EE D Fourier Transform

Bahadir K. Gunturk EE Image Analysis I 2 Summary of Lecture 2 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 Image Analysis I 3 Fourier Transform What is ahead?  1D Fourier Transform of continuous signals  2D Fourier Transform of continuous signals  2D Fourier Transform of discrete signals  2D Discrete Fourier Transform (DFT)

Bahadir K. Gunturk EE Image Analysis I 4 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 Image Analysis I 5 Fourier Transform 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. A complex number: z = x + j*y A complex exponential signal: r*exp(j*a) =r*cos(a) + j*r*sin(a)

Bahadir K. Gunturk EE Image Analysis I 6 Fourier Transform: 1D Cont. Signals ■ Fourier Transform of a 1D continuous signal ■ Inverse Fourier Transform “Euler’s formula”

Bahadir K. Gunturk EE Image Analysis I 7 Fourier Transform: 2D Cont. Signals ■ Fourier Transform of a 2D continuous signal ■ Inverse Fourier Transform ■ F and f are two different representations of the same signal.

Bahadir K. Gunturk EE Image Analysis I 8 Examples Magnitude: “how much” of each component Phase: “where” the frequency component in the image

Bahadir K. Gunturk EE Image Analysis I 9 Examples

Bahadir K. Gunturk EE Image Analysis I 10 Fourier Transform: Properties ■ Linearity ■ Shifting ■ Modulation ■ Convolution ■ Multiplication ■ Separable functions

Bahadir K. Gunturk EE Image Analysis I 11 Fourier Transform: Properties ■ Separability 2D Fourier Transform can be implemented as a sequence of 1D Fourier Transform operations.

Bahadir K. Gunturk EE Image Analysis I 12 Fourier Transform: Properties ■ Energy conservation

Bahadir K. Gunturk EE Image Analysis I 13 Fourier Transform: Properties ■ Remember the impulse function (Dirac delta function) definition ■ Fourier Transform of the impulse function

Bahadir K. Gunturk EE Image Analysis I 14 Fourier Transform: Properties ■ Fourier Transform of 1 Take the inverse Fourier Transform of the impulse function

Bahadir K. Gunturk EE Image Analysis I 15 Fourier Transform: 2D Discrete Signals ■ Fourier Transform of a 2D discrete signal is defined as where ■ Inverse Fourier Transform

Bahadir K. Gunturk EE Image Analysis I 16 Fourier Transform: Properties ■ Periodicity: Fourier Transform of a discrete signal is periodic with period Arbitrary integers

Bahadir K. Gunturk EE Image Analysis I 17 Fourier Transform: Properties ■ Linearity, shifting, modulation, convolution, multiplication, separability, energy conservation properties also exist for the 2D Fourier Transform of discrete signals.

Bahadir K. Gunturk EE Image Analysis I 18 Fourier Transform: Properties ■ Linearity ■ Shifting ■ Modulation ■ Convolution ■ Multiplication ■ Separable functions ■ Energy conservation

Bahadir K. Gunturk EE Image Analysis I 19 Fourier Transform: Properties ■ Define Kronecker delta function ■ Fourier Transform of the Kronecker delta function

Bahadir K. Gunturk EE Image Analysis I 20 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 Image Analysis I 21 Impulse Train ■ Define a comb function (impulse train) as follows where M and N are integers

Bahadir K. Gunturk EE Image Analysis I 22 Impulse Train Fourier Transform of an impulse train is also an impulse train:

Bahadir K. Gunturk EE Image Analysis I 23 Impulse Train

Bahadir K. Gunturk EE Image Analysis I 24 Impulse Train In the case of continuous signals:

Bahadir K. Gunturk EE Image Analysis I 25 Impulse Train

Bahadir K. Gunturk EE Image Analysis I 26 Sampling

Bahadir K. Gunturk EE Image Analysis I 27 Sampling No aliasing if

Bahadir K. Gunturk EE Image Analysis I 28 Sampling If there is no aliasing, the original signal can be recovered from its samples by low-pass filtering.

Bahadir K. Gunturk EE Image Analysis I 29 Sampling Aliased

Bahadir K. Gunturk EE Image Analysis I 30 Sampling Anti-aliasing filter

Bahadir K. Gunturk EE Image Analysis I 31 Sampling ■ Without anti-aliasing filter: ■ With anti-aliasing filter:

Bahadir K. Gunturk EE Image Analysis I 32 Anti-Aliasing a=imread(‘barbara.tif’);

Bahadir K. Gunturk EE Image Analysis I 33 Anti-Aliasing a=imread(‘barbara.tif’); b=imresize(a,0.25); c=imresize(b,4);

Bahadir K. Gunturk EE Image Analysis I 34 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)=1; Da=fft2(a); Da=fftshift(Da); Dd=Da.*H; Dd=fftshift(Dd); d=real(ifft2(Dd));

Bahadir K. Gunturk EE Image Analysis I 35 Sampling

Bahadir K. Gunturk EE Image Analysis I 36 Sampling No aliasing if and

Bahadir K. Gunturk EE Image Analysis I 37 Interpolation Ideal reconstruction filter:

Bahadir K. Gunturk EE Image Analysis I 38 Ideal Reconstruction Filter