Efficient single-pixel multispectral imaging via non-mechanical spatio-spectral modulation

Combining spectral imaging with compressive sensing (CS) enables efficient data acquisition by fully utilizing the intrinsic redundancies in natural images. Current compressive multispectral imagers, which are mostly based on array sensors (e.g, CCD or CMOS), suffer from limited spectral range and relatively low photon efficiency. To address these issues, this paper reports a multispectral imaging scheme with a single-pixel detector. Inspired by the spatial resolution redundancy of current spatial light modulators (SLMs) relative to the target reconstruction, we design an all-optical spectral splitting device to spatially split the light emitted from the object into several counterparts with different spectrums. Separated spectral channels are spatially modulated simultaneously with individual codes by an SLM. This no-moving-part modulation ensures a stable and fast system, and the spatial multiplexing ensures an efficient acquisition. A proof-of-concept setup is built and validated for 8-channel multispectral imaging within 420~720 nm wavelength range on both macro and micro objects, showing a potential for efficient multispectral imager in macroscopic and biomedical applications.

Scientific RepoRts | 7:41435 | DOI: 10.1038/srep41435 Although these strategies enable faster and more photon-efficient acquisitions, the use of mechanical modulation leads to unstable and complex implementations, especially in high speed operation.
Here, we propose a multispectral imaging scheme under a single-pixel CS architechture with an all-optical multiplexing technique instead of mechanical modulation. Single-pixel CS systems use a spatial light modulator (SLM) to modulate spatio-resolved light into structured pattern, whose resolution is the same as that of the reconstructed image. Inspired by the spatial resolution gap between current SLMs (e.g., 1920 × 1080 pixels for Texas Instruments device: DLPNIRSCANEVM,.9XGA DMD) and target reconstructions (whose dimension are mostly in the order of 100 × 100 pixels), we integrate CS technique of different spectral bands in a spatial multiplexing manner. Specifically, the light emitted from the scene is first divided and spatially mapped to different subregions after travelling through an optical splitter introduced by refs 21 and 22, with each counterpart of distinguishing spectral distribution. Separated channels then enter an SLM and get coded simultaneously by independent patterns. The summation of spectral information for all channels is collected as one measurement by a single-pixel detector, and this ensures a high signal-to-noise ratio (SNR) compared to systems that record spectral information onto separate detectors or sequentially onto a single detector. To fully utilize the sparsity both in spatial domain and along spectral direction, 3D compressive sensing algorithm is adopted to reconstruct the multispectral information.

Results
Schematic and acquisition. The optical configuration of our system is schematically illustrated in Fig. 1.
The configuration is essentially a single-pixel system operating on multiple channels simultaneously. Spatially and spectrally distinguished signals are individually modulated on the SLM plane, and the coded multispectral information is effectively recorded without sequential scanning.
Illumination light generated by a broadband white light source is first filtered to the wavelength range of 420~720 nm by a band-pass filter, and then gets converged by a collective lens to illuminate the target object. Emitted light coming from the object plane then propagates through an objective lens (convex lens for macro objects or microscopic objective for micro samples), gets colliminated at a proper beam size, and reaches the spectral splitting device. Instead of mechanical translation or filtering device, we spatially split the multispectral components by an optical splitter. The splitter is assembled with a polarizer and a cascade of alternatively placed phase retarders and Wollaston polarizing beam splitters, as shown in the topleft inset of Fig. 1. The polarizer first converts colliminated light into linearly polarized light. The following phase retarder changes the polarization state of outgoing light into a function of the wavelength. Considering a phase retarder with thickness d, and the ordinary and extraordinary refractive indexes n o and n e , the phase shift for wavelength λ is given by Δ = π|n o − n e |d/λ. When propagates through a Wollaston prism, light is splitted into two orthogonal linearly polarized beams (noted as parallel and orthogonal polarizations for simplicity), and separated in propagation directions by a small separation angle θ. The spectral transmission functions of the parallelly and orthogonally polarized beams through a phase retarder and Wollaston prism pair can be respectively expressed as propagates through an all-optical spectral splitter, which separates light into eight parts with different spectral distributions. The splitter is composed of a polarizer (P) and three alternatively placed Wollaston prisms (WP) and phase retarders (PR). The 1920 × 1080-pixel SLM encodes each channel with independent patterns and a single-pixel detector records multiplexed multi-channel information simultaneously. The system is applicable to both macroscopic scenes and microscopic specimens, differentiating in objective lens. Adjacent Wollaston prisms are placed with their splitting directions perpendicular to each other, and this layout splits light along vertical and horizontal directions alternately. For a three level retarder-prism pairs employed in our work, eight channels are splitted in 2 × 4 layout with distinguishing spectral distributions as where Δ k = π|n o − n e |d k /λ, k = 1, 2, 3 and Ω = x x ( ) cos ( ) i k 2 or sin 2 (x) depending on the polarization state of ith channel when it propagates through the kth prism. The thicknesses for the quartz phase retarders are optimally chosen in our experiments to produce eight spectral bandpasses approximately equally spaced within the wavelength range 420~720 nm. Corresponding spectral response curves are plotted in Fig. 1, with the main lobes for all channels distributing nearly uniformly. Minimizing overlap among main lobes ensures reliable demultiplexing.
