2-D Finite Impulse Response (FIR) Filters. h(k,l)x(m k,n l)

Transcription

1 C. A. Bouman: Digital Image Processing - January 8, 8 -D Finite Impulse Response (FIR) Filters Difference equation N y(m,n) = k= N N l= N h(k,l)x(m k,n l) ForN = - input points; output point Number of multiplies per output point Multiplies = (N +) Transfer function H(z,z ) = H(e jµ,e jν ) = N k= N N k= N N l= N N l= N h(k,l)z k z l h(k,l)e j(kµ+lν)

2 C. A. Bouman: Digital Image Processing - January 8, 8 Spatial FIR Smoothing Filtering Filter point spread function (PSF) or impulse response: The box, X, indicates the center element of the filter. 4 6 Apply filter using free boundary condition: Assume that pixels outside the image are Input Image Output Image

3 C. A. Bouman: Digital Image Processing - January 8, 8 3 PSF for FIR Smoothing Filter 4 6

4 C. A. Bouman: Digital Image Processing - January 8, 8 4 Spatial FIR Horizontal Derivative Filtering Filter point spread function (PSF) or impulse response: The box, X, indicates the center element of the filter Apply filter using free boundary condition: Assume that pixels outside the image are Input Image Output Image

5 C. A. Bouman: Digital Image Processing - January 8, 8 5 PSF of FIR Horizontal Derivative Filter 4 4 6

6 C. A. Bouman: Digital Image Processing - January 8, 8 6 Spatial FIR Vertical Derivative Filtering Filter point spread function (PSF) or impulse response: The box, X, indicates the center element of the filter Apply filter using free boundary condition: Assume that pixels outside the image are Input Image Output Image

7 C. A. Bouman: Digital Image Processing - January 8, 8 7 PSF of FIR Vertical Derivative Filter 4 4 6

8 C. A. Bouman: Digital Image Processing - January 8, 8 8 Example : -D FIR Filter Consider the impulse response h(m,n) = h (m)h (n) where 4h (n) = (,,,,,, ) h (n) = (δ(n+)+δ(n)+δ(n ))/4 Then h(m,n) is a separable function with 6h(m,n) = The DTFT of h (n) is H (e jω ) = 4 ( e jω ++e jω) = (+cos(ω)) The DSFT ofh(m,n) is H(e jµ,e jν ) = H (e jµ )H (e jν ) = 4 (+cos(µ))(+cos(ν))

9 C. A. Bouman: Digital Image Processing - January 8, 8 9 Example : Frequency Response of -D FIR Filter Plot of frequency response H(e jµ,e jν ) = 4 (+cos(µ))(+cos(ν)) 3 D Plot of H(e jµ,e jν ) ν axis 4 4 µ axis This is a low pass filter with H(e j,e j ) =

10 C. A. Bouman: Digital Image Processing - January 8, 8 Example : -D FIR Filter Consider the impulse response h(m,n) = h (m)h (n) where 4h (n) = (,,,,,, ) h (n) = (δ(n+) δ(n)+δ(n ))/4 Then h(m,n) is a separable function with 6h(m,n) = The DTFT of h (n) is H (e jω ) = 4 ( e jω +e jω) = ( cos(ω)) The DSFT ofh(m,n) is H(e jµ,e jν ) = H (e jµ )H (e jν ) = 4 ( cos(µ))( cos(ν))

11 C. A. Bouman: Digital Image Processing - January 8, 8 Example : Frequency Response of -D FIR Filter Plot of frequency response H(e jµ,e jν ) = 4 ( cos(µ))( cos(ν)) 3 D Plot of H(e jµ,e jν ) ν axis 4 4 µ axis This is a high pass filter with H(e j,e j ) =

12 C. A. Bouman: Digital Image Processing - January 8, 8 Ordering of Points in a Plane Recursive filter implementations require the ordering of points in the plane. Lets = (s,s ) Z and r = (r,r ) Z. Quarter plane - then s < r means: (s < r ) and (s < r ) and s r Symmetric half plane - then s < r means: (s < r ) Nonsymmetric half plane - then s < r means: (s < r ) or ((s = r ) and (s < r ))

