Monday, June 17, 2013: 1:30 PM
- 3:30 PM

1900

Monday, June 17 & Tuesday, June 18

Linear time-invariant (LTI) filters, most suitable when
the signal of interest is (approximately) restricted to a known
frequency band, are widely used in science, engineering, and general
time series analysis [3]. At the same time, the effectiveness of an
alternate approach to signal filtering, most suitable when the signal of
interest either is itself sparse or admits a sparse representation, has
been increasingly recognized [2]. However, signals that arise in
functional neuroimaging applications often are more complex: neither
isolated to a specific frequency band nor admitting a sparse
representation. The problem addressed here is filtering the latter type
of signal, where neither denoising approach is appropriate by itself.
The utility of two computationally efficient algorithms will be
demonstrated.

The mathematical model adopted for the noisy data (*y*) is that the underlying signal comprises a low-frequency component (*f*) and a sparse-derivative component (*x*): *y* = *f* + *x* + *w*, where *w*
is stationary Gaussian noise. We have formulated an optimization
approach, involving the minimization of a non-differentiable, strictly
convex cost function, that enables simultaneous use of low-pass
filtering and sparsity-based denoising to estimate *f* and *x*.
The specific optimization problem is a total variation regularized
inverse problem. As is standard, the cost function consists of
data-fidelity and penalty terms. However, in contrast to standard
formulations, the data-fidelity term measures the energy of the output
of a high-pass filter **H** that is complementary to a low-pass filter **L**, where **L** is selected to match the *f* component. The penalty term is the total variation of *x*, defined as the *L*_{1} norm of the derivative of *x*.
Two algorithms have been derived, based on majorization-minimization
(MM) in one case and on the alternating direction method of multipliers
(ADMM) in the other.

For an initial demonstration of the performance of the MM
algorithm, we worked with the synthetic time series shown in Fig. 1,
which consists of a low-frequency sinusoid (*f*), two additive step discontinuities (*x*), and additive white Gaussian noise (*w*) [4]. The solution (convergence after 30 iterations, in ~0.1 s) successfully resolves *f* and *x*, preserves the discontinuities in *x*
without introducing Gibbs-like phenomena, and smoothes the data
substantially. Data for demonstrating the performance of the ADMM
algorithm was obtained from a functional near infrared spectroscopy
(fNIRS) time series measurement on a dynamic tissue-simulating phantom
[1], as its NIR absorption was varied in a manner that mimics the
hemodynamic response of a human brain to intermittently delivered
stimuli. Thus we closely approximated human-subject fNIRS measurement
conditions, while preserving the ability to assess the accuracy of the
computed solution. In Fig. 2 it is seen that the separation of the *f* and *x*
components (convergence after 50 iterations, in 0.14 s) is nearly as
good as that seen in the synthetic-data case, and that the shapes of the
hemodynamic pulses are well preserved. Conventional band-pass filtering
has edge-spreading and plateau-rounding effects, and it obscures the
amplitude of the hemodynamic pulses relative to the baseline.

Neuroimaging modalities (fNIRS and others) are prone to
producing data that are well described by the noisy-data model
considered here. Our novel algorithms are designed for data of this
type. If comparable performance is consistently obtained, possible
methodological consequences include: the ability to consider inter-epoch
variability in hemodynamic responses could be enhanced, as it would be
less necessary to average multiple responses to achieve high SNR;
alternatively, less measurement time may be required for accurate
determination of the average response.

Motion Correction and Preprocessing

2. Elad, M. (2010), Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer).

3. Parks, T.W. (1987), Digital Filter Design (John Wiley and Sons).

4. Selesnick, I.W. (2012), 'Polynomial smoothing of time series with additive step discontinuities', IEEE Trans. Signal Process., vol. 60, no. 12, pp. 6305-6318.