nips nips2008 nips2008-30 knowledge-graph by maker-knowledge-mining

30 nips-2008-Bayesian Experimental Design of Magnetic Resonance Imaging Sequences

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Author: Hannes Nickisch, Rolf Pohmann, Bernhard Schölkopf, Matthias Seeger

Abstract: We show how improved sequences for magnetic resonance imaging can be found through optimization of Bayesian design scores. Combining approximate Bayesian inference and natural image statistics with high-performance numerical computation, we propose the ﬁrst Bayesian experimental design framework for this problem of high relevance to clinical and brain research. Our solution requires large-scale approximate inference for dense, non-Gaussian models. We propose a novel scalable variational inference algorithm, and show how powerful methods of numerical mathematics can be modiﬁed to compute primitives in our framework. Our approach is evaluated on raw data from a 3T MR scanner. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 de Abstract We show how improved sequences for magnetic resonance imaging can be found through optimization of Bayesian design scores. [sent-5, score-0.562]

2 Combining approximate Bayesian inference and natural image statistics with high-performance numerical computation, we propose the ﬁrst Bayesian experimental design framework for this problem of high relevance to clinical and brain research. [sent-6, score-0.369]

3 We propose a novel scalable variational inference algorithm, and show how powerful methods of numerical mathematics can be modiﬁed to compute primitives in our framework. [sent-8, score-0.222]

4 1 Introduction Magnetic resonance imaging (MRI) [7, 2] is a key diagnostic technique in healthcare nowadays, and of central importance for experimental research of the brain. [sent-10, score-0.215]

5 To select the optimum sequence for a given problem, and to tune its parameters, is a difﬁcult task even for experts, and even more challenging is the design of new, customized sequences to address a particular question, making sequence development an entire ﬁeld of research [1]. [sent-13, score-0.339]

6 The main drawbacks of MRI are high initial and running costs, since a very strong homogeneous magnetic ﬁeld has to be maintained, moreover long scanning times due to weak signals and limits to gradient amplitude. [sent-14, score-0.129]

7 Beyond reduced costs, faster imaging also leads to higher temporal resolution in dynamic sequences for functional MRI (fMRI), less annoyance to patients, and fewer artifacts due to patient motion. [sent-16, score-0.286]

8 In this paper, we employ Bayesian experimental design to optimize MRI sequences. [sent-17, score-0.157]

9 Image reconstruction from MRI raw data is viewed as a problem of inference from incomplete observations. [sent-18, score-0.226]

10 For most sequences used in hospitals today, reconstruction is done by a single fast Fourier transform (FFT). [sent-20, score-0.244]

11 However, natural and MR images show stable low-level statistical properties,1 which allows them to be reconstructed from 1 These come from the presence of edges and smooth areas, which on a low level deﬁne image structure, and which are not present in Gaussian data (noise). [sent-21, score-0.149]

12 A similar idea is known as compressed sensing (CS), which has been applied to MRI [8]. [sent-24, score-0.125]

13 It has been proposed to design sequences by blindly randomizing aspects thereof [8], based on CS theoretical results. [sent-28, score-0.302]

14 Our proposal requires efﬁcient Bayesian inference for MR images of realistic resolution. [sent-31, score-0.109]

15 We present a novel scalable variational approximate inference algorithm inspired by [16]. [sent-32, score-0.187]

16 Finally, we are not aware of Bayesian or classical experimental design methods for dense non-Gaussian models, scaling comparably to ours. [sent-36, score-0.157]

17 Our model and experimental design framework are described in Section 2, a novel scalable approximate inference algorithm is developed in Section 3, and our framework is evaluated on a large-scale realistic setup with scanner raw data in Section 4. [sent-38, score-0.487]

18 Under ideal conditions, the raw data y ∈ Rm from the scanner is a linear map3 of u, motivating the likelihood y = Xu + ε, ε ∼ N (0, σ 2 I). [sent-41, score-0.186]

19 In the context of this paper, the problem of experimental design is how to choose X within a space of technically feasible sequences, so that u can be best recovered given y. [sent-43, score-0.157]

20 The posterior has the form q τ e−˜j |sj | , P (u|y) ∝ N (y|Xu, σ 2 I) s = Bu, τj = τj /σ, ˜ (1) j=1 the prior being a product of Laplacians on linear projections sj of u, among them the image gradient and wavelet coefﬁcients. [sent-46, score-0.175]

