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75 nips-2008-Estimating vector fields using sparse basis field expansions

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Author: Stefan Haufe, Vadim V. Nikulin, Andreas Ziehe, Klaus-Robert Müller, Guido Nolte

Abstract: We introduce a novel framework for estimating vector ﬁelds using sparse basis ﬁeld expansions (S-FLEX). The notion of basis ﬁelds, which are an extension of scalar basis functions, arises naturally in our framework from a rotational invariance requirement. We consider a regression setting as well as inverse problems. All variants discussed lead to second-order cone programming formulations. While our framework is generally applicable to any type of vector ﬁeld, we focus in this paper on applying it to solving the EEG/MEG inverse problem. It is shown that signiﬁcantly more precise and neurophysiologically more plausible location and shape estimates of cerebral current sources from EEG/MEG measurements become possible with our method when comparing to the state-of-the-art. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Estimating vector ﬁelds using sparse basis ﬁeld expansions Stefan Haufe1, 2, * Vadim V. [sent-1, score-0.371]

2 de Abstract We introduce a novel framework for estimating vector ﬁelds using sparse basis ﬁeld expansions (S-FLEX). [sent-6, score-0.371]

3 The notion of basis ﬁelds, which are an extension of scalar basis functions, arises naturally in our framework from a rotational invariance requirement. [sent-7, score-0.654]

4 We consider a regression setting as well as inverse problems. [sent-8, score-0.125]

5 All variants discussed lead to second-order cone programming formulations. [sent-9, score-0.111]

6 While our framework is generally applicable to any type of vector ﬁeld, we focus in this paper on applying it to solving the EEG/MEG inverse problem. [sent-10, score-0.212]

7 It is shown that signiﬁcantly more precise and neurophysiologically more plausible location and shape estimates of cerebral current sources from EEG/MEG measurements become possible with our method when comparing to the state-of-the-art. [sent-11, score-0.414]

8 Such “truly” vectorial functions are called vector ﬁelds and become manifest for example in optical ﬂow ﬁelds, electromagnetic ﬁelds and wind ﬁelds in meteorology. [sent-16, score-0.257]

9 The ﬁrst type are direct samples (xn , yn ), xn ∈ RP , yn ∈ RQ , n = 1, . [sent-22, score-0.28]

10 The second case occurs, if only indirect measurements zm ∈ R, m = 1, . [sent-26, score-0.127]

11 , M are available, which we assume to be generated by a known linear1 transformation of the vector ﬁeld outputs yn belonging to nodes xn , n = 1, . [sent-29, score-0.368]

12 This kind of estimation problem is known as an inverse problem. [sent-33, score-0.125]

13 , yN )T the N × Q matrix of vector ﬁeld outputs and vec(Y ) a column vector containing the stacked transposed rows of Y . [sent-40, score-0.153]

14 1 As an example of an inverse problem consider the way humans localize acoustic sources. [sent-43, score-0.162]

15 Here z comprises the signal arriving at the ears, v is the spatial distribution of the sound sources and F is given by physical equations of sound propagation. [sent-44, score-0.186]

16 regularization) is indeed the most effective strategy for solving inverse problems [13], which are inherently ambiguous. [sent-52, score-0.158]

17 , regression may be applied to cope with the ambiguity of inverse problems. [sent-55, score-0.125]

18 For the estimation of scalar functions, methods that utilize sparse linear combinations of basis functions have gained considerable attention recently (e. [sent-56, score-0.342]

19 Apart from the computational tractability that comes with the sparsity of the learned model, the possibility of interpreting the estimates in terms of their basis functions is a particularly appealing feature of these methods. [sent-59, score-0.339]

20 While sparse expansions are also desirable in vector ﬁeld estimation, lasso and similar methods cannot be used for that purpose, as they break rotational invariance in the output space RQ . [sent-60, score-0.466]

21 This is easily seen as sparse methods tend to select different basis functions in each of the Q dimensions. [sent-61, score-0.298]

