Hough Transform Voting Scheme for Detection of Parabolas and Open Conics in Images

The Hough transform (HT) is a popular tool for line detection due to its robustness to noise and missing data. Its main use is for transformation of raster images like PNGs, GIFs, and JPEGs to scalable vector graphics. However, the apparent complexities of equations, the difficulties to understand the voting scheme and importance of discretization parameter have prevented HT algorithm developments for generic polynomials. The main concern of this work is to describe the importance of the process of turning discrete the parameters of the equations that describe different curves to be identified in the images. Although it applies to any type of equation, the influence of discretization of open and closed conics, such as parabolas and ellipses, is specifically discussed. The proposed approach identifies the parameters and the possible dependence among them to define the search order of them. The discrete limits and the incremental value of each parameter are commented as well. They are fundamental for the voting scheme, playing a key role on the accuracy of this and promoting a considerable impact on the HT result. Both the number of parameters (that is, the size of the solution space) and the discretization values used are directly proportional to the number of accumulator cells created. Thus, both the computational cost and the processing time may compromise the implementation process. A correct order to calculate the values of the parameters in the algorithm and the appropriate limits for each one of them guarantees a reduction and simplification of operations, as well as an improvement in the efficiency of the results obtained.