G16.4428 Practical Magnetic Resonance Imaging II Sackler Institute of Biomedical Sciences New York University School of Medicine Compressed Sensing Ricardo Otazo, PhD ricardo.otazo@nyumc.org
Compressed sensing: The big picture Exploit image sparsity/compressibility to reconstruct undersampled data Original Gradient (sparse) Which image would require fewer samples? Nyquist: same FOV, same number of samples Common sense: fewer non-zero pixels, fewer samples
Image compression Essential tool for modern data storage and transmission Exploit pixel correlations to reduce number of bits First reconstruct, then compress Fully-sampled acquisition (Nyquist rate) Sparsifying transform (e.g. wavelets) Recover image from sparse coefficients 3 MB 300 kb Store or transmit non-zero coefficients only 10-fold compression
Image compression is inefficient Question for image compression Why do we need to acquire samples at the Nyquist rate if we are going to throw away most of them? Answer We don t. Do compressed sensing instead Build data compression in the acquisition First compress, then reconstruct Candès E,Romberg J,Tao T. IEEE Trans Inf Theory 2006; 52(2): 489-509 Donoho D. IEEE Trans Inf Theory 2006; 52(4): 1289-1306.
Compressed sensing components Sparsity Represent images with a few coefficients Transform: wavelets, gradient, etc. Incoherence Sparsity is preserved Noise-like aliasing artifacts Non-linear reconstruction Denoise aliasing artifacts
Simple compressed sensing example k-space No undersampling Image space Regular undersampling Coherent aliasing Random undersampling Incoherent aliasing Courtesy of Miki Lustig
Simple compressed sensing example Undersampled signal Threshold - Threshold Final sparse solution Courtesy of Miki Lustig
Sparsifying transforms Finite differences y ( n) = x( n) x( n 1) Inmatrix form: y = Dx 1 1 D = 1 1 1 1 Total variation TV N ( x) = x( n) x( n 1 ) mintv ( x) = min Dx 1 n= 2
Sparsifying transforms Wavelets Space-frequency localization More efficient than sines and cosines to represent discontinuities
Sparsifying transforms Wavelets Multiresolution image representation Recursive application of the wavelet function modified by the scaling function (each resolution is twice of that of the previous scale) Brain image Daubechies 4-tap wavelet transform
Incoherence Incoherence property: the acquisition domain and the sparse domain must have low correlation E: acquisition matrix W: sparsifying transform matrix <a i,w j > is small for all basis functions
Transform Point Spread Function (TPSF) Encoding model: s = Em Representation model: m = Wp (p is sparse) s = EWp Transform point spread function for position r TPSF( r) = W H E H EW ( r )
Transform Point Spread Function (TPSF) Tool to check incoherence in the sparse domain Computation Apply inverse FFT to sampling mask (1=sampled, 0=otherwise) Apply sparsifying transform Incoherence = ratio of the main peak to the std of the pseudo-noise (incoherent artifacts) Sampling mask PSF TPSF (Wavelet)
Incoherence of k-space trajectories Lustig M, Donoho DL, Santos JM, Pauly JM. Compressed Sensing. MRI, IEEE Signal Processing Magazine, 2008; 25(2): 72-82
Incoherent sampling patterns What are the best samples?? Sampling pattern Image domain Sparse domain Random undersampling (Cartesian) Variable-density random undersampling (Cartesian) Regular undersampling (radial)
Dimensionality and incoherence Sampling pattern (R=4) PSF 64x64 space I=71 I=140 128x128 space
Compressed sensing reconstruction Acquisition model: s = Em s : acquired data E : undersampled Fourier transform m : image to reconstruct Sparsifying transform: W min Wm 1 m subject to Em = s Data consistency Sparsity N x = (l 1 - norm of x) 1 n= 1 x i
Compressed sensing reconstruction Unconstrained optimization (in practice) min m Em s 2 2 + λ Wm 1 Regularization parameter λ Trade-off between data fidelity and removal of aliasing artifacts High λ: artifact removal and denoising at the expense of image corruption (blurring, ringing, blocking, etc) Low λ : no image corruption, but residual aliasing
Compressed sensing reconstruction Gradient descent Cost function: Iterations: m C( m) = Em s + λ Wm n+ 1 = mn α C( mn 2 2 ) 1 C( m n ) = E H ( ) H 1 Emn s + λw M Wmn M is a diagonal matrix: Classwork: compute the gradient of C(m) * Hint: x = x x + µ