22 Only few attempts have been made on rotation-invariant sparse vector ﬁeld expansions so far. [sent-62, score-0.187]

23 In [8] a dense expansion is discussed, which could be modiﬁed to a sparse version maintaining rotational invariance. [sent-63, score-0.258]

24 In section 3 we will apply the (appropriately customized) method for solving the EEG/MEG inverse problem. [sent-67, score-0.158]

25 2 Method Our model is based on the assumption that v can be well approximated by a linear combination of some basis ﬁelds. [sent-69, score-0.184]

26 A basis ﬁeld is deﬁned here (unlike in [8]) as a vector ﬁeld, in which all output vectors point in the same direction, while the magnitudes are proportional to a scalar (basis) function b : RP → R. [sent-70, score-0.362]

27 1, this model has an expressive power which is comparable to a basis function expansion of scalar functions. [sent-72, score-0.284]

28 Given a set (dictionary) of basis functions bl (x), l = 1, . [sent-73, score-0.337]

29 , L, the basis ﬁeld expansion is written as L cl bl (x) , v(x) = (1) l=1 with coefﬁcients cl ∈ RQ , l = 1, . [sent-76, score-0.582]

30 Note that by including one coefﬁcient for each output dimension, both orientations and proportionality factors are learned in this model (the term “basis ﬁeld” thus refers to a basis function with learned coefﬁcients). [sent-80, score-0.184]

31 In order to select a small set of ﬁelds, most of the coefﬁcient vectors cl have to vanish. [sent-81, score-0.117]

32 However, care has to be taken in order to maintain rotational invariance of the solution. [sent-83, score-0.242]

33 We here propose to use a regularizer that imposes sparsity and is invariant with respect to rotations, namely the 1 -norm of the magnitudes of the coefﬁcient vectors. [sent-84, score-0.241]

34 bL (xN ) the basis functions evaluated at the xn . [sent-100, score-0.339]

35 2 1 2 3 SUM Figure 1: Complicated vector ﬁeld (SUM) as a sum of three basis ﬁelds (1-3). [sent-106, score-0.238]

36 One has to distinguish invariance in input- from invariance in output space. [sent-109, score-0.218]

37 The former requirement may arise in many estimation settings and can be fulﬁlled by the choice of appropriate basis functions bl (x). [sent-110, score-0.337]

38 The latter one is speciﬁc to vector ﬁeld estimation and has to be assured by formulating a rotationally invariant cost function. [sent-111, score-0.178]

39 for an orthogonal matrix R ∈ RQ×Q , RT R = I L L Rcl 2 l=1 L tr(cT RT Rcl ) = l = l=1 cl 2 . [sent-117, score-0.117]

40 (4) l=1 For the same argument, additional regularizers R∗ (C) = vec(D∗ C) 2 (the well-known Tikhonov 2 regularizer) or R+ (C) = D+ C 1,2 (promoting sparsity of the linearly transformed vectors) may be introduced without breaking the rotational invariance in RQ . [sent-118, score-0.342]

41 5 is an instance of second-order cone programming (SOCP), a standard class of convex programs, for which efﬁcient interior-point based solvers are available. [sent-133, score-0.111]

42 This approach also requires complex coefﬁcients, by which it is then possible not only to optimally scale the basis functions, but also to optimally shift their phase. [sent-142, score-0.184]

43 Similarly, it is possible to reconstruct complex vector ﬁelds from complex measurements using real-valued basis functions. [sent-143, score-0.367]

44 3 3 Application to the EEG/MEG inverse problem Vector ﬁelds occur, for example, in form of electrical currents in the brain, which are produced by postsynaptic neuronal processes. [sent-144, score-0.211]

45 Invasive measurements allow very local assessment of neuronal activations, but such procedure in humans is only possible when electrodes are implanted for treatment/diagnosis of neurological diseases, e. [sent-146, score-0.201]

