Image Processing

where the sums of the discrete image along the rays are denoted by v_t = v_4,1(t), t = 0 : 35. With the new variables

$a_n={\displaystyle \sum _m{v}_{n+8m}={\displaystyle \sum _m{v}_{4,1,n+8m},n=0:7,}}$

the complete system of linear equations, which defines the subset P(4, 1) of components of the 2-D paired transform with the triplet-numbers of the subset U(4, 1), can be written as follows:

$\begin{array}{l}{f^{\prime }}_{4,1,0}=a_0-a_4\hfill \\ {f^{\prime }}_{4,1,1}=a_1-a_5\hfill \\ {f^{\prime }}_{4,1,2}=a_2-a_6\hfill \\ {f^{\prime }}_{4,1,3}-a_3-a_7\hfill \\ {f^{\prime }}_{0,2,0}=\left(a_0+a_4\right)-\left(a_2+a_6\right)\hfill \\ {f^{\prime }}_{0,2,2}=\left(a_1+a_5\right)-\left(a_3+a_7\right)\hfill \\ {f^{\prime }}_{0,4,0}=\left(a_0+a_4\right)-\left(a_1+a_5\right)+\left(a_2+a_6\right)-\left(a_3+a_7\right)\hfill \\ {f^{\prime }}_{0,0,0}=\left(a_0+a_4\right)+\left(a_1+a_5\right)+\left(a_2+a_6\right)+\left(a_3+a_7\right).\hfill \end{array}$

(3.71)

These calculations correspond to the 8-point discrete paired transform of the vector a = (a₀, a₁, a₂, …, a₇)′,

$P\left(4,1\right)=\left[{{\chi }^{\prime }}_8\right]a.$

To express the sums v(t) of the discrete image f_n,m through the line-integrals w(t), ...