1. Introduction

There has been recent interest in the use of multiplex sensing techniques in applications at the atomic scale [1–8]. Traditionally, measurements using x-rays or electrons systematically interrogate a system at a single point in space, time, or photon/electron energy at once before moving on to the next measurement. In contrast, multiplex methods measure a combination of space/time/energy points simultaneously. Many repeated multiplexed measurements can convey the same information as a systematic scan if the points combined are chosen wisely. In some cases a multiplex scheme has clear advantages, either fundamental or practical. In other cases, systematic scanning is preferable. The purpose of this paper is to provide a framework for reasoning about which is best in the context of x-ray and electron science.

Through a very general analysis, we are able to conclude that for signals linear in the x-ray or electron intensity:

To build some intuition for how multiplex sensing techniques work, consider acquiring an x-ray fluorescence image of a sample. We could go about this in the usual way: by rastering a focused x-ray probe over the sample, thereby illuminating one “pixel” at a time, and recording a spectrum at each position. At the end of the experiment, the fluorescence image is simply the set of all acquired spectra, with each spectrum assigned to a spatial location on the image.

Alternatively, we could de-focus the x-ray beam and modulate the transverse intensity distribution with a random mask. This mask might illuminate half of the pixels from our previous raster scan, chosen at random. Then, a measurement acquires the summed spectra from all the illuminated pixels, which by itself contains no spatial information. Imagine, however, we fabricate a large number of such masks and repeat the experiment many times. Every time a specific pixel $i$ is illuminated, it will contribute the $k$ points in its spectrum ${\boldsymbol {\mathbf {x}}}_i$ to the spectral sum; for mask $m$, let’s write that measured sum

In the example presented, we specifically designed the masks and therefore knew exactly how to write our matrix $A$, with 1s and 0s for illuminated or shadowed pixels for each mask. The scheme would work just as well, however, if we had random masks we did not choose specifically, so long as we were able to measure and record which pixels were illuminated. This gives us ability to use the randomness inherent in the experiment to our advantage – so long as we can measure that random patterning to high resolution. Figure 1 shows an illustration of all three methods discussed for acquiring a signal: raster scanning, multiplexing with known patterning, and multiplexing using measured patterning. The example envisioned is spatial in nature. The same procedure works just as well, however, in the time or energy (spectral) domains.

 

Fig. 1. Three different sampling schemes for a single imaging task. Top left: in a raster scan, small regions of the object are illuminated and recorded one at a time. Top right: in a multiplex scan, many selected regions are illuminated and recorded at the same time, and their sum recorded. The patterns of illumination are chosen, so they are known. The procedure is repeated many times and the final object reconstructed mathematically. Bottom: a second example of a multiplex scan, ghost imaging. In a ghost imaging setup, a randomly patterned illumination is used and the sum recorded. The patterning, though random and uncontrolled, is measured via a diagnostic, typically a beam splitter and area detector. The reconstruction procedure is exactly the same as for the multiplex scan.

Download Full Size | PDF

When is multiplexing advantageous? In this paper we provide a framework for analyzing this question, then discuss three specific cases where multiplex sensing may be useful. Our focus is on applications involving the use of x-rays and electrons, where measurements are in many ways expensive: they are often flux-limited, cause irreversible damage to the sample and require time at costly facilities. In some cases, multiplexing will offer a way to alleviate these concerns, but in other situations it may exaggerate them. Our goal is to understand when multiplexing is useful for x-ray and electron experiments and when it should be avoided.

An important note on the use of prior information in imaging experiments is necessary. In general, if one knows something about the signal about to be acquired, for example if it is smooth or always-positive, that fact can be used to improve the measurement process. These priors can be implemented equally well in any sampling scheme, whether rastering or multiplexed. There is one powerful prior, however, for which this equivalence is known to not be true. If the signal is sparse in some representation (i.e. compressible) then multiplex sensing can offer serious advantages over raster scanning [9,10]. Because of this unique feature, sparsity will be the only prior discussed in this paper. We will first consider sampling schemes that do not make use of sparsity, then show how if the signal is known to be sparse, that information can be readily incorporated into the analysis of different imaging schemes.