Separated channels then arrive at a tube lens and get imaged on the SLM. Instead of temporal multiplexing which requires long acquisition time, we conduct spatial multiplexing by modulating different channels with different patterns. Figure 1 shows an exemplar binary pattern displayed on the SLM. Eight sub-patterns are placed correspondingly in 2 × 4 layout, with the positions determined by the locations and separation angles of the Wollaston prisms. For each round of acquisition, the eight sub-patterns are updated simultaneously. To meet the incoherent sampling condition raised by CS theory 6 , sub-patterns for different channels are required to be independent of each other. Binary Gaussian random codes are utilized in our work for its high incoherency.

Multispectral imaging results.
To demonstrate the effectiveness of the proposed system, we conduct experiments on a number of colored objects. The resolution of the target objects is fixed to 128 × 128 pixels, and a 1920 × 1080-resolution fast SLM (Texas Instrument DLP Discovery 4100,.95XGA DMD) is used to spatially modulate eight channels. The SLM works at 10 kHz with 25% blank time (displaying an all-zero pattern) for synchronization, and the sampling rate of the single-pixel detector (Thorlabs PDA100A-EC Silicon photodiode, 340~1100 nm) is set to be 50 MHz. A self-synchronized sampling scheme 23 is utilized to synchronize the SLM and the detector. For each object, 4915 high-resolution patterns (equivalently, 39320 low-resolution Gaussian random binary patterns for 8 channels) are used for multispectral image reconstruction, and the corresponding compression ratio is 0.3 comparing to the full measurement of the spectral cube (N = 128 × 128 × 8). It takes around 0.5 s for the data acquisition and the reconstruction is conduct offline. The working spectral range is set to be 420~720 nm by filtering the light from the white light source (Energetiq LDLS EQ-1500) with a bandpass filter.
The first experiment is conducted on a printed color checker film with nine patches (see Fig. 2(a)). The size of the film is around 20 × 20 mm 2 and each single channel imaged on the SLM is around 2.7 × 2.7 mm 2 . To fit the size of the images, 256 × 256 pixels on the SLM are used to display a 128 × 128 sub-pattern with 4 neighboring pixels binned as one. We also calibrare the spherical aberration of the imaging lens and correct the distortion of the bordering channels. Eight multispectral images discretized in the range 420~720 nm are succesfully reconstructed from the multiplexed measurements, as presented in Fig. 2(b). Nine color patches are recovered with high quality. The promising results demonstrate the effectiveness of the proposed multispectral imaging approach.
To further evaluate the accuracy of our approach, we conduct quantitative analysis on the reconstructed images. Each patch on the target film is monochrome, thus we take the average of all pixels within each color patch as the reconstructed spectrum for the certain color. Figure 2(c) shows the reconstructed spectral response curves linearly interpolated on eight discrete wavelengths for all nine tested colors. Ground truth spectrums are measured by a compact spectrometer (Thorlabs CCS200/M, 200~1000 nm) and presented at an interval of 10 nm. The spectral responses are both normalized to the range [0, 1] and the standard deviation of each color patch is calculated and illustrated with error bars. As shown, our reconstructed spectrums are of quite close intensity distributions with the ground truth. The root mean square errors (RMSE) between reconstructed spectrums and ground truth are 0.041, 0.024, 0.057, 0.046, 0.028, 0.054, 0.055, 0.054 and 0.034 for nine color patches, respectively, as illustrated in bottomright inset of Fig. 2(c). Small deviations from the ground truth demonstrate that our system is of high spectral accuracy.
