Image Processing

where v_2,1(−1) = 0. The sums of the image along the arithmetical rays can be calculated from ten line-integrals along the geometrical rays, by using the following equation:

$\begin{array}{ll}{v}_{2,1}\left(t\right)\hfill & ={\left(-1\right)}^t\left[w_{2,1}\left(0\right)-w_{2,1}\left(1\right)+w_{2,1}\left(2\right)-\mathrm{\dots }+{\left(-1\right)}^t{w}_{2,1}\left(t\right)\right]\hfill \\ =w_{2,1}\left(t\right)-v_{2,1}\left(t-1\right),\left(v_{2,1}\left(-1\right)=0\right)\hfill \\ t=0,1,2,\mathrm{\dots },9,\hfill \end{array}$

(3.27)

where the factor of $1/\left(2\sqrt{5}\right)$ is omitted.

We also can define two new binary orthogonal matrices

$\left[{{\chi }^{\prime }}_{2,1,0+1}\right]=\left[\begin{array}{rrrr}\hfill 1& \hfill -1& \hfill 1& \hfill -1\\ \hfill 1& \hfill -1& \hfill 1& \hfill -1\\ \hfill -1& \hfill 1& \hfill -1& \hfill 1\\ \hfill -1& \hfill 1& \hfill -1& \hfill 1\end{array}\right],\left[{{\chi }^{\prime }}_{2,1,0-1}\right]=\left[\begin{array}{rrrr}\hfill 1& \hfill -1& \hfill 1& \hfill -1\\ \hfill -1& \hfill 1& \hfill -1& \hfill 1\\ \hfill -1& \hfill 1& \hfill -1& \hfill 1\\ \hfill 1& \hfill -1& \hfill 1& \hfill -1\end{array}\right]$

and consider new components of superposition of these functions with the image,

${f^{\prime }}_{2,1,0+1}={\displaystyle \sum _{m=0}^3{\displaystyle \sum _{n=0}^3{{\chi }^{\prime }}_{2,1,0+1}\left(n,m\right)f_{n,m}}}$

and

${f^{\prime }}_{2,1,0-1}={\displaystyle \sum _{m=0}^3{\displaystyle \sum _{n=0}^3{{\chi }^{\prime }}_{2,1,0-1}\left(n,m\right)f_{n,m}}}.$

These components can be calculated from the line-integrals along the geometrical rays as

$\begin{array}{ll}{f^{\prime }}_{2,1,0+1}\hfill & =(v_{2,1,0}-\hfill \end{array}$