Transform Fourier (Chapter 4)
CS474/674 Fall 2008 Prof. Bebis
Mathematical Background: Complex Numbers
A complex number x has the form:
a: real part, b: imaginary part Addition
Multiplication
Mathematical Background: Complex Numbers (cont'd)
Magnitude-Phase (i.e.,vector) representation
Magnitude: Phase:
Mathematical Background: Complex Numbers (cont'd)
Multiplication using magnitude-phase representation
Complex conjugate
Properties
Mathematical Background: Complex Numbers (cont'd)
Euler's formula Properties
j
Mathematical Background: Sine and Cosine Functions
Periodic functions General form of sine and cosine functions:
Mathematical Background: Sine and Cosine Functions
Special case: A=1, b=0, =1
Period equal to T=2 Frequency is defined as f=1/T
Mathematical Background: Sine and Cosine Functions (cont'd)
Shifting or translating the sine function by a const b
Cosine is a shifted sine function:
i.e., shift to the right and take reflection along x-axis
Mathematical Background: Sine and Cosine Functions (cont'd)
Changing the amplitude A
Mathematical Background: Sine and Cosine Functions (cont'd)
Changing the period 2/||
Replace t with t: y=sin(t)
=4 period 2/4=/2 shorter period higher frequency (i.e., oscillates faster)
Fourier Series
Theorem: Any periodic signal can be expressed as a weighted sum (infinite) of sine and cosine waves of varying frequency
is called the "fundamental frequency" and are the weights of the expansion
Fourier Series (cont'd)
Illustration
1 2 3
Continuous Fourier Transform (FT)
Transforms a signal (i.e., function) from the spatial domain to the frequency domain.
(IFT) where
Why is FT Useful?
Remove undesirable frequencies from a signal. Easier and faster to perform certain operations in the frequency domain than in the spatial domain.
Example: Removing undesirable frequencies
noisy signal frequencies
To remove certain frequencies, set their corresponding F(u) coefficients to zero!
remove high frequencies
reconstructed signal
How do frequencies show up in an image?
High frequencies correspond to quickly varying information (i.e., edges) Low frequencies correspond to slowly varying information (i.e., continuous surface)
Original Image
High-passed
Low passed
Steps in Frequency Filtering
1. Take the FT of f(x): 2. Remove undesired frequencies: 3. Convert back to a signal:
We'll talk more about this later .....
Definitions
F(u) is a complex function: of Magnitude FT (spectrum): Phase of FT: Magnitude-Phase representation: Power of f(x): P(u)=|F(u)|2=
Example: rectangular pulse
rect(x) function
sinc(x) function or sin(x)/x
Example: impulse or "delta" function
Definition of delta function: Properties:
Example: impulse or "delta" function (cont'd)
FT of delta function
Example: sine and cosine functions
FT of the sine function
jF(u)
sin(2u0x)
Example: sine and cosine functions (cont'd)
FT of the cosine function
cos(2u0x)
1/2
F(u)
Extending FT in 2D
Forward FT
Inverse FT
Example: 2D rectangle function
FT of 2D rectangle function
2D sinc()
Discrete Fourier Transform (DFT)
Discrete Fourier Transform (DFT) (cont'd)
Forward DFT
Inverse DFT
1/Nx
Example
Extending DFT to 2D
Assume that f(x,y) is M x N image. Forward DFT
Inverse DFT:
Extending DFT to 2D (cont'd)
Special case: f(x,y) is N x N image. Forward DFT
Inverse DFT What is the complexity of 2D DFT?
O(N2)
Visualizing DFT
Typically, we display |F(u,v)| The dynamic range of |F(u,v)| is typically very large Apply scaling: (c is const)
original image
before scaling
after scaling
FT Properties: (1) Separability
The 2D DFT can be computed using 1D transforms only:
Forward DFT:
Inverse DFT:
FT Properties: (1) Separability (cont'd)
Rewrite F(u,v) as follows:
Let's set:
Then:
FT Properties: (1) Separability (cont'd)
)
FT Properties: (2) Periodicity and Symmetry
The DFT and its inverse are per...