Appendices - Feature Extraction and Image Processing - page 330

So a 3× 3 template is:

0.111 0.111 0.111

!
!

averaging_template(3)=

0.111 0.111 0.111

0.111 0.111 0.111

and to apply it: smoothed:=tm_conv(eye,averaging_template(3))
Since there is a duality between convolution in the time domain and multiplication in
the frequency domain, we can implement template convolution by using the Fourier transform.
Template convolution is the inverse Fourier transform of the product of Fourier transform
of the image with the transform of the template. First we need a picture of the template, this
picture must be the same size as the image we want to convolve it with. For averaging, we
need a 3× 3 square in the centre of an image of the same size as the eye image:

for x ∈ 0..cods(eye)-1
for y ∈ 0..rows(eye)-1

square:=

rows(eye)
2

rows(eye)
2

1
9 if y

y,x ←

≤

pic -1 y +1

cols(eye)
2

cols(eye)
2

⋅

≤

x -1 x +1

pic y,x ← 0 otherwise

pic

Then, template convolution is given by:

conv(pic,temp):= pic_spectrum ← Fourier(pic) Take transform of image

temp_spectrum ← Fourier(temp) Transform template

convolved_spectrum ←
(pic_spectrum temp_spectrum)

Form product

result ← inv_Fourier
(rearrange(convolved_spectrum)) Inverse transform
result Supply result

Let's see what happens: Fsmoothed:=conv(eye,square)
To check the result, we need to scale its magnitude:
sc_smooth:= cols(eye)· |Fsmoothed|
Now, let's see the difference