Digital Image Analysis and Processing CPE 0907544 Filtering in the - PowerPoint PPT Presentation

Digital Image Analysis and Processing CPE 0907544 Filtering in the Frequency Domain – Part I Introduction Chapter 4 Sections : 4.1-4.6 Dr. Iyad Jafar

Outline  Background  Preliminary Concepts  Sampling  The 1D Discrete Fourier Transform  The 2D Discrete Fourier Transform  Essential Properties of the 2D Fourier Transform 2

Background  In 1807, the French mathematician Jean Fourier proposed that any periodic function that satisfies some mild mathematical conditions can be expressed as the sum of sines and/or cosines of different frequencies and amplitudes  Now, we know this as Fourier series  Even aperiodic functions that have finite energy can be represented as an integral of sines and/or cosines multiplied weighting function. This formulation is known as Fourier transform  Both formulations have an important feature  A function represented in Fourier series or transform can be fully reconstructed into its original form without any loss of information 3

Background  The use of Fourier analysis was in the field of heat diffusion.  The usefulness of Fourier transform is of greater importance and applicability in many fields  The availability of digital computers and the discovery of the Fast Fourier Transform (FFT) revolutionized the field of signal processing  Our focus in this course will be on Fourier Transform as we will be working with images of finite duration (energy).  Specifically, we will use Fourier transform as a tool to study filtering in the frequency domain 4

Preliminaries  Complex Numbers  A complex number C is represented by C = R + jI Real Imaginary  Complex numbers can be viewed as a point in the complex plane  Representation of complex number in polar coordinates C = |C|e j θ = |C| (cos θ + j sin θ ) where |C| = (R 2 +I 2 ) 1/2 Magnitude θ = tan -1 (I/R) Phase 5

Preliminaries  Impulses  A unit impulse of continuous variable located at t=t 0 is defined as     , t t 0    δ(t t )  0  0 , t t   0 constrained to    δ(t t )dt = 1 0   Sifting property    f (t )δ(t t )dt = f(t ) 0 0  6

Preliminaries  Impulses  In the discrete domain    1 , t t 0    δ(t t )  0  0 , t t   0 constrained to    δ(t t ) = 1 0  and sifting property becomes    f (t )δ(t t ) = f(t ) 0 0  7

Preliminaries  Impulses  Impulse Train  A set of infinite, periodic impulses that are ∆T apart     s (t ) = δ(t n T )  T  n 8

Preliminaries  The Fourier Series of Periodic Functions  The Fourier series of a continuous periodic function f(t) with period T is  2 πn  j t T f (t ) = c e n  n where T / 2 2 πn  1 j t    T c = f (t ) e dt , n = 0, 1, 2,... n T  T / 2 9

Preliminaries  The Fourier Transform of Functions of One ContinuousVariable  The Fourier transform of a continuous function f(t) of a continuous variable t      j πμt 2  f (t ) = F(μ) = f (t ) e dt   We can reconstruct f(t) back using      1 j πμt 2  F( μ) = f (t ) = F( μ) e dμ  10

Preliminaries  The Fourier Transform of One Continuous Variable  Example 4.1. Compute the Fourier transform of the function f(t) shown below 11

Preliminaries  The Fourier Transform of One Continuous Variable  Example 4.1 – continued sin(πμW ) F(μ) = AW = AW sinc(πμW ) πμW  Usually we work with the magnitude of F(u) sin(πμW ) F(μ) = AW = AW sinc(πμW ) πμW 12

Preliminaries  The Fourier Transform of One Continuous Variable  Example 4.2. Compute the Fourier transform of the unit impulse located at t=0        δ(t )  j πμt 2   δ(t ) = F(μ) = δ(t ) e dt 1  13

Preliminaries  The Fourier Transform of One Continuous Variable  Example 4.3. Compute the Fourier transform of the unit impulse located at t=t0     δ(t t ) 0       j π μ t 2 j π μ t 2     0 δ(t t ) = F(μ) = δ(t t ) e dt e 0 0   j π au 2   δ(t a ) e 14