Our analysis builds on progress from a variety of fields. One specific implementation of multiplex sensing is classical “ghost imaging” (Fig. 1), which refers to a class of experiments where a randomly patterned wavefront is split into two branches [11]. One branch measures the total transmission through a sample of interest, called the bucket signal. The other branch images the wavefront. A moment of thought reveals this process to be a case of spatial multiplexing, identical to the imaging fluorescence spectroscopy scan discussed previously, only using random beam patterning instead of known masks. In the ghost imaging community, multiplexing with known masks is conversely known as “computational ghost imaging” [12]. Similarly, Fourier transform spectroscopy is also a multiplexing scheme, where many wavelengths are measured simultaneously per measurement [13,14]. Our presentation generalizes these examples and many others into a single framework. Our treatment is completely classical in nature, ignoring any information gain that might be achieved via performing measurements with entangled particles. Entanglement has been focus of specific studies in ghost imaging – for perspective see [11] – but is not heavily used in current applications.

While the idea to use multiplexing in x-ray and electron experiments has only recently started to recieve significant attention, demonstrations at synchrotrons [1–3], FELs [7], and electron sources [6] have already been conducted, paving the way for multiplexing to start to impact applied x-ray and electron science.

2. Framework

Consider a measurement process where we perform $m$ measurements with $n$ unknown channels, for example the pixels in an image or amplitudes for given frequencies in a spectrometer. The channel values are the unknowns we wish to infer. We measure the weighted sum of all channels; weights may be zero. Crucially, we assume the system response is linear in all aspects, including as a function of beam intensity and the number/character of the channels. Further, we consider the general case where each channel’s output can be described by a $k$-length vector, for example $k=3$ if a pixel gives a simultaneous scalar output for red, green, and blue when measured, and these colors can be recorded separately. Finally, we assume some additive noise in each measurement.

Our goal is choose a weighting scheme that will allow us to infer each individual channel’s $k$ dimensional response. We write this symbolically as follows. Let $R$ be the $m \times k$ matrix of recorded outcomes, $A$ be the $m \times n$ sensing matrix, and $X$ the $n \times k$ signal of interest. Note that multi-dimensional signals can be written as a flattened one-dimensional signal – in the following formalism, we consider each pixel independent and do not consider shared information between adjacent pixels. In practice, it is possible to extend the method to include shared information, see for example [15], but the details depend on the domain of application. Then we consider a measurement process in the presence of uncorrelated, additive noise $\epsilon$ (an $m \times k$ matrix, but where all $k$ pixels are assumed to have equivalent noise) with finite variance $\sigma ^2$,

The solution to (4) is given by the ordinary least squares solution,

 

Fig. 2. Simulations showing the performance of different sampling schemes: raster, Bernoulli, half-Gaussian, and Hadamard. A 1023-length waveform with iid random samples from $\mathcal {N}(0,1)$ smoothed using a size-15 median filter was used as the signal of interest. This signal was reconstructed using either the ordinary least squares estimator (Eq. (5)) for which the mean-squared error [left] and the dose-error product is shown [middle], or the traditional ghost imaging correlation algorithm (Eq. (19)) [right]. Top: in the absence of noise, Bottom: in the presence of 20% ($\sigma _m = 0.2$) per-measurement additive Gaussian noise, not per-channel noise, highlighting Felgett’s advantage (see Section 4.2). Each simulation was repeated 10 times with new random values for the signal, sensing matrices, and noise, with average results shown. Black line shows $m = n$. Observations: (1) the dose and error predictions of Table 1 are reproduced, (2) the Felgett advantage is clearly visible in the simulation containing noise [bottom left], (3) the performance of the traditional ghost imaging algorithm is inferior to the OLS solution as predicted by theory (Appendix B). Python code to reproduce simulations is available in Code File 1 (Ref. [22]).

Download Full Size | PDF

Table 1. Quantitative comparison between the sampling schemes discussed, produced using Eqs. (6) and (9). Recall $m$ is the number of measurements, $n$ is the number of channels (i.e. unknowns), $a$ is the average dose per channel, $\sigma$ is the standard deviation of the additive Gaussian noise per measurement. The factor $1.66$ in the half Gaussian distribution approximates $\left ( 1 - 2/ \pi \right )^{-1/2}$. Note that all these values assume $A^T A$ is not singular, which can only occur when $m \geq n$. $^\ddagger$Hadamard error estimate derived in [14].

