calculate displacement between 2 images

Hello,

I would like to ask a question about normalized cross correlation.

Indeed,the steps of calculation of normalized cross correlation (NCC) are :

 1. Calculation of FFT or convolution
 2. Calculation of local sums

The formula of NCC is available in the bottom of this documentation https://fr.mathworks.com/help/images/ref/normxcorr2.html

In the first step, I took:

a =

      0.4693    0.2015    0.3220    0.8843
      0.9090    0.1872    0.7254    0.8845
      0.6666    0.6917    0.4871    0.3952
      0.1845    0.9348    0.2403    0.9641

b =

      0.5244    0.0120    0.0678    0.8644
      0.4334    0.1436    0.4166    0.7587
      0.8684    0.7909    0.5884    0.9340
      0.4638    0.0562    0.5183    0.6035

Then

a2 =

      0.4693    0.2015    0.3220    0.8843         0         0
      0.9090    0.1872    0.7254    0.8845         0         0
      0.6666    0.6917    0.4871    0.3952         0         0
      0.1845    0.9348    0.2403    0.9641         0         0
           0         0         0         0         0         0
           0         0         0         0         0         0

b2 =

    0.5244    0.0120    0.0678    0.8644         0         0
    0.4334    0.1436    0.4166    0.7587         0         0
    0.8684    0.7909    0.5884    0.9340         0         0
    0.4638    0.0562    0.5183    0.6035         0         0
         0         0         0         0         0         0
         0         0         0         0         0         0

I found :

conv2(rot90(a,2),b)

ans =

    0.5055    0.1376    0.5584    0.9575    0.2733    0.8205    0.1595
    0.6250    0.5028    1.2366    1.7782    1.0741    1.4291    0.7161
    1.4722    1.6299    2.2702    4.1249    2.3243    2.0076    1.4637
    1.6373    1.5220    2.8883    5.0546    3.0854    2.4246    1.8292
    1.3345    1.8440    2.3115    4.0665    2.5920    1.8209    1.6073
    1.1781    1.3650    1.5358    2.9240    1.3762    1.0484    0.9869
    0.4101    0.1990    0.5698    0.9295    0.3251    0.3648    0.2832

conv2(rot90(a2,2),b2)

ans =

         0         0         0         0         0         0         0         0         0         0         0
         0         0         0         0         0         0         0         0         0         0         0
         0         0    0.5055    0.1376    0.5584    0.9575    0.2733    0.8205    0.1595         0         0
         0         0    0.6250    0.5028    1.2366    1.7782    1.0741    1.4291    0.7161         0         0
         0         0    1.4722    1.6299    2.2702    4.1249    2.3243    2.0076    1.4637         0         0
         0         0    1.6373    1.5220    2.8883    5.0546    3.0854    2.4246    1.8292         0         0
         0         0    1.3345    1.8440    2.3115    4.0665    2.5920    1.8209    1.6073         0         0
         0         0    1.1781    1.3650    1.5358    2.9240    1.3762    1.0484    0.9869         0         0
         0         0    0.4101    0.1990    0.5698    0.9295    0.3251    0.3648    0.2832         0         0
         0         0         0         0         0         0         0         0         0         0         0
         0         0         0         0         0         0         0         0         0         0         0

So, adding zeros will not change the result of FFT /convolution. But after calculating local sums, the result changes because it takes into consideration the size of a (or a2). (normxcorr2(a,b) is different from normxcorr2(a2,b2))

So when I want to calculate the displacement between two grayscale images using cross correlation, the result is not the same than if I translated the image (by padding with zeros)

Could you explain me why the displacement changes even if the translation (padding) is the same?