Hough Transform can be used to extract lines from images. It can also be used to extract curves - this is a little harder though because higher dimensional Hough transforms are resource consuming. I was wondering whether how one restricts the Hough transform to 2D voting space for a curve of order 3 i.e. x^{3}+ax^{2}+bx+c ?
Anyone know any good sites explaining this (can't seem to find any). Or an explanation here if there isn't one :).
The essence of the Generalised Hough Transform that the "sides" of the accumulator is the answer you are looking for. If you are trying to match ellipses or arbitrary curves - in your case a, b, c parameters then you should build 3D accumulator and look for maximum there. Google "ellipse detection using hough transform" or "arbitrary shape detection using hough transform".
There are many way to optimise your search in multi dimensional accumulator, so don't be afraid to build multidimensional HT parameterised space - it can give you good overview of your problem.
You may want to split your search into two stage - for example build a classic 2D for your a and b parameters, then use very simple 1D accumulator for finding c, this has been done in edge detection, but be aware that this split can introduce large errors if you a,b,c interdependent.
Ways to optimise multidimensional Hough Transform: (Probabilistic) Randomised Hough transform, Hybrid and Multidimensional Hough Transform.
Also Generalised Hough Transform and Radon Transform are nearly synonymous, so for arbitrary shape detection "Radon transform" may give you better ideas: Hough Transform is a discrete version of continuous Radon Transform.
Try googling "Generalized Hough Transform" and you'll find a lot of stuff on this, including the original paper by Ballard, which seems quite readable. Which is the best of these for you depends on where you're starting from with this, so google is probably your best option.
scholar.google.com gives many papers, but few of them are free (though if you have access, it's probably the best start).
Do you only need to locate the curve for which you already know your parameters a,b,c? Using GHT you can create a discrete voting space from your eq. Use it to vote in a 2d space and you will find your curve. If your are trying to determine a,b,c from the Hough Transform it will be harder :)
来源：https://stackoverflow.com/questions/5502726/hough-transform-for-finding-curve-segments