Compressed Sensing for Rapid MR Imaging

Transcription

1 Compressed Sensing for Rapid Imaging Michael Lustig1, Juan Santos1, David Donoho2 and John Pauly1 1 Electrical Engineering Department, Stanford University 2 Statistics Department, Stanford University rapid I RL 1

2 SRL rapid I RL 2

3 SRL People rapid I RL 3

4 Imaging Non-radiation Non toxic imaging modality. Flexible tissue contrast Arbitrary plane imaging Many applications Still Too Slow rapid I RL 4

5 Outline A short tour of k-space Compressed Sensing CS and I + applications rapid I RL 5

6 k A Tour in k-space RL

7 Signal Equation iφ ( r,t ) s( t ) = m( r ) e dr V Phase Magnitude (known and (Unknown) controlled by linear gradient magnetic fields) rapid I RL 7

8 k-space k-space image-space ky Fourier kx rapid I RL 8

9 k-space k-space image-space ky Fourier kx rapid I RL 9

10 k-space k-space image-space ky Fourier kx rapid I R L 10

11 k-space Applying linear magnetic field gradients controls the phase. Like taking a tour in kspace rapid I R L 11

12 k-space Trajectories Cartesian: Used routinely in almost 99% of sequences. Extremely robust. 2D Fourier Transform Echo-Planar Whenever possible, use this! rapid I R L 12

13 k-space Trajectories Cartesian: Used routinely in almost 99% of sequences. Extremely robust. 2D Fourier Transform Echo-Planar Whenever possible, use this! Spirals: Used mainly for real-time and rapid imaging Spiral Fast & hardware efficient. rapid I R L 13

14 k-space Trajectories Cartesian: Used routinely in almost 99% of sequences. Extremely robust. 2D Fourier Transform Echo-Planar Whenever possible, use this! Spirals: Used mainly for real-time and rapid imaging Spiral Fast & hardware efficient. Radial: Specific applications (Hi res, flow ) Almost always under-sampled rapid I radial R L 14

15 Pulse Sequence Cartesian Trajectory rapid I Spiral Trajectory R L 15

16 Hardware constraints & Physical Imperfections Hardware issues: Gradients limited by maximum amplitude and slew rate. k < G t max k(t) ktt < Smax Physical issues: Nerve stimulation Eddy currents Trajectory deviations Trajectory dependent artifacts rapid I R L 16

17 k-space Sampling It s all about finite support! Fourier Transform rapid I R L 17

18 K-space Sampling Finite support in frequency Resolution Fourier Transform rapid I R L 18

19 K-space Sampling Finite support in frequency Resolution Fourier Transform rapid I R L 19

20 K-space Sampling Finite support in frequency Resolution Fourier Transform rapid I R L 20

21 K-space Sampling Finite support in Space k=1/fov Fourier Transform FOV rapid I R L 21

22 K-space Sampling Finite support in Space 3 k= /FOV Fourier Transform FOV rapid I R L 22

23 Non Cartesian K-space Sampling NonUniform Fourier Transform k=1/fov FOV rapid I R L 23

24 Non Cartesian K-space Sampling NonUniform Fourier Transform rapid I 3 k= /FOV R L 24

25 Here are the Important points: In I, sampled are collected in the Fourier domain (k-space). Sampling is Controllable ( but with restrictions) scan time # samples RL

26 Compressed Sensing RL

27 Sparsity But what if the image is sparse? Sparse in Space Sparse in time Sparse in a transform domain wavelets rapid I Finite differences R L 27

28 Compressibility Lossless or visually lossless compression rapid I R L 28

29 Compressed Sensing Lossless or visually lossless compression rapid I R L 29

30 A Surprising Experiment * FT Randomly throw away 83% of samples * E.J. Candes, J. Romberg and T. Tao. rapid I R L 30

31 A Surprising Result * FT Minimum - norm Non-linear recon. Min. Total Variation (TV) rapid I * E.J. Candes, J. Romberg and T. Tao. R L 31

32 Compressed Sensing First Presented by: E.J. Candes, J. Romberg and T. Tao. Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information. D. Donoho Compressed Sensing rapid I R L 32

33 Compressed Sensing Basic idea: Sparse/Compressible signals can be accurately recovered from highly incomplete random Fourier samples. Recovery by solving a non-linear convex optimization problem. rapid I R L 33

