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

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Familiar Fourier Facts Fundamental convolution theorem

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Question? What is Butterworth? 80p a pound !! ? ?

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

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Use 1D profile data profile

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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

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

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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

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Count Profile - Fourier fit

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Amplitude - Frequency plot

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What Happens to Noisy Data?

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

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Tomography back projection

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

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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

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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!

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

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Butterworth Settings

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Butterworth Filter

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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

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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

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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

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

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

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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

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Bone images WienerButterworth

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Heart Images Butterworth filterWiener filter

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2D Fourier

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

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Partial Volume Effect =FWHMx2x0.5

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Cylinder phantom low pass Wiener

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

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

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

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Heart phantom Study

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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

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