View Table

Equation (6) naturally separates the error contributions from the noise, given by $\sigma$, and the stability of the inversion of $A$ represented by the trace term $\mathrm {Tr} (A^T A)^{-1}$. As we will see, both depend on the choice of $A$, and are therefore determined by the experimental design. The noise factor, $\sigma$, depends also on the noise characteristics of the detection method, which turns out to be important for the comparison between different sampling schemes.

2.1 Noise model and transferred noise

While in general the presented equations are applicable in any case of additive noise, in real experiments the noise can often be broken into three standard types:

Typically, one source of noise will be dominant in the system. In that case, the other sources will be negligible and we can approximate,

More interesting is the case where the error is proportional only to the number of measurements. This occurs for example if there is some significant constant readout noise for a detector – then, the number of channels doesn’t matter, only the number of readouts (measurements). In this case, the overall error is reduced by increasing the number of channels and scales as $\mathcal {E} \sim n^{-1/2}$. The advantage comes from each channel contributing additional intensity, and that intensity boosts the signal higher and higher above the constant noise floor. This is well known in the field of spectroscopy, where the $n^{-1/2}$ reduction of the noise as a function of multiple channels is known as the Felgett advantage, one motivation behind Fourier transform and Hadamard spectroscopies. See [14] for a comprehensive study of noise in multiplex spectroscopy.

The question is if this kind of noise is applicable in modern x-ray and electron imaging setups. Current x-ray detectors are able to routinely count individual photons with high quantum efficiency, which effectively eliminates readout noise of this sort [20]. Electron detectors have also progressed into the quantum counting regime recently [21]. The major source of noise in modern setups, therefore, is quantum (Poisson) in nature, and occurs on a per-channel basis. The traditional Felgett advantage does not apply in such situations, and no signal-to-noise advantage is achieved by multiplexing.

2.2 Inversion stability: trace term

Equation (5) assumes that $A^T A$ is non-singular and can be inverted, which in general is not true, especially if $A$ is random. In such cases approximate – but often very good – solutions may be found by solving Eq. (4) via a matrix factorization or a minimization algorithm. Reasonable solutions will only be obtained if $A^T A$ is high rank, and good design of a sampling scheme should ensure that $A^T A$ is non-singular with high probability in the limit of a reasonable number of samples. The schemes we discuss here are all of this variety. In the case of random sensing matrices, however, $A^T A$ may only be strictly non-singular in the limit of a large number of random samples, and errors computed using (6) will be lower bounds.

2.3 Radiation dose

In this model, the total dose on the sample $\mathcal {D}$ for a set of measurements is given by the sum of all the elements of $A$,

3. The analysis of some examples

We consider four sampling schemes for a demonstration comparison: a raster setup, two random multiplexed schemes, and one deterministic multiplexing scheme. For each, we compute the error of a recovered signal as a function of the number of measurements performed and radiation dose incurred. Results are summarized in Table 1.

3.1 Raster scans

In a raster scan, each pixel is sequentially illuminated with intensity $a$ so that $A \equiv aI$, where $I$ is the identity matrix with $1$ on the diagonals and zeros elsewhere. Usually the number of measurements equals the number of channels ($m = n$). In the presence of noise, it might be productive to repeat measurements. In that case $A$ once again becomes an $m \times n$ matrix where rows of the identity matrix start to repeat.

The solution to the reconstruction problem (Eq. (5)) is simply $\hat {X} = R/a$, reflecting the intuition that in raster scanning no reconstruction is necessary. The radiation dose of the experiment is $am$ while the expected error is $(\sigma \sqrt {n}) / ( a \sqrt {m} )$, showing how increasing the beam intensity overcomes background noise at the cost of possibly damaging the sample.

