Reconstruction methods for under-sampled MR data aka: Constrained reconstruction methods Jeffrey A. Fessler EECS Department, BME Department, Dept. of Radiology University of Michigan ISMRM Imaging Strategies Course May 8, 2011 Acknowledgments: Doug Noll, Sathish Ramani 1 Disclosure Declaration of Relevant Financial Interests or Relationships Speaker Name: Jeff Fessler I have the following relevant financial interest or relationship to disclose with regard to the subject matter of this presentation: Company name: GE Healthcare and GE Global Research Type of relationship: X-ray CT image reconstruction collaborations I have no conflicts of interest with regards to MR topics. 2 Introduction Reasons for under-sampling: • Static imaging: reduce scan time • Dynamic imaging: inherent ◦ dynamic contrast studies (microscopic motion?) ◦ bulk motion All such situations require assumptions / constraints / models. 3 Under-Sampled K-space: Examples K−space Partial/Half Under−sampled by 2x ky kx Variable density Random Radial 4 Is This Under-Sampled K-space? ky kx 1/FOV Note: k-space sample spacing is 1/FOV (Nyquist sample spacing). Answers (audience response system): 1. No 2. Yes 3. Unsure 4. Will this be on the final exam? 5 Is This Under-Sampled K-space? 1/FOV 6 Basic MRI Signal Model Ignoring many physical effects, the baseband signal in lth receive coil is approximately: ~ s (t) = f (~r)c (~r)exp −ı2πk(t)·~r d~r. (1) l l (cid:16) (cid:17) Z • ~r: spatial position • c (~r): receive sensitivity of the lth coil, l = 1,...,L l ~ • k(t): k-space trajectory • f (~r): (unknown) transverse magnetization of the object MR scan data is noisy samples thereof: y = s (t )+ε , i = 1,...,M, l = 1,...,L (2) li l i li • y : ith sample of lth coil’s signal li • εεε : additive complex white gaussian noise, li • M: number of k-space samples. Goal: reconstruct object f (~r) from measurement vector yyy = (yyy ,...,yyy ), 1 L where yyy = (y ,...,y ) is data from lth coil. l l1 l,M 7 MR Image Reconstruction is Ill-Posed ~ y = f (~r)c (~r)exp −ı2πk(t )·~r d~r+ε li l (cid:16) i (cid:17) li Z • Unknown object f (~r) is a continuous space function • Measurement vector yyy is finite dimensional . .. All MRI data is under-sampled Uncountably infinitely many objects f (~r) fit the data yyy exactly, even for “fully sampled” data, even if there were no noise. For “fully sampled” Cartesian k-space data, ˆ how shall we choose one reconstructed image f(~r) from among those? 1. Impose some assumptions / constraints / models 2. Just take an inverse FFT of the data 3. Both of the above 4. None of the above 8 Inverse FFT for MR Image Reconstruction Using an inverse FFT for reconstruction from “fully sampled” single-coil data is equivalent to assuming the object lies in a finite-dimensional subspace: N−1M−1 f (~r) = f (x,y) = ∑ ∑ f [n,m]b(x−n△ )b(y−n△ ). X Y n=0 m=0 What choice of basis function b(·) is implicit in IFFT reconstruction? 1. Dirac impulse 2. Rectangle (pixel) 3. Sinc 4. Dirichlet (periodic sinc) . .. The use of assumptions / constraints / models is ubiquitous in MR. In particular, constraining the estimate to lie in a finite-dimensional sub- space is nearly ubiquitous. (All models are wrong but some models are useful...) 9 Conventional Approach: Partial K-space Conventional solution: Homodyning Partial/Half Noll et al., IEEE T-MI, June 1991 Constraint: object phase is smooth Related iterative methods Fessler & Noll, ISBI 2004 Bydder & Robson, MRM, June 2005 10