Example: Line Fitting

Figure 5.5 shows how the FHT works for $k=2$, when parameter space is a plane, hyperplanes are straight lines and hypercubes are squares whose associated hyperspheres are circles passing through the vertices of the squares (figure 5.5).

This is applicable to the problem of finding a straight line through points on a plane. If the plane has coordinates $(u,v)$ the line can be written

$$v=\alpha u + \beta$$

where $\alpha$ and $\beta$ are constant. Each point $(u_j , v_j )$ votes for a line in parameter space:

$$\beta=v_j - \alpha u_j .$$

Let the initial ranges of $\alpha$ and $\beta$, defining the root hypercube, be $L_{\alpha}$ and $L_{\beta}$ centred around $\alpha_0$ and $\beta_0$ respectively. Then the above equation can put in the form of equation 5.15 using the transformation

$$X_1 = \frac{\alpha}{L_\alpha} ,\; X_2 = \frac{\beta}{L_\be... ...{1j} = \frac{L_\alpha u_j}{q} ,\; a_{2j} = \frac{L_\beta}{q}$$

where $q = \sqrt{L_\alpha^2 u_j^2 + L_\beta^2}$.