CN109948529B - fMRI data space source phase mapping method from real number domain to complex number domain - Google Patents

fMRI data space source phase mapping method from real number domain to complex number domain Download PDF

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CN109948529B
CN109948529B CN201910205693.0A CN201910205693A CN109948529B CN 109948529 B CN109948529 B CN 109948529B CN 201910205693 A CN201910205693 A CN 201910205693A CN 109948529 B CN109948529 B CN 109948529B
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林秋华
张超颖
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Dalian University of Technology
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Abstract

The invention discloses a method for mapping fMRI data space source phase from a real number domain to a complex number domain, and belongs to the field of biomedical signal processing. Firstly, carrying out spatial ICA separation on fMRI real number domain phase data, carrying out polarity correction on a source component obtained by ICA according to a spatial reference network of an interested component, and selecting the interested component; then, selecting a spatial source phase of the interested component, and carrying out positive value taking and spatial smoothing pretreatment on the spatial source phase; then, estimating a distribution histogram of the space source phase value, generating symmetrical data points, and solving a function fitting the data points; and finally, transforming the value domain of the fitting function to [0, pi ] to obtain a mapping function, and using the mapping function for mapping the space source phase from a real number domain to a complex number domain. The invention expands the noise elimination advantage of the complex field space source phase to the real number field, overcomes the performance disadvantage of the traditional threshold noise elimination method, and provides a new means for post-processing noise elimination of fMRI data analysis.

Description

fMRI data space source phase mapping method from real number domain to complex number domain
Technical Field
The invention relates to the field of biomedical signal processing, in particular to a mapping method of functional magnetic resonance imaging (fMRI) data space source phase from a real domain to a complex domain.
Background
fMRI is a medical imaging technique for measuring brain neuron activity based on the Blood Oxygen Level Dependent (BOLD) effect. Experimentally acquired fMRI data is complex, comprising both magnitude and phase components. The existing research shows that fMRI phase data contains unique brain function information with definite physiological significance. With the data-driven Independent Component Analysis (ICA) algorithm, a series of maximally mutually independent spatially activated brain (SM) components and their time series (TC) components can be extracted from fMRI data without any a priori knowledge.
In particular, fMRI magnitude and phase data both belong to the real number domain, and both constitute fMRI complex field data. The SM and TC components separated from fMRI data using the ICA method belong to the source signal. Thus, two SM phase components (referred to as spatial source phases) can be acquired from fMRI data. (1) Complex field spatial source phase: the phase, value range, of the complex SM component separated by ICA from fMRI complex data is determined as [ -pi, pi ]; (2) real number domain spatial source phase: the SM component, separated from fMRI real-number domain phase data using ICA, is value-domain random.
The existing research shows that the complex field space source phase has strong noise elimination capability. In all the voxels of SM, the BOLD related voxels exhibit a small phase, i.e., the BOLD related voxels correspond to small phase values within the range of [ - π/4, π/4], while the noisy voxels correspond to large phase values outside of [ - π/4, π/4] (Yu MC, Lin QH, Kuang LD, Gong XF, Cong F, Calhoun VD. ICA of full complex-valued fMRI data using phase information of spatial maps. journal of neurological Methods 249, 75-91,2015). By utilizing the characteristic of complex domain space source phase, which is called as small-phase characteristic, voxels related to BOLD can be positioned, and a large number of noise voxels which cannot be removed by a threshold method, especially brain edge noise, can be removed.
In contrast, the real number domain space source phase position shows high application value in brain network identification and brain function connection analysis. First, combining the real-number domain spatial source phase with the spatial source amplitude (the SM component extracted by the ICA from fMRI real-number domain amplitude data) can improve the accuracy of healthy person and patient classification. Secondly, the brain function connection mode of the real number domain space source phase and the source amplitude is different, and a new biomarker is provided for the identification of brain diseases. Finally, at higher ICA model orders, real number domain spatial source amplitude tends to split, and spatial source phase still has complete components, which is more advantageous in the classification of healthy people and patients.
