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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).

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...he same result for lines like $A$ and
$B$, but not for line $C$}
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 $q = \sqrt{L_\alpha^2 u_j^2 + L_\beta^2}$.