Master Thesis - Hochschule Bonn-Rhein-Sieg

5. Algorithms Master Thesis Björn Ostermann page 69 of 126
result of this addition has to be divided by the combined values of the Gaussian matrix. Therefore, in
praxis, the values in a Gaussian matrix used in a computer program add up to one.
�
1
4
6
4
1
1 4 6 4 1
1 4 6 4 1
4 16 24 16 4
6 24 36 24 6
4 16 24 16 4
1 4 6 4 1
Figure 46: Gaussian matrix, build by multiplying a Gaussian vector with its transpose
– image created in collaboration with [81]
The disadvantage of this approach is the high computational effort. A filter with the width and height
n
n
of n uses n multiplications and n �1
additions. With the 5 times 5 filter in the given example,
each centre element needs 25 multiplications and 24 additions.
Due to the fact that a Gaussian matrix is build by multiplying a Gaussian vector with its transpose (see
Figure 46), the computational effort can be reduced by applying each vector separately.
The optimized algorithm, applied in this project, uses the Gaussian vector, to convert the original
matrix into a temporary matrix (see Figure 47a), which is then turned into the result matrix by
applying the transpose of the Gaussian vector (see Figure 47b).
a) b)
1
4
6
4
1
1 4 6 4 1
Figure 47: a) Converting the original matrix (blue) to a temporary matrix (green)
b) and converting the temporary matrix into the result matrix (yellow)
– images created in collaboration with [81]