Optimal compressive multiphoton imaging at depth using single-pixel detection
Philip Wijesinghe, Adrià Escobet-Montalbán, Mingzhou Chen, Peter R T Munro, Kishan Dholakia
OOptimal compressive multiphoton imaging atdepth using single-pixel detection
Philip Wijesinghe , Adri`a Escobet-Montalb´an , Mingzhou Chen ,Peter R T Munro , and Kishan Dholakia SUPA, School of Physics and Astronomy, University of StAndrews, North Haugh, St Andrews, KY16 9SS, UK Department of Medical Physics and Biomedical Engineering,University College London, Gower Street, London WC1E 6BT, UKJuly 5, 2019
Abstract
Compressive sensing can overcome the Nyquist criterion and recordimages with a fraction of the usual number of measurements required.However, conventional measurement bases are susceptible to diffractionand scattering, prevalent in high-resolution microscopy. Here, we explorethe random Morlet basis as an optimal set for compressive multiphotonimaging, based on its ability to minimise the space-frequency uncertainty.We implement this approach for the newly developed method of wide-fieldmultiphoton microscopy with single-pixel detection (TRAFIX), which al-lows imaging through turbid media without correction. The Morlet basisis well-suited to TRAFIX at depth, and promises a route for rapid acqui-sition with low photodamage.
Optical imaging at depth has gained a strong impetus in the past decade as itallows access to rich and intricate molecular information in three dimensions,even within living animals. This is now a burgeoning need in several fields,including neuroscience [1] and histopathology [2]. Researchers are particularlydrawn to multiphoton microscopy (MPM), specifically two-photon and latterlythree-photon modes, whose near-infrared excitation wavelengths allow deeperpenetration into biological tissues [3–6]. At these depths, to overcome the degra-dation of beam quality through scattering biological tissues, a range of wavefrontshaping methods have been demonstrated [7]. However, these methods are chal-lenging, slow, and are typically single-point correction schemes, requiring rapidrecalibration when considering any form of wide-field or volumetric imaging.Rapid MPM may be enabled by the concept of temporal focusing (TF),where the axial localisation is performed by focusing a pulse in time rather thanin space [8, 9], alleviating the need for point-scanning with a facile use of a1 a r X i v : . [ phy s i c s . op ti c s ] J u l iffracting element. Recently, TF has come to the forefront with the realisationthat spectrally dispersed light preserves spatial fidelity throughout scatteringmedia due to the temporal pulse compression being supported only by the in-phase, minimally scattered photons [10–13]. Wide-field TF MPM has beendemonstrated as a novel option for correction-free imaging at a depth of upto seven scattering mean-free-path lengths with two-photon [10] excitation andmay go further using three-photon [14] excitation modes. This advance (termedTRAFIX) was enabled with single-pixel detection, wherein structured patternsare sequentially projected onto the sample plane. The total diffuse signal isrecorded by a single-pixel detector, and a minimisation algorithm is used torecover the image. This alleviates the need for spatial coherence in the detectionpath [15], enables compressive sensing [11], and can also be performed in parallel,supporting fast acquisition times [12].Compressive sensing (CS) has led to remarkable achievements [16, 17], witha primary advantage of being able to reconstruct images with sampling wellbelow that required by the Nyquist criterion [18–20]. CS, however, has been ap-plied primarily to macro imaging. To date, little consideration has been given tohigh-resolution microscopy. CS may have a number of advantages in this area,including a reduction in photodamage [21]. CS requires the use of appropriatelychosen structured illumination patterns as a measurement basis set. Patternspossessing high spatial frequencies are challenging to relay through microscopysystems due to the diffraction limit, resulting in degraded resolution and fidelityand, ultimately, the loss of information about the sample. In selecting the opti-mal compression basis, one must consider optimisation of the spatial frequencycontent, beyond simply using a basis that best suits CS algorithms.Optimal spatial and frequency sampling has been described by Gabor [22],presenting the minimal trade-off between spatial and frequency localisation,which reaches the uncertainty limit. These real-valued Gabor filters (Morlets)have been used to generate a randomised basis for compressive imaging, op-timising information transfer to typical frequencies found in photographs [23].However, this principle is ideally suited for microscopy, where the