21 In the variant of Bayesian sequential experimental design used here, an extension of X by X∗ ∈ Rd,n is scored by the entropy difference ∆(X∗ ) := H[P (u|y)] − EP (y∗ |y) [H[P (u|y, y∗ )]] , (2) where P (u|y, y∗ ) is the posterior after including (X∗ , y∗ ). [sent-52, score-0.235]

22 After each extension, a new scanner measurement is obtained for the single extended sequence only. [sent-58, score-0.224]

23 Our Bayesian predictive approach allows us to score many candidates (X∗ , y∗ ) without performing costly MR measurements for them. [sent-59, score-0.115]

24 First, MR sequences naturally decompose in a sequential fashion: they describe a discontinuous path of several smooth trajectories (see Section 4). [sent-61, score-0.219]

25 Finally, the computational complexity of optimizing over complete sequences is staggering. [sent-63, score-0.127]

26 3 Scalable Approximate Inference In this section, we propose a novel scalable algorithm for the variational inference approxτ imation proposed in [3]. [sent-65, score-0.154]

27 This algorithm is also essential for scoring many candidates in each design step of our method (see Section 3. [sent-89, score-0.19]

28 It was found in [12] that aggressive sparsiﬁcation, notwithstanding being computationally convenient, hurts experimental design (and even reconstruction) for natural images. [sent-102, score-0.157]

29 For nc candidates of d rows, computing scores would need d · nc LCG runs, which is not feasible. [sent-113, score-0.146]

30 An MR scanner acquires Fourier coefﬁcients Y (k) at spatial frequencies5 k (the 2d Fourier domain is called k-space), along smooth trajectories k(t) determined by magnetic ﬁeld gradients g(t). [sent-131, score-0.362]

31 In Cartesian sampling, trajectories are parallel equispaced lines in k-space, so the FFT can be used for image reconstruction. [sent-135, score-0.287]

32 It is often used for dynamic studies, such as cardiac imaging and fMRI. [sent-138, score-0.125]

33 As for other reconstruction methods, most of our running time is spent in the gridding (MVMs with X, X T , and X∗ ). [sent-143, score-0.18]

34 gy in [mT/m] gx in [mT/m] For our experiments, we acquired data on an equispaced grid. [sent-144, score-0.148]

35 7 In r−space: U(r) k−space: Y(k) gradients: g(t) theory, the image u is real-valued; 50 n 1/2 in reality, due to resonance fre0 quency offsets, magnetic ﬁeld inhomogeneities, and eddy currents [1, −50 0 2 4 6 0 50 ch. [sent-145, score-0.278]

36 8 Note Figure 1: MR signal acquisition: r-space and k-space representhat |utrue |, against which recon- tation of the signal on a rectangular grid as well as the trajectory structions are judged below, is not al- obtained by means of magnetic ﬁeld gradients tered by this correction. [sent-150, score-0.186]

37 From the corrected raw data, we simulate all further measurements under different sequences using gridding interpolation. [sent-151, score-0.296]

38 While no noise is added to these measurements, there remain signiﬁcant highfrequency erroneous phase contributions in utrue . [sent-152, score-0.226]

39 Interleaved outgoing Archimedian spirals employ trajectories k(t) ∝ θ(t)ei2π[θ(t)+θ0 ] , θ(0) = 0, where the gradient g(t) ∝ dk/dt grows to maximum strength at the slew rate, then stays there [1, ch. [sent-153, score-0.195]

40 For equispaced offset angles θ0 , the Nyquist spiral (respecting the limit radially) has Nshot = 16. [sent-160, score-0.49]

41 Our goal is to design spiral sequences with smaller Nshot , reducing scan time by a factor 16/Nshot . [sent-161, score-0.505]

42 Since utrue is approximately real-valued, measurements at k and −k are quite redundant, which is why we restrict9 ourselves to offset angles θ0 ∈ [0, π). [sent-164, score-0.357]

43 We score candidates (π/256)[0 : 255] in each round, comparing to equispaced placements jπ/Nshot , and to drawing θ0 uniformly at random. [sent-165, score-0.214]