3.2 Multiplex scans

Any non-raster scan is a multiplex scan. To highlight the differences, we will discuss situations where the sensing matrix $A$ is fairly dense, i.e. about half or more of the elements are non-negligible. Consider the following three concrete multiplex scans:

3.3 Comparison

A quantitative comparison between these four sampling schemes is presented in Table 1, containing analytical results, and Fig. 2 which contains numerical results. Some important observations are immediately noticeable:

4. Experimental opportunities for multiplex sensing

The presented framework gives us the tools for assessing the fundamental capabilities and costs associated with a particular sampling scheme. Often, however, the benefits of multiplexing are practical. The ability to use flexible illumination in either space or time can alleviate engineering challenges. For example, Kabra and colleagues recently used a multiplexing approach to measure the quantum efficiency map of copper photocathodes using the natural fluctuations in the drive laser [15] parasitically. Previously, the cathode would have to have been taken offline and raster-scanned, but using multiplexing the mapping can be done in the background durign normal operations. Next, we discuss situations where multiplexing can give performance advantages due to either practical or fundamental concerns.

4.1 Resolution improvement

In standard, raster-style imaging, the resolution of the measurement is limited by either the probe or detector resolution. For instance, in the x-ray fluorescence imaging application discussed in the introduction, the final resolution of the hyperspectral image obtained will be limited by the x-ray focus size.

Multiplexing enables one to trade the resolution of the probe or detection system for the resolution at which the probe can be patterned or measured. This trade may make it possible to collect data at higher resolution. Alternatively, it may be a simpler or cheaper option than increasing the probe resolution.

As an example, we recently suggested along with our colleagues that by measuring the random temporal fluctuations inherent in the power of an XFEL pulse, pump-probe experiments with time resolution faster than the pulse duration would be possible. A typical XFEL pulse has a duration on the order of 10-100 fs, but its power over time can be readily measured to $\sim 1$ fs resolution using well established diagnostics such as the LCLS’s XTCAV [23]. This should enable x-ray pump-probe measurements to be conducted down to 1 fs resolution. The alternative of decreasing the pump and probe durations – preparing 1 fs x-ray pulses – is possible, but challenging, and requires new hardware to implement [24–26]. Therefore for some experiments time-domain multiplexing should offer an attractive alternative.

This highlights an interesting aspect of multiplex sensing. In cases where an experimental setup naturally results in random fluctuations in the measurement, it may be possible to use that randomness rather than fight it.

4.2 Felgett’s advantage: more signal per measurement

Multiplex scans have multiple channels illuminated at once. Thus, for a system that has a fixed flux-per-channel, multiplexed illumination offers a higher overall flux on the sample. In experiments with per-measurement noise, this enables one to achieve a specific signal-to-noise ratio with fewer measurements. Given the large investments in making brighter x-ray and electron sources and the expense associated with operating those sources, the ability to use more of the incident flux is attractive.

It should be emphasized that the experiment time is reduced precisely because the sample is exposed to more incident radiation. This does not necessarily translate into more information acquired for a given radiation dose. Moreover, the multiplex advantage only exists with per-measurement noise (such as $\sigma _m$ in Eq. (8)), and there is no gain when using photon-counting detectors.

4.3 Signal-to-noise per dose

Can multiplex sensing enable one to obtain more information for a given radiation dose? Many atomic resolution imaging tasks are radiation-damage limited, and the possibility to use multiplex sensing to overcome that limit has been discussed in recent literature [3–5,7]. This discussion has revolved around ghost imaging setups, where it has been suggested that damage might be avoided by decreasing the flux on the object branch and increasing the flux into the reference branch. Since the reference branch radiation does not interact with the object, then it follows damage will be reduced.

This argument needs to consider errors in the measurements performed on both branches. The analysis presented so far corresponds to the case of an infinitely bright reference branch with no error on the matrix $A$. In the case where such error is significant, one expects regression dilution [16] and our work only provides a lower bound error estimate. Here, we consider only the limiting case, where the reference branch is made so bright that the error on the measurement of $A$ is negligible.

Now we only need to consider error on the object branch (as written in Eq. (6)). Even with an ideal detector, there will at least be Poisson noise due to the fact we image with quanta (photons or electrons).

Even in this best case scenario, our analysis shows there is no fundamental way classical ghost imaging – or any multiplexing scheme – can alleviate radiation damage (in the absence of priors, see Section 5). Multiplexing with $n$ channels, corresponding to pixels on the reference branch, reduces the error by at most a factor of $\sqrt {n}$ (Eq. (8)) which occurs in the presence of per-measurement noise. From a signal-to-noise perspective, this is equivalent to reducing the number of measurements, and therefore the radiation dose, by a factor of $n$ (since error scales with the square root of the number of measurements, $\mathcal {E} \sim m^{-1/2}$). The $n$ additional channels, however, increase the dose by a factor of $n$ (Eq. (9)). Thus, the additional dose used to obtain Felgett’s advantage exactly offsets the dose reduction possible by conducting fewer measurements (see Table 1). In the best case scenario, multiplexing does not improve the error-to-dose ratio.

