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In this paper we present a Compressed Sensing CS approach for compressing this measurement process, showing that the time needed to reconstruct HeSE spectra can be reduced by several orders of magnitude compared to standard Discrete Fourier Transform DFT reconstruction techniques. Recently, compressed sensing has also seen applications focusing on Raman spectroscopy measurements 18 and in molecular dynamics simulations Spin-echo spectroscopy shares clear similarities with these fields, such as Fourier transforms arising naturally in data acquisition, however there are also significant differences.

In particular, one of the goals of spin-echo spectroscopy is to determine dynamical processes by monitoring the change of polarisation data. Here we consider the whole process of data processing, from polarisation data measurements to the extraction of the molecular dynamics information. Unlike NMR and many other spectroscopy-based applications, after we have performed compression on the initial Fourier transform step we cannot directly use the output data, it must instead undergo several further transforms.

This precludes the use of standard DFT-based CS techniques as this transform distorts the necessarily discrete set of values that we can solve for. Instead a new continuous CS approach, recently introduced by Adcock, Hansen et al.


With this method one has the freedom of evaluating the reconstructed function at any point they desire while still having the speed-up benefits of compressive sampling. Such an approach could be used to handle other inverse problems that require further transformations after reconstruction.

The stages highlighted in red correspond to the target data we wish to reconstruct. The plotted intensities are in consistent arbitrary units a. Moreover, in standard NMR experiments the smallest group of data that is taken in one measurement is typically a line or path of data points in k-space, with the exception of new realisations such as electron spin echo envelope modulation However, in HeSE each measurement corresponds to a single point. This gives us an additional degree of freedom in the data acquisition process which makes the application of CS particularly effective, as one can utilize the new approach of structured multilevel sampling in 21 to its fullest to boost performance.

In particular, by also taking the structure of the signal into account when designing the sampling strategy one can outperform the classical compressed sensing results see refs 24 and 25 for experimental validation that are dictated by the estimate on the number of samples m to be.

Typically, the coefficients corresponding to M 1 are the most important, and most of the energy in the signal is contained in these. This is very convenient as we do not have to pay a log factor for these coefficients. In practice this means substantial gain over the standard approaches as demonstrated in recent puplications 24 , This is convenient because it permits changing the problem to one of handling a Fourier transform to that of handling a Fourier series expansion:. Typically the next step is to truncate the Fourier series expansion, meaning that one makes the approximation.

The problem is now feasible as only finitely many data points l are required to determine. The Fourier series approximation could have been evaluated at any point in the interval [ a, b ] and suddenly the best we can do is reconstruct the , even though we are still working with the same number of Fourier samples.

Continuous Compressed Sensing for Surface Dynamical Processes with Helium Atom Scattering

Do we really have to pay this price in order to be able to compress this problem? The answer is: no. Note that this is similar in spirit to the work on finite rate of innovation 26 as well as to the concept suggested by Markovich et al. This is achieved by working with the infinite change of basis matrix for the two bases:. The approximation is now a continuous function as opposed to discrete that can be evaluated at any point and hence this allows the non-linear change of variables going from the wavelength distribution to the scattering function S as shown in Fig.

Such a transform is not possible with conventional discrete CS techniques. The approximation is computed with the actual coefficients in the new expansion of the wavelength function, and hence this approximation has the characteristics of the approximation in the new basis rather than the truncated Fourier series.

This means reducing Gibbs ringing and other artefacts coming from Fourier approximations see refs 20 , 21 , 22 for details. Details on the theoretical background of CS, what basis to use, the convex optimisation problems we solve and how to subsample the Fourier data are provided later on in the paper.

CS is used to demonstrate the compressibility of phonon detection. Thereafter we elaborate on the advantages of the continuous CS technique in order to measure processes that appear on different energy scales. The principle of the Helium spin-echo apparatus is the following: A beam of thermal 3 He is generated from the source in a fixed direction.

The nuclear spins are polarised and then rotated by the initial or incoming solenoid before being scattered from the target crystal surface. Afterwards any scattered He atoms heading in the direction of the detector are then rotated by the final or outgoing solenoid and passed through another polarisation filter. While a schematic sketch of the machine used at the Cavendish laboratory can be found in the Supplementary Information further details can be found in Jardine et al.

The currents I i , I f that run through the initial and final solenoid respectively. Here, r denotes position and t is time. Upon scattering from a dynamic surface, the wavevector k and the energy of the He atom E before and after the scattering will typically change. By measuring the probability of a He atom to go from an initial state i to a final state f information about the surface dynamical processes can be gained.

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Therefore, we need to measure the properties of the initial and final He beam which is in practice done by changing the current through the solenoids. This section follows closely the review of Alexandrowicz and Jardine 5 and a more detailed description can be found in the Supplementary Information.

Recall that we have two solenoids that generate magnetic fields which rotate the polarisation of the He beam.

The polarisation of the He beam in terms of amplitude and phase can be conveniently written as a complex number. The change in wavelength can be caused by the creation or annihilation of surface phonons.

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A short description of the scattering upon surface vibrations can be found in the Supplementary Information. An example of a wavelength intensity function is displayed on the right-hand side of Fig. The right panel shows a demonstration of the CS for a 1D example shown on the left. The Fourier slice phenomenon is demonstrated by the green lines on the right which indicate the direction of integration with the corresponding projections shown as lines in blue.

The green lines on the left hand side represent the one-dimensional Fourier transforms of the projections shown on the right.

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The polarisation data intensity is shown on a log scale for the sake of readability. Samples are taken from the Fourier data according to the sampling histograms shown. Sampling pattern A is unreliable in reconstructing the rightmost feature as it is the least sparse of the four peaks while sampling pattern B remedies this by taking more of the lower frequency values that it depends upon.

Reconstructions are at a resolution of data points. Looking at Fig.

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By restricting the measurement of P along the line the Fourier transform 10 becomes:. For a more detailed derivation of the Fourier slice theorem please refer to the Supplementary Information. Figure 2a shows how the Fourier slice theorem applies to the wavelength intensity function. Notice that different angles of integration produce different results, especially when it comes to discerning different features. If we only had knowledge of a fraction of the entries of f we can no longer use 14 to determine g directly as the problem is now underdetermined.

Therefore, the problem is not well posed and has to be modified. The matrix equation 14 can be inverted to give. The classical CS approach is to solve this problem via the now well established recovery problem. Typically this is a wavelet transformation. We can then solve this kind of problem quickly and conveniently using convex solvers such as the SPGL1 algorithm In this case, as well as many others, uniform random sampling may give suboptimal results and one has to sample with structured variable density sampling, see 21 and references therein.

The key problem is that the optimality of variable density sampling depends on the signal itself 21 , 24 , 25 , and thus designing the best sampling pattern is a very delicate task. We will give a short demonstration below. The key to understanding structured sampling is to understand the structure of the signal. For example, the coefficients of a signal in a wavelet basis typically have a very specific level structure. This is known as sparsity in levels.

Let x be a vector. This known structure can be utilised when designing the sampling strategy and is the motivation behind multilevel sampling.