In the following experiment, we illuminate a approximately vertically placed 20 × 20 mm 2 scene which includes a layer of multi-color beads (seen in Fig. 3(a)) with the spectrally filtered light source, and collect reflective light for spectral splitting, mapping and modulation. Reconstructed multispectral images are presented in Fig. 3(b). Since the beams are placed slightly tilted from the vertical imaging plane, defocusing appears at the top and bottom edges. It is clear that the responses of green beads are higher in the range of 495~545 nm, while the responses of pink beads are higher in the range of 450~475 nm and 580~670 nm. The high quality reconstruction demonstrates our ability to achieve multispectral imaging on real scenes.
To validate the effectiveness of our approach in microscopy, we further conduct experiments on a multicolor dyed Volvox specimen. To visualize the micro-scale structures of the specimen, a microscope objective lens (10 × 0.25 NA) is utilized in place of the convex lens in above experiments. The successive achromatic lens (L = 30 mm) colliminates the light from the image at the back focal plane of the objective. Multi-channel images are then focused to the SLM by a tube lens. We captured a reference color image by an RGB CCD camera (PointGrey FL2-08s2c), as illustrated in Fig. 4(a), and the eight 128 × 128-pixel spectral images reconstructed by our approach are presented in Fig. 4(b). It is evident that the grand red Volvox colony appears dimmer at low frequency spectrum and brighter at high frequency spectrum, while the green structures are dimmer at medium frequency spectrum and brighter at high and low frequency spectrums. The results validate the applicability of our approach for microscopic imaging.

Discussion
This paper proposes an efficient multispectral imaging scheme with a single-pixel detector. Utilizing the spatial resolution redundancy of current SLMs, we extend conventional 2D spatial reconstruction to 3D spatio-spectral reconstruction via spatial multiplexing of multiple channels, without introducing additional acquisition time. For the spatial multiplexing, spatial splitting and spectral separation are achieved in an all-optical manner, which avoids the instability and limited response speed of commonly used mechanical modulation techniques. In addition, the single-pixel detector collects the coded accumulation of spectral bands, so the imaging is of high photon efficiency. Experiments validate the ability of our system to resolve spatial and spectral information effectively and accurately. The system holds great potential for developing stable, compact and photon efficient multispectral cameras in the future.
Although we acquire eight channels within the 420~720 nm range in the presented experiments, the system is of high flexibility for extention to 16 or 32 channels by adding another one or two levels of phase retarder-prism pair. It should be mentioned that as the spectral channel number increases, larger Wollaston prisms (e.g, 30 × 30 mm 2 -size) are required. Moreover, the spectral range of our system is adjustable by simply changing the  thicknesses of the phase retarders. This shows enormous promise for macroscopy and microscopy imaging within non-visible spectral ranges, under which conventional array sensors may be unavailable or of high cost.
The current prototype is designed to split spectral components with an average bandwidth of 25 nm. Nevertheless, the technique can be utilized for finer spectral resolution imaging by adjusting the composition of the spectral splitter. Experimental results have been demonstrated for retinal imaging with spectral resolution of 5 ± 2 nm within 570~620 nm using the same Lyot filter 21 . Spatial resolution depends on two factors: the NA of the system and the pixel resolution of modulation. In our system setting, the number of modulation pixels for each channel is the dominant limit for spatial resolution. With the effective modulation pixels fixed to be 128 × 128, we can calculate the spatial resolution by dividing the FOV with the pixel number, resulting in 0.16 mm for macroscopic imaging and approximately 10 μm for microscopic imaging. Higher spatial resolution can be achieved by more modulation pixels, however at the expense of increased acquisition time and computational cost.
Different from other spectral imagers that conduct spatial multiplexing with dispersive devices such as gratings 24 and prisms 25 which separate light along one dimension, our system enables a two-dimensional dispersion. This ensures a better use of the SLM's spatial resolution redundancy. Besides, the single-pixel architecture may produce a more compact camera design. Compared with single-pixel spectral imaging schemes recently proposed by Bian et al. 17 and Wang et al. 16 , an important advantage of our system is the elimination of mechanical modulation. These approaches utilized a rolling disk for frequency modulation or spectral separation, which could largly limit the working speed and introduce mechanical vibrations. For example, the frequency multiplexing scheme by Bian et al. takes around 1 minute to acquire a 64 × 64 × 10 datacube. As a contrast, our technique can achieve good performance at 2 Hz acquisition for the 128 × 128 × 8 data. Considering the total transmission efficiency, both reported works are limited to no more than 40%, with dominant light losses caused by the grating and frequency modulation for Bian et al.'s approach and the color filters for Wang et al.'s. However, our system reaches a total efficiency up to 50% with half of the light excluded by the polarizer.
