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I understood the concept behind Hough transform. One thing i am still unsure is the rho and theta range of hough transform. The range is given below.

If theta is in the range [0,pi), then rho values can have negative values. If theta is in the range[0,2pi), then rho can have only positive values.

How can rho be negative ?. Is the range standard for all the images ?. I have referred questions on Hough Transform under stack overflow as well as dsp-stackoverflow, but it wasn't helpful. Can someone explain me bit more clearly please ?.

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The Hough transform seeks to replace finding colinear points by finding concurrent lines. The two problems are mathematically equivalent. But dealing with all discrete pixels and a large number of digital lines is complicated and computationally expensive.

Most implementations of the Hough transform discretize the set of distances and angles. With respect to some origin, one can describe a line in polar coordinates by:

  • a positive distance and an angle in $[0,\,2\pi[$
  • a signed distance and an angle in $[0,\,\pi[$

This is explained in Polar coordinate system/Uniqueness of polar coordinates:

any polar coordinate is identical to the coordinate with the negative radial component and the opposite direction

History: the rho-theta parametrization universally used today was first described in Duda, R.O.; Hart, P. E. (1972). "Use of the Hough Transformation to Detect Lines and Curves in Pictures".

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