frequencytransfer function is well-known. In this paper, we examine the random Morletbasis (a convolution of a Gabor filter with a random matrix) as an optimal basisset for CS in TRAFIX. We demonstrate that the Morlet basis provides superiorMPM performance to the conventional Hadamard and Random bases, with CSand when imaging through scattering media. The elegant formation and opti-mal performance of the Morlet basis are likely to stimulate its wider adoptionfor compressive wide-field microscopy.Pattern efficiency in TRAFIX relies on two principles: the suitability ofthe basis for CS, and the resilience of the patterned wavefront to propagationand scattering, which we describe in turn. CS is achieved by recognising thatmost signals are close-to sparse in some domain [20]. Briefly, we consider ourlinearised image vector x to comprise a sparse signal s in some domain, Ψ(here, the discrete cosine transform domain), such that x = Ψ s . Given thatthe majority of images are compressible, it is likely that many coefficients of s are close to zero [20]. Thus, images with N total pixels, can be acquired2ith M < N measurements, where M exceeds the number of non-zero coef-ficients of s . We do this by constructing a measurement basis Φ, comprising M rows of linearised patterns with N columns. Projecting each row sequen-tially onto the sample generates measurements on the photodetector given by y = Φ x = ΦΨ s = Θ s . We estimate the image ˆ x using l -norm minimisation:ˆ x = Ψˆ s = Ψ · arg min || s || , s.t. Θ s = y , i.e. , by finding the most sparse s thatcan generate the measurements in y . The efficacy of CS lies in designing Φ suchthat any linear combination of columns of Θ = ΦΨ are mutually incoherent ( i.e. , posses low correlation between any two columns) [24]. Orthonormal bases,such as the Hadamard, perform well when fully sampled ( M = N ); however,when under-sampled ( M < N ), lead to a Θ that is not mutually incoherent.Interestingly, random matrices are mutually incoherent and, as such, can beused for a substantially higher compression [20].For high-resolution applications, patterns to be imaged into the sample spaceare susceptible to degradation arising from spatial filtering in the objective’sback focal plane. Conventional CS patterns, such as the random pattern, possessvery broad spatial frequency spectra. This leads to the loss of high-frequencycomponents in both illumination and detection, a discrepancy between the gen-erated and the projected Ψ, and thus the loss of image quality. The proposedrandom Morlet patterns can be contained primarily within the entrance pupilof the objective, thus, they will be transmitted faithfully.The Morlet wavelet is described by a real-valued, centred, zero-mean Gaborfilter: g ( x, y ) = N · exp (cid:26) − x + y σ (cid:27) × cos (cid:110) πn p σ [ x cos θ + y sin θ ] (cid:111) , (1)where the first exponential term is a Gaussian with a given space-frequencybandwidth, σ , and the second term sets a modulation along a given direction θ that shifts the wavelet in the frequency domain; n p is the number of peaks ofthe Morlet wavelet; and, N in a normalisation factor, chosen such that | g | = 1.A basis is generated from a set of Morlet wavelets with σ and n p chosenrandomly from a normal distribution, convolved with an array of normally dis-tributed random values [23]. The resultant basis, inspired by Gabor’s filters [22],allows for fine spatial features to be sampled, whilst minimising the required spa-tial frequency bandwidth required. In particular, it is important to confine thefrequency content of the basis to that supported by the imaging system. Fora Morlet pattern illuminating the sample, we can approximate its field in theFourier plane as the Fourier transform of g ( x, y ) evaluated at spatial frequencies( x /λf, y /λf ). A Morlet wavelet in the Fourier plane is described by: G ( x , y ) ∝ exp (cid:40) − σ f (cid:2) ( x − a f cos θ ) + ( y − a f sin θ ) (cid:3)(cid:41) , (2)where: σ f = f λ/ πσ and a f = f λn p / σ represent the frequency bandwidthand the frequency shift, respectively. Given a particular back aperture radius,3 = f · NA, it is trivial to select σ and n p that fit, for instance, a f + 2 σ f < R .By contrast, a binary Random basis will uniformly overfill the back aperture tothe Nyquist spatial frequency.Fig. 1 illustrates the experimental set up of TRAFIX, where a pulsed lasersource (Chameleon Ultra II, Coherent) illuminates a digital micromirror device(DMD; DLP9000, TI). The DMD is imaged onto a blazed reflective diffractiongrating (DG; 600 lines/mm, Thorlabs), which spatially disperses the light inthe back aperture of the objective. The dispersed pulse is spatio-temporallyrefocused in the sample plane by an air immersion objective (20 × , 0.75 