed:
Every point in Image Space is transformed to a line on Hough Space and all lines are placed through grid bins(horizontal combined with vertical bins) and their intersections are counted as votes for each bin.
The bin with the most votes(intersected lines) is the bin which contains the point which will be represented as a line back in Image Space
Possible Extensions for Hough Transform:
- Take the gradient and use it as θ or a range θ +- 10 degrees, and solve instead of looping through all the points
- Count stronger edges with more votes
- Change resolution hierarchically that is:
Start with few large bins and find the maximum(peaks) lines in them and then do it again for each smaller area found, but this time take many bins on them.