In the case of a perfect detector with only quantum noise, the scaling is even worse – multiplexing does not decrease the error at all, but still incurs the dose penalty. Alone, multiplexing cannot limit radiation damage, only make it worse.

We can understand this intuitively. Consider the limiting case in a ghost imaging setup, where for each ghost image exposure the object branch is so weak that exactly a single photon travels along that branch (zero photons would provide no information, as the bucket would always read zero). That photon is either absorbed by the object or transmitted and detected. An infinitely bright reference branch provides a probability distribution for where the photon interacted with the sample. What is the most informative possible distribution? Clearly, a delta function – telling us exactly where the imaging photon went – would be most useful. This corresponds precisely to a raster scan, where we send the photon in a pre-determined and known direction and measure the result. Ghost imaging imparts at most an equal amount of information per object-branch photon as a raster scan, and in general less.

4.4 An example of applied multiplexing: XFEL self-seeding

We propose an application of ghost imaging to solve a real problem currently facing the LCLS XFEL at SLAC. Many spectroscopy experiments require a high intensity concentrated in a narrow bandwidth of x-ray energies. XFEL self-seeding has been pursued to produce XFEL pulses with a higher spectral brilliance than simple SASE pulses [29–31], but at soft x-ray wavelengths these seeded beams come with a large “pedestal” of undesired wavelengths that dilute the spectral purity of the beam [27]. For spectroscopy experiments that require the smallest bandwidths, this means the beam must still be monochromated, which inevitably reduces the total flux due to losses in the monochromator [32].

Multiplexing provides a possible way to overcome this issue. In a standard soft x-ray absorption spectroscopy experiment, a monochromatic beam, generated either using a monochromator or self-seeding, is passed into the sample and the absorbed fraction of the beam (or some proxy of absorbance) is recorded. The wavelength of the beam is scanned to measure the absorbance at all relevant wavelengths. If the incoming beam is not purely monochromatic, the spectrum obtained is convolved by the beam bandwidth.

If, however, we measure the incident x-ray spectrum across many randomly fluctuating exposures, we can use the multiplexing algorithm (3) to recover the absorption spectrum at the resolution of the incident x-ray spectrum measurement. Seeded or SASE beams with a large bandwidth but high intensity can be used without suffering from the loss of resolution caused by the large bandwidth. Measurement of the XFEL spectrum is easier than controlling it precisely, and in the specific case of soft x-ray self seeding, the desired level of control is not currently possible [27]. A similar application of multiplexing to soft x-ray spectroscopy has recently been experimentally demonstrated to overcome large bandwidths caused by the use of transform-limited attosecond XFEL pulses [33].

To demonstrate the ability of multiplexing to overcome the pedestal present in current soft x-ray self-seeding at LCLS, we have conducted a simulation of the results expected for an absorption spectroscopy experiment performed in both the traditional fashion and using multiplexing, for both SASE and seeded beams. The XFEL spectra used in the simulation are real LCLS spectra recorded as part of self-seeding tests [32]. The sample absorption spectrum is a phantom generated to include two broad features separated by a narrow bandwidth on a sloping baseline, capturing possible challenges associated with different samples [32]. The results, presented in Fig. 3, show that by applying the multiplexing approach, weak spectral features of 200 meV linewidth can be resolved by the multiplexing analysis, but not the standard binning approach. Additionally, we see that the seeded beam, which is similar to a true raster scan, recovers the signal with less error than the broader-bandwith SASE beam. Nonetheless, spectroscopy with the SASE beam on this sample would be possible with the multiplexing analysis combined with a sufficient quantity of data.