NA,Nikon). The resultant multiphoton signal is diffusely collected by a photo-multiplier tube (PMT; PMT2101, Thorlabs). The laser is tuned to a centrewavelength of 800-nm, with a 140-fs pulse duration and an 80-MHz repetitionrate. In-house built MATLAB and C l -norm minimisation is performed using the open source ‘ l -magic’ toolbox [18]and approximated using the ‘NESTA’ algorithm [25].Fig. 1(b–d) illustrates representative patterns from Hadamard, Random andMorlet bases, respectively, evaluated using Fourier optics in free space. An idealpattern is shaped by the DMD (location (1)), and a Fourier plane (FP) is formedby lens, L, at the back focal plane of the objective (location (2)). The DC com-ponent is omitted for clarity. The periodic structure of the Hadamard patternin Fig. 1(b) leads to a broad, structured, field intensity in the FP, whilst theRandom pattern in Fig. 1(c) leads to a speckle pattern in the FP. It is evidentthat both patterns greatly overfill the entrance pupil of the objective, which ismarked by the blue circle (15-mm diameter). This leads to a substantial portionof the field being filtered out by the aperture before reaching the sample. Thus,the patterns look considerably distorted at the sample image plane (location(3)). Such distortion is consistent with the qualitative observations in experi-ment that unless the pattern pixel size is made substantially greater than thediffraction limit, resolution is compromised. Fig. 1(c) illustrates that the Morletfield at the FP can be effectively transmitted through the objective, leading tothe image plane at the sample and the DG being nearly identical.We experimentally evaluate the bases on compressive imaging of 4.8- µ mgreen fluorescent polystyrene beads (G0500, Thermo Scientific). The beadswere suspended in water, dried onto a microscope cover slip and sealed usingUV-curing optical adhesive (68, Norland), minimising photobleaching to lessthan 5% in 2 hours of continuous imaging. Fig. 2(a-c) shows the beads imagedwith 64 × µ m field-of-view with variouslevels of compression, compared to a reference image taken by an EMCCDcamera (iXon EM + 885, Andor) (Fig. 2(d)). The illumination intensity andPMT integration time were constant for all recordings; thus, the images arescaled to the same system noise floor. Image quality is quantified as the peaksignal-to-noise ratio (PSNR) (Fig. 2(e)), defined as: 10 log (max( I ) /M SE ),where I is the image intensity and M SE is the mean squared error between theimage and the camera reference. Qualitatively, and from the PSNR, we can seethat the Hadamard basis performs well without compression; however, image4igure 1: Principle of TRAFIX. (a) Optical set-up. BE: beam expander; DMD:digital micromirror device; RL: relay lenses; DG: diffraction grating; L: lens;DM: dichroic mirror; EP: entrance pupil; Obj: objective; S: sample; and, PMT:photomultiplier tube. Numbered locations correspond to (1) the image on theDG, (2) the Fourier plane of the Obj, and (3) the sample image plane. (b-d) Simulated field intensity in free space of a Hadamard, Random and Morletpattern, respectively, at locations (1-3). Clipping by the EP is illustrated by ablue circle. 5delity declines rapidly with compression. Even at 25% compression ( i.e. , using3/4 of the total patterns), image quality drops more than two fold. The Randombasis performs consistently with compression; however, since it comprises thehighest spatial frequency bandwidth of all bases, the maximum achievable PSNRis reduced overall. The most significant benefit is observed using the Morletbasis, demonstrating the highest PSNR and a high resilience of image quality tocompression. Remarkably, even at 87.5% compression, individual beads can beresolved and localised, with a dynamic range above the Hadamard and Randombases at 25% compression. It is important to note that since the Randomand Morlet bases are not orthonormal, CS recovery at none to low compressionleads to an overdetermined measurement matrix whose inverse is ill-conditioned.To overcome this, the no-compression acquisitions Figs. 2(b-1) and (c-1) aredecomposed into two 50% compression datasets, which are averaged together.Figure 2: TRAFIX of 4.8- µ m beads using 64 × µ m. (e) Image quality (PSNR) as afunction of compression.We further evaluate the capacity of these bases to image through scatteringmedia. Fig. 3 shows 4.8- µ m beads imaged through a 360- µ m thick scattering6hantom (mean-free-path length, l s = 115 µ m), described in [10]. At this thick-ness, multiple scattering of the two-photon signal scrambles spatial informationsuch that no discernible image can be formed at the camera. However, usingsingle-pixel detection alleviates the need to preserve spatial information, thus,beads can still be resolved (Figs. 