fMRI activations via low-complexity second-order inverse-sparse-transform blind separation
Introduction
Functional Magnetic Resonance Imaging (fMRI) has been widely used in disease diagnosis and brain science research owing to its non-radiation and non-trauma, higher spatial resolution and signal-to-noise ratio than Electroencephalogram (EEG) and Magnetoencephalography (MEG). In the cortex area of brain activity, the relative decrease of deoxygenated hemoglobin concentration will lead to the enhancement of local magnetic field unevenness of blood vessels and peripheral tissues [1]. From this, fMRI can detect Blood Oxygenation Level Dependent (BOLD) in a subject's brain area. Collecting fMRI data requires designing experimental tasks first. When the subject performs a given task, the BOLD signal in the brain area will change accordingly [2]. From the changing signal, a task-related (CTR) brain active zone can be located using a related algorithm. Therefore, how to quickly and accurately locate the active area is of great significance for the corresponding psychological behaviors and diseases.
To address fMRI activation location, many scholars have proposed a variety of algorithms, where Statistical Parametric Maps (SPMs) [3], [4] are widely used. SPMs is a statistical method, and convolves a stimulus function corresponding to a task with a Hemodynamic Response Function (HRF) to obtain a general linear model (GLM) [5] design matrix. A restricted maximum likelihood (REML) [6] or Bayesian algorithm [7], [8] can be used to estimate the GLM parameters. Under a null hypothesis, then, the GLM parameters statistically analyzed through methods such as T test and F test, so as to obtain the statistical maps of the parameters about brain activation regions. However, this method requires the GLM design matrix [9] to be known in advance, and the matrix design has some uncertainties. Thus, how to design the matrix will directly affect the locating result. In addition, in order to make an fMRI voxel's probability distribution closer to the Gaussian distribution, this method also needs to perform temporal and spatial smoothing [10]. This will degrade the temporal and spatial resolution of fMRI data.
In fact, the observed fMRI signals can be seen as a linear combination of spatial maps (SMs) of activations and time courses (TCs) [11]. Therefore, locating fMRI activations can be regarded as a blind source separation (BSS) [12] issue. Most BBS algorithms use the statistics of observed signals to complete the separation. From this, blind separation algorithms can be roughly divided into higher-order-statistics [13], [14], [15] and second-order-statistics algorithms [16]. In higher-order-statistics algorithms, the most widely used one is independent component analysis (ICA) [17], [18], [19], [20], [21], [22], which uses maximized kurtosis, likelihood or negative entropy to separate signals that are statistically independent and satisfy non-Gaussian distribution characteristics. The earlier application of ICA to fMRI separation is proposed in [17]. Without the estimation of TC, the method uses Comon's ICA algorithm [18] and Bell's ICA algorithm [19] to achieve an accurate separation of CTR activations. However, the both show slow convergence. In many ICA methods, FastICA [20] adopts batch computation and fixed-point iteration and thus has fast convergence and high robustness. GIFT (Group ICA of fMRI Toolbox) method [21] uses FastICA, where multiple subjects are combined into a group to reduce noise and a principal component analysis (PCA) algorithm performs dimensionality reduction [23]. It can effectively find the CTR activations and have better computation speed. Even for the faster FastICA in ICA algorithms, however, its computational complexity is still high due to its fourth-order statistics and much iteration for convergence. Compared with higher-order ICA algorithms, second-order blind identification (SOBI) algorithms [24], [25], [26] only calculate a correlation matrix of observed signals. Thus, they are less computationally complex and also receive many attentions. Unfortunately, the fMRI activation signals appear sparsity [27], each of which only has values in its activation regions and are close to zero in the other regions. Therefore, the information that the autocorrelation matrix of the fMRI signals contains is not rich. This will cause that the application of SOBI to fMRI separation will produce more separation errors. In order to solve the difficulty for SOBI to separate sparse signals, a frequency-domain SOBI (f-SOBI) algorithm [28] is proposed. The algorithm performs inverse Fourier transform on the signals to obtain a rich autocorrelation matrix, which can achieve better separation of sparse signals. However, the algorithm has to be performed in the case with small noise. When the noise of the fMRI signal is large, the separation performance of f-SOBI will decrease.
In this paper, we propose an inverse-sparse transformation method to solve the difficulty of second-order blind separation of fMRI signals. The idea of this method is to transform the sparse fMRI signals into another domain, so that the transformed signals no longer have sparsity. Thus, the autocorrelation matrix of the transformed signals contains more information and the observed fMRI signals can be better separated. From the idea, we derive that the transformation matrix of inverse-sparse transformation should be orthogonal, for example, orthogonal wavelet transform, dictionary learning and other orthogonal transformation methods can be [29], [30], [31]. From the view of the transformation matrix, the f-SOBI algorithm is only an inverse-sparse algorithm using an orthogonal inverse-Fourier transform matrix. In order to make f-SOBI can be used for the separation of noisy fMRI signals, we perform group-dimension reduction and whitening on the fMRI signals of multiple subjects with same tasks and thereby propose a called group f-SOBI (GFS) algorithm. To further reduce the complexity of GFS, we use an orthogonal cosine transform matrix to perform an inverse-sparse transform, and propose a called group-cosine SOBI (GCS) algorithm. Because GFS and GCS both use second-order statistics, their complexity is lower than that of higher-order ICA algorithms.