 

Fig. 3. Simulation of a soft x-ray absorption spectroscopy experiment performed at LCLS, using both traditional and multiplexing approaches. Simulations were performed using 175,000 individual XFEL spectra measured at LCLS, either in SASE mode at undulator 19 or after soft x-ray self seeding after undulator 24 [27]. Spectra were randomly shuffled and energy-shifted before use. These spectra ($A$) were multiplied by a putative spectrum ($\mathbf {x}$, top left panel) to produce a measured bucket signal $\mathbf {r} = A \mathbf {x}$. Teal traces show recovery using the mean energy of the individual x-ray spectra, while the blue traces show recovery using least-squares regression as in eqn. 4 along with an additional regularization term $\alpha || \mathbf {x} ||^2_2$ (ridge regression [28]) that stabilizes the solution. The right panels show the mean spectra (bold) and five representative spectra (background) for both the seeded and SASE cases.

Download Full Size | PDF

5. Compressed sensing

Throughout our discussion we have assumed that no prior information is used in the signal reconstruction process. There is, however, one important situation where prior information about the structure of the signal can – in combination with multiplex scanning – dramatically improve the imaging process.

If the signal $X$ to be measured is sparse in some basis, then techniques of compressed sensing can be applied [9,10]. Here we quickly review compressed sensing theory, in order to show how multiplexed measurements can enable a general application of compressed sensing in x-ray and electron science.

Compressed sensing presumes that our signal $X$ of interest has a sparse representation. Concretely, this means that there exists some orthonormal basis $\Psi$ such that $X_s \equiv \Psi ^{-1} X$ is $s$-sparse, meaning it has only $s$ non-negligible ($\approx 0$) entries. If $s \ll n$ we may be able to determine the signal $X$ of interest with order $s$ measurements instead of $n$ (Fig. 4). If we can do so, we can obtain the signal with less time, radiation dose, and error.

 

Fig. 4. Compressed sensing uses sparsity to enable inference with fewer samples than unknowns. Top: when solving linear systems, if the number of unknown variables $n$ is greater than the number of known measurements $m$, then the problem is underdetermined and no unique solution exists. Middle: if, however, the signal is known to be sparse in some basis $\Psi$, then the number of knowns is effectively reduced and the problem may become solvable. It is essential, however, that the majority of the measurements provide information about the non-sparse elements of the sparse signal. Bottom: if the matrices $A$ and $\Psi$ are CS-incoherent, then this is ensured. If, however, $A$ and $\Psi$ are highly correlated, then there is a high chance any measurement only informs on zero components of the sparse signal and provides no information. The required number of samples in this case is expected to return to order $n$, as this number of samples are required to ensure all components of the sparse signal are measured.

Download Full Size | PDF

The basis $\Psi$ can be chosen to maximize the sparsity. For example, if the signal is composed of a waveform with only a few frequencies, choosing $\Psi$ as a DFT matrix will transform the signal into the frequency domain, where it will be sparse. Further, it is absolutely possible for the signal to be sparse in the original domain of interest (for example, an image with a uniform background), in which case $\Psi = I$ is a reasonable choice.

If we knew precisely which elements in $X_s$ were sparse, the procedure to use would be straightforward. We could simply truncate $X_s$ to the non-negligible components and perform our inference as before with $s$ instead of $n$ unknowns, under the image formation model $R = A \Psi X_s$. We assume a complete column of $k$ elements in $X_s$ is zero or not for the purposes of counting sparse elements. If one element is non-zero, that entire column is counted as one of the $s$ datapoints we have to infer. If, however, we do not know which components will be sparse, it is essential that each measurement provides some new information about each non-sparse component. This is where multiplexed measurements enter the picture. To ensure that each of our measurements are highly likely to give information about all $s$ components, the measurement scheme $A$ must be constructed in a particular way. It turns out random multiplexing satisfies this requirement for any signal and any sparsity basis, as we will now discuss.