However, no real number domain spatial source phase has been reported in the literature in terms of noise cancellation capability. In effect, the real-domain spatial source phase and the complex-domain spatial source phase delineate the phase variation of the brain activation voxels from different sides. Existing fMRI analysis shows that the spatial source amplitude of the real domain component of interest belongs to a hyperstatic distribution (corean, AdaliT, LiYO, calhound. complex of noise associated with geographic information for fmriusingannew matlab toolbox: gift. ieeee international convergence acoustics,2005), while the spatial source phase of the real domain component of interest is relatively similar to the source amplitude (ChenZ, CaprihanA, DamarajuE, rachakoands, calhound. functional analysis of nuclear synthesis data. journal of nuclear synthesis center methods293, 299-309,2018), so that it also obeys the hyperstatic distribution, as well as the actual fMRI analysis experiments verify this. In a super-gaussian distribution, a significant portion of the source phase values are concentrated near the peak and correspond to noisy voxels, which match the small phase characteristic of the complex-domain spatial source phase. Therefore, if the small phase characteristic of the complex number domain space source phase is used as a priori, a mapping function is obtained according to the distribution of the real number domain space source phase, and a mapping method of the space source phase from the real number domain to the complex number domain is found, the noise elimination capability of the real number domain space source phase can be discovered, so that the method plays a great role in space source amplitude and source phase noise elimination (currently, a threshold method is adopted, the threshold value is set subjectively) and amplitude-phase combined analysis.
Disclosure of Invention
The invention provides a mapping method of a space source phase from a real number domain to a complex number domain, which maps the real number domain space source phase of an interested component value domain to a complex number domain range of [ -pi, pi ] according to the small phase characteristic of the complex number domain space source phase.
A method for mapping fMRI data spatial source phase from real domain to complex domain, comprising the steps of:
firstly, carrying out spatial ICA separation on fMRI real number domain phase data, carrying out polarity correction on source components obtained by ICA according to a spatial reference network of the interested components, and selecting the interested components; the method comprises the following steps:
input single tested fMRI real number domain phase data
Figure BDA0001997470260000031
Wherein T represents time points and V represents brain endosome hormone number;
second, PCA (principal component analysis) dimension reduction; carrying out PCA dimension reduction on Z, recording the model order as N, wherein N is less than or equal to T, and obtaining dimension-reduced data
Figure BDA0001997470260000032
(III) separating ICA components, correcting polarity and selecting interested components;
(1) infmax pair adopting real number domain ICA classical algorithm
Figure BDA0001997470260000033
Separating to obtain N SM components
Figure BDA0001997470260000034
(2) Spatial reference network s using components of interestrefAnd (3) carrying out polarity correction on the N SM components to enable positive values to correspond to a brain activation region:
Figure BDA0001997470260000035
where N is 1, …, N, "corr" represents a correlation coefficient operation;
(3) according to corr(s)n,sref) From large to small pairs snOrdering, snIs the corrected SM component; the first C are selected as the candidates of the interesting components, which are abbreviated as sc(ii) a C is 1, …, C; the identification of the components of interest using the method in "Qiu Y, Lin QH, Kuang LD, Gong XF, Cong FY, Wang YP, Calhoun VD. spatial source phase: A new feature for identification of spatial differential on complex-valued stopping-state fMRI data. human Brain Brand, 1-15,2019" is as follows:
Figure BDA0001997470260000041
wherein "#" represents an intersection and "Vox0.5"indicates the number of voxels with amplitude greater than 0.5, and the final selected component of interest is denoted as sc*；
Secondly, selecting a spatial source phase of the interested component, and performing positive value taking and spatial smoothing pretreatment on the spatial source phase; the method comprises the following steps:
(IV) extracting the spatial source phase of the interested component of the best run; repetition ofStep (III) ICA component separation, polarity correction and interested component selection for H times to obtain H interested components sc*,hH is 1, …, H; the method described in "Kuang LD, Lin QH, Gong XF, Cong F, Sui J, Calhoun VD. model order effects on ICA of stopping-state complex-value fMRI data: application to schizoopening. journal of Neuroscience Methods 304, 24-38,2018" is used from sc*,hThe best primary ICA result is obtained to obtain the real number domain space source phase s of the interested componentc*,h*Abbreviated as s;
(V) taking a positive value for s to obtain s+；
(VI) spatial smoothing; to s+Smoothing is carried out by using a spatial smoothing strategy in "Chen Z, Caprihan A, Damaraju E, Rachakonda S, Calhoun VD. functional bridge connection in suppression-state fMRI using phase and magnetic data. journal of Neuroscience Methods293, 299-309, 2018" to obtain
Figure BDA0001997470260000042
Figure BDA0001997470260000043
Wherein "smooth" represents a 3D spatial smoothing operation, using a gaussian kernel with full width at half maximum (FWHM);
thirdly, estimating a distribution histogram of the spatial source phase values, generating symmetrical data points, and solving a function fitting the data points; the method comprises the following steps:
(seventhly) histogram estimation; is provided with
Figure BDA0001997470260000044
Contains V voxels, i.e.
