Real-time functional magnetic resonance imaging (rt-fMRI) is a technique that enables us to observe human brain activations in real time. However, some unexpected noises that emerged in fMRI data collecting, such as acute swallowing, head moving and human manipulations, will cause much confusion and unrobustness for the activation analysis. In this paper, a new activation detection method for rt-fMRI data is proposed based on robust Kalman filter. The idea is to add a variation to the extended kalman filter to handle the additional sparse measurement noise and a sparse noise term to the measurement update step. Hence, the robust Kalman filter is designed to improve the robustness for the outliers and can be computed separately for each voxel. The algorithm can compute activation maps on each scan within a repetition time, which meets the requirement for real-time analysis. Experimental results show that this new algorithm can bring out high performance in robustness and in real-time activation detection.

Functional magnetic resonance imaging (fMRI) offers a noninvasive method in studying human brain functions by recording blood-oxygen-level-dependent (BOLD) signal changes related to neuronal activity across the brain with high spatial resolution [

In common fMRI experiments, MRI scanner acquires whole brain data at an interval of 2 seconds, also called repetition time. To meet the real-time requirements, all the processing steps of real-time fMRI need to be completed within a repetition time. Simple real-time fMRI processing steps consist of data reconstruction, spatial realignment (head motion correction), and statistical analysis. Among them, incremental statistical analysis on each voxel of the fMRI dataset will result in huge computational costs. To overcome the computational costs of the statistical analysis, a number of incremental activation detection algorithms have been developed for rt-fMRI applications.

Cox et al. [

The fMRI time series has low signal to noise ratio [

In recent years, with the developments of convex optimization, Mattingley and Boyd [

Real-time fMRI signal is three-dimensional volume data at each scan during a repetition time and the intensity of each voxel represents the blood oxygen level associated with neural activities. Data of each incoming fMRI scans is spatially aligned with the first scan of the series. Voxel values at different scan time points are arranged to time sequence, forming the measured time series. Each voxel will form a time series, and the length of time series is growing with the time increasing. The time series of each voxel can be calculated independently, so in the following discussion we only consider the situation of a single voxel time series.

Roche et al. [

The general linear model (GLM) explains the measured time series

The design matrix is

In the GLM-AR model, it is assumed that

Alexis Roche proved that the maximum likelihood estimate of

First, they combine the parameters to be estimated

Secondly, linearize the error function

Finally, they solved the (nonlinear) least-squares regression problem by means of an EKF. The EKF updates the parameters using the following recursion:

At time

The number of explanatory variables is

The sensor failures or measurement outliers will cause the sparse measurement noise and they may cause rapid degrade on the detection performance. We derive a new algorithm to detect the outliers in order to eliminate the effect on the kalman filter algorithm.

We suppose that there is a sparse term

Then linearize the

The linearized constraint equation is approximate as follows:

The measurement update step of standard kalman filter algorithm is essentially an optimization problem, and the linearized parameter optimization problem can be described as follow:

We use the robust kalman filter method to detect the outlier hidden in the measurement by replacing the measurement update with the solution of a similar convex minimization problem, which includes an

To (approximately) handle the additional sparse noise term

In the optimization problem (

To solve this convex optimization problem, we adopt a fast transform method proposed by Mattingely and Boyd [

After the transformation, the original problem was transformed into an equivalent convex quadratic program problem:

With variable

The standard kalman filter consists of alternating time and measurement updates. Since

After solving the problem, we achieve an estimate outlier value

Furthermore, we combine the

For the real-time

Finally, the algorithm recursion is summarized as follows:

The algorithm is tested on a single run from an fMRI experiment involving a visual and auditory task. The protocol is block design, and the run consisted of 10 blocks, each block including one activation epoch (20 s) and one control epoch (10 s). We aim at finding the voxels associated with the visual and auditory function; each activation epoch, the visual stimulus, and auditory stimuli are present, but the intensity is different. The data has an abrupt head motion during the scan in the 45th repetition time (TR) and the head motion caused severe motion artifacts. The repetition time was 2 seconds for a total of 152 scans. Functional images have

As is shown in Figure

In Figure

Cox method derived a threshold for active detection, so there is no

We also tested the algorithm on an inactivate voxel. In Figure

The

In Figure

Figure

We have applied our detection algorithm to the whole brain data, in the case where threshold value the same, the final activation maps are shown in Figure

We have a C++ version of the algorithm, and we test it on a 16-core processor workstation, in which computing hole brain (

In our algorithm, parameter

The fMRI signal has a low signal to noise ratio, so we cannot give an exact threshold value and this algorithm provides a method to threshold the signal, and we can even achieve the value of the outlier and use it to modify the detection algorithm. fMRI experiment is complex, which needs cooperation with subject and operator; any problem of them may cause the experiment to fail. Our algorithm can detect this kind of failure and it can detect the outliers, which may have a great effect on follow processes, where we can stop the experiment to check the problem or just mark it and eliminate the outlier data after the experiment.

In future works, we will explore a proper method to determine

The robust kalman filter is introduced in the activation detection of fMRI experiment. Convex optimization method is used to modify the extended kalman filter by introducing a sparse noise term. The robustness of our method to the sparse noise is improved. Moreover, the performance of the proposed method does not degrade rapidly when disturbances are involved. When applied to the time series voxels, our method can obtain more stable

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National High Technology Research and Development Program of China (no. 2012AA011603).