In the theory of compressed sensing, the ability of a sensing scheme to yield information about all the non-zero signal components is quantified by the coherence $\mu$

Given a pair $A, \Psi$, we can replace Eq. (4) with the lasso regression algorithm,

 

Fig. 5. Example of compressed sensing using the multiplexing schemes discussed. A signal (top left) of 1023 samples was constructed by summing 16 cosine functions with random frequencies (chosen uniformly over the band limit but excluding the DC component) and positive amplitudes from $10 \times |\mathcal {N}(0,1)|$, making the signal sparse in the Fourier domain (top right). The signal was then reconstructed from a variable number of samples using either the ordinary least squares algorithm (bottom left) or the lasso algorithm with $\lambda = 10^{-2}$ for all sensing matrices, except the identity case (which has a factor of $\approx 1024 / 2$ smaller amplitude) for which $\lambda = 1 \cdot 10^{-5}$. Values of $\lambda$ were hand-tuned. Shown are the averages of 10 randomized repeats of the simulation process. On the bottom panels, the leftmost black line indicates $m = s \log n$, and the rightmost line is $m = n$. The Hadamard reconstruction requires additional samples because it is partially coherent with the Fourier basis ($\mu = 20.3$), as compared to the incoherent identity, Bernoulli, and half-Gaussian sensing bases ($\mu = 1.4, 3.4, 3.0$ respectively). Simulations were measurement noise-free, variation seen is a function of the randomness of the simulated true signal only. Python code to reproduce simulations is available in Code File 1 (Ref. [22]).

Download Full Size | PDF

Consider instead trying to use a raster scan ($A=I$) to reveal a naturally sparse signal ($\Psi = I$). Here, $A \Psi = I$ and so $\mu ^2 (A, \Psi ) = n$, confirming the expectation that $n$ independent measurements are required to uniquely infer a signal of size $n$.

Since we wish to presume as little as possible about the sample of interest and its sparsity structure, it is natural to ask: are there sensing matrices $A$ that are CS-incoherent with nearly all sparsity representations $\Psi$? It turns out the answer is yes, and randomly multiplexed schemes, such as the half-Gaussian and Bernoulli matrices discussed in this paper, are examples. It can be shown the random nature of such sensing matrices make it almost certain $\mu \ll n$. In other words, signals that are randomly multiplexed are guaranteed to be CS-incoherent. This holds for any sparsity basis $\Psi$ (specifically $\mu ^2\;<\;c \log n$, and often $\mu =$ const.) [9,10]. This theory shows why multiplexing is so useful in compressed sensing applications – random multiplexing enables us to use any sparsity basis of our choice, thereby maximizing sparsity and the associated advantages of compressed sensing.

As an intuitive example, consider an example where the signal of interest consists of an image with only a few localized features on blank background, but the feature locations are unknown (here, $\Psi = I$). Raster scanning for the features will be slow. In fact for $n$ pixels, $s$ of which are non-blank, we get useful information only a fraction ($s/n$) of the time we perform a raster measurement. Further, we have to scan all the pixels to ensure we don’t miss any feature. Instead, consider conducting measurements where each pixel has a $50\%$ probability of being illuminated in each measurement. On average, we obtain information on fraction $s/2$ of the information-containing pixels per measurement, as compared to the average fraction $s/n$ in the raster scan. Since it is likely $n \gg 2$, this is a more efficient measurement scheme. Further, the probability of missing a given feature goes as $2^{-m}$, which means it is unlikely we miss any feature entirely, even for a modest number of measurements ($<\;0.1\%$ for $m=10$).

We emphasize that by reducing the number of samples below $n$, compressed sensing is the only multiplexing approach which reduces the total dose needed to achieve a given signal-to-noise ratio.

6. Summary

We have presented a framework for assessing the performance of different sampling schemes. In addition to the common rastering method, where a single point in space, time or energy is measured, we have considered multiplex schemes where sets of points are measured simultaneously. In specific cases, the latter can offer advantages in experiment time, sample use, and measurement resolution:

The advantages offered, however, are idiosyncratic to each application. When considering a multiplex sampling scheme, we recommend answering four questions first:

Large capital investments are currently being made in x-ray and electron facilities and instrumentation. It is our belief that compressed sensing offers an under-utilized opportunity to push the capabilities of these new radiation sources, making current experiments more efficient and opening new experimental possibilities. Since compressed sensing and multiplexing go hand-in-hand, this in turn means we anticipate an increased use of multiplex sensing in x-ray and electron science in the near future. Future work is necessary to develop general ways to apply compressed sensing to the wide variety of imaging currently being performed with electrons and x-rays.

Even in cases where sparsity cannot be leveraged, we believe multiplexing offers significant advantages for very specific x-ray and electron imaging experiments. Our hope is that the work presented here helps clearly identify those advantageous cases and implement them.