Figure BDA0001997470260000045
Estimating its frequency distribution histogram
Figure BDA0001997470260000046
Each element of y is calculated as follows:
Figure BDA0001997470260000051
wherein, Ui＝[ui,ui+1) Denotes the ith interval, Uwidth＝ui+1-uiIndicates the interval width, i is 1, M is the number of intervals,
Figure BDA0001997470260000052
Figure BDA0001997470260000053
representing a top function;
Figure BDA0001997470260000054
to represent
Figure BDA0001997470260000055
Falls in UiVoxels within the interval, "Num" represents the voxel number statistics; u shapewidthIt is preferably 0.01 to 0.3.
(eight) data point generation; let xiIs a section UiIs the right boundary point of (1), i.e. xi＝ui+1M, M data points (x) are generated from the histogrami,yi)；
(nine) data point symmetric supplementation; finding yiThe maximum value of (2) is recorded as the corresponding index i0(ii) a Retention of i0All data points on the right
Figure BDA0001997470260000056
Δ i ═ 1, …, Q, Q is i0Total number of right data points, Q ═ M-i0(ii) a With i0Centered, left side Q data points are generated symmetrically
Figure BDA0001997470260000057
Combination of
Figure BDA0001997470260000058
Figure BDA0001997470260000059
And
Figure BDA00019974702600000510
new 2Q +1 data points were obtained and recorded as
Figure BDA00019974702600000511
k＝i0-Q,…,i0-1,i0,i0+1,…,i0+Q；
(ten) fitting a function; for data points
Figure BDA00019974702600000512
The function fitting is performed by using a nonlinear least squares method, and the function family f is selected as follows:
Figure BDA00019974702600000513
wherein β ═ β1,β2,β3,β4]A parameter vector corresponding to the function f; solving the fitting parameter betafitThe following were used:
Figure BDA00019974702600000514
and (3) solving the formula (6) by adopting an L-M (Levenberg-Marquardt) nonlinear least square method, wherein the updating rule of the parameter vector beta is as follows:
Figure BDA00019974702600000515
wherein J is Jacobian matrix, the k row and J column elements of which are
Figure BDA00019974702600000516
j is 1, …,4, λ is a non-negative damping factor, and equation (6) is the mostThe final β is βfit＝[βfit,1,βfit,2,βfit,3,βfit,4]Fitting a function
Figure BDA00019974702600000517
Completing the solution;
fourthly, transforming the value domain of the fitting function to [0, pi ] to obtain a mapping function, and using the mapping function for mapping the space source phase from a real number domain to a complex number domain; the method comprises the following steps:
(eleven) value domain transformation; known fitting function
Figure BDA0001997470260000061
Has a value range of [0, beta ]fit,1]Will be
Figure BDA0001997470260000062
Is transformed into [0, pi ]]To obtain a transformed mapping function
Figure BDA0001997470260000063
The following were used:
Figure BDA0001997470260000064
wherein, betamap＝[π,βfit,2,βfit,3,βfit,4]As a function of the mapping
Figure BDA0001997470260000065
The parameter vector of (2);
(twelve) spatial source phase mapping; (iv) spatially smoothing the result of step (six)
Figure BDA0001997470260000066
As
Figure BDA0001997470260000067
The independent variable of (a) is selected,
Figure BDA0001997470260000068
the dependent variable is the mapped complex field space source phase
Figure BDA0001997470260000069
(thirteen) outputting the mapped complex field space source phases
Figure BDA00019974702600000610
The method maps the space source phase from the real number domain to the complex number domain, discovers the denoising capability of the real number domain space source phase, and obtains more and more continuous activated voxels than the traditional threshold method when the method is used for denoising the real number domain space source phase. For example, for 82 tested resting state fMRI complex data, complex domain space source phase obtained after mapping by the method and small phase characteristics of [ -pi/4, pi/4 ] range thereof are applied to perform noise elimination on single tested real number domain space source phase Auditory (AUT) components (see fig. 2D), and 57% more activated voxels are extracted than a traditional threshold method (see fig. 2E); the group AUT component (phase range 0, pi/4) of 82 tested also extracted 41% more active voxels (3860vs 2729) than the thresholding (threshold 0.5). Therefore, the invention popularizes the noise elimination advantage of the complex field space source phase to the real number field, overcomes the performance disadvantage of the traditional threshold noise elimination method, and provides a new means for post-processing noise elimination of fMRI data analysis.