Compressed sensing reconstruction Proximal gradient descent (iterative soft-thresholding) Soft-thresholding operation 4 3 S( x, λ) = 0, if x x x λ ( x λ), if x > λ S(x, 1 ) 2 1 0-1 -2-3 -4-5 -4-3 -2-1 0 1 2 3 4 5 x Our gradient gradient-descent algorithm becomes m [ ( [ H S W m E ( Em )],λ 1 n+ 1 = W n n s
Iterative soft-thresholding Sparse signal denoising Original With noise
Iterative soft-thresholding Sparse signal denoising Original After 5 iterations
Iterative soft-thresholding Sparse signal denoising Original After 15 iterations
Iterative soft-thresholding in Matlab function m=cs_ista(s,lambda,nite) % W and Wi are the forward and inverse sparsifying % transforms m = ifft2_mri(s); % initial solution sm = s~=0; % sampling mask for ite=1:nite, % enforce sparsity m = Wi (SoftT(W(m),lambda)); % enforce data consistency m = m ifft2_mri( (fft2_mri(m).*sm s).*sm ); end % soft-thresholding function function y=softt(x,p) y = (abs(x)-p).*x./abs(x).*(abs(x)>p); end
Iterative soft-thresholding Initial solution Iterations Inverse FT of the zero-filled k-space Sparse representation S(, λ) w i After soft-thresholding After data consistency k-space representation
Iterative soft-thresholding (R=3) Initial solution Inverse FT of the zero-filled k-space After 30 iterations Iterative soft-thresholding
How many samples do we need in practice? ~4K samples for a K-sparse image N = 6.5x10 4 K = 6.5x10 3 Compression ratio C = 10 R = 2.5 (safe undersampling factor) Rule of thumb: about 4 incoherent samples for each sparse coefficient
Image quality in compressed sensing SNR is not a good metric Loss of small coefficients in the sparse domain Loss of contrast Blurring Blockiness Ringing Images look more synthetic R=2 R=4
Combination of compressed sensing and parallel imaging
Why would CS & PI make sense? Image sparsity and coil-sensitivity encoding are complementary sources of information Compressed sensing can regularize the inverse problem in parallel imaging Parallel imaging can reduce the incoherent aliasing artifacts
Why would CS & PI make no sense? CS requires irregular k-space sampling while PI requires regular k-space sampling Non-linear reconstruction in CS can prevent high g- factors due to irregular sampling Incoherence along the coil dimension is very limited because the sampling pattern for all coils is the same It does not matter, because we are not exploiting incoherence along the coil dimension
Approaches for CS&PI CS with SENSE parallel imaging model Muticoil radial imaging with TV minimization Block T et al. MRM 2007 SPARSE-SENSE Liang D et al. MRM 2009 Otazo R et al. MRM 2010 CS with GRAPPA parallel imaging model l 1 -SPIRiT Lustig M et al. MRM 2010
CS with SENSE parallel imaging model Enforce joint sparsity on the multicoil system coil sensitivities E sparsifying transform W Coil 1 s 1 Coil 2 s 2 s CS & PI m Coil N s N min m { 2 Em - s + λ Wm } 2 1 SENSE data consistency Joint sparsity m represents the contribution from all coils
How does PI help CS? Reduces incoherent aliasing artifacts Sampling pattern (R=4) Point spread function (PSF) 1 coil 8 coils k y k x y y
How does PI help CS? Reduces incoherent aliasing artifacts 1 coil 8 coils Which one would be easier to reconstruct?
Limits of Performance Compressed sensing alone ~4K samples for a K-sparse image Compressed sensing & parallel imaging Close to K samples for large number of coils Noise-free simulation K = 32 N c = number of coils Otazo et al. ISMRM 2009; 378 UISNR coil geometry
CS & PI for 2D imaging Siemens 3T Tim Trio 12-channel matrix coil array 4-fold acceleration GRAPPA Coil-by-coil CS Joint CS & PI
Application to musculoskeletal imaging Conventional clinical protocol for knee imaging Multislice 2D-FSE (axial) Multislice 2D-FSE (coronal) 10-12 minutes Multislice2D-FSE (sagittal) New compressed sensing protocol (Sparse-SPACE) 3D-FSE 4-5 minutes NYU-Siemens collaboration to be presented at RSNA 2014
Application to musculoskeletal imaging First clinical study 50 patients >90% agreement between conventional 2D-FSE and Sparse- SPACE 2D-FSE (10:56 min) Sparse-SPACE (4:36 min) NYU-Siemens collaboration to be presented at RSNA 2014
Summary Compressed sensing New sampling theorem Information rate (sparsity) rather than pixel rate (bandwidth) Ingredients Sparsity Incoherence Non-linear reconstruction Fast imaging tool for MRI MR images are naturally compressible Data acquisition in k-space facilitates incoherence Can be combined with parallel imaging