46 The reconstruction of the current density from such measurements is an inverse problem. [sent-150, score-0.396]

47 1 Method speciﬁcation In the following the task is to infer the generating cerebral current density given an EEG measurement z ∈ RM . [sent-152, score-0.138]

48 The current density is a vector ﬁeld v : R3 → R3 assigning a vectorial current source to each location in the brain. [sent-153, score-0.308]

49 Inside the brain, we arranged 2142 nodes in a regular grid of 1 cm distance. [sent-155, score-0.12]

50 The forward mapping F ∈ RM ×2142·3 from these nodes to the electrodes was constructed according to [9] – taking into account the realistic geometry and conductive properties of brain, skull and skin. [sent-156, score-0.155]

51 Dictionary In most applications the “true” sources are expected to be small in number and spatial extent. [sent-157, score-0.186]

52 However, many commonly used methods estimate sources that almost cover the whole brain (e. [sent-158, score-0.197]

53 Another group of methods delivers source estimates that are spatially sparse, but usually not rotationally invariant (e. [sent-161, score-0.169]

54 Both the very smooth and the very sparse estimates are unrealistic from a physiological point of view. [sent-165, score-0.114]

55 For achieving a similar effect we here propose a sparse basis ﬁeld expansion using radial basis functions. [sent-167, score-0.493]

56 More speciﬁcally we consider spherical Gaussians 1 −3 2 −2 bn,s (x) = (2πσs ) 2 exp − x − xn 2 σs (6) 2 s = 1, . [sent-168, score-0.11]

57 5 cm, σ4 = 2 cm and being centered at nodes xn , n = 1, . [sent-173, score-0.23]

58 Using this redundant dictionary our expectation is that sources of different spatial extent can be reconstructed by selecting the appropriate basis functions. [sent-178, score-0.417]

59 Figure 2: Gaussian basis functions with ﬁxed center and standard deviations 0. [sent-180, score-0.229]

60 Normalization Our 1,2 -norm based regularization is a heuristic for selecting the smallest possible number of basis ﬁelds necessary to explain the measurement. [sent-182, score-0.184]

61 It is therefore important to normalize the basis functions in order not to a-priori prefer some of them. [sent-184, score-0.229]

62 Let Bs be the N × N matrix containing the basis functions with standard deviation σs . [sent-185, score-0.229]

63 Due to volume conduction, the signal captured at the sensors is much stronger for superﬁcial sources compared to deep sources. [sent-192, score-0.142]

64 This can be done by either penalizing activity at locations with high variance or by penalizing basis functions with high variance in the center. [sent-195, score-0.229]

65 We here employ the former approach, as the latter may be problematic for basis functions with large extent. [sent-196, score-0.229]

66 Therefore, we restrict ourselves here to nodes xn , n = 1, . [sent-198, score-0.149]

67 Let Wn ∈ R3×3 denote the inverse matrix square root of ˆ the part of S belonging to node xn . [sent-202, score-0.27]

68 WN ˆ the coefﬁcients are estimated using C = arg min C C ˆ estimated current density at node xn is v(xn ) = Wn 3. [sent-218, score-0.215]

69 Experiments Validation of methods for inverse reconstruction is generally difﬁcult due to the lack of a “ground truth”. [sent-223, score-0.199]

70 The measurements z cannot be used in this respect, as the main goal is not to predict the EEG/MEG measurements, but the vector ﬁeld v(x) as accurately as possible. [sent-224, score-0.146]

71 Therefore, the only way to evaluate inverse methods is to assess their ability to reconstruct known functions. [sent-225, score-0.162]

72 We do this by reconstructing a) simulated current sources and b) sources of real EEG data that are already well-localized by other studies. [sent-226, score-0.433]

73 we perform 25 inverse reconstructions based on different training sets containing 80 % of the electrodes. [sent-229, score-0.16]

74 Most important ˆ ˆ is the reconstruction error, deﬁned as Cy = vec(Y )/ vec(Y ) 2 − vec(Y tr )/ vec(Y tr ) 2 2 , ˆ where Y tr are the vector ﬁeld outputs at nodes xn , n = 1, . [sent-231, score-0.493]