In experiments, we use a public simulation software named SimTB [32] to generate a group of simulation data, and also use a group of real fMRI data to test the algorithms. For the simulation data, the separation error of the proposed algorithms is less than half that of traditional SOBI and close to that of ICA, but their running time is only 1/30 of ICA. For the real data, the locations of activations by the proposed algorithms are consistent with ICA, but their running time is only 1/5 of ICA.
Section snippets
SPM method
SPM method [3], [4] is a widely used method for localizing fMRI activations. This method generally uses the GLM model. As shown in Fig. 1, let BOLD signals be expressed as [5] where
T is the number of scans, i.e. a time factor,
is a design matrix consisting of reference signals and basis functions that can be obtained in experiments,
is a scale coefficient vector of regression,
P is the number of regression coefficients,
e is an error vector.
Therefore, given the
Sparsity of FMRI signal
The traditional GIFT uses FastICA algorithm to separate fMRI BOLD signals. However, it requires the computation of fourth-order statistics and many iterations and the computational complexity is not optimal. Thus, this paper considers SOBI algorithm with less computational complexity. However, the traditional SOBI algorithm does not consider signal sparsity. From the fMRI linear mixture model in (3), the source signal matrix S consists of brain component activations, and it will be sparse
Separation matrix of inverse-sparse transform
The second-order blind separation algorithm of anti-sparse transform needs to perform inverse-sparse transform on the signals of sparse BOLD signals, and then solve the correlation matrix to obtain the separation matrix. This section will explain how to solve the separation matrix after inverse-sparse transform. Let be a linear inverse-sparse transform matrix. Performing a linear inverse-sparse transform on each row vector in the dimension-reduction whitened Z matrix will obtain the
Experimental setup
The experimental data used in this experiment includes simulated data and measured data, which are generated as follows. The simulation data is obtained by SimTB software [32], and its download address is http://trendscenter.org/trends/software/simtb/index.html. Some relevant parameters of the data are given in Table 3. The other parameters are set by a file named by “experiment_params_ aod.m”. It should be noted that the fMRI simulation data is assumed to have been aligned, so the
Discussion
For the blind source separation of fMRI sparse signals, this paper uses inverse Fourier transform and inverse cosine transform to process the signals so that the signals are no longer in sparse form, and then are separated. The experimental results show that compared with the existing FastICA algorithm, the brain activations separated by the proposed method can correspond to those by FastICA while the proposed method's running time is reduced very much. Compared with the existing SOBI
Conclusion
Traditional fMRI brain region separation uses blind source separation algorithms such as ICA, but this algorithm needs to compute high-order statistics, and the computational complexity is not low. To reduce the computational complexity, this paper proposes an inverse-sparse transform method to achieve the second-order blind separation of fMRI signals, which transforms sparse BOLD signals to an inverse-sparse domain and then compute the autocorrelation matrix to complete the separation. In
CRediT authorship contribution statement
Haifeng Wu: Conceptualization, Methodology, Writing - Reviewing and Editing. Dong Li: Software, Writing - Original draft preparation. Mingzhi Lu: Visualization, Investigation. Yu Zeng: Software, Supervision, Validation.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (61762093), the 17th Batch of Young and Middle-aged Leaders in Academic and Technical Reserved Talents Project of Yunnan Province (2014HB019), Program for Innovative Research Team (in Science and Technology) in University of Yunnan Province, the Key Applied and Basic Research Foundation of Yunnan Province (2018FA036), and the Scientific Research Fund project of in Education Department of Yunnan Province (2020Y0238).
Haifeng Wu received the M.S. degree in electrical engineering from Yunnan University, Kunming, China, in 2004, and the Ph.D. degree in electrical engineering from Sun Yat-Sen University, Guangzhou, China, in 2007. He is currently an professor at the Department of Information Engineering at the Yunnan Minzu University. Prior to that, he was postdoctoral scholar at the Kunchuan Institute of Technology from 2007 to 2009. His research interests include machine learning, neural signal processing and
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Haifeng Wu received the M.S. degree in electrical engineering from Yunnan University, Kunming, China, in 2004, and the Ph.D. degree in electrical engineering from Sun Yat-Sen University, Guangzhou, China, in 2007. He is currently an professor at the Department of Information Engineering at the Yunnan Minzu University. Prior to that, he was postdoctoral scholar at the Kunchuan Institute of Technology from 2007 to 2009. His research interests include machine learning, neural signal processing and mobile communications.
Dong Li is now pursuing the M.S. degree in electrical engineering from Yunnan Minzu University, Kunming, China. His interests include neural system and machine learning.
Mingzhi Lu received the M.S. degree in electrical engineering from Yunnan Minzu University, Kunming, China. His interests include neural system and machine learning.
Yu Zeng received the M.S. degree in electrical engineering from Yunnan University, Kunming, China, in 2006. She is currently an assistant professor at the Department of Information Engineering at the Yunnan Minzu University. Prior to that, she was an electrical engineer in Kunming Institute of Physics from 2006 to 2009. Her research interests include wireless network and mobile communications.