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### Transcript of Fourier Transform (Chapter 4) CS474/674 – Prof. Bebis

• Slide 1
• Fourier Transform (Chapter 4) CS474/674 Prof. Bebis
• Slide 2
• Mathematical Background: Complex Numbers A complex number x is of the form: : real part, b: imaginary part Addition: Multiplication:
• Slide 3
• Mathematical Background: Complex Numbers (contd) Magnitude-Phase (i.e.,vector) representation Magnitude: Phase: Magnitude-Phase notation:
• Slide 4
• Mathematical Background: Complex Numbers (contd) Multiplication using magnitude-phase representation Complex conjugate Properties
• Slide 5
• Mathematical Background: Complex Numbers (contd) Eulers formula Properties j
• Slide 6
• Mathematical Background: Sine and Cosine Functions Periodic functions General form of sine and cosine functions:
• Slide 7
• Mathematical Background: Sine and Cosine Functions Special case: A=1, b=0, =1 /2 3/2
• Slide 8
• Mathematical Background: Sine and Cosine Functions (contd) Note: cosine is a shifted sine function: Shifting or translating the sine function by a const b
• Slide 9
• Mathematical Background: Sine and Cosine Functions (contd) Changing the amplitude A
• Slide 10
• Mathematical Background: Sine and Cosine Functions (contd) Changing the period T=2/|| consider A=1, b=0: y=cos(t) period 2/4=/2 shorter period higher frequency (i.e., oscillates faster) =4 Frequency is defined as f=1/T Alternative notation: cos(t)=cos(2t/T)=cos(2ft)
• Slide 11
• Basis Functions Given a vector space of functions, S, then if any f(t) S can be expressed as the set of functions k (t) are called the expansion set of S. If the expansion is unique, the set k (t) is a basis.
• Slide 12
• Image Transforms Many times, image processing tasks are best performed in a domain other than the spatial domain. Key steps: (1) Transform the image (2) Carry the task(s) in the transformed domain. (3) Apply inverse transform to return to the spatial domain.
• Slide 13
• Transformation Kernels Forward Transformation Inverse Transformation inverse transformation kernel forward transformation kernel
• Slide 14
• Kernel Properties A kernel is said to be separable if: A kernel is said to be symmetric if:
• Slide 15
• Notation Continuous Fourier Transform (FT) Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT)
• Slide 16
• Fourier Series Theorem Any periodic function f(t) can be expressed as a weighted sum (infinite) of sine and cosine functions of varying frequency: is called the fundamental frequency
• Slide 17
• Fourier Series (contd) 11 22 33
• Slide 18
• Continuous Fourier Transform (FT) Transforms a signal (i.e., function) from the spatial (x) domain to the frequency (u) domain. where
• Slide 19
• Why is FT Useful? Easier to remove undesirable frequencies. Faster perform certain operations in the frequency domain than in the spatial domain.
• Slide 20
• Example: Removing undesirable frequencies remove high frequencies reconstructed signal frequencies noisy signal To remove certain frequencies, set their corresponding F(u) coefficients to zero!
• Slide 21
• How do frequencies show up in an image? Low frequencies correspond to slowly varying information (e.g., continuous surface). High frequencies correspond to quickly varying information (e.g., edges) Original Image Low-passed
• Slide 22
• Example of noise reduction using FT Input image Output image Spectrum Band-pass filter
• Slide 23
• Frequency Filtering Steps 1. Take the FT of f(x): 2. Remove undesired frequencies: 3. Convert back to a signal: Well talk more about these steps later.....
• Slide 24
• Definitions F(u) is a complex function: Magnitude of FT (spectrum): Phase of FT: Magnitude-Phase representation: Power of f(x): P(u)=|F(u)| 2 =
• Slide 25
• Example: rectangular pulse rect(x) functionsinc(x)=sin(x)/x magnitude
• Slide 26
• Example: impulse or delta function Definition of delta function: Properties:
• Slide 27
• Example: impulse or delta function (contd) FT of delta function: 1 u x
• Slide 28
• Example: spatial/frequency shifts Special Cases:
• Slide 29
• Example: sine and cosine functions FT of the cosine function cos(2u 0 x) 1/2 F(u)
• Slide 30
• Example: sine and cosine functions (contd) FT of the sine function sin(2u 0 x) -jF(u)
• Slide 31
• Extending FT in 2D Forward FT Inverse FT
• Slide 32
• Example: 2D rectangle function FT of 2D rectangle function 2D sinc()
• Slide 33
• Discrete Fourier Transform (DFT)
• Slide 34
• Discrete Fourier Transform (DFT) (contd) Forward DFT Inverse DFT 1/Nx
• Slide 35
• Example
• Slide 36
• Extending DFT to 2D Assume that f(x,y) is M x N. Forward DFT Inverse DFT:
• Slide 37
• Extending DFT to 2D (contd) Special case: f(x,y) is N x N. Forward DFT Inverse DFT u,v = 0,1,2, , N-1 x,y = 0,1,2, , N-1
• Slide 38
• Extending DFT to 2D (contd) 2D cos/sin functions
• Slide 39
• Visualizing DFT Typically, we visualize |F(u,v)| The dynamic range of |F(u,v)| is typically very large Apply streching: (c is const) before stretchingafter stretching original image |F(u,v)| |D(u,v)|
• Slide 40
• DFT Properties: (1) Separability The 2D DFT can be computed using 1D transforms only: Forward DFT: kernel is separable:
• Slide 41
• DFT Properties: (1) Separability (contd) Rewrite F(u,v) as follows: Lets set: Then:
• Slide 42
• How can we compute F(x,v)? How can we compute F(u,v)? DFT Properties: (1) Separability (contd) ) N x DFT of rows of f(x,y) DFT of cols of F(x,v)
• Slide 43
• DFT Properties: (1) Separability (contd)
• Slide 44
• DFT Properties: (2) Periodicity The DFT and its inverse are periodic with period N
• Slide 45
• Symmetry Properties
• Slide 46
• DFT Properties: (4) Translation f(x,y) F(u,v) ) N Translation in spatial domain: Translation in frequency domain:
• Slide 47
• DFT Properties: (4) Translation (contd) Warning: to show a full period, we need to translate the origin of the transform at u=N/2 (or at (N/2,N/2) in 2D) |F(u-N/2)| |F(u)|
• Slide 48
• DFT Properties: (4) Translation (contd) To move F(u,v) at (N/2, N/2), take ) N ) N
• Slide 49
• DFT Properties: (4) Translation (contd) no translation after translation
• Slide 50
• DFT Properties: (5) Rotation Rotating f(x,y) by rotates F(u,v) by
• Slide 51
• DFT Properties: (6) Addition/Multiplication but
• Slide 52
• DFT Properties: (7) Scale
• Slide 53
• DFT Properties: (8) Average value So: Average: F(u,v) at u=0, v=0:
• Slide 54
• Magnitude and Phase of DFT What is more important? Hint: use the inverse DFT to reconstruct the input image using magnitude or phase only information magnitude phase
• Slide 55
• Magnitude and Phase of DFT (contd) Reconstructed image using magnitude only (i.e., magnitude determines the strength of each component!) Reconstructed image using phase only (i.e., phase determines the phase of each component!)
• Slide 56
• Magnitude and Phase of DFT (contd)