44 For the former, favoured by MRI practitioners right now, the maximum k-space distance between samples is minimized, while the latter is aligned with compressed sensing recommendations [8]. [sent-166, score-0.125]

45 For a given sequence, we consider different image reconstructions: the posterior mode (convex MAP estimation) [8], linear least squares (LS; linear conjugate gradients), and zero ﬁlling with density compensation (ZFDC; based on Voronoi diagram) [1, ch. [sent-167, score-0.162]

46 We selected the τ scale parameters (there are two of them, as in [12]) optimally for the Nyquist spiral Xnyq , and set σ 2 to the variance of Xnyq (utrue − |utrue |). [sent-172, score-0.203]

47 10 We report L2 distances between reconstruction and true image |utrue |. [sent-174, score-0.209]

48 8 We sample the center of k-space on a p × p Cartesian grid, obtaining a low-resolution reconstruction ˜ by FFT, whose phase ϕ we use to correct the raw data. [sent-181, score-0.231]

49 While reconstruction errors generally decrease somewhat with larger p, the relative differences between all settings below are insensitive to p. [sent-183, score-0.117]

50 9 Dropping this restriction disfavours equispaced {θ0 } setups with even Nshot . [sent-184, score-0.185]

51 28 slices 2,4,6,10,12,14 from design of slice 8 MAPop MAPeq LSop LSeq 9. [sent-293, score-0.293]

52 27 Figure 3: Results for spiral interleaves on slices 8, 12 (table left). [sent-329, score-0.387]

53 Offset angles θ0 ∈ [0, π): op (optimized; our method), rd (uniformly random; avg. [sent-331, score-0.117]

54 errors for slices 2,4,6,10,14, measured with sequences optimized on slice 8. [sent-335, score-0.343]

55 Table lower right: Results for Nyquist spiral eq[Nshot = 16]. [sent-336, score-0.203]

56 The standard reconstruction method ZFDC is improved upon strongly by LS (both are linear, but LS is iterative), which in turn is improved upon signiﬁcantly by MAP. [sent-337, score-0.15]

57 This is true even for the Nyquist spiral (Nshot = 16). [sent-338, score-0.203]

58 Results such as ours, together with the availability of affordable highperformance digital computation, strongly motivate the transition away from direct signal processing reconstruction algorithms to modern iterative statistical estimators. [sent-341, score-0.15]

59 Note that ZFDC (and, to a lesser extent, LS) copes best with equispaced designs, while MAP works best with optimized angles. [sent-342, score-0.195]

60 This is because the optimized designs leave larger gaps in k-space (see Figure 4). [sent-343, score-0.143]

61 Nonlinear estimators can interpolate across such gaps to some extent, using image sparsity priors. [sent-344, score-0.137]

62 Methods like ZFDC merely interpolate locally in k-space, uninformed about image statistics, so that violations of the Nyquist limit anywhere necessarily translate into errors. [sent-345, score-0.137]

63 It is clearly evident that drawing the spiral offset angles at random does not work well, even if MAP reconstruction is used as in [8]. [sent-346, score-0.459]

64 Our 6 results strongly suggest that randomizing MR sequences is not a useful design principle. [sent-353, score-0.335]

65 Reasons why CS theory as yet fails to guide measurement design for real images, are reviewed there, see also [15]. [sent-355, score-0.175]

66 The outcome of our Bayesian optimized design is stable, in that sequences found in several repetitions gave almost identical reconstruction performance. [sent-357, score-0.415]

67 Since utrue is close to real, both Slice 8, Nshot=8 Slice 12, Nshot=8 attain close to Nyquist performance up from Nshot = 8. [sent-359, score-0.169]

68 Breaking up such regular designs seems to be the major role of random- Figure 4: Spirals found by our algorithm. [sent-371, score-0.141]

69 The ordering ization in CS theory, but our results show that is color-coded: dark spirals selected ﬁrst. [sent-372, score-0.106]

70 We see that approximate Bayesian experimental design is useful to optimize measurement architectures for subsequent MAP reconstruction. [sent-374, score-0.241]

71 To our knowledge, no similar design optimization method based purely on MAP estimation has been proposed (ours needs approximate inference), rendering the beneﬁcial interplay between our framework and subsequent MAP estimation all the more interesting. [sent-375, score-0.157]