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FIG. 1 is a flow chart of the implementation of the present invention.
Fig. 2 is a comparison of the noise cancellation results for a single AUT component tested. (A) AUT component spatial reference; (B) single tested AUT component (before noise cancellation); (C) the invention eliminates noise and masks (phase range [0, pi/4 ]); (D) the invention eliminates the noise result; (E) the conventional thresholding noise-canceling result (threshold ═ 1.5) gives the correlation coefficient with (a) and the number of voxels included in each sub-graph in sub-graphs (B) - (E).
Detailed Description
The following describes in detail a specific embodiment of the present invention with reference to the drawings.
There is a healthy person who has fMRI complex data acquired at rest. T-146 scans were performed in the time dimension, and 53 × 63 × 46 whole brain data were obtained for each scan, and the number of voxels in the brain V-62336.
The first step is as follows: inputting single-subject fMRI real number domain phase data
Figure BDA0001997470260000071
The second step is that: and (5) reducing the dimension by PCA. Carrying out PCA dimension reduction on Z, recording the model order as N-80, and obtaining dimension-reduced data
Figure BDA0001997470260000072
The third step: ICA component separation, polarity correction and component of interest selection. (1) Infmax pair adopting real number domain ICA classical algorithm
Figure BDA0001997470260000073
Separating to obtain 80 SM components
Figure BDA0001997470260000074
(2) Substituting equation (1) into spatial reference network s using AUT componentsref(see fig. 2A) polarity correction was performed on 80 SM components to make positive values correspond to brain activation regions; (3) according to corr(s)n,sref) From large to small pairs snSorting(s)nFor corrected SM component), the first 10 candidates are selected as AUT components, abbreviated as s c1, …,10, identifying the component of interest using the method of "Qiu Y, Lin QH, Kuang LD, Gong XF, Cong FY, Wang YP, Calhoun VD. spatial source phase: A new feature for identifying spatial differential on complex-valued retained-state of MRI data. HumBranin Mapping, 1-15,2019", substituting into equation (2), the final selected AUT component is expressed as sc*。
The fourth step: the spatial source phase extraction of the components of interest of best run. Repeating the third step H10 times to obtain 10 components of interest sc*,hH is 1, …, 10. Adopts "Kuang LD, Lin QH, Gong XF, Cong F, Sui J, Calhoun VD. model order effects on ICA of stopping-state complex-value fMRI data: application to schizoopening. journalof Neuroscience Methods 304, 24-38,2018 ", from sc*,hThe best primary ICA result is obtained to obtain the real number domain space source phase s of AUT componentc*,h*Abbreviated as
Figure BDA0001997470260000075
The fifth step: taking a positive value for s to obtain
Figure BDA0001997470260000081
And a sixth step: the space is smooth. Using a Gaussian kernel of 15X 15mm3FWHM, for s+Smoothing by using a spatial smoothing strategy in "Chen Z, Caprihan A, Damaraju E, Rachakonda S, Calhoun VD. functional bridge connection in suppressing-state fMRI using phase and magnetic data. journal of Neuroscience Methods293, 299-309, 2018", substituting into formula (3) to obtain
Figure BDA0001997470260000082
The seventh step: and (6) histogram estimation.
Figure BDA0001997470260000083
Contains 62336 voxels, i.e.
Figure BDA0001997470260000084
Substituting into formula (4), estimating its frequency distribution histogram
Figure BDA0001997470260000085
UwidthTake 0.21, M56.