13 C. A. Bouman: Digital Image Processing - January 8, 8 3 -D Infinite Impulse Response (IIR) Filters Difference equation N y(m,n) = k= N N l= N b(k,l)x(m k,n l) + P k= P P a(k,l)y(m k,n l) l= + Simplified notation y s = r P a(k,)y(m k,n) k= b r x s r + r>(,) a r y s r For nonsymetric half plane with N = and P = Number of multiplies per output point Multiplies = (N +) FIR Part +(P +)P IIR Part

14 C. A. Bouman: Digital Image Processing - January 8, 8 4 -D IIR Filter Transfer Functions Transfer function in Z-transform domain is H(z,z ) = N N k= N l= N b(k,l)z k z l P P k= P l= a(k,l)z k z l P k= a(k,)z k Transfer function in DSFT domain is H(e jµ,e jν ) = N P k= P k= N N l= N b(k,l)e j(kµ+lν) P l= a(k,l)e j(kµ+lν) P k= a(k,)e j(kµ)

15 C. A. Bouman: Digital Image Processing - January 8, 8 5 Example 3: -D IIR Filter Consider the difference equation y(m,n) = x(m,n)+ay(m,n)+ay(m,n ) Spatial dependencies - previous value; curent value Taking the Z-transform of the difference equation Y(z,z ) = X(z,z )+az Y(z,z )+az Y(z,z ) The transfer functions is then H(z,z ) = Y(z,z ) X(z,z ) = az H(e jµ,e jν ) = ae jµ ae jν az

16 C. A. Bouman: Digital Image Processing - January 8, 8 6 Example 3: -D IIR Filter in Space Domain Fora = / y(m,n) = x(m,n)+ y(m,n)+ y(m,n ) Looks like / / Apply filter in raster scan order. 64 Input Image Output Image

17 C. A. Bouman: Digital Image Processing - January 8, 8 7 Example 3: Frequency Response of -D IIR Filter Plot of frequency response fora =.4. H(z,z ) = az az 3 Contour Plot of MTF for (.4*Z.4*Z ) Mesh Plot of MTF for (.4*Z.4*Z ) ν 4 ν 4 4 µ µ

18 C. A. Bouman: Digital Image Processing - January 8, 8 8 Example 4: -D IIR Filter Consider the difference equation y(m,n) = x(m,n)+ay(m,n)+ay(m,n ) +ay(m+,n ) Spatial dependencies - previous value; curent value The transfer functions is then H(z,z ) = az az az +z H(e jµ,e jν ) = ae jµ ae jµ ae +jµ jν

19 C. A. Bouman: Digital Image Processing - January 8, 8 9 Example 4: -D IIR Filter in Space Domain Fora = /4 y(m,n) = x(m,n)+ 4 y(m,n)+ 4 y(m,n ) + y(m+,n ) Looks like /4 / /4 Apply filter in raster scan order. 64 Input Image }{{ 47 8 } Output Image

20 C. A. Bouman: Digital Image Processing - January 8, 8 Example 4: Frequency Response of -D IIR Filter Plot of frequency response fora =.. H(z,z ) = az az az z 3 Contour Plot of MTF for (.*Z.*Z.4*Z *Z ) Mesh Plot of MTF for (.*Z.*Z.4*Z *Z ) ν 4 ν 4 4 µ µ Notice that transfer function has a diagonal orientation.

21 C. A. Bouman: Digital Image Processing - January 8, 8 Example 5: -D IIR Filter Consider the difference equation y(m,n) = x(m,n)+ay(m,n)+ay(m,n ) +ay(m+,n)+ay(m,n+) Spatial dependencies - previous value; curent value Theoretically, the transfer functions is then H(z,z ) = H(e jµ,e jν ) = THIS DOESN T WORK az az az az ae jµ ae jµ ae jµ ae jν