NUFFT for Medical and Subsurface Image Reconstruction

Transcription

1 NUFFT for Medical and Subsurface Image Reconstruction Qing H. Liu Department of Electrical and Computer Engineering Duke University Duke Frontiers 2006 May 16, 2006

2 Acknowledgment Jiayu Song main contributor for MRI and subsurface imaging applications Nhu Nguyen collaborator for forward NUFFT Susan X. Tang collaborator for inverse NUFFT. Discussion w/ X. Sun A. Johnson s group MRI data W. Scott s and L. Collins groups subsurface imaging data Research supported by NIH and DARPA

3 Outline Introduction: History and Motivation NUFFT algorithm (Liu/Nguyen, 1998) MRI and CT applications Subsurface imaging applications Summary

4 What is NUFFT? Nonuniform Fast Fourier Transform Q.H. Liu and N. Nguyen, An accurate algorithm for nonuniform fast Fourier transforms (NUFFT), IEEE Microwave Guided Wave Lett., vol. 8, pp , Fast algorithm for discrete Fourier transform sum when sample points are nonuniform

5 Brief History of NUFFT Many slow summation methods before Cost: O(MN) Dutt and Rokhlin (1995) first fast algorithm with cost of Other fast algorithms: Beylkin (1995); Dutt & Rokhlin (1996); Anderson & Dahleh (1996) NUFFT coined in 1998 (Liu & Nguyen, 1998; Nguyen & Liu, 1999) Fessler/Sutton (2003); Greengard/Lee (2005) MRI application: Sha, Guo and Song (2003) and others

6 Motivation K-space Image space Medical Imaging: MRI, CT Raw data is acquired in nonuniform Fourier space (MRI) or can be related to Fourier space (CT)

7 MRI Magnetization Process Transverse Magnetization Decay M () t = M e xy 0 t/ T * 2 z M0 Mz y RF B1 Mxy x RF Vertical Magnetization Recovery M t M e z t / T1 () = (1 ) 0 B0

8 Bz z Phase and Encoding y K-space trajectory Bx By x γ k() t = G( τ ) dτ 2 t π τ = 0 t φ( x, t) = γb( x, τ) dτ τ = 0 = ω t 2πxk 0 B0 Signal Model iω0 ( k) = m ( x,0) t 2 S π xk FOV xy e e d = iω0t 2π xk e m ( x,0) e d FOV xy x x

9 Non-Cartesian Trajectories Radial Rosette Spiral Lissajou

10 Motivation (2) Nonuniform K-space Ground penetrating radar (GPR) or seismic sensors Image space Raw data acquired can be related to nonuniform Fourier space data

11 A Phenomenological Model Example: Radar Waves Air Tx Rx Radar freq. Soil 2 m 2 m Air 1.5 m

12 0.96 ns interval Movie of Radar Waves First arrival: Direct head waves with speed in air Second arrival: Wave speed in soil Third arrival: Reflection from tunnel

13 Waveforms at Receivers Air Soil Tunnel

14 GPR Sensor Scan Sensor Scan (x,y) Wavenumber Space (kx,ky) uniform Wave travel time (t) Nonlinear dispersion relation kz nonuniform

15 Common problem: Nonuniform Samples in K Space 1D Uniform versus Non-Uniform Samples Fast Fourier Transform: O(N log N) CPU time NUFFT: Fast algorithm to achieve O(N log N) CPU time

16 Outline Introduction: History and Motivation NUFFT algorithm (Liu/Nguyen, 1998) MRI and CT applications Subsurface imaging applications Summary

17 NUFFT Algorithm (Liu/Nguyen) Optimal Uses least-square interpolation of exponential function A new class of matrices regular Fourier matrices have been discovered Independent of nonuniform points Closed-form solution for an excitation (data) vector

18 NUFFT Procedures Preprocessing: Calculates interpolation coefficients for exponential functions Interpolation: Calculates interpolated Fourier coefficients at an oversampled grid Regular FFT: Uses regular FFT to calculate the DFT on the oversampled grid Scaling: Evaluates the NUDFT from the DFT on the oversampled grid

19 Comparison of NUFFT with previous algorithm [Dutt/Rokhlin, 1993]: More than one order of magnitude inprovement Computation cost

20 Outline Introduction: History and Motivation NUFFT algorithm MRI and CT applications Subsurface imaging applications Summary

21 MRI Applications Scanner Phantom 2T 7T Source: Center for In Vivo Microscopy

22 Computer Simulation 256X256 64X64X64 IDFT IDFT INUFFT 1000X 1000X IDFT-INUFFT e = 0.042% X IDFT-INUFFT e = 0.034% 1000X 1000X

23 Accuracy-Speed Curve NUFFT w/o prep NUFFT w/ prep Kaiser-Bessel w/o prep Kaiser-Bessel w/ prep Time L2 Error

24 Phantom Scan 2D 3D I NUFFT z I NUFFT y x

25 In Vivo Scan 2D I NUFFT Source: cardiac data from Elizabeth Bucholz 3D

26 Outline Introduction: History and Motivation NUFFT algorithm MRI and CT applications Subsurface imaging applications Summary

27 NUFFT Imaging of 3D Objects x y 3D Configuration (W. Scott, Georgia Tech)

28 3D Imaging Results (GT plywood) Raw Data

29 A Field Example of IED Imaging Ground Return

30 NUFFT Imaging Target NUFFT Side View Target NUFFT Top View

31 3D View Target 1 Target

32 Summary NUFFT has widespread applications in medical imaging and subsurface sensing It allows arbitrary sampling patterns Improved accuracy enables faster data acquisition important for functional imaging