Eighth step: data points are generated. Let xiIs a section UiIs the right boundary point of (1), i.e. xi＝ui+1I 1, …,56 data points (x) were generated from the histogrami,yi)。
The ninth step: data points were supplemented symmetrically. Finding yiGet the corresponding index as i 01 is ═ 1; retention of i0Total number on right sideBased on the fact that
Figure BDA0001997470260000086
Δ i ═ 1, …, Q, Q is i0Total number of right data points, Q55. With i0Centered, the left 55 data points were generated symmetrically
Figure BDA0001997470260000087
Combination of
Figure BDA0001997470260000088
Figure BDA0001997470260000089
And
Figure BDA00019974702600000810
obtain new 111 data points, and record as
Figure BDA00019974702600000811
k＝-54,…,56。
The tenth step: and (6) fitting a function. For 111 data points
Figure BDA00019974702600000812
The function fitting is performed using a non-linear least squares method. According to the function shown in formula (5), will
Figure BDA00019974702600000813
Substituting the formula (6) and the formula (7) to obtain the parameter vector of the fitting function as betafit＝[3.10,0.15,0.82,0.21]The corresponding fitting function is
Figure BDA00019974702600000814
The eleventh step: and (4) value domain transformation. Known fitting function
Figure BDA00019974702600000815
Has a value range of [0, beta ]fit,1]Will be
Figure BDA00019974702600000816
Value domain transformation of [0, pi ]]Then, the transformed mapping function is obtained by substituting the formula (8)
Figure BDA00019974702600000817
βmap＝[π,0.15,0.82,0.21]As a function of the mapping
Figure BDA00019974702600000818
The parameter vector of (2).
The twelfth step: spatial source phase mapping. Mixing the obtained in the sixth step
Figure BDA00019974702600000819
As
Figure BDA00019974702600000820
The independent variable of (a) is selected,
Figure BDA00019974702600000821
the dependent variable is the mapped complex field space source phase
Figure BDA00019974702600000822
The thirteenth step: outputting the mapped complex field spatial source phase
Figure BDA00019974702600000823
When the spatial source phase s (see fig. 2B) of the best run component of interest obtained in the fourth step ICA is denoised, the method in patent 201410189199.7 is first applied to the method obtained in the thirteenth step
Figure BDA0001997470260000091
The noise-eliminating mask of the present invention is constructed (see FIG. 2C), and the phase range is [0, π/4]](ii) a The result of denoising s (see fig. 2B) by the denoising mask of the present invention is shown in fig. 2D, the result of denoising by the conventional thresholding method is shown in fig. 2E, and the threshold is 1.5. The correlation coefficients between FIGS. 2B-E and FIG. 2A are: 0.30, 0.40, 0.38, 0.29, the number of activated voxels of figures 2B-E respectively: 62336, 10116, 10116, 6442; figure 2D extracts 57% more activated voxels than figure 2E.

Claims (2)

1. A method for mapping spatial source phase of fMRI data from real domain to complex domain, characterized by:
firstly, carrying out spatial ICA separation on fMRI real number domain phase data, carrying out polarity correction on source components obtained by ICA according to a spatial reference network of the interested components, and selecting the interested components; the method comprises the following steps:
input single tested fMRI real number domain phase data
Figure FDA0003326522760000011
Wherein T represents time points and V represents brain endosome hormone number;
second, PCA (principal component analysis) dimension reduction; carrying out PCA dimension reduction on Z, recording the model order as N, wherein N is less than or equal to T, and obtaining dimension-reduced data
Figure FDA0003326522760000012
(III) separating ICA components, correcting polarity and selecting interested components;
(1) infmax pair adopting real number domain ICA classical algorithm
Figure FDA0003326522760000013
Separating to obtain N SM components
Figure FDA0003326522760000014
(2) Spatial reference network s using components of interestrefAnd (3) carrying out polarity correction on the N SM components to enable positive values to correspond to a brain activation region:
Figure FDA0003326522760000015
where N is 1, …, N, "corr" represents a correlation coefficient operation;
(3) according to corr (s n,sref) From big to smalls nThe order is given to the user,s nis the corrected SM component; the first C are selected as the candidates of the interesting components, which are abbreviated as sc(ii) a C is 1, …, C; the components of interest are identified as follows:
Figure FDA0003326522760000016
wherein "#" represents an intersection and "Vox0.5"indicates the number of voxels with amplitude greater than 0.5, and the final selected component of interest is denoted as sc*；
Secondly, selecting a spatial source phase of the interested component, and performing positive value taking and spatial smoothing pretreatment on the spatial source phase; the method comprises the following steps:
(IV) extracting the spatial source phase of the interested component of the best run; repeating the step (III) of ICA component separation, polarity correction and interested component selection for H times to obtain H interested components sc*,hH is 1, …, H; from sc*,hThe best primary ICA result is obtained to obtain the real number domain space source phase s of the interested componentc*,h*Abbreviated as s;
(V) taking a positive value for s to obtain s+；
(VI) spatial smoothing; to s+Smoothing by adopting a spatial smoothing strategy as follows to obtain
Figure FDA00033265227600000217
Figure FDA0003326522760000021
Wherein "smooth" represents a 3D spatial smoothing operation, using a full-width-at-half-maximum gaussian kernel;
thirdly, estimating a distribution histogram of the spatial source phase values, generating symmetrical data points, and solving a function fitting the data points; the method comprises the following steps:
(seventhly) histogram estimation; is provided with
Figure FDA0003326522760000022
Contains V voxels, i.e.
