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Lecture 14 Properties of Fourier Transform for 2D Signal
- 1. Properties of Fourier Transform for 2D Signal Subject: Image Procesing & Computer Vision Dr. Varun Kumar Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 1 / 13
- 2. Outlines 1 Properties of DFT Periodicity and Conjugate Rotation Distributivity and Scaling Convolution and Correlation 2 Fast Fourier transform 3 References Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 2 / 13
- 3. Properties of Discrete Fourier Transform 3 Periodicity Properties Let f (x, y) is a 2D image signal and its Fourier transform is F(u, v), where the number of discrete sample (N) remain same in x and y direction, then F(u, v) = F(u + N, v) = F(u, v + N) = F(u + N, v + N) (1) Proof F(u, v) = 1 N N−1 x=0 N−1 y=0 f (x, y)e−j 2π N (ux+vy) F(u + N, v) = 1 N N−1 x=0 N−1 y=0 f (x, y)e−j 2π N ((u+N)x+vy) = 1 N N−1 x=0 N−1 y=0 f (x, y)e−j 2π N ((ux+vy) e−j2πx ⇒ x ∈ Integer ⇒ e−j2πx = 1 Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 3 / 13
- 4. Continued– 4. Conjugate Properties Let f (x, y) is a 2D image signal and its Fourier transform is F(u, v), where the number of discrete sample (N) remain same in x and y direction, then F∗ (u, v) = F(−u, −v) (2) Proof Since f (x, y) is a real valued signal. Now F∗ (u, v) = 1 N N−1 x=0 N−1 y=0 f (x, y)e−j 2π N (ux+vy) ∗ = 1 N N−1 x=0 N−1 y=0 f (x, y)ej 2π N (ux+vy) = 1 N N−1 x=0 N−1 y=0 f (x, y)e−j 2π N (−ux−vy) = F(−u, −v) Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 4 / 13
- 5. Continued– 5 Rotation Propety: Cartesian to polar conversion 1 x = r cos θ and y = r sin θ ⇒ f (x, y) ⇒ f (r, θ) 2 u = w cos φ and v = w sin φ ⇒ F(u, v) ⇒ F(w, φ) As per the rotation property f (r, θ + θ0) ⇐⇒ F(w, φ + θ0) (3) Proof F(w, φ) = 1 N rN−1 r=r0 θN θ1 f (r, θ)e−j 2π N (rw(cos θ cos φ+sin θ sin φ=cos(θ−φ))) F = 1 N rN−1 r=r0 θN +θ0 θ=θ1+θ0 f (r, θ + θ0)e−j 2π N (rw(cos(θ+θ0−φ))) = F(w, φ + θ0) Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 5 / 13
- 6. Continued– 6 Distributive property If two 2D signals are f1(x, y) and f2(x, y) and their Fourier transforms are F1(u, v) and F2(u, v), then F a1f1(x, y) + a2f2(x, y) = a1F f1(x, y) + a2F f2(x, y) = a1F1(u, v) + a2F2(u, v) (4) Proof F = 1 N N−1 x=0 N−1 y=0 a1f1(x, y) + a2f2(x, y) e−j 2π N (ux+vy) = a1 N N−1 x=0 N−1 y=0 f1(x, y)ej 2π N (ux+vy) + a2 N N−1 x=0 N−1 y=0 f2(x, y)e−j 2π N (ux+vy) = a1F1(u, v) + a2F2(u, v) Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 6 / 13
- 7. Continued– 7 Scaling property If a 2D signal is f (x, y) and its Fourier transforms is F(u, v). For a new signal f (ax, by) the Fourier transform is as f (ax, by) F −→ 1 |ab| F u a , v b (5) Proof F = 1 N N−1 x=0 N−1 y=0 f (ax, by)e−j 2π N (ux+vy) let ax = z1 and by = z2 = 1 abN a(N−1) z1=0 b(N−1) z2=0 f (z1, z2)ej 2π N (u z1 a +v z2 b ) = 1 ab F u a , v b Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 7 / 13
- 8. Averaging of 2D signal Average Let f (x, y) is a 2D discrete signal and ˆf (x, y) is its average ˆf (x, y) = 1 N2 N−1 x=0 N−1 y=0 f (x, y) DC component of Fourier transform F(0, 0) = 1 N N−1 x=0 N−1 y=0 f (x, y) Relation between average of 2D signal and DC component of its Fourier transform ˆf (x, y) = 1 N F(0, 0) Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 8 / 13
- 9. Convolution/Correlation property Convolution ⇒ g1(x) ∗ g2(x) = G1(u)G2(u) Convolution of 1D space domain signal ⇒ g1(x)g2(x) = G1(u) ∗ G2(u) Multiplication of 1D signal ⇒ g1(x, y) ∗ g2(x, y) = G1(u, v)G2(u, v) Convolution of 2D signal ⇒ g1(x, y)g2(x, y) = G1(u, v) ∗ G2(u, v) Multiplication of 2D signal Correlation ⇒ g1(x, y) o g2(x, y) = G∗ 1 (u, v)G2(u, v) Convolution of 2D signal ⇒ g∗ 1 (x, y)g2(x, y) = G1(u, v) o G2(u, v) Multiplication of 2D signal Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 9 / 13
- 10. Fast Fourier transform FFT Computational observation 1D signal F(u) = 1 N N−1 x=0 f (x)e−j 2π N ux ⇒ N2 2D signal F(u, v) = 1 N N−1 x=0 N−1 y=0 f (x, y)e−j 2π N (ux+vy) ⇒ N4 Note: This massive amount of data processing is not fruitful for sending image in a band-limited channel. Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 10 / 13
- 11. Complexity reduction 1D signal Let N = 2n and N = 2M, where M = 2n−1 F(u) = 1 N N−1 x=0 f (x)e−j 2π N ux = 1 N N−1 x=0 f (x)W ux N ⇒ WN = e−j 2π N = 1 2M 2M−1 x=0 f (x)W ux 2M = 1 2 1 M M−1 x=0 f (2x)W u2x 2M + 1 M M−1 x=0 f (2x + 1)W u(2x+1) 2M = 1 2 1 M M−1 x=0 f (2x)W u2x 2M + 1 M M−1 x=0 f (2x + 1)W ux) M W u 2M = 1 2 Feven(u) + Fodd (u)W u 2M Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 11 / 13
- 12. Continued– Conclusion: In single step processing N2 computation order reduces to (N2 4 + N2 4 ) = N2 2 In similar fashion even and odd terms can be further reduced. At the end of computation order reduction, i.e, N2 → N log2(N) Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 12 / 13
- 13. References M. Sonka, V. Hlavac, and R. Boyle, Image processing, analysis, and machine vision. Cengage Learning, 2014. D. A. Forsyth and J. Ponce, “A modern approach,” Computer vision: a modern approach, vol. 17, pp. 21–48, 2003. L. Shapiro and G. Stockman, “Computer vision prentice hall,” Inc., New Jersey, 2001. R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital image processing using MATLAB. Pearson Education India, 2004. Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 13 / 13