Microscopic Slice Image Processing and Automatic Labeling - Biomedical Signal Analysis: Contemporary Methods and Applications - page 364

Depending on the image patch type - for example, patches containing
higher-order structures such as axon-dendrite networks of the neurons -
a more elaborate network structure has to be found in order to avoid the
symmetry. In the following, we introduce a preprocessing method that
allows us to orient the image patches in a default way, thus enabling the
shape classification of arbitrarily oriented image patches. We will denote
trained neural classifiers employing this preprocessing as directional
neural networks .
The idea, similar to PCA, is to orient an image I along its principal
axis, where the image pixels themselves are interpreted as samples of
a two-dimensional random vector x I . Hence I is a two-dimensional
histogram of x I ,andthedensityof x I at the pixel ( x, y ) can be estimated
by p x I ( x, y ) I ( x, y ) /T with T := x,y I ( x, y ). This yields estimates ≈

for the mean

x
y

I ( x, y )
T

≈ μ I :=
x,y

E ( x I ) (13.13)

and the covariance

C I :=
x,y

x 2 I ( x, y )

T

xy

− μ I μ I . Cov( x I ) ≈ (13.14)

y 2

xy

For a given image I ,let ρ ( I ) be the rotated image I of the same size such
that the eigenvector of C ρ(I) corresponding to the largest eigenvalue
is parallel to the x-axis (1 , 0). Applying the neural network training
from section 13.4 to the “normalized” training set ( ρ ( x λ ) ,f ( x λ )) (after
adding possible reflections of the patches at the x-axis) yields the desired
directional neural network, which now is directionally selective. Any
possibly rotated input image patch is applied to the composed classifier
f ρ .

Figure 13.7(a) is an example of the application of the eigenvector-
based rotation. In the top row, five 15

◦

15 input patches displaying
the character A in various rotations and typed in various fonts are
displayed. The corresponding normalized images are given in the second
row; clearly all characters, except for the last one, were oriented such
that their main axis is parallel to the x-axis. Apparently due to aliasing
effects in the original image, in the last character the horizontal bar
of the A contributed most to the covariance, and hence could not be
rotated correctly.

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