Figure FDA0003326522760000023
Estimating its frequency distribution histogram
Figure FDA0003326522760000024
Each element of y is calculated as follows:
Figure FDA0003326522760000025
wherein, Ui＝[ui,ui+1) Denotes the ith interval, Uwidth＝ui+1-uiIndicates the interval width, i is 1, M is the number of intervals,
Figure FDA0003326522760000026
Figure FDA0003326522760000027
representing a top function;
Figure FDA0003326522760000028
to represent
Figure FDA0003326522760000029
Falls in UiVoxels within the interval, "Num" represents the voxel number statistics;
(eight) data point generation; let xiIs a section UiIs the right boundary point of (1), i.e. xi＝ui+1M, M data points (x) are generated from the histogrami,yi)；
(nine) data point symmetric supplementation; finding yiThe maximum value of (2) is recorded as the corresponding index i0(ii) a Retention of i0All data points on the right
Figure FDA00033265227600000210
1, Q is i0Total number of right data points, Q ═ M-i0(ii) a With i0Centered, left side Q data points are generated symmetrically
Figure FDA00033265227600000211
Combination of
Figure FDA00033265227600000212
Figure FDA00033265227600000213
And
Figure FDA00033265227600000214
new 2Q +1 data points were obtained and recorded as
Figure FDA00033265227600000215
k＝i0-Q,...,i0-1,i0,i0+1,...,i0+Q；
(ten) fitting a function; for data points
Figure FDA00033265227600000216
The function fitting is performed by using a nonlinear least squares method, and the function family f is selected as follows:
Figure FDA0003326522760000031
wherein β ═ β1,β2,β3,β4]A parameter vector corresponding to the function f; solving the fitting parameter betafitThe following were used:
Figure FDA0003326522760000032
and (3) solving the formula (6) by adopting an L-M (Levenberg-Marquardt) nonlinear least square method, wherein the updating rule of the parameter vector beta is as follows:
Figure FDA0003326522760000033
wherein J is Jacobian matrix, the k row and J column elements of which are
Figure FDA0003326522760000034
λ is a non-negative damping factor, and the finally obtained β of the formula (6) is βfit＝[βfit,1,βfit,2,βfit,3,βfit,4]Fitting a function
Figure FDA0003326522760000035
Completing the solution;
fourthly, transforming the value domain of the fitting function to [0, pi ] to obtain a mapping function, and using the mapping function for mapping the space source phase from a real number domain to a complex number domain; the method comprises the following steps:
(eleven) value domain transformation; known fitting function
Figure FDA0003326522760000036
Has a value range of [0, beta ]fit,1]Will be
Figure FDA0003326522760000037
Is transformed into [0, pi ]]To obtain a transformed mapping function
Figure FDA0003326522760000038
The following were used:
Figure FDA0003326522760000039
wherein, betamap＝[π,βfit,2,βfit,3,βfit,4]As a function of the mapping
Figure FDA00033265227600000310
The parameter vector of (2);
(twelve) spatial source phase mapping; (iv) spatially smoothing the result of step (six)
Figure FDA00033265227600000311
As
Figure FDA00033265227600000312
The independent variable of (a) is selected,
Figure FDA00033265227600000313
the dependent variable is the mapped complex field space source phase
Figure FDA00033265227600000314
(thirteen) outputting the mapped complex field space source phases
Figure FDA00033265227600000315
2. A method for mapping fMRI data space source phase from real domain to complex domain as claimed in claim 1, UwidthTaking 0.01-0.3.
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