Procedure of the Hough Transform

The discrete line is represented only by the set of the discrete sampling points $\{(\hat{x_i},\hat{y_i})\vert \ i=1,2,\cdots,n \ \}$, and the parameters for the line are unknown. The standard procedure of the line detection by the Hough Transform is summarized as the following. Let u denote the scanning parameter, let $\hat{u}_k$ denote the k-th sampling point of u, and let $\Delta u$ denotes the sampling interval of u. In the same way, $\hat{v}_m$ denotes the m-th sampling point of v, and $\Delta v$ denotes the sampling interval of v. First, we put an accumulator cell(k, m), which holds the votes, on each of the sampling points $(\hat{u}_k, \hat{v}_m)$ in the parameter space. For every point $(\hat{x}_i, \hat{y}_i)$ on the discrete figure and for every $\hat{u}_k$, $v_{ik}=F_v(\hat{x}_i, \hat{y}_i ; \hat{u}_k)$ is calculated. The quantized value of v_ik is given by $\hat{v}_m (=\lfloor\frac{v_{ik}}{\Delta v}+\frac{1}{2}\rfloor\cdot \Delta v)$. By varying i and k, votes are accumulated at the corresponding cell(k, m). Finally, a local maximum of the votes (shown as the peak A in Figure 2) is found among the cells. The estimated parameters are given by $(\hat{u}_k, \hat{v}_m)$.

  

Figure 2: Transformation error in the Hough Transform