Medical Image Processing Using Transforms

Transcription

1 Medical Image Processing Using Transforms Hongmei Zhu, Ph.D Department of Mathematics & Statistics York University MRcenter.ca Outline Image Quality Gray value transforms Histogram processing Filters in image space Filters in Fourier space Filters in Time-frequency space 2 5. Filters in time-frequency space 5.1. Time-Frequency analysis Fields, 08, Zhu 1

2 The Fourier Transform (1807) ( ) = ( ) ( 2π ) F k f t exp i kt dt () ( ) ( 2π ) f t = F k exp i kt dt Joseph Fourier 4 The Fourier Transform (FT) music : Amplit ude Tub a N ois e Fl ute FT: F ( ) Tim φ ( k ) e 5 The Fourier Transform (FT) 6 2

3 Most signals are non-stationary Finite duration Time/Spatial varying Corrupted by noise How can we characterize a signal simultaneously in time and frequency? ---- the aim of time-frequency analysis 7 Atomic decomposition Linearly decompose a signal over a set of elementary building blocks which would be reasonably localized in both time and frequency λ * ( τ ) = () τ (),k f t b t dk f,k () = f ( ) τ,k () * b () i f t λ τ,k b t dτ dk where τ,k is some analysis fucntion deduced from the "synthesis" function b τ,k, making f,k a (linear) time-frequency representation of () i λ ( τ ) f () t. 8 The Gabor Transform (1946) Also called the short-time or windowed FT * ( τ ) = ( ) ( τ) ( 2π ) G,k f t w t exp i kt dt where, e.g., w can be the Gaussian function w( t τ)= 1 2πb exp (t τ ) 2 2 2b 2 9 3

4 Time-Frequency Representation (Gabor) 10 How accurate can the GT be? Width of time window Smaller 11 Heisenberg Inequality Also called the Uncertainty Principle: Resolution in time and frequency cannot be arbitrarily small, because their product is bounded below: 1 t k 4π Here, given the window functions FT wt, () t 2 = t 2 wt () 2 dt wt () 2 dt k 2 = W( k) k 2 W() k 2 dk W() k 2 dk 12 4

5 Next Step t k 1 4π There always is a trade off between t and k. Fortunately, many signals consist of low frequencies of long duration and/or high frequencies of short duration The next logical step is to use a windowing technique with variable sizes: long time window for better k at low frequencies, short time window for better t at highfrequencies. 13 The Continuous Wavelet Transform (CWT) Wavelets: small waves The CWT decomposes a signal into the scaled and shifted replica of the Mother wavelet (a waveform of effectively limited duration and zero mean) 14 The Continuous Wavelet Transform Wavelets: small waves (1984) CW ( τ,s)= 1 s f t ()w * t τ dt s where 1s w t τ s is the scaled and shifted replica of the Mother wavelet, a waveform satisfying () 2 W k c w = dk < k Effectively, W(0) = 0 and W(k) 0 as k 15 5

6 The Continuous Wavelet Transform 29 Amplitu de Scale 29 Tim e 1 29 Marr (x 2-1) exp(-x 2 /2) 1 Scale Morlet π(-1/4)s -1/2 exp(j k x /s) exp(-(x / s) 2 /2) Scale 1 Paul (1- x s ) -m-1 16 Gabor Transform: Time-frequency Representation Wavelet Transform: Multi-scale Analysis The ST is a Multi-scale Time-frequency Analysis 17 The Stockwell Transform ( τ ) = ( ) ( τ 1 ) ( 2π ) S,k f t g t, / k exp i kt dt where the window function g is the Gaussian function with frequency-dependent window width, gt ( τ,1/k)= k 2π exp (t τ )2 k 2 2 Stockwell (1996) IEEE T Signal Processing, V

7 The Stockwell Transform (ST) Signal: Amplit ude The ST: 0 Frequen cy Time Time 19 The ST and Morlet wavelets With the complex Morlet mother wavelet 2 ν 1 t 0 ψ () t = exp exp ( i2πν0t ), 2π 2 the Morlet wavelet transform (MWT) is defined as 1 ν 0* t τ MW,a f t ψ dt. ( τ ) () = a a ν 0 where a =. We can show that k i2πkτ 1 S ( τ,k) = k e M 1 τ, ψ. Du, Wong, Zhu (2006) k Gibson, Lamoureux, Margrave (2006) 20 The ST and Morlet wavelets (a) Small oscillations occur for small frequencies (b) The absolute referenced phase information is retained in the ST, while the MWT gives relative referenced phase information. Liu, Zhu (2007) 21 7

8 Many different time-frequency transforms FT created FT published Joseph Fourier 1996 ST orth WT WT 1964, A. Calderon, harmonic analysis FFT GT 1909 Alfred Haar Haar wavelets 1998 HHT MeyerMorlet Cooley, Grossman Tukey 1946 Dennis Gabor 22 Applications of Time-frequency Representations ( ) TF ( τ,k) f t Object Time-frequency Analyze the raw signal in the (τ, k) domain to identify its local characteristics Remove noise from signal or separate and analyze specific components Extract Features from its time-frequency representation Extension to two or higher dimensions; 23 Correct motion artifacts in fmri 8

9 Functional MRI (fmri) Visual Stimulation Test Artifact corrections Statistical analysis Flashing Checkerboard Time Neural activation MR signal changes visual cortex Activation Map 25 fmri Signal 5% of collected MR data is related to neural activities triggered by fmri experiment Limited data is also corrupted by noise fmri signal Time Problem How can we correct unpredictable motion artifacts to improve the accuracy and reproducibility in fmri analysis? 26 fmri Visual Stimulation Experiment Experiment paradigm fmri signal Expected fmri signal of the activated pixels Time Zhu, Goodyear, et al. Med Phys 30: (2003) 27 9

10 ST Correction for Motion Artifacts 1 fmri data signal filter_signal Frequen cy Time Before filtered Zhu, Goodyear, et al. Med Phys 30: (2003) Time After filtered 28 Filtering using wavelet transforms Discrete Wavelet Transforms If the high-pass and low-pass filters satisfy certain conditions, we can downsample the details by two. This is because max freq is halved according to Nyquist s rule 30 10

11 DWT: Multi-level decomposition wavelet decomposition tree The DWT gives samples of the CWT 31 2D-DWT 32 2D-DWT 33 11

12 Thermal Noise White Noise Gaussian prob. distribution ft ()= 1 2πσ exp t m 2 2σ 2 ( )2 MR Intensity Time Time 34 Denoising 35 Wavelet-based Wiener Filter 36 12

13 Wavelet-based Wiener Filter

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