The acousto-optic tunable filter（AOTF）is a programmable optic filter based on the acousto-optic diffraction effect，easily controlled via radio frequency（RF）signals. Its advantages of flexible wavelength selection，high-frequency optical switching，and solid-state construction provide excellent environmental adaptability. Spectrometers with AOTF as the core spectroscopic element have been successfully applied in the field of in-situ planetary exploration^［1-6］，astronomical observation^［7］，urban and industrial pollution monitoring^［8］，aurora observation^［9］，atmospheric remote sensing^［10-11］，surface plasma resonance sensing^［12］，plume observation，online inspection in the food industry，etc.

When the target light radiation intensity is sufficient，AOTF can take full advantage of quick tuning to achieve fast and effective spectral information acquisition of targets. But under normal circumstances，the intensity of the reflected light from the object is relatively weak，and the energy is relatively low during the measurement. It is necessary to ensure that the detector has sufficient integration time for each measurement. With the conventional AOTF-based spectral-domain scanning sampling method，i.e. wavelength-by-wavelength narrow-band filtering，it takes a long time to acquire the entire spectral data when the number of spectral channels is large^［13］. While in some special application scenarios，the acquisition time of a complete spectral data set is strictly limited to ensure the authenticity and validity of the data. For example，in scenarios where there is relative motion between the target and the instrument，the system design usually requires that the full spectral information of the spatial target corresponding to an image element in the image plane needs to be obtained within the time that the image element moves a distance. Again，in a class of application scenarios such as plume observation where the target spectral features change rapidly with time，the system design requires the acquisition of instantaneous spectral data with a guaranteed data signal-to-noise ratio（SNR）. In addition，in the field of deep space exploration，due to channel transmission capacity limitations，the massive data obtained by the spectral imaging system often need to be compressed and transmitted. The ground receiving system needs to decompress the data after receiving it for data analysis. Conventional AOTF spectrometers often struggle to meet the needs of these types of applications.

To enable the AOTF spectrometer to increase the system luminous flux，some researchers have developed a multi-frequency mode of operation using the AOTF，where multiple frequencies of RF signals are emitted into the AOTF at the same time，allowing it to output multiple wavelengths of light signals. For a given frequency，the diffraction efficiency（DE）has Gaussian behavior with respect to the applied power. With a multi-channel drive，the AOTF generates a multi-channel passband，resulting in a significant increase in luminous flux stacking. However，Multiplexed AOTF operation is limited by the RF power density that can be accepted by the transducer^［14］. There is a limit to the luminous flux increase，and the number of drivers and the share of power per frequency division are mutually constraining. In this multi-channel configuration，different wavelengths of light are coupled together and output to the detector's photosensitive surface. The data obtained directly is the value of a multi-wavelength superposition and cannot be decoupled directly.

In previous researches，various multi-frequency drive methods were proposed. Spanish autonomous tunable filtering system（ATFS）can flexibly configure the spectral resolution and other performance parameters of the AOTF-based spectral imaging system utilizing digital to analog convertor（DAC）multi-frequency drive or direct digital synthesizer（DDS）single-frequency sweep^［15］. This approach improves the system's luminous flux，but sacrifices part of the system's spectral resolution because the direct data is obtained after superimposed coupling. The introduction of computational reconstruction mechanisms has guided the strategy of wavelength combination，such as the Hadamard-transform（HT）mechanism^［16-17］ of combined sampling and back-end decomposition in some studies and appreciable signal-to-noise enhancement is demonstrated. However，this Hadamard encoding method requires the same number of samples as the traditional acquisition method，and this mechanism becomes tricky in the case of the application scenario when the number of wavelength-points to be sampled is large，and does not make good use of the benefits of computational reconstruction.

Since the compressed sensing（CS）theory was put forward^［18-19］，a variety of computational spectra measurement schemes have emerged. At present，there have been many studies using liquid crystal devices^［20］，Fabry–Pérot resonant cavities^［21］，quantum dots^［22］，and random filter arrays^［23］，etc. to achieve compressed sensing spectral measurements，with good results. This type of computational spectral measurement system requires back-end inversion recovery reconstruction to obtain the original data after the data are acquired. Its features and great benefits in reducing the number of measurements as well as sampling while performing data compression provide new ideas for improving the performance of conventional systems to suit the needs of specific application scenarios. Compared with the conventional wavelength-by-wavelength scanning sampling method，the sampling method of compressed sensing encoding measurement can reduce the number of samples without reducing the spectral resolution and data SNR，resulting in a reduction in spectral data sampling time.

This paper proposes an AOTF spectral measurement method based on a compressed sensing mechanism. The AOTF multi-passband transmittance encoding curve is used to simulate the compressive sampling process and to reconstruct the obtained data for recovery using three algorithms. The feasibility and effectiveness of the method are demonstrated by means of simulation experiments and numerical analysis.

In spectroscopy，the relationship between the spectrum vector x∈ℝN×1 of the light source and the modulated signal y∈ℝM×1 collected by the detector can be expressed as：

where A∈ℝM×N is a description of the signal sensing process in the spectroscopic system，which is called the sensing matrix. Without loss of generality，the transmittance matrix of a spectral element is denoted by T∈ℝN×N. One of the rows is the spectral transmittance curve of the element in one of the measurements，and is expressed as：

In conventional spectroscopic instruments，one of the measurement processes in the wavelength-by-wavelength spectroscopic measurement method can be expressed as the inner product of a row of the sensing matrix and the spectral vector，considered as the measured value yi of a wavelength point λi. At this point，the row Ti is generally equivalent to the spectral transmittance curve of a narrow-band filter with a central wavelength λi. The number of spectral measurements is also the number of spectral passbands，i.e.，M=N. Thus，the sensing matrix for conventional measurements can be expressed as：Ac=IT，where I is the identity matrix.

Conventionally，narrow bandpass filtering is chosen to perform a wavelength decomposition task. To achieve sufficient spectral resolution，measurements using a certain number of sufficiently narrow passband filters are required. This strategy introduces a trade-off between spectral resolution and light throughput. Meanwhile，the narrow spectral response curve，which limits the light throughput，leads to a limited signal-to-noise ratio^［24］.

The most distinctive difference between computational and traditional spectroscopic instruments is the sensing matrix. Based on compressed sensing theory，the sensing matrix A∈ℝM×N,M<N，implying fewer measurements，is the transmittance matrix of the multi-passband optical devices. By applying multiplexing and computational reconstruction methods^{［20］［21］}，the fact that multiple spectral peaks are transmitted for each measurement means higher optical throughput，leading to a higher SNR，compared to conventional band-selective spectrometer systems that transmit a single peak.

To solve CS problems，several algorithms were developed^［26-28］. A class of reconstruction algorithms estimated the sparse representation s∈ℝN×1 of the spectrum vector x by solving the l1minimization problem：

where τ is a regularization parameter and ||⋅||p the lp norm.

The reconstructed original spectra can be obtained from the sparse inverse transform：

where Φ∈ℝN×N is the sparse inverse transform matrix.

Φ depends on the sparsity characteristics of the spectral signal and needs to be specified before the measurement is performed. The sparsity of signals and images in a certain transform domain or dictionary has been exploited in many applications in signