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- 1. Transforms
- 2. A sine wave 8 5*sin (2π4t) 6 Amplitude = 5 4 Frequency = 4 Hz 2 0 -2 -4 -6 -8 0 0.1 0.2 0.3 0.4 0.5 seconds 0.6 0.7 0.8 0.9 1
- 3. A sine wave signal 8 5*sin(2π4t) 6 Amplitude = 5 4 Frequency = 4 Hz 2 Sampling rate = 256 samples/second 0 -2 Sampling duration = 1 second -4 -6 -8 0 0.1 0.2 0.3 0.4 0.5 seconds 0.6 0.7 0.8 0.9 1
- 4. An undersampled signal sin(2π8t), SR = 8.5 Hz 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
- 5. The Nyquist Frequency • The Nyquist frequency is equal to one-half of the sampling frequency. • The Nyquist frequency is the highest frequency that can be measured in a signal.
- 6. Fourier series • Periodic functions and signals may be expanded into a series of sine and cosine functions
- 7. The Fourier Transform • A transform takes one function (or signal) and turns it into another function (or signal)
- 8. The Fourier Transform • A transform takes one function (or signal) and turns it into another function (or signal) • Continuous Fourier Transform: close your eyes if you don’t like integrals
- 9. The Fourier Transform • A transform takes one function (or signal) and turns it into another function (or signal) • Continuous Fourier Transform: ∞ H ( f ) = ∫ h ( t ) e 2πift dt −∞ ∞ h ( t ) = ∫ H ( f ) e −2πift df −∞
- 10. The Fourier Transform • A transform takes one function (or signal) and turns it into another function (or signal) • The Discrete Fourier Transform: N −1 H n = ∑ hk e 2πikn N k =0 1 N −1 hk = ∑ H n e −2πikn N N n =0
- 11. Fast Fourier Transform • The Fast Fourier Transform (FFT) is a very efficient algorithm for performing a discrete Fourier transform • FFT principle first used by Gauss in 18?? • FFT algorithm published by Cooley & Tukey in 1965 • In 1969, the 2048 point analysis of a seismic trace took 13 ½ hours. Using the FFT, the same task on the same machine took 2.4 seconds!
- 12. Famous Fourier Transforms 2 1 Sine wave 0 -1 -2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 300 250 200 Delta function 150 100 50 0 0 20 40 60 80 100 120
- 13. Famous Fourier Transforms 0.5 0.4 Gaussian 0.3 0.2 0.1 0 0 5 10 15 20 25 30 35 40 45 50 6 5 4 Gaussian 3 2 1 0 0 50 100 150 200 250
- 14. Famous Fourier Transforms 1.5 1 Sinc function 0.5 0 -0.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 6 5 4 Square wave 3 2 1 0 -100 -50 0 50 100
- 15. Famous Fourier Transforms 1.5 1 Sinc function 0.5 0 -0.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 6 5 4 Square wave 3 2 1 0 -100 -50 0 50 100
- 16. Famous Fourier Transforms 1 0.8 Exponential 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 30 25 20 Lorentzian 15 10 5 0 0 50 100 150 200 250
- 17. FFT of FID 2 1 0 f = 8 Hz SR = 256 Hz T2 = 0.5 s -1 -2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 70 60 50 40 30 20 10 0 0 20 40 60 80 100 ( t ) = sin( 2πft ) exp − t F T 2 120
- 18. FFT of FID 2 f = 8 Hz SR = 256 Hz T2 = 0.1 s 1 0 -1 -2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 14 12 10 8 6 4 2 0 0 20 40 60 80 100 120
- 19. FFT of FID 2 1 0 -1 -2 f = 8 Hz SR = 256 Hz T2 = 2 s 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 200 150 100 50 0 0 20 40 60 80 100 120
- 20. Effect of changing sample rate 2 1 0 -1 -2 f = 8 Hz T2 = 0.5 s 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 70 35 60 30 50 25 40 20 30 15 20 10 10 5 0 0 10 20 30 40 50 60 0
- 21. Effect of changing sample rate 2 SR = 256 Hz SR = 128 Hz 1 0 -1 -2 f = 8 Hz T2 = 0.5 s 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 70 35 60 30 50 25 40 20 30 15 20 10 10 5 0 0 10 20 30 40 50 60 0
- 22. Effect of changing sample rate • Lowering the sample rate: – Reduces the Nyquist frequency, which – Reduces the maximum measurable frequency – Does not affect the frequency resolution
- 23. Effect of changing sampling duration 2 1 0 -1 -2 f = 8 Hz T2 = .5 s 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 14 16 18 20 70 60 50 40 30 20 10 0
- 24. Effect of changing sampling duration 2 1 ST = 2.0 s ST = 1.0 s 0 -1 -2 f = 8 Hz T2 = .5 s 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 14 16 18 20 70 60 50 40 30 20 10 0
- 25. Effect of changing sampling duration • Reducing the sampling duration: – Lowers the frequency resolution – Does not affect the range of frequencies you can measure
- 26. Effect of changing sampling duration 2 1 0 -1 -2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 200 150 100 50 0 f = 8 Hz T2 = 2.0 s 0 2 4 6 8 10 12 14 16 18 20
- 27. Effect of changing sampling duration 2 ST = 2.0 s ST = 1.0 s 1 0 f = 8 Hz T2 = 0.1 s -1 -2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 14 16 18 20 14 12 10 8 6 4 2 0
- 28. Measuring multiple frequencies 3 f = 80 Hz, T2 = 1 s 1 2 1 f = 90 Hz, T2 = .5 s 2 2 f = 100 Hz, T2 = 0.25 s 1 3 3 0 -1 -2 -3 SR = 256 Hz 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 120 100 80 60 40 20 0 0 20 40 60 80 100 120
- 29. Measuring multiple frequencies 3 f = 80 Hz, T2 = 1 s 1 2 1 f = 90 Hz, T2 = .5 s 2 2 f = 200 Hz, T2 = 0.25 s 1 3 3 0 -1 -2 -3 SR = 256 Hz 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 120 100 80 60 40 20 0 0 20 40 60 80 100 120
- 30. L: period; u and v are the number of cycles fitting into one horizontal and vertical period, respectively of f(x,y).
- 31. Discrete Fourier Transform 1 w−1 h −1 2π (ux + vy ) 2π (ux + vy ) F (u , v) = ∑∑ f ( x, y) cos wh ÷+ j sin wh ÷ wh x =0 y =0 1 w−1 h −1 − j 2π ( ux + vy ) / wh F (u , v) = ∑∑ f ( x, y)e wh x =0 y =0
- 32. Discrete Fourier Transform (DFT). • When applying the procedure to images, we must deal explicitly with the fact that an image is: – Two-dimensional – Sampled – Of finite extent • These consideration give rise to the The DFT of an NxN image can be written:
- 33. Discrete Fourier Transform • For any particular spatial frequency specified by u and v, evaluating equation 8.5 tell us how much of that particular frequency is present in the image. • There also exist an inverse Fourier Transform that convert a set of Fourier coefficients into an image. 1 f ( x, y ) = N N −1 N −1 F (u , v)e j 2π ( ux +vy ) / N ∑∑ x =0 y =0
- 34. 2 P (u , v) = F (u , v) = R (u , v) + I (u , v) 2 2 PSD • The magnitudes correspond to the amplitudes of the basic images in our Fourier representation. • The array of magnitudes is termed the amplitude spectrum (or sometime ‘spectrum’). • The array of phases is termed the phase spectrum. • The power spectrum is simply the square of its amplitude spectrum:
- 35. FFT • The Fast Fourier Transform is one of the most important algorithms ever developed – Developed by Cooley and Tukey in mid 60s. – Is a recursive procedure that uses some cool math tricks to combine sub-problem results into the overall solution.
- 36. DFT vs FFT
- 37. DFT vs FFT
- 38. DFT vs FFT
- 39. Periodicity assumption • The DFT assumes that an image is part of an infinitely repeated set of “tiles” in every direction. This is the same effect as “circular indexing”.
- 40. Periodicity and Windowing • Since “tiling” an image causes “fake” discontinuities, the spectrum includes “fake” highfrequency components Spatial discontinuities
- 41. Discrete Cosine Transform Real-valued π( 2i + 1) m π( 2k + 1) n Gc ( m, n) = α ( m)α ( n)∑ ∑ g( i, k ) cos cos 2 N 2N i= 0 k = 0 with an inverse N −1 N −1 π( 2i + 1) m π ( 2k + 1) n gc ( i, k ) = ∑ ∑ α ( m)α ( n)Gc ( m, n) cos cos 2 N 2N m= 0 n = 0 where N −1 N −1 α ( 0) = 1 N and α ( m) = 2 N for 1≤ m ≤ N
- 42. DCT in Matrix Form G c = CgC where the kernel elements are Ci , m π( 2i + 1) m = α( m) cos 2N
- 43. Discrete Sine Transform Most Convenient when N=2 p - 1 2 N −1 N −1 π( i + 1) ( m + 1) π( k + 1) ( n + 1) Gsin ( m, n) = g( i, k ) sin ∑∑ sin N + 1 i=0 k =0 N +1 N +1 with an inverse 2 N −1 N −1 π( i + 1) ( m + 1) π( k + 1) ( n + 1) gsin ( i, k ) = ∑0 ∑ Gs ( m, n) sin N + 1 sin N + 1 N + 1 m= n = 0
- 44. DST in Matrix Form G c = TgT where the kernel elements are Ti , k = 2 π( i + 1) ( k + 1) sin N +1 N +1
- 45. DCT Basis Functions*
- 46. (Log Magnitude) DCT Example*
- 47. Hartley Transform • Alternative to Fourier • Produces N Real Numbers • Use Cosine Shifted 45o to the Right cas( θ) = cos( θ) + sin( θ) π = 2 cos θ − 4
- 48. Square Hartley Transform 2π ( im + kn ) 1 N −1 N −1 GHartley ( m, n ) = ∑0 ∑= 0 g H ( i, k ) cas N N * N i= k with an inverse 2π ( im + kn ) g Hartley ( i, k ) = ∑ ∑ GH ( m, n ) cas N m= 0 n= 0 N −1 N −1
- 49. Rectangular Hartley Transform 1 2π mx 2π ny GHartley ( m, n ) = ∑ ∑ g H ( x, y ) cas + wh y = 0 x = 0 w h h − 1 w− 1 m ∈ [ 0..h ] , n ∈ [ 0..w] with an inverse 2π mx 2π ny g Hartley ( m, n ) = ∑ ∑ GH ( x, y ) cas + w h y= 0 x= 0 h − 1 w− 1
- 50. Hartley in Matrix Form G Hartley = TgT where the kernel elements are Ti , k 1 2πik = cas N N
- 51. What is an even function? • the function f is even if the following equation holds for all x in the domain of f:
- 52. Hartley Convolution Theorem • Computational Alternative to Fourier Transform • If One Function is Even, Convolution in one Domain is Multiplication in Hartley Domain g( x ) = f ( x )* h( x ) ⇔ G( ν) = F ( ν) H even ( ν) + F ( − ν) H odd ( ν)
- 53. Rectangular Wave Transforms • Binary Valued {1, -1} • Fast to Compute • Examples – – – – Hadamard Walsh Slant Haar
- 54. Hadamard Transform • Consists of elements of +/- 1 • A Normalized N x N Hadamard matrix satisfies the relation H Ht = I 1 1 1 H2 = 2 1 −1 Walsh Tx can be constructed as H 2N 1 H N = 2 H N HN −H N
- 55. Walsh Transform, N=4 Gonzalez, Wintz *
- 56. Non-ordered Hadamard Transform H8 +1 +1 +1 +1 H8 = +1 +1 +1 +1 +1 −1 +1 −1 +1 −1 +1 −1 +1 +1 −1 −1 +1 +1 −1 −1 +1 −1 −1 +1 +1 −1 −1 +1 +1 +1 +1 +1 −1 −1 −1 −1 +1 −1 +1 −1 −1 +1 −1 +1 +1 +1 −1 −1 −1 −1 +1 +1 +1 −1 −1 +1 −1 +1 +1 −1
- 57. Sequency • In a Hadamard Transform, the Number of Sign Changes in a Row Divided by Two • It is Possible to Construct an H matrix with Increasing Sequency per row
- 58. Ordered Hadamard Transform 1 F ( u, v ) = N N −1 N −1 ∑ ∑ F ( j, k )( −1) q ( j , k , u, v ) j =0 k =0 where N −1 q( j, k , u, v ) = ∑ [ g i ( u) ji + g i ( v ) k i ] i=0 and g 0 (u ) ≡ un −1 g1 (u) ≡ un −1 + un − 2 g 2 ( u ) ≡ un − 2 + un − 3 g n-1 (u ) ≡ u1 + u0
- 59. Ordered Hadamard Transform* Gonzalez, Wintz *
- 60. Haar Transform • Derived from Haar Matrix • Sampling Process in which Subsequent Rows Sample the Input Data with Increasing Resolution • Different Types of Differential Energy Concentrated in Different Regions – Power taken two at a time – Power taken a power of two at a time, etc.
- 61. 1 1 2 0 H4 = * 1 1 2 0 Castleman Haar Transform*, H4 1 1 −1 −1 0 0 2 − 2
- 62. Karhunen-Loeve Transform • Variously called the K-L, Hotelling, or Eignevector • Continuous Form Developed by K-L • Discrete Version Credited to Hotelling • Transforms a Signal into a Set of Uncorrelated Representational Coefficients • Keep Largest Coefficients for Image Compression
- 63. Discrete K-L N −1 N −1 F (u, v ) = ∑ ∑ F ( j, k ) A( j, k ; u, v ) j =0 k =0 where the kernel satisfies N −1 N −1 λ( u, v ) A( j, k ; u, v ) = ∑ ∑ K F ( j, k ; j ′, k ′) A( j ′, k ′; u, v ) j =0 k =0 where K F ( j, k ; j ′, k ′ ) is the image covariance function λ( u, v ) is a constant for a fixed ( u, v ) , the eigenvalues of K F
- 64. Singular Value Decomposition An NxN matrix A can be expressed as A = UΛV t where Columns of U are Eigenvectors of AA t Columns of V are Eigenvectors of A t A L is the NxN diagonal matrix of singular values Λ = U t AV
- 65. Singular Value Decomposition • If A is symmetric, then U=V • Kernel Depends on Image Being Transformed • Need to Compute AAt and AtA and Find the Eigenvalues • Small Values can be Ignored to Yield Compression
- 66. Transform Domain Filtering • Similar to Fourier Domain Filtering • Applicable to Images in which Noise is More Easily Represented in Domain other than Fourier – Vertical and horizontal line detection: Haar transform produces non-zero entries in first row and/or first column

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