2. Filters All will be made clear Bill Thomson, City hospital Birmingham

3. Familiar Fourier Facts Fundamental convolution theorem

4. Question?  What is Butterworth? 80p a pound !! ? ?

5. Filters Simple overview only No Maths What is Butterworth? Why Frequency? - Fourier (Well, only a little!)

6. Use 1D profile data profile

7. Who to blame? Fourier zJean Baptiste Joseph Fourier zAuxerre, 1768 - 1830 znearly became a priest zstudied with Lagrange, Laplace zarrested twice, nearly guillotined zscientific adviser to Napoleon in Egypt ztheory of heat transfer used series of sines, cosines z15 years before accepted and published

8. Fourier analysis zRepresent a function by sums of sin and cos terms zeasier maths zneed to consider frequencies

9. sines and cosines A wavelength A= size (amplitude) Wavelength= distance (cm, pixels etc) frequency = 1 / wavelength (cm -1, pixels -1 ) Amplitude = same wavelength = 1/2 frequency = double A wavelength

10. Count Profile - Fourier fit

11. Amplitude - Frequency plot

12. What Happens to Noisy Data?

13. Power Spectrum Normally plot (Amplitude) 2 against frequency, on log scale

14. Tomography back projection

15. Blurring of back projection zEach true count is ‘blurred’ by 1/r function

16. How can we remove the 1/r blurring? True data ‘blurred’ by 1/r = Back projection data taking Fourier transform, F ‘blurring’ becomes simple multiplication F (1/r) becomes 1 / so converting to Fourier, F i.e. in frequency terms F (true data) x 1/ = F (back projection data) F (true data) = F (back projection data) x

17. Ramp Function  F (true data) = F (back projection data) x Problem ! Amplifies higher frequencies Noise at higher frequencies Need to stop at a frequency which contains most signal and little noise In theory, all done!

18. Butterworth Filter zUsed to ‘cut-off’ the ramp effect zhas two components - order cut-off

19. Butterworth Settings

20. Butterworth Filter

21. Wiener Filter zRestorative filter zamplifies mid range frequencies zdepends on resolution (MTF) and noise zmust have MTF file for isotope and collimator zfactor ‘tweaks’ noise component ztrust computer selection! MTF( ) MTF( ) 2 + 1/SNR( ) SNR( )= Object power Noise power x multiplier

22. Maximum cut-off ? zdata sample at 2x highest freq in the data zpixel is smallest sample so, freq max = 0.5 pixel -1 zhighest freq in patient data  1 cm -1 zsample at 2cm -1, pixel size 5mm z64x64 = 8mm 128 x128 = 3.6mm Sample = FWHM / 3 resolution 15 - 18mm sample at 5 - 6 mm

23. Automatic Filter zUses power spectrum z10% noise means ‘rejects’ 90% of noise zbased on analysis of four images  suggest use it for most clinical imaging

24. Software Revision 128 x 128 Butterworth, power spectrum changed Spatial Frequency (cycles/ pixel) Spatial Frequency (cycles/ 2pixels)

25. Resolution effects zResolution depends on detector resolution cut-off frequency of filter zif filter cutoff is low, filter determines resolution

26. Tomographic Noise zCannot ignore noise in sampling zNot simple ‘Poisson’ - complex zproportional to square root of cts per pixel (N) 1/2 fourth root of total pixels (P) 1/4 zfor the same ‘signal to noise’ improve spatial resolution by 2 counts must increase by 8

27. Bone images WienerButterworth

28. Heart Images Butterworth filterWiener filter

29. 2D Fourier

30. 2D Filter of a Duck 2D Fourier transform Inverse Fourier

31. Partial Volume Effect =FWHMx2x0.5

32. Cylinder phantom low pass Wiener

33. Phantom Study Two holes separated by diameter ‘building blocks’ join together fill with Tc99m, tomo scan in water 8mm9mm10mm11mm12mm13mm

34. Twin hole phantom WienerBest Butterworth Cut –off 0.45 pixel -1 Poor Butterworth Cut off 0.26 pixel -1

35. Multi Hole Phantom Hi-res coll Butterworth Gen purpose Butterworth Hi-res Wiener

36. Heart phantom Study

37. Conclusions Filter choice still very user dependent essentially a balance of noise / detail frequency needed for the maths behind the scenes check if cycles/cm, cycles/pixel, cycles/(2 pixels) higher frequencies needed for detail / resolution Remember partial volume effect