72 Our implementation requires about 5 hours on a single standard desktop machine to optimize 11 angles sequentially, 256 candidates per extension, with n and d as above. [sent-377, score-0.145]

73 It is neither feasible nor desirable on most current MR scanners to optimize the sequence during the measurement, so an important question is whether sequences optimized on some slices work better in general as well (for the same contrast and similar objects). [sent-379, score-0.317]

74 13 Two spirals found by our method are shown in Figure 4 (2 of 8 interleaves, Nshot = 8). [sent-382, score-0.106]

75 5 Discussion We have presented the ﬁrst scalable Bayesian experimental design framework for automatically optimizing MRI sequences, a problem of high impact on clinical diagnostics and brain research. [sent-385, score-0.251]

76 Any spiral interleave samples more closely around the origin. [sent-388, score-0.237]

77 In fact, the sampling density as a function of spatial frequency |k(t)| does not depend on the offset angles θ0 . [sent-389, score-0.139]

78 12 In another set of experiments (not shown), we compared optimization, randomization, and equispacing of θ0 ∈ [0, 2π), in disregard of the approximate real-valuedness of utrue . [sent-390, score-0.244]

79 Artiﬁcial phantoms of extremely simple structure, often used in MR sequence design, are not suitable in that respect. [sent-393, score-0.107]

80 Real MR images are much more complicated than simple phantoms, even in low level statistics, and results obtained on phantoms only should not be given overly high attendance. [sent-394, score-0.12]

81 We demonstrated the power of our approach in a study with spiral sequences, using raw data from a 3T MR scanner. [sent-396, score-0.26]

82 The sequences found by our method lead to reconstructions of high quality, even though they are faster than traditionally used Nyquist setups by a factor up to two. [sent-397, score-0.219]

83 They improve strongly on sequences obtained by blind randomization. [sent-398, score-0.16]

84 Moreover, across all designs, nonlinear Bayesian MAP estimation was found to be essential for reconstructions from undersamplings, and our design optimization framework is especially useful for subsequent MAP reconstruction. [sent-399, score-0.179]

85 Our results strongly suggest that modiﬁcations to standard sequences can be found which produce similar images at lower cost. [sent-400, score-0.217]

86 Namely, with so many handles to turn in sequence design nowadays, this is a high-dimensional optimization problem dealing with signals (images) of high complexity, and human experts can greatly beneﬁt from goal-directed machine exploration. [sent-401, score-0.168]

87 Randomizing parameters of a sequence, as suggested by compressed sensing theory, helps to break wasteful symmetries in regular standard sequences. [sent-402, score-0.17]

88 As our results show, many of the advantages of regular sequences are lost by randomization though. [sent-403, score-0.213]

89 The optimization of Bayesian information leads to irregular sequences as well, improving on regular, and especially on randomized designs. [sent-404, score-0.127]

90 Finally, our framework seems also promising for real-time imaging [1, ch. [sent-407, score-0.125]

91 4], where the scanner allows for on-line adaptations of the sequence depending on measurement feedback. [sent-409, score-0.224]

92 This will come with new problems not addressed in Section 4, such as phase or image errors that depend on the sequence employed14 (which could be accounted for by a more elaborate noise model). [sent-412, score-0.193]

93 In our experiments in Section 4, the choice of different offset angles is cost-neutral, but when a larger set of candidates is used, respective costs have to be quantiﬁed in terms of real scan time, error-proneness, heating due to rapid gradient switching, and other factors. [sent-413, score-0.256]

94 FLASH imaging: Rapid NMR imaging a using low ﬂip-angle pulses. [sent-450, score-0.125]

95 RARE imaging: A fast imaging method for clinical MR. [sent-459, score-0.16]

96 Image formation by induced local interactions: Examples employing nuclear magnetic resonance. [sent-466, score-0.169]

97 Sparse MRI: The application of compressed sensing for rapid MR imaging. [sent-472, score-0.125]

98 Bayesian inference and optimal design for the sparse linear model. [sent-492, score-0.209]

99 Large scale variational inference and experimental design for sparse generalized linear models. [sent-502, score-0.285]

100 14 Some common problems with spirals are discussed in [1, ch. [sent-520, score-0.106]

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