3(a-d)). Since the phantom is not perfectlyflat, not all beads are in the focal plane. Fig. 3(d) visualises the intensity acrossa bead in focus for all bases. We quantify the image quality as the contrast-to-noise ratio (CNR), calculated as the difference in the mean intensity of thebead and the background, over the standard deviation of the background noise, i.e. , CN R = ( ¯ I bead − ¯ I bg ) /σ bg . With no compression, Hadamard, Random andMorlet bases generate an CNR of 6.5, 10 and 13, respectively. Unlike the PSNRin Fig. 2(e), the Hadamard performs poorly. In Fig. 3(a), we can see that theHadamard basis generates a non-uniform point-spread funtion (PSF). The struc-tured, periodic nature of the Hadamard basis may lead to a discrete proportionof the patterns being either transmitted or lost. In the image in Fig. 3(a), thisis manifested as larger scale pixelation. Interestingly, with 87.5% compression,no discernible image can be reconstructed from the Hadamard, and the CNRbecomes 0.2, 4.9 and 9.8, for each respective basis. It is evident that the Morletbasis generates superior image quality, with and without compression throughscattering. Fig. 3(h) demonstrates that even at high compression and throughscattering, the bead is clearly identified.Figure 3: TRAFIX images of 4.8- µ m beads through 360 µ m of a scatteringphantom, with (a-d) no compression and (e-h) 87.5% compression; using 64 × µ m.A particular advantage of CS in MPM is that the use of fewer patterns min-imises photobleaching, which is an important consideration for sensitive mark-ers, for in vivo and for long-term imaging applications. If the time of exposure7s limited, the Morlet basis should permit an image with the highest samplingresolution for a given image quality. Fig. 4 demonstrates this by imaging 10- µ m beads without scattering over a 220- µ m field-of-view with a constrainedacquisition of 3072 patterns. Equivalent noise performance is obtained fromHadamard, Random and Morlet bases with 25%, 67% and 82%, respectively,as can be estimated from the PSNR in Fig. 2(e). Within these limits, we areable to employ 64-, 96- and 128-pixel wide bases, respectively. Given the largersampling, we employ the substantially more efficient NESTA algorithm [25] thatapproximately solves the l -minimisation problem, whilst adhering to a set spa-tial smoothness, || y − Φ x || l < (cid:15) . Figs. 4(d–f) show a close up of the beads.It is evident that there is a progressive increase in tolerance to higher sam-pling resolutions, with the Morlet basis clearly delineating beads with the leastpixelation.Figure 4: TRAFIX images of 10- µ m beads without scattering over a 220- µ mfield-of-view using 3072 patterns (equal acquisition time). (a) 64 × × × µ m for (a,b,c) and 10 µ m for (d,e,f).In this paper, we have demonstrated that the random Morlet basis, basedon Gabor’s minimisation of the uncertainty criteria, presents an elegant andoptimal solution to compressive imaging in microscopy. Very recently, we haveseen a flurry of demonstrations combining wide-field temporal focusing withsingle-pixel detection to achieve correction-free multiphoton imaging at im-8roved depths [10–12, 14]. Fortuitously, this geometry lends itself to CS, withprospects to increase acquisition speed and minimise photobleaching, which initself TRAFIX minimises compared to standard point-scanning approaches. Wehave demonstrated that unlike many other conventional CS applications, a sub-stantial consideration must be given to the fidelity of the measurement basis asit is relayed through the focusing system and scattered by the sample.The ideal space-frequency control of the Morlet basis, designed to optimisewavefront propagation through a microscopy system, leads to an overall superiorperformance in image quality and a high resilience to compression. Furthermore,the Morlet basis minimises power loss, which is important to MPM, wherehigh illumination power is difficult to achieve over a wide field-of-view withaffordable laser sources. The high compression achievable with the Morlet basisresults in a substantial reduction in illumination time, and thus photodamageof the sample [21]. This is a particular area of concern for two-photon andlatterly three-photon microscopy. In this area, as well as due to its optimalperformance, the Morlet basis is well-positioned to make a major impact forcompressive wide-field single-pixel multiphoton imaging. Acknowledgements
We thank the UK Engineering and Physical SciencesResearch Council (grant EP/P030017/1) and the European Union’s Horizon2020 Framework Programme (H2020) (675512, BE-OPTICAL). We thank VanyaMetodieva, Wardiya Afshar Saber and Federico M. Gasparoli for helpful discus-sions and support.
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