Frequency Decomposition The base frequency or the fundamental - PowerPoint PPT Presentation

IIT Bombay Slide 21 Frequency Decomposition The base frequency or the fundamental frequency is the lowest frequency. All multiples of the fundamental frequency are known as harmonics . A given signal can be constructed back from its frequency decomposition by a weighted addition of the fundamental frequency and all the harmonic frequencies GNR607 Lecture 18-19 B. Krishna Mohan

IIT Bombay Slide 22 Click to edit Master text styles Input signal Second level ● Third level Sinusoid 1 ● Fourth level ● Fifth level Sinusoid 2 Sinusoid 3 Sinusoid 4 Reconstructed signals Image reconstruction by weighted summation of sinusiodal functions GNR607 Lecture 18-19 B. Krishna Mohan

IIT Bombay Slide 23 Different forms of Fourier Transform • Continuous Fourier Transform 1 +∞ ∫ − π = j 2 ux F u ( ) f x e ( ) dx π 2 −∞ +∞ ∑ = + π + π f x ( ) a a cos(2 nx ) b sin(2 nx ) • Fourier Series 0 n n =−∞ n where 1 π ∫ = π a f x ( )cos(2 nx dx ) n π 2 − π 1 π ∫ = π b f x ( )sin(2 nx dx ) n π 2 − π GNR607 Lecture 18-19 B. Krishna Mohan

IIT Bombay Slide 24 Continuous Fourier Transform • In the continuous domain, the basis functions of the Fourier transform are the complex exponentials e-j2 π ux • These functions extend from - ∞ to + ∞ • These are continuous functions, and exist everywhere Please recall that e-j2 π ux = cos(2 π ux) – j.sin(2 π ux) (according to Euler’s identity ) GNR607 Lecture 18-19 B. Krishna Mohan

IIT Bombay Slide 25 Real and Imaginary Parts of Fourier Transform 1 +∞ ∫ − π = j 2 ux F u ( ) f x e ( ) dx π 2 −∞ 1 1 +∞ +∞ ∫ ∫ = π − π F u ( ) f x ( )cos(2 ux dx ) j f x ( )sin(2 ux dx ) π π 2 2 −∞ −∞ Real part Imaginary Part GNR607 Lecture 18-19 B. Krishna Mohan

IIT Bombay Slide 26 Fourier Series • Fourier series is computed for periodic functions, where the coefficients of the sinusoids are computed over one period of the function. • It is assumed that the function has finite energy within one period so that the integral can be computed GNR607 Lecture 18-19 B. Krishna Mohan

IIT Bombay Slide 27 f(x) 1 T /2 ∫ − π 1 = j 2 ux F u ( ) ( ) f x e ( ) dx T − T /2 − π j 2 ux 1 1 e T /2 ∫ = − π = ∈ − j 2 ux ( ) 1. e dx ( ) , x { / 2, T T / 2} − π T T j 2 u − T /2 x  − π − π −  j 2 uT /2 j 2 u ( T /2) 1 e e 1   = − = π − − π j 2 uT j 2 uT ( ) e e -T/2 T/2     − π − π π T j 2 u j 2 u j 2 uT   |F(u)| 1 [ ] = π + π − π + π cos uT j sin uT cos uT j sin uT π j 2 uT π sin uT = = π s inc( uT ) π uT 1-D FT u GNR607 Lecture 18-19 B. Krishna Mohan u

IIT Bombay Slide 27a 1-D FT GNR607 Lecture 18-19 B. Krishna Mohan u

IIT Bombay Slide 28 Input Image GNR607 Lecture 18-19 B. Krishna Mohan

IIT Bombay Slide 29 Fourier Transform Magnitude GNR607 Lecture 18-19 B. Krishna Mohan

IIT Bombay Slide 30 Observations Points to be noted  The basis functions are continuous and extend all over the input domain  An abrupt cut-off in one domain leads to infinite extent of the transformed function, with the waves or ripples decreasing in magnitude from zero-frequency  The width of the rectangle in one domain is inversely proportional to the spacing of the ripples in the other domain GNR607 Lecture 18-19 B. Krishna Mohan

IIT Bombay Slide 31 Discrete Fourier Transform • In the discrete domain, the Discrete Fourier Transform (DFT) is defined as − 1 N 1 ∑ − π = j 2 un N / F u ( ) f n e ( ) N = n 0 • u is the frequency variable. weights e-j2 π ux • F(u) one term of Fourier Transform • The input sequence finite length N • u=0 zero-frequency term, and u=N-1 highest frequency GNR607 Lecture 18-19 B. Krishna Mohan

IIT Bombay Slide 32 Inverse Fourier Transform • Inverse Fourier transform is given by = ∫ ∞ π j 2 ux f x ( ) F u e ( ) du −∞ • function f(x) has contributions from ALL frequencies F(u). • If F(u) = 0 for u ≥T , then the function f(x) is said to be band-limited . GNR607 Lecture 18-19 B. Krishna Mohan

IIT Bombay Slide 33 Inverse Discrete Fourier Transform • Inverse discrete Fourier transform (IDFT) is given by − N 1 = ∑ + π j 2 un N / f n ( ) F u e ( ) = n 0 • Each element in the input domain is a weighted combination of ALL elements of the frequency domain GNR607 Lecture 18-19 B. Krishna Mohan

IIT Bombay Slide 34 Real and Imaginary Parts of Fourier Transform 1 +∞ ∫ − π = j 2 ux F u ( ) f x e ( ) dx π 2 −∞ 1 +∞ ∫ = π − F u ( ) f x ( )cos(2 ux dx ) π 2 −∞ 1 +∞ ∫ π j f x ( )sin(2 ux dx ) π 2 −∞ GNR607 Lecture 18-19 B. Krishna Mohan

IIT Bombay Slide 35 Magnitude and Phase of Fourier Transform • Given the real and imaginary parts of the Fourier transform of f(x), we can write • F(u) = FR(u) + j FI(u) • Fourier magnitude = [FR(u)2 + FI(u)2]0.5 • Fourier phase = Arctan[FI(u)/FR(u)] GNR607 Lecture 18-19 B. Krishna Mohan

IIT Bombay Slide 36 Fourier Magnitude and Phase • Fourier magnitude denotes strength of Fourier components • Fourier phase controls the relative positioning of features • If parts of the image are positionally interchanged, the Fourier magnitude remains unchanged, while the phase will be different • Fourier phase is more important for recon-structing the input image from the transformed version This is an important property to be noted GNR607 Lecture 18-19 B. Krishna Mohan

IIT Bombay Slide 37 Two-dimensional Fourier Transform • In case of two-dimensional data (e.g., images), the 2-D Fourier transform is defined by 1 ∞ ∫ − π + = j 2 ( ux vy ) F u v ( , ) f x y e ( , ) dxdy π 2 −∞ • The basis functions in the 2-D Fourier transform are defined by e-j2 π (ux+vy) • For each pair of (u,v), we have a basis image generated from different values of (x,y). GNR607 Lecture 18-19 B. Krishna Mohan

IIT Bombay Slide 37a Basis Images Basis images or basis matrices are two-dimensional (2-D) versions of basis vectors The process of transforming an image from the spatial domain into another domain, or mathematical space, amounts to projecting the image onto the basis images The transform coefficient is obtained by taking the inner product of the image with the basis image Frequency transforms can be applied to the entire image or smaller blocks GNR607 Lecture 18-19 B. Krishna Mohan

IIT Bombay Slide 38 2-D Discrete Fourier Transform • The 2-D DFT is defined by π   um vn − − 1 M 1 N 1 − + j 2 ∑∑  ÷ =  M N  F u v ( , ) f m n e ( , ) MN = = m 0 n 0 Basis • The inverse 2-D DFT is defined by image π   um vn − − M 1 N 1 = ∑∑ + + j 2  ÷   M N f m n ( , ) F u v e ( , ) = = m 0 n 0 GNR607 Lecture 18-19 B. Krishna Mohan

IIT Bombay Slide 38a Image Transform • Input Image Basis Images … GNR607 Lecture 18-19 B. Krishna Mohan