75 This is deﬁned as Cz = zte − F te vec(Y tr ) 2 , where zte and F te are 2 the parts of z and F belonging to the test set. [sent-239, score-0.27]

76 We compared the sparse basis ﬁeld expansion (S-FLEX) approach using Gaussian basis functions (see section 3. [sent-240, score-0.538]

77 All three competitors correspond to using unit impulses as basis functions while employing different regularizers. [sent-242, score-0.272]

78 , is a Tikhonov regularized least-squares estimate while MCE is equivalent to applying lasso to each dimension separately, yielding current vectors that are biased towards being axes-parallel. [sent-245, score-0.105]

79 Interestingly, FVR can be interpreted as a special case of S-FLEX employing the rotation-invariant regularizer R+ (C) to enforce both sparsity and smoothness. [sent-248, score-0.117]

80 Simulated data We simulated current densities in the following way. [sent-252, score-0.149]

81 Finally, each yn was shortened by the 90th percentile of the magnitudes of all yn – leaving only 10% of the current vectors active. [sent-259, score-0.318]

82 We simulated ﬁve densities and computed respective pseudo-measurements for 118 channels using the forward model F . [sent-263, score-0.125]

83 Real data We recorded 113-channel EEG of one healthy subject (male, 26 years) during electrical median nerve stimulation. [sent-265, score-0.121]

84 Artifactual trials as well as artifactual electrodes were excluded from the analysis. [sent-275, score-0.115]

85 Finally, a single measurement vector was constructed by averaging the EEG amplitudes at 21 ms across 1946 trials (50% left hand, 50% right hand). [sent-277, score-0.153]

86 3 shows a simulated current density along with reconstructions according to LORETA, MCE, FVR and S-FLEX. [sent-282, score-0.221]

87 From the ﬁgure it becomes apparent, that LORETA and MCE do not approximate the true current density very well. [sent-283, score-0.105]

88 The MCE solution consists of eight spikes scattered across the whole somatosensory area. [sent-298, score-0.132]

89 This is due to the fact that the parameter of FVR controlling the tradeoff between sparsity and smoothness was ﬁxed here to a value promoting “maximally sparse sources which are still smooth”. [sent-304, score-0.319]

90 60 Table 1: Ability of LORETA, FVR, S-FLEX and MCE to reconstruct simulated currents (Cy SIM) and generalization performance with respect to the EEG measurements (Cz SIM/REAL). [sent-331, score-0.253]

91 6 SIM LORETA FVR S-FLEX MCE Figure 3: Simulated current density (SIM) and reconstruction according to LORETA, FVR, S-FLEX and MCE. [sent-333, score-0.179]

92 LORETA FVR S-FLEX MCE Figure 4: Localization of somatosensory evoked N20 generators according to LORETA, FVR, S-FLEX and MCE. [sent-335, score-0.125]

93 7 4 Conclusion and Outlook This paper contributes a novel and general methodology for obtaining sparse decompositions of vector ﬁelds. [sent-337, score-0.123]

94 Interestingly, the latter constraint together with sparsity leads to a second-order cone programming formulation. [sent-339, score-0.176]

95 We have focussed here on solving the EEG/MEG inverse problem, where our proposed S-FLEX approach outperformed the state-of-the-art in approximating the true shape of the current sources. [sent-340, score-0.226]

96 Combining sparsity and rotational invariu ance in EEG/MEG source reconstruction. [sent-371, score-0.198]

97 From basis functions to basis ﬁelds: vector ﬁeld approximation from sparse data. [sent-417, score-0.536]

98 Low resolution electromagnetic tomography: a new method for localizing electrical activity in the brain. [sent-445, score-0.107]

99 Penalized least squares methods for solving the EEG inverse problem. [sent-486, score-0.158]

100 An empirical bayesian strategy for solving the simultaneous sparse approximation problem. [sent-495, score-0.102]

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