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Chapter14
 

Chapter14

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    Chapter14 Chapter14 Presentation Transcript

    • 240-373 Image Processing 240-373: Chapter Montri Karnjanadecha montri@coe.psu.ac.t h http:// fivedots.coe.psu.ac.t h/~montri 1
    • Chapter 14 The Frequency Domain 240-373: Chapter 2
    • The Frequency Domain • Any wave shape can be approximated by a sum of periodic (such as sine and cosine) functions. • a--amplitude of waveform • f-- frequency (number of times the wave repeats itself in a given length) • p--phase (position that the wave starts) • Usually phase is ignored in image processing 240-373: Chapter 3
    • 240-373: Chapter 4
    • 240-373: Chapter 5
    • The Hartley Transform • Discrete Hartley Transform (DHT) – The M x N image is converted into a second image (also M x N) – M and N should be power of 2 (e.g. .., 128, 256, 512, etc.) – The basic transform depends on calculating the following for each pixel in the new M x N array 1 H (u , v) = MN M −1 N −1 ∑∑ x =0 y =0   ux vy   ux vy  f ( x, y ) ⋅ cos(2π ) −  + sin( 2π ) −  M N   M N   240-373: Chapter 6
    • The Hartley Transform where f(x,y) is the intensity of the pixel at position (x,y) H(u,v) is the value of element in frequency domain – The results are periodic – The cosine+sine (CAS) term is call “the kernel of the transformation” (or ”basis function”) 240-373: Chapter 7
    • The Hartley Transform • Fast Hartley Transform (FHT) – M and N must be power of 2 – Much faster than DHT – Equation: H (u, v) = { T (u , v) + T ( M − u, v) + T (u , N − v) − T ( M − u , N − v)} / 2 240-373: Chapter 8
    • The Fourier Transform • The Fourier transform – Each element has real and imaginary values – Formula: F (u , v) = ∫∫ f ( x, y )e −2iπ ( ux +vy ) dxdy – f(x,y) is point (x,y) in the original image and F(u,v) is the point (u,v) in the frequency image 240-373: Chapter 9
    • The Fourier Transform • Discrete Fourier Transform (DFT)  ux vy  − 2 iπ  +  1 M −1 N −1 F (u, v) = f ( x, y )e  M N  ∑∑ MN x =0 y =0 – Imaginary part 1 M −1 N −1  ux vy  Fi (u, v) = − f ( x, y ) sin 2π  +  ∑∑ MN x =0 y =0 M N  – Real part 1 Fr (u , v) = MN M −1 N −1 ∑∑ x =0 y =0  ux vy  f ( x, y ) cos 2π  +  M N  – The actual complex result is Fi(u,v) + Fr(u,v) 240-373: Chapter 10
    • Fourier Power Spectrum and Inverse Fourier Transform • Fourier power spectrum F (u , v) = Fr (u , v) 2 + Fi (u , v) 2 • Inverse Fourier Transform 1 f ( x, y ) = MN 240-373: Chapter M −1 N −1 ∑∑ F (u, v)e  ux vy  2 iπ  +  M N  x =0 y =0 11
    • Fourier Power Spectrum and Inverse Fourier Transform • Fast Fourier Transform (FFT) – Much faster than DFT – M and N must be power of 2 – Computation is reduced from M2N2 to MN log2 M . log2 N (~1/1000 times) 240-373: Chapter 12
    • Fourier Power Spectrum and Inverse Fourier Transform • Optical transformation – A common approach to view image in frequency domain A D B C Original image 240-373: Chapter C B D A Transformed image 13
    • Power and Autocorrelation Functions • Power function: [ 1 P (u , v) = F (u , v) = H (u , v) 2 + H (−u ,−v) 2 2 • Autocorrelation function F (u , v) 2 – Inverse Fourier transform of or 1 [ H (u, v) 2 + H (−u,−v) 2 ] – Hartley transform of 2 240-373: Chapter 14 ]
    • Hartley vs Fourier Transform 240-373: Chapter 15
    • Interpretation of the power function 240-373: Chapter 16
    • Applications of Frequency Domain Processing • Convolution in the frequency domain 240-373: Chapter 17
    • Applications of Frequency Domain Processing – useful when the image is larger than 1024x1024 and the template size is greater than 16x16 – Template and image must be the same size 1 1 1 1 ⇒ 1 1 0 0 240-373: Chapter 1 1 0 0 0 0 0 0 0 0 0 0 18
    • – Use FHT or FFT instead of DHT or DFT – Number of points should be kept small – The transform is periodic • zeros must be padded to the image and the template • minimum image size must be (N+n-1) x (M+m-1) 240-373: Chapter 19
    • 0 1 2 0 4 5 6 0 8 9 10 0 0 0 0 0 240-373: Chapter 1 2 0 0 ⊗ 45 58 44 12 7 3 0 0 97 110 76 32 = 0 0 0 0 26 29 10 16 0 0 0 0 3 13 14 0 20
    • – Convolution in frequency domain is “real convolution” Normal convolution 0 1 2 0 4 5 6 1 2 0 0 0 ⊗ 8 9 10 0 0 0 0 0 7 3 0 0 0 0 0 0 45 = 0 0 0 0 58 44 12 97 110 76 32 26 29 10 16 3 13 14 0 Real convolution 0 4 8 0 1 2 0 5 6 0 9 10 0 0 0 0 240-373: Chapter 0 0 ⊗ 0 0 0 0 0 0 0 0 3 2 0 0 1 4 4 0 4 20 33 18 = 7 36 72 85 38 1 56 87 97 30 21
    • Convolution using the Fourier transform Technique 1: Convolution using the Fourier transform USE: To perform a convolution OPERATION: – zero-padding both the image (MxN) and the template (m x n) to the size (N+n-1) x (M+m-1) – Applying FFT to the modified image and template – Multiplying element by element of the transformed image against the transformed template 240-373: Chapter 22
    • Convolution using the Fourier transform OPERATION: (cont’d) – Multiplication is done as follows: F(image) (r1,i1) F(template) F(result) (r2, i2) (r1r2 - i1i2, r1i2+r2i1) i.e. 4 real multiplications and 2 additions – Performing Inverse Fourier transform 240-373: Chapter 23
    • Hartley convolution Technique 2: Hartley convolution USE: To perform a convolution OPERATION: – zero-padding both the image (MxN) and the template (m x n) to the size (N+n-1) x (M+m-1) image 0 1 2 template 0 1 2 0 0 4 5 6 0 8 9 10 0 0 0 0 0 7 3 0 0 0 0 0 0 0 0 0 0 240-373: Chapter 24
    • Hartley convolution – Applying Hartley transform to the modified image and template image template 11.25 2.25 3.75 − 5.25 − 2.25 − 3.75 − 0.75 2.75 3.25 3.25 3.75 0.75 1.25 − 1.75 − 9.75 − 0.25 − 3.25 1.25 - 1.75 - 1.75 - 1.25 - 1.25 - 1.75 - 0.25 - 1.25 - 2.75 240-373: Chapter 25 3.25 1.75 0.75 0.75 0.75 2.25
    • Hartley convolution – Multiplying them by evaluating:  I (u , v)T (u, v) + I (u , v)T ( N − u , M − v) NM   R (u , v) = + I ( N − u , M − v)T (u , v)  2  − I ( N − u , M − v)T ( N − u, M − v)    240-373: Chapter 26
    • Hartley convolution: Cont’d 585 − 33 45 − 213 − 417 75 − 49 39 − 105 − 11 − 25 45 − 27 Giving: − 59 25 125 – Performing Inverse Hartley transform, gives: 0 1 4 4 4 20 33 18 36 72 85 38 56 87 97 30 240-373: Chapter 27
    • Hartley convolution: Cont’d 240-373: Chapter 28
    • Deconvolution • Convolution R = I * T • Deconvolution I = R *-1 T • Deconvolution of R by T = convolution of R • by some ‘inverse’ of the template T (T’) 240-373: Chapter 29
    • Deconvolution • Consider periodic convolution as a matrix operation. For example 1 2 3 4 5 6 7 8 9 1 1 0 * 1 1 0 0 0 0 240-373: Chapter 12 16 14 = 24 28 26 18 22 20 30
    • Deconvolution is equivalent to A 1  1 0  1 (1 2 3 4 5 6 7 8 9) 1  0  0 0  0  B C 0 1 0 0 0 1 0 1  1 0 0 0 0 1 1 0 1 1 0 0 0 0 1 1  0 1 1 0 1 0 0 0 1 0 1 1 0 0 0 0  = (12 16 14 24 28 26 18 22 20 )  1 1 0 1 1 0 0 0  0 0 1 0 1 1 0 1 0 0 1 1 0 1 1 0  0 0 0 1 1 0 1 1  240-373: Chapter AB =C ABB-1 = CB-1 A = CB-1 31