So far, the proposed single-pixel spectral system is not fast enough for dynamic scenes, even though spatial and intra-spectral redundancy is utilized to reduce required measurements. We can further accelerate the speed by using advanced pattern designs and content-adaptive schemes 26 . Besides, current spatial resolution remains insufficient for large-scale scenes (especially in biological microscopy) due to the trade-off between spatial resolution and computation cost of CS reconstruction. Fortunately, recent studies have reported on less computation consuming techniques for high resolution reconstruction of single-pixel imaging architectures, such as the phase-shifting sinusoid structured illumination scheme 27 and the sub-pixel displacement microscanning approach 28 . These strategies provide a possibility for improving the spatial resolution of our system.

Methods
Spectral splitter. The mapping of 3D spatio-spectral data to spatially splitted 2D spectral channels is achieved by an all-optical splitting device consisting of successive pairs of alternatively placed birefringent phase retarders and Wollaston polarizers. Input linearly polarized broadband light is transformed into nonlinearly polarized by the phase retarder, with the polarization state defined as a function of the wavelength. The nonlinearly polarized beam is then splitted into two orthogonal linearly polarized beams by the polarized prism, differentiating in splitting intensity ratio according to the polarization state. Hence we can conclude that the splitting ratio is wavelength-dependent and the phase retarder-prism pair achieves simultaneous spatial splitting as well as spectral splitting. A schematic drawing illustrating the polarization states for the light before and after passing through each optical element is shown in Fig. 5. In our experiments, calcite Wollaston polarized prisms (20 × 20 mm 2 in size) with the separation angle θ = 10° and quartz phase retarders with the thicknesses d = 0.10, 0.145, 0.29 mm are used to assemble the spectral splitter. Spectral transmission functions of each retarder-prism pair result in the forms of cos 2 (·) and sin 2 (·) depending on whether the output polarization state is parallel with or orthogonal to the input (see Eq. (1)), as illustrated in the bottomleft inset of Fig. 5. We can then deduce the transmission functions for eight output beams from the splitter to be the cumulative products of three levels of cos 2 (·) or sin 2 (·) functions, as presented in the right inset of Fig. 5. For each channel, the spectral distribution concentrates in one narrow band, and the main lobes for all channels distribute nearly uniformly in the spectral range, with center frequencies locating at 450, 475, 495, 525, 545, 580, 620 and 670 nm respectively. The slightly nonuniform distribution is inevitable due to the nonlinear relationship between transmissive intensity and wavelength λ.
Multispectral reconstruction. The reconstruction of spatial multiplexed multispectral data is conducted on a three dimensiton (3D) compressive sensing scheme to achieve high reconstruction efficiency. Denoting the sub-patterns for the mth measurement as p m,1 , p m,2 , … , p m,8 , the total intensity value of eight modulated channels recorded by the single-pixel detector can be writen as N represent the vectorized spectral image of the ith channel, N x , N y denote the resolutions along x and y axis, and T represents the transpose operator, after M samplings, we record a measurement vector Scientific RepoRts | 7:41435 | DOI: 10.1038/srep41435 where ∇ represents a two-dimensional TV operator in spatial domain and μ is a weighting parameter to balance the TV regulation terms along spatial and spectral dimensions. We fix μ to be 0.5 in the following experiments. Equation (5) is a standard convex optimization problem and can be mathmatically solved by multiple algorithms.
In our experiments, we adopt Augmented Lagrangian Multiplier algorithm due to its high efficiency for noisy data input 29 and write the code in Matlab. To reconstruct a 128 × 128 × 8 datacube, approximately 30 seconds is required on a computer with Intel i7-5960X CPU @ 3.0 GHz and 32 GB RAM. The reconstruction process can be further accelerated by using GPUs. Figure 5. Illustration of the spatial and spectral splitting process for a broadband light beam. The bottomleft shows the associated spectral transmission functions for each pair pf phase retarder and prism, and the right inset shows the spectral response for each bandpass channel. P, polarizer; PR, phase retarder; WP, Wollaston prism; PSin, input polarization state; PSout, output polarization state.