EE 7730 2 D Fourier Transform Bahadir K
EE 7730 2 D Fourier Transform Bahadir K. Gunturk EE 7730 - Image Analysis I
Summary of Lecture 2 n n We talked about the digital image properties, including spatial resolution and grayscale resolution. We reviewed linear systems and related concepts, including shift invariance, causality, convolution, etc. Bahadir K. Gunturk EE 7730 - Image Analysis I 2
Fourier Transform n What is ahead? q q 1 D Fourier Transform of continuous signals 2 D Fourier Transform of discrete signals 2 D Discrete Fourier Transform (DFT) Bahadir K. Gunturk EE 7730 - Image Analysis I 3
Fourier Transform: Concept ■ A signal can be represented as a weighted sum of sinusoids. ■ Fourier Transform is a change of basis, where the basis functions consist of sines and cosines. Bahadir K. Gunturk EE 7730 - Image Analysis I 4
Fourier Transform n n Cosine/sine signals are easy to define and interpret. However, it turns out that the analysis and manipulation of sinusoidal signals is greatly simplified by dealing with related signals called complex exponential signals. n A complex number: z = x + j*y n A complex exponential signal: r*exp(j*a) =r*cos(a) + j*r*sin(a) Bahadir K. Gunturk EE 7730 - Image Analysis I 5
Fourier Transform: 1 D Cont. Signals ■ Fourier Transform of a 1 D continuous signal “Euler’s formula” ■ Inverse Fourier Transform Bahadir K. Gunturk EE 7730 - Image Analysis I 6
Fourier Transform: 2 D Cont. Signals ■ Fourier Transform of a 2 D continuous signal ■ Inverse Fourier Transform ■ F and f are two different representations of the same signal. Bahadir K. Gunturk EE 7730 - Image Analysis I 7
Examples Magnitude: “how much” of each component Phase: “where” the frequency component in the image Bahadir K. Gunturk EE 7730 - Image Analysis I 8
Examples Bahadir K. Gunturk EE 7730 - Image Analysis I 9
Fourier Transform: Properties ■ Linearity ■ Shifting ■ Modulation ■ Convolution ■ Multiplication ■ Separable functions Bahadir K. Gunturk EE 7730 - Image Analysis I 10
Fourier Transform: Properties ■ Separability 2 D Fourier Transform can be implemented as a sequence of 1 D Fourier Transform operations. Bahadir K. Gunturk EE 7730 - Image Analysis I 11
Fourier Transform: Properties ■ Energy conservation Bahadir K. Gunturk EE 7730 - Image Analysis I 12
Fourier Transform: Properties ■ Remember the impulse function (Dirac delta function) definition ■ Fourier Transform of the impulse function Bahadir K. Gunturk EE 7730 - Image Analysis I 13
Fourier Transform: Properties ■ Fourier Transform of 1 Take the inverse Fourier Transform of the impulse function Bahadir K. Gunturk EE 7730 - Image Analysis I 14
Fourier Transform: 2 D Discrete Signals ■ Fourier Transform of a 2 D discrete signal is defined as where ■ Inverse Fourier Transform Bahadir K. Gunturk EE 7730 - Image Analysis I 15
Fourier Transform: Properties ■ Periodicity: Fourier Transform of a discrete signal is periodic with period 1. 1 1 Arbitrary integers Bahadir K. Gunturk EE 7730 - Image Analysis I 16
Fourier Transform: Properties ■ Linearity, shifting, modulation, convolution, multiplication, separability, energy conservation properties also exist for the 2 D Fourier Transform of discrete signals. Bahadir K. Gunturk EE 7730 - Image Analysis I 17
Fourier Transform: Properties ■ Linearity ■ Shifting ■ Modulation ■ Convolution ■ Multiplication ■ Separable functions ■ Energy conservation Bahadir K. Gunturk EE 7730 - Image Analysis I 18
Fourier Transform: Properties ■ Define Kronecker delta function ■ Fourier Transform of the Kronecker delta function Bahadir K. Gunturk EE 7730 - Image Analysis I 19
Fourier Transform: Properties ■ Fourier Transform of 1 To prove: Take the inverse Fourier Transform of the Dirac delta function and use the fact that the Fourier Transform has to be periodic with period 1. Bahadir K. Gunturk EE 7730 - Image Analysis I 20
Impulse Train ■ Define a comb function (impulse train) as follows where M and N are integers Bahadir K. Gunturk EE 7730 - Image Analysis I 21
Impulse Train n Fourier Transform of an impulse train is also an impulse train: Bahadir K. Gunturk EE 7730 - Image Analysis I 22
Impulse Train Bahadir K. Gunturk EE 7730 - Image Analysis I 23
Impulse Train n In the case of continuous signals: Bahadir K. Gunturk EE 7730 - Image Analysis I 24
Impulse Train Bahadir K. Gunturk EE 7730 - Image Analysis I 25
Sampling Bahadir K. Gunturk EE 7730 - Image Analysis I 26
Sampling No aliasing if Bahadir K. Gunturk EE 7730 - Image Analysis I 27
Sampling If there is no aliasing, the original signal can be recovered from its samples by low-pass filtering. Bahadir K. Gunturk EE 7730 - Image Analysis I 28
Sampling Aliased Bahadir K. Gunturk EE 7730 - Image Analysis I 29
Sampling Anti-aliasing filter Bahadir K. Gunturk EE 7730 - Image Analysis I 30
Sampling ■ Without anti-aliasing filter: ■ With anti-aliasing filter: Bahadir K. Gunturk EE 7730 - Image Analysis I 31
Anti-Aliasing a=imread(‘barbara. tif’); Bahadir K. Gunturk EE 7730 - Image Analysis I 32
Anti-Aliasing a=imread(‘barbara. tif’); b=imresize(a, 0. 25); c=imresize(b, 4); Bahadir K. Gunturk EE 7730 - Image Analysis I 33
Anti-Aliasing a=imread(‘barbara. tif’); b=imresize(a, 0. 25); c=imresize(b, 4); H=zeros(512, 512); H(256 -64: 256+64, 256 -64: 256+64)=1; Da=fft 2(a); Da=fftshift(Da); Dd=Da. *H; Dd=fftshift(Dd); d=real(ifft 2(Dd)); Bahadir K. Gunturk EE 7730 - Image Analysis I 34
Sampling Bahadir K. Gunturk EE 7730 - Image Analysis I 35
Sampling No aliasing if Bahadir K. Gunturk and EE 7730 - Image Analysis I 36
Interpolation Ideal reconstruction filter: Bahadir K. Gunturk EE 7730 - Image Analysis I 37
Ideal Reconstruction Filter Bahadir K. Gunturk EE 7730 - Image Analysis I 38
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