Hypersampling of pseudo-periodic signals by analytic phase projection
Introduction
Many signals in the biological and biomedical sciences are of a pseudo-periodic nature with irregularly spaced, stretched, or otherwise distorted variations of a repeating cycle. An example for a pseudo-periodic cycle is the characteristic QRS complex observed in electric recordings of the heart [1]. Another example are patterns of electrical activity of the brain observed in electroencephalographic (EEG) surface recordings [2]. Those signals usually can be measured with a sufficient sampling rate to resolve their underlying pseudo-periodic cycles (QRS-complex, EEG waveform, respectively). However, often it is not possible to measure the effects of the pseudo-periodic dynamics in parts of the body that cannot be accessed so easily, for example deep within the brain. The method of choice to obtain signals from anywhere in the brain is magnetic resonance imaging (MRI). Dynamic or functional MRI of the brain is typically sampled at an insufficient rate to resolve the cardiac cycle or EEG patterns [3]. In order to investigate the pseudo-periodic signal in a particular location within the brain, one solution is to upsample the MRI signal at that point with an effective sampling time that is much smaller than the average cardiac cycle or EEG waveform period. The cardiac or EEG recordings then can serve as a reference used to define the pseudo-periodicity of the dynamics of interest.
Here, an efficient upsampling procedure, called hypersampling, is described. Hypersampling consists of an upsampling of the undersampled time series by using the method of analytic phase projection (APP). Hypersampling can be seen as a generalization of retrospective gating [4,5]. Whereas in retrospective gating a recurring template is identified from the reference signal, in hypersampling the continuous phase underlying the pseudo-periodic reference signal is identified from the reference signal. This phase estimate is then used for APP.
The organization of this manuscript is as follows: First, hypersampling by APP is described in Section 2. Hypersampling is demonstrated on simulations in Section 3. In Section 4, these concepts are applied to an MRI of the brain in order to visualize the very fast pulse waves traversing the brain, which normally cannot be resolved with MRI. A discussion including a comparison with retrospective gating and possible further applications to other hybrid systems with fast and slow time scales concludes the manuscript. An appendix provides code for hypersampling via APP, and the supplementary data a video of the pulse waves observed in the human brain.
The following is the supplementary data related to this article: .
Section snippets
Overview
Hypersampling by analytic phase projection (APP) is summarized in Fig. 1.
Two signals are acquired from the system under study: An undersampled pseudo-periodic time series x(t) and a sufficiently sampled pseudo-periodic reference signal y(t). The reference signal and the time series are acquired during the same time interval and are assumed to have the same pseudo-periodicity. Then, a monocomponent signal yM(t) is obtained by low-pass filtering the reference signal y(t) (box “Monocomponent
Simulation
A signal is being simulated as a nonlinearly transformed chirp signal with linearly increasing frequency from the beginning to the end of the signal time series in order to simulate pseudo-periodic data. Added to the data is 10% white noise. The Appendix contains a Matlab script defining the used signals and the analysis in detail. Fig. 2, row A, contains the given signal. The sampling time of the signal is 2 s. (Artificial physical units are introduced here in order to facilitate comparison
Application: Cerebral pulse waves
As an example, hypersampling via APP is applied to pulse waves in the human brain. Studying the properties of cerebral pressure waves and their possible effects on human health is a very active area of research [[9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]]. Here, a functional MRI scan from a publicly available data base [22] is used to demonstrate that hypersampling can resolve pulse waveforms in the brain.
The MRI data consists of image volumes covering parts of
Comparison with retrospective gating
The proposed method is generalizing well-known retrospective gating approaches. In most retrospective gating approaches, templates of the reference signal (for example, the R peak of the cardiac cycle) are used to define gating points. A pseudo-period, or cycle, is then defined by the interval from one template of the reference signal to the next, for example, RR intervals [4,5]. Signal sampling times are then linearly projected on cycle times. The analytic phase projection method differs in
Conclusion
An algorithm to upsample an insufficiently sampled pseudo-periodic signal with the help of a reference signal, “hypersampling” has been presented. Hypersampling is based on a projection of the signal time series to the analytic phase of the reference signal. It generalizes the widely used retrospective gating approach. Hypersampling has been validated in numerical simulations of pseudo-periodic signals, and it has been discussed that in the case of nonlinear phase evolution, hypersampling can
Acknowledgments and disclosure
The author would like to thank M. Hanke, A. Brechmann, and J. Stadler for help in using their database (M. Hanke Scientific Data 1, 140003, 2014). The author is partially funded by NIH grant # 5 R21 EY027568 02. The study sponsor was not involved in the study design, in the collection, analysis and interpretation of data; in the writing of the manuscript; or in the decision to submit the manuscript for publication. This research uses an algorithm described in a patent application by the Center
Henning U. Voss studied physics in Hamburg and Potsdam, Germany. He received the diploma and a Ph.D. in physics both from Potsdam University in 1994 and 1998, respectively. Since 2009 he has been Associate Professor of Physics in Radiology at Weill Cornell Medical College, New York City. From 2000 to 2003 he was Assistant Professor at Freiburg University, Germany. From 2003 to 2009 he was Assistant Professor of Physics in Radiology at his present institution. His research interests are imaging
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Henning U. Voss studied physics in Hamburg and Potsdam, Germany. He received the diploma and a Ph.D. in physics both from Potsdam University in 1994 and 1998, respectively. Since 2009 he has been Associate Professor of Physics in Radiology at Weill Cornell Medical College, New York City. From 2000 to 2003 he was Assistant Professor at Freiburg University, Germany. From 2003 to 2009 he was Assistant Professor of Physics in Radiology at his present institution. His research interests are imaging techniques, modeling dynamical systems, and applications of these fields to biomedical research.