1) The document describes a numerical method for image registration using nonlinear geometric transforms. It aims to decrease geometric deformation in sets of images and find the best estimate of mean values. 2) The method uses phase correlation to determine pixel movement vectors between images, then applies interpolation to align pixels. It calculates the arithmetic mean of the corrected image set. 3) Maps of pixel movement vectors between original and mean images are generated to visualize geometric corrections made by the registration process. The final output is a sharpened mean image providing the best estimate of values while correcting for geometric distortions.

1. Numerical method of image registration using
nonlinear geometric transform
Michael Rára
Faculty of Mechanical Engineering, Brno University of Technology
2.11.2018
Michael Rára
Numerical method of image registration using nonlinear geometric

2. Goals
1 Describe numerical methods for image analysis with
special aim to nonlinear geometric transform.
2 Create software to decrease geometric deformation in set
of images.
3 Find the best estimate of mean value from a set of images
defected by geometric deformation.
Michael Rára
Numerical method of image registration using nonlinear geometric

3. Entry data set
Michael Rára
Numerical method of image registration using nonlinear geometric

4. Light refraction
This phenomenon is typical on border of two different
environments (typical example is border between water and
air).
Michael Rára
Numerical method of image registration using nonlinear geometric

5. Astronomical seeing
Refraction index is influenced by atmospheric pressure,
which is different in every moment. Lets consider bigger
half-circle is border of atmosphere and the second is
surface of Earth.
Thanks to refraction index observer (point P) see source of
light (point Q) in different position (point R) than it really is.
Thanks to time variability of refraction index we have to
deal with problem known as astronomical seeing.
Astronomical seeing causes geometric deformations in
images, that is because refraction index is different in
every moment and that means we see point Q in different
position in every moment.
Michael Rára
Numerical method of image registration using nonlinear geometric

6. Arithmetic mean of entry data set
We simply calculate arithmetic value of brightness of pixel
at coordinates i,j. Index k means k-th image.
Thanks to arithmetic mean we have good estimate of
mean value of the data set. Unfortunately gain image is
blurred, see next slide.
¯ai,j =
1
n
n
k=1
ai,j;k
Michael Rára
Numerical method of image registration using nonlinear geometric

7. Michael Rára
Numerical method of image registration using nonlinear geometric

8. Edge detection in image
Gradient method is used to detect edges in image. f(i, j) is
value of brightness of pixel at coordinates i,j.
| f(i, j)| =
∂f(i, j)
∂i
2
+
∂f(i, j)
∂j
2
Approximation of equations
above for pixels which lie on the
borders of image.
∂f(i, j)
∂i
≈ f(i, j) − f(i − 1, j)
∂f(i, j)
∂i
≈ f(i + 1, j) − f(i, j)
Approximation of equations
above for pixels which do not lie
on the borders of image.
∂f(i, j)
∂i
≈ f(i + 1, j) − f(i − 1, j)
Analogously for ∂f(i,j)
∂j
Michael Rára
Numerical method of image registration using nonlinear geometric

9. Edges in image gain by arithmetic mean
Michael Rára
Numerical method of image registration using nonlinear geometric

10. Net of pixels from image on previous slide
Michael Rára
Numerical method of image registration using nonlinear geometric

11. Phase correlation
Now we investigate movement of every pixel of every image
due to image gain by arithmetic mean of entry data set.
r =
1
n
n
i=1(xi − ¯x)(yi − ¯y)
σ(x)σ(y)
r =
1
n
n
i=1(xi − ¯x)(yi − ¯y)
σ(x)σ(y)
Michael Rára
Numerical method of image registration using nonlinear geometric

12. Interpolation map
Thanks to phase correlation we know move vectors of all white
pixels. Move vector of green pixels is zero, see the next slide.
Michael Rára
Numerical method of image registration using nonlinear geometric

13. Michael Rára
Numerical method of image registration using nonlinear geometric

14. Linear interpolation of function f(x, y)
Function values at points
A, B, C, D are move vectors at
these points. We want to get
move vector at point D with use
of interpolation plane.
Equation of interpolation plane
can be written as determinant of
this matrix:
xD − xA yD − yA f(D)-f(A)
xB − xA yD − yA f(B)-f(A)
xC − xA yD − yA f(C)-f(A)
= 0
Michael Rára
Numerical method of image registration using nonlinear geometric

15. These images show idea of looking suitable group of pixels to
create matrix from previous slide. We want to interpolate
function value in orange pixel..
In this case we do not
need third pixel. We will
use linear interpolation
over line.
It is obvious we want to
find the nearest pixels to
the orange one, but the
red one can not be used.
In this situation we have
to find first, third and
fourth nearest pixel to the
orange one.
Michael Rára
Numerical method of image registration using nonlinear geometric

16. Bilinear interpolation of values of brightness.
f(P) = f(Q1,1)
(x2 − x)(y2 − y)
(x2 − x1)(y2 − y1)
+ f(Q2,1)
(x − x1)(y2 − y)
(x2 − x1)(y2 − y1)
+
f(Q1,2)
(x2 − x)(y − y1)
(x2 − x1)(y2 − y1)
+ f(Q2,2)
(x − x1)(y − y1)
(x2 − x1)(y2 − y1)
.
Thanks to phase correlation we know vector
of move of every pixel between original
image and image gain by arithmetic mean
(call it A). This vector can be for example
(0, 01; 1, 4). It means that pixel in image A
with coordinates i, j, can be found in original
image at coordinates i + 0, 01; j + 1, 4 (point
P). To get value of brightness of this pixel
we use bilinear interpolation of surrounded
values (points Q).
Michael Rára
Numerical method of image registration using nonlinear geometric

17. Corrected set of entry data
Michael Rára
Numerical method of image registration using nonlinear geometric

18. Arithmetic mean of corrected images
Michael Rára
Numerical method of image registration using nonlinear geometric

19. Sharpen image
We use discrete 2D convolution.
¯f(i, j) =
1
m=−1
1
n=−1
f(i − m, j − n)h(m, n)
Convolution matrix
h =


−1 −1 −1
−1 9 −1
−1 −1 −1


f(i, j) is value of brightness of pixel with coordination (i, j) from
image in previous slide. Result is sharpened image.
The sharpened image ¯f(i, j) is main output of the software. This
image is the best estimate of the mean value of the entry set,
see next slide.
Michael Rára
Numerical method of image registration using nonlinear geometric

20. Michael Rára
Numerical method of image registration using nonlinear geometric

21. Map of movements of pixels
−−−−−−−−→
MoveVector := (Px, Py)
Brightness := ||
−−−−−−−−→
MoveVector||
255:=max{Brightness}
Michael Rára
Numerical method of image registration using nonlinear geometric

22. Maps of movements of pixels between images from
original set and image gain by arithmetic mean of
them.
Please, see the next slide.
Michael Rára
Numerical method of image registration using nonlinear geometric

23. Michael Rára
Numerical method of image registration using nonlinear geometric

24. Thank you for attention.
Michael Rára
Numerical method of image registration using nonlinear geometric