34 CS Formulation minimize Ψm 1 s.t. Fm-y 2 ε m image F Random Fourier Sampling Operator y Measurements Ψ- Sparsifying transform rapid I R L 34

35 Reconstruction Formulation minimize Ψm 1 s.t. Fm-y 2 < ε Randomly undersampled Fourier Transform Enforces Data Consistency m image F Random Fourier Sampling Operator y Measurements Ψ- Sparsifying transform rapid I R L 35

36 Reconstruction Formulation minimize Ψm 1 Sparsity transform s.t. Fm-y 2 < ε x 1= xi - Crucial for the reconstruction! m image Ψ- Sparsifying transform (finite differences, wavelet) rapid I R L 36

37 Why Does it Work?.. And why random? Let s think simple, 1D signal. Sparse in signal domain rapid I 256 R L 37

38 Equispaced VS Random 4 Signal: K-space: Equi-spaced sampling rapid I random sampling R L 38

39 Equispaced VS Random 4 Signal: K-space: Equi-spaced sampling random sampling Inherent ambiguity rapid I R L 39

40 Equispaced VS Random 4 Signal: K-space: Equi-spaced sampling random sampling Burried in interference Inherent ambiguity rapid I Random noise R L 40

41 Equispaced Vs Random 4 Equispaced under-sampling 0 Inherent ambiguity Burried in interference Random undersampling 0-2 rapid I Random noise R L 41

42 Sparse Recovery from Random Sampling 4 Threshold Threshold 0-21 rapid I 256 R L 42

43 Sparse Recovery from Random Sampling 4 Threshold Threshold Get coefficients rapid I 256 R L 43

44 Sparse Recovery from Random Sampling 4 Threshold Threshold Get coefficients Compute interference Interference rapid I 256 R L 44

45 Sparse Recovery from Random Sampling 4 Threshold Threshold Get coefficients Compute interference Compute residual Interference rapid I 256 R L 45

46 Sparse Recovery from Random Sampling 4 Threshold Threshold Get coefficients Compute interference Compute residual 4 Threshold Interference Threshold rapid I 256 R L 46

47 Sparsity Revisited Sparse in Space 256 Sparse in a transform domain wavelets Finite differences rapid I R L 47

48 Here are the Important points: For Compressed Sensing to work we need: Sparsity Incoherent interference between the transform coefficients (random partial Fourier). Sparsity based reconstruction. rapid I R L 48

49 Compressed Sensing and I Practical schemes for CS in Magnetic Resonance Imaging RL

50 CS and I CS Signals are sparse/compressible. Random partial Fourier sampling. Small number of measurements. I Images are often compressible. Control on Fourier sampling. Scan time proportional to # samples. rapid I R L 50

51 CS and I CS Signals are sparse/compressible. Random partial Fourier sampling. Small number of measurements. I Images are often compressible. Control on Fourier sampling. Scan time proportional to # samples. rapid I R L 51

52 CS and I CS Signals are sparse/compressible. Random partial Fourier sampling. Small number of measurements. I Images are often compressible. Control on Fourier sampling. Scan time proportional to # samples. rapid I R L 52

53 Practical Schemes for random sampling Randomness is too important to be left to chance * *Robert R. Coveyou, Oak Ridge National Laboratory rapid I R L 53

54 Practical Schemes for random sampling Pure random sampling is impractical in I. Instead, design effectively random sampling. Incoherent aliasing interference Efficient for hardware and application Robust Tailor trajectory for application (Cartesian, spiral ) Randomly perturb to be effectively random. rapid I R L 54

55 Talk about.. Randomized 2D spirals Whole heart imaging with a single breathhold Randomized 3D Cartesian Angiography Teaser: Dynamic cardiac imaging rapid I R L 55

56 Perturbed Spirals Spirals are: Fast and efficient. Irregular sampling. Coherent interference rapid I R L 56

57 Perturbed Spirals Spirals are: Fast and efficient. Irregular sampling. Introduce randomness: Deviating from the analytic spiral costs a little. Perturbing angle for free Can also use Variable density more irregular rapid I R L 57

58 Coherency Randomness is GOOD! rapid I R L 58

59 Phantom Scans Min norm CS: Total Variation CS: L1 Wavelet 19/34 interleaves perturbed spiral Nominal FOV 16cm Resolution 1mm 3ms readout, GRE sequence rapid I R L 59