Preliminaries  The Fourier Transform of One Continuous Variable  Example 4.4. Compute the inverse Fourier transform of the following function  F( μ) = δ( μ a)      1 j π μ t 2 j π a t 2   F( μ) = δ( μ a ) e dμ = e   j π a t 2  e δ( μ  a) 15

Preliminaries  The Fourier Transform of One Continuous Variable  Example 4.5. Compute the Fourier transform of sin(2 π at) and cos(2 π at) 1       sin(2 πat) δ(μ a ) δ(μ a ) 2 j 1       cos(2 πat) δ(μ a ) δ(μ a ) 2 16

Preliminaries  The Fourier Transform of One Continuous Variable  Example 4.6. Compute the Fourier transform of the impulse train with period ∆T     s (t ) = δ(t n T )  T  n   1 n     S (t ) = S(μ) = δ( μ )  T   T T  n 17

Preliminaries  ConvolutionTheorem  The convolution of two continuous functions f(t) and h(t) where t is a continuous variable is given by    f (t) h(t) = f (τ ) h(t-τ )dτ   It can be easily shown that convolution in the spatial/time domain is equivalent to multiplying the Fourier transforms of the two functions in the frequency domain   f(t) h(t) F(μ) H(μ) 18

Preliminaries  ConvolutionTheorem – cont’d  Similarly, convolving two functions in the frequency domain is defined as    F(μ) H(μ) = F(τ ) H(μ-τ )dτ   It can be easily shown that convolution in the frequency domain is equivalent to multiplying the two functions in the original domain   f(t) h(t) F(μ) H(μ) 19

Sampling  Sampling is required to convert continuous signals into discrete form  Collecting samples of a signal/function f(t) that are spaced by ∆T can be viewed as multiplying the signal by an impulse train s ∆T (t) with period ∆T ~  f(t ) f(t ) s (t ) Δ T   x  = f(t )δ(t n T ) Δ  n 20

Sampling  The value of each sample f k can be computed by integration    f f ( t ) δ(t-n T)dt = f (k T ) Δ Δ k  21

Sampling  The Fourier Transform of Sampled Function  Let F(u) denotes the Fourier transform of f(t), then according to convolution theorem, the Fourier transform is the convolution between F(u) and the Fourier transform of s ∆T (t)   ~ ~        F( μ ) f ( t ) f ( t )s ( t ) = F( μ ) S( μ )   Δ T   where   1 n     s (t ) = S(μ) = δ( μ )  T   T T  n 22

Sampling  The Fourier Transform of Sampled Function  Carrying out convolution in frequency domain ~   F( μ ) F( μ ) S( μ )     F( τ )S( μ τ )dτ     1 n     = F( τ ) δ( μ τ )dτ Δ T T  n      1 n    = F( τ )δ( μ τ )dτ Δ T T   n   1 n   = F( μ ) Δ T T 23  n

Sampling  The Fourier Transform of Sampled Function  The previous formulation implies that the Fourier transform of a sampled function is simply an infinite, periodic sequence of copies of F(u)that are centered at multiples of 1/∆T Sampling 24

Sampling  The Sampling Theorem  If f(t) represents a function whose Fourier transform F(u) is band-limited, i.e. F(u) = 0 , u >= |u max |, then the original spectrum can be recovered from the sampled spectrum if the sampling rate is at least twice the maximum frequency in F(u) 1  2μ max  T  We have three different cases  Under-sampling (1/∆T < 2 u max )  Critically sampling; sampling at Nyquest rate (1/∆T = 2 u max )  Over-sampling (1/∆T > 2 u max ) 25

Sampling  The Sampling Theorem Over-Sampling Critical-Sampling Under-Sampling 26

Sampling  Recovering f(t) from its SampledVersion  To recover f(t) we can multiply the sampled spectrum with a function    T , -μ Δ μ μ  max max   H( μ)  0 , otherwise    Accordingly ~  F(u ) F( μ)H( μ)  And f(t) can be computed using the inverse Fourier transform      1 j πμt 2  f (t ) = F( μ) = F( μ) e dμ  27

Sampling  Recovering f(t) from its Sampled Version 28

Sampling  Aliasing  When we sample with a rate less than the Nyquest rate , the replicas of the function transform will overlap. This affects the recovery of the original function. 29