Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image...
Transcript of Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image...
![Page 1: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/1.jpg)
Complex numbers and image analysis
Henrik Skov Midtiby
Maersk McKinney Moller [email protected]
2015-05-28
1/ 50
![Page 2: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/2.jpg)
2/ 50
What is a complex number?What is a complex number?
How to use complex numbers to detect repeating patterns?
Pattern in time
Direction of edges
Marker detection
Outro
![Page 3: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/3.jpg)
3/ 50
What is a complex number?
Complex numbers let us solve certain equations like
x2 = −4
by introducing the number i , with the property i2 = −1.
![Page 4: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/4.jpg)
4/ 50
Matlab code example
>> x = 2i
>> x^2
x =
0.0000 + 2.0000i
ans =
-4
>> x = 1 + 1i
>> x^2
x =
1.0000 + 1.0000i
ans =
0.0000 + 2.0000i
![Page 5: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/5.jpg)
5/ 50
Dual representation
Cartesian form
z = a + ib
Polar form
z = r · e iθ
![Page 6: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/6.jpg)
6/ 50
Eulers identity
Eulers identitye iθ = cos(θ) + i sin(θ)
describes how to calculate complex exponentials.
![Page 7: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/7.jpg)
7/ 50
Oscillating exponentials
e ikt k ∈ R
Complex exponentials ”oscillate” and are simpler to do calculationswith than sine and cosine.
![Page 8: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/8.jpg)
8/ 50
Matlab code example
To plot the oscillating function
f (t) = e it
The following code is used
x = linspace(0, 2*pi, 1000);
x = x(2:end);
ex = exp(1i * x);
plot(x, real(ex)); hold on;
plot(x, imag(ex)); hold off;
legend('Real', 'Imag')
![Page 9: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/9.jpg)
9/ 50
Oscillating function
![Page 10: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/10.jpg)
10/ 50
Two oscillating functions
![Page 11: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/11.jpg)
11/ 50
What is a complex number?
How to use complex numbers to detect repeating patterns?The Fourier transform
Pattern in time
Direction of edges
Marker detection
Outro
![Page 12: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/12.jpg)
12/ 50
Representations of a periodic function
A periodic function, f (t), with period 2π can be represented af asum of oscillating functions with different frequencies.Sine and cosine based
f (t) = a1 · sin(1t) + a2 · sin(2t) + a3 · sin(3t) + . . .
b0 + b1 · cos(1t) + b2 · cos(2t) + . . .
Based on complex exponentials
f (t) = c1 · e1ti + c2 · e2ti + c3 · e3ti + . . .
c0 + c−1 · e−1ti + c−2 · e−2ti + c−3 · e−3ti + . . .
![Page 13: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/13.jpg)
13/ 50
Example of a periodic function
Question: How to locate the center of the peak?
![Page 14: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/14.jpg)
14/ 50
Alternative representationFourier series with five terms
![Page 15: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/15.jpg)
15/ 50
Alternative representationFourier series with one term
![Page 16: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/16.jpg)
16/ 50
Orthogonal functions
The complex exponentials enti and emti are orthogonal withrespect to the inner product
〈f (t), g(t)〉 =
∫ 2π
0f (t) · (g(t))∗ dt
if and only if n 6= m.
〈enti , emti 〉 =
∫ 2π
0enti · (emti )∗ dt
=
∫ 2π
0e(n−m)ti dt
For example
〈e2ti , e2ti 〉 = 2π 〈e2ti , e3ti 〉 = 0
![Page 17: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/17.jpg)
17/ 50
How to determine cn?
The orthogonal property of the complex exponentials allows us tocalculate cn as follows:
f (t) = cn−1 · e(n−1)ti + cn · enti + cn+1 · e(n+1)ti
〈enti , f (t)〉 = cn−1 · 〈enti , e(n−1)ti 〉+cn · 〈enti , enti 〉+ cn+1 · 〈enti , e(n+1)ti 〉
= cn · 〈enti , enti 〉
cn =〈enti , f (t)〉〈enti , enti 〉
![Page 18: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/18.jpg)
18/ 50
Matlab code example
x = linspace(0, 2*pi, 1001);
x = x(1:(end - 1));
fx = sign(sin(x - 0.6));
ex1 = exp(1i * x );
a1 = fx * ex1' / 1000;
exm1 = exp(-1i * x );
am1 = fx * exm1' / 1000;
peaklocation = -angle(a1)
peaklocation = angle(am1)
plot(x, fx); hold on;
plot(x, real(a1 * ex1 + am1 * exm1))
![Page 19: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/19.jpg)
19/ 50
Determined peak location
![Page 20: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/20.jpg)
20/ 50
What is a complex number?
How to use complex numbers to detect repeating patterns?
Pattern in timePattern in time
Direction of edges
Marker detection
Outro
![Page 21: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/21.jpg)
21/ 50
Image of experimental setup
![Page 22: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/22.jpg)
22/ 50
Polarization angles
![Page 23: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/23.jpg)
23/ 50
Matlab code example
Load image stack
imagestack = [];
for k=1:36
filename = sprintf('img/%02d.png', k);
img = imread(filename);
imagestack(:, :, k) = img;
end
![Page 24: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/24.jpg)
24/ 50
Matlab code example
Plot intensity variations of a region in the image sequence
vals = [];
for k = 1:36
temp = imagestack(181:207, 162:193, k);
vals(k) = sum(temp(:));
end
plot(vals);
![Page 25: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/25.jpg)
25/ 50
Intensity variations
![Page 26: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/26.jpg)
26/ 50
Matlab code example
x = (1:36) * 2 * pi / 36
ex2 = exp(2i * x );
a2 = vals * ex2' / 36
exm2 = exp(-2i * x );
am2 = vals * exm2' / 36
peaklocation = -angle(a2) / 2
plot(x, real(a2 * ex2 + am2 * exm2))
![Page 27: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/27.jpg)
27/ 50
Second harmonics
![Page 28: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/28.jpg)
28/ 50
Located peaks
![Page 29: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/29.jpg)
29/ 50
Matlab code example
phaseimage = imagestack(:, :, 1) * 0;
for k = 1:36
phaseimage = phaseimage + ...
imagestack(:, :, k) * exp(1i * k * 2 * 2*pi / 36);
end
absPhaseImage = abs(phaseimage);
absPhaseImage = absPhaseImage / max(absPhaseImage(:));
argPhaseImage = angle(phaseimage);
rgbimage = hsv2rgb((argPhaseImage + pi)/(2*pi), ...
1+0*argPhaseImage, 1 + 0*argPhaseImage);
image(rgbimage);
![Page 30: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/30.jpg)
30/ 50
Located orientations
![Page 31: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/31.jpg)
31/ 50
What is a complex number?
How to use complex numbers to detect repeating patterns?
Pattern in time
Direction of edgesDirection of edges
Marker detection
Outro
![Page 32: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/32.jpg)
32/ 50
Building block of an algorithm from my phd
![Page 33: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/33.jpg)
33/ 50
Fourier transform around points in an image
![Page 34: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/34.jpg)
34/ 50
What is a complex number?
How to use complex numbers to detect repeating patterns?
Pattern in time
Direction of edges
Marker detectionMarker localisationBending spaceMarker localization
Outro
![Page 35: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/35.jpg)
35/ 50
Marker localisation
![Page 36: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/36.jpg)
36/ 50
Bending space
![Page 37: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/37.jpg)
37/ 50
Bending space
![Page 38: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/38.jpg)
38/ 50
Bending space
![Page 39: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/39.jpg)
39/ 50
Bending space
![Page 40: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/40.jpg)
40/ 50
Bending space
![Page 41: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/41.jpg)
41/ 50
Bending space
![Page 42: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/42.jpg)
42/ 50
Final marker
![Page 43: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/43.jpg)
43/ 50
Matlab code example
kernel = genSymDetectorKernel(4, 150);
function kernel = genSymDetectorKernel(order, kernelsize)
stepsize = 2 / (kernelsize-1);
temp1 = meshgrid(-1:stepsize:1);
kernel = temp1 + 1i*temp1';
magni = abs(kernel);
kernel = kernel.^order;
kernel = kernel.*exp(-8*magni.^2);
abs(kernel);
end
![Page 44: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/44.jpg)
44/ 50
Marker localization
Real part of kernel
Absolute value of kernel
Imaginary part of kernel
Argument of kernel
![Page 45: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/45.jpg)
45/ 50
Input image
![Page 46: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/46.jpg)
46/ 50
Magnitude response of convolution
![Page 47: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/47.jpg)
47/ 50
Input image
![Page 48: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/48.jpg)
48/ 50
What is a complex number?
How to use complex numbers to detect repeating patterns?
Pattern in time
Direction of edges
Marker detection
OutroOutro
![Page 49: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/49.jpg)
49/ 50
Conclusion
Complex numbers can be used to detect repeating patterns inimages.The patterns might repeat over
I time
I orientation
![Page 50: Henrik Skov Midtibyhenrikmidtiby.github.io/downloads/2015-05-28... · Complex numbers and image analysis Henrik Skov Midtiby Maersk McKinney Moller Institute hemi@mmmi.sdu.dk 2015-05-28](https://reader033.fdocuments.us/reader033/viewer/2022043002/5f7dbbce4dbd3d1fc461dda8/html5/thumbnails/50.jpg)
50/ 50
Questions