60 Whole Heart, Single Breath-hold Coronary Arteries Angiography* 2005: Heart related diseases #2 killer in the US. (#1 till 2004) and #1 in Israel. High-res cardiac imaging is very challenging Unfortunately (.. Or not), the heart moves. People insist on breathing! Many approaches, however, Long scan time (10-30 min). Partial coverage rapid I * with Juan Santos R L 60

61 Whole Heart Single Breath-Hold Imaging Trigger Trigger Trigger delay delay delay delay Real-time localization Trigger Start breathhold Slice 1 Slice 2 Slice N Short time frame rapid I R L 61

62 Coronary Imaging single breath-hold whole heart at 3T Min. Norm CS - Total Variation 17 Itlv Variable density spirals ~50% 5.6ms readout Nominal FOV mm resolution J. Santos, C. Cunningham, M. Lustig, B. Hargreaves, B. Hu, D. Nishimura, J. Pauly Single Breath-Hold Whole-Heart A Using Variable-Density Spirals at 3T 2005, Accepted Mag Res Med rapid I R L 62

63 Talk about.. Randomized 2D spirals Whole heart imaging with a single breathhold Randomized 3D Cartesian Angiography Teaser: Dynamic cardiac imaging rapid I R L 63

64 3D randomized Cartesian Ky Ky Ky Kx Kz rapid I Kz Kz R L 64

65 3D randomized Cartesian Randomly decimating phaseencodes is free. Purely random in 2D. Scan time in 3D is an issue. Robust trajectory. Easy to implement. Separability of kz-ky and kx. Many applications. rapid I Ky Kx Kz Ky Kz R L 65

66 3D randomly under sampled Cartesian angiography Angiograms are truly sparse. Very bright blood vessels, low background perfect! Blood vessels are piece-wise constant, use Total-Variation Speed essential, especially for contrast enhanced rapid I R L 66

67 Results Angiography Non-contast angiograms T2 prep with fat saturation. Data under-sampled retrospectively to 33%. 128x128x384 matrix, each slice reconstructed separately Recon with Total Variation. rapid I 100% 33% CS R L 67

68 Results Angiography Non-contast angiograms T2 prep with fat saturation. Data under-sampled retrospectively to 33%. 128x128x384 matrix, each slice reconstructed separately Recon with Total Variation. rapid I R L 68

69 More Angiography Low Res 5% 5% Min Norm Random Sampling TV 5% Random Sampling 9% 9% 9% 13% 13% 13% 20% 20% 20% 30% 30% 30% 50% 50% 50% 80% 80% 80% a. b. rapid I c. R L 69

70 More Angiography Narrowing Small vessels disappear Low Res 5% 5% Min Norm Random Sampling TV 5% Random sampling 9% 9% 9% 13% 13% 13% 20% 20% 20% 30% 30% 30% 50% 50% 50% 80% 80% 80% a. b. rapid I c. R L 70

71 Teaser RL

72 Hi Frame-Rate Dynamic Imaging Imaging a dynamic scene, results in aliasing in spatial temporal frequency space Sequential phase encode ordering ky kx t Very hard application rapid I R L 72

73 Hi Frame-Rate Dynamic Imaging But, Smooth& periodic signals are sparse in wavelet-fourier space space time wavelet Fourier Can we exploit sparsity? rapid I R L 73

74 Hi Frame-Rate Dynamic Imaging Cartesian 3D can be a 2D spatial 1D temporal. Randomize ordering! (For Free) Random line ordering ky kx t space time wavelet Fourier rapid I R L 74

75 Hi Frame-Rate Dynamic Imaging Reconstruction at 4-fold acceleration (16x4ms / frame) Spatial transform: wavelets Temporal transform: Fourier Ground Truth Sliding window L1 Ground Truth L1 wavelet-fourier space Sliding window time rapid I R L 75

76 Hi Frame-Rate Dynamic Imaging L1 wavelet-fourier Sliding window TR = 4.5ms Res = 2.5mm Slice = 9mm 4.5 sec acquisition (1152*4.5ms) 7-fold acceleration (25FPS) rapid I (b) (a) R L 76

77 Main Points Find an impossible/important application. Make sure the signal is sparse. Design a robust & efficient randomized sampling scheme tailored to the application. rapid I R L 77

78 Conclusions Promising applications. Exploit sparsity where sparsity exists. Choose application wisely. Transformations. Recon. Algorithms. Clinical trials the Doc test. rapid I R L 78

79 The END rapid I R L 79

80 The END rapid I R L 80