The triangular prism surface area (TPSA) algorithm [1][2] consideres a greyscale image pixel P and a square environment with edges at the pixels A, B, C and D (fig. 9).
Figure 9: Triangular Prism Surface Area Method
The connections of the pixels's greyscale values ,
,
,
and
produces four triangles from which the area is taken. This is done for
every image pixel that is far enough away from the image margin.
A unique central pixel exists only in those cases where the size of
the square is an odd pixel number. If
is even, the mean of the greyscale
value of the four pixel in the center is taken to define the triangles
(fig. 10).
Figure 10: Odd and even sizes of the environment
Instead of four triangles it is also possible to define eight triangles as depicted in fig. 11.
Figure 11: Tap 4 and tap 8
In this case the hypotenuse of the ground triadic area is choosen as .
Thus it is possible to mix tap 4 and tap 8 analysis.
The areas of all triangles for every central pixels are summed up to
the entire area for different scales
. The double logarithmic
Richardson-Mandelbrot plot of
against
should again yield a linear line
whose slope is used to determine the TPSA dimension
:
The Richardson-Mandelbrot plot of the TPSA analysis for fig. 2 is shown in fig. 12, where only the even environments (comp. fig. 10) and tap 4 (comp. fig. 11) is used for the analysis.
Figure 12: Richardson-Mandelbrot plot for the TSPA analysis of a greyscale image
According to equ. (9) the TPSA dimension is: