ECE468_10 - ECE 468: Digital Image Processing Lecture 10...

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Unformatted text preview: ECE 468: Digital Image Processing Lecture 10 Prof. Sinisa Todorovic [email protected] Outline • • Sampling theorem of functions in 2D (Textbook 4.5.3) Filtering of images in the frequency domain (Textbook 4.7) Sampling Theorem 1 ¯ F (µ, ν ) = TZ m n F (µ − , ν − ) T T m=−∞ n=−∞ ∞ ￿ ∞ ￿ Aliasing Aliasing Due to Subsampling Centering DFT f (x, y )(−1) x+y M N → F (u − ,v − ) 2 2 Example f (x, y ) F (u, v ) ￿ ￿ ￿ M N￿ ￿F (u− , v − )￿ ￿ 2 2￿ ￿ ￿￿ ￿ M N￿ log 1 + ￿F (u− , v − )￿ ￿ 2 2￿ ￿ Translation in Space -> No change in Spectrum Rotation in Space -> Rotation in Spectrum Example Shape !!! ? Filtering in the Frequency Domain Filtering in the Frequency Domain f (x, y ) |F (u, v )| H (u, v ) = ￿ 0 1 , , u = 0, v = 0 otherwise F −1 {H (u, v )F (u, v )} Filtering in the Frequency Domain Steps in Frequency Domain Filtering Steps in Frequency Domain Filtering 1. Input: f(x,y) of size MxN 2. Compute padding fp(x,y) of size PxQ, where P = 2M, Q = 2N 3. Multiply fp(x,y)(-1)x+y to center its DFT 4. Compute DFT of fp(x,y)(-1)x+y --> F(u,v) 5. Use filter H(u,v) of size PxQ, with center at coordinates (P/2,Q/2) 6. Multiply element-wise G(u,v) = H(u,v)F(u,v) 7. Compute the real part of IDFT, gp(x,y) = real[ IDFT(G(u,v)) ] (-1)x+y 8. Crop the top left MxN region to get g(x,y) Ideal Lowpass Filter H (u, v ) = ￿ 1 0 , , D(u, v ) ≤ D0 D(u, v ) > D0 ￿2 ￿ ￿2 D(u, v ) = ￿￿ P u− 2 Q + v− 2 Example: Lowpass Filtering Example: Lowpass Filtering Butterworth Lowpass Filter H (u, v ) = 1+ ￿￿ ￿ 1 D (u,v ) D0 ￿2n ￿2 D(u, v ) = P u− 2 ￿2 Q + v− 2 ￿ Butterworth Lowpass Filter Ringing for large n Butterworth Lowpass Filter Ringing for large n Example: Butterworth Lowpass Filter Lowpass Gaussian Filter D (u, v ) H (u, v ) = exp − 2 2D0 2 ￿ ￿ D(u, v ) = ￿￿ P u− 2 ￿2 Q + v− 2 ￿ ￿2 Example: Lowpass Gaussian Filter Example: Lowpass Gaussian Filter Example: Lowpass Gaussian Filter After this goes sharpening for Hollywood!! Highpass Filtering in the Frequency Domain HHP (u, v ) = 1 − HLP (u, v ) Highpass Filters Example: Highpass Filtering Ideal Butterworth Gaussian Bandpass and Bandreject Filtering HBP = 1 − HBR Laplacian Filter (Homework) H (u, v ) = −4π (u + v ) 2 2 2 Image Sharpening in the Frequency Domain g (x, y ) = f (x, y ) ± c∇ f (x, y ) 2 G(u, v ) = F (u, v ) − H (u, v )F (u, v ), c = −1 needs scaling g (x, y ) = F −1 {[1 + 4π (u + v )]F (u, v )} 2 2 Image Sharpening in the Frequency Domain Unsharp Masking ¯ g (x, y ) = f (x, y ) + c[f (x, y ) − f (x, y )] ¯(x, y ) = F −1 {HLP (u, v )F (u, v )} f g (x, y ) = F −1 {[1 + c(1 − HLP (u, v ))]F (u, v )} g (x, y ) = F −1 {[1 + cHHP (u, v )]F (u, v )} high frequency emphasis filter Example: Lowpass Gaussian Filter After this goes sharpening for Hollywood!! Unsharp Masking in the Frequency Domain Highpass Filtering in the Frequency Domain HHP (u, v ) = 1 − HLP (u, v ) Highpass Filters Example: Highpass Filtering Ideal Butterworth Gaussian Next Class • Preparation for Exam 1 ...
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This note was uploaded on 12/05/2010 for the course ECE 468 taught by Professor Lucchese during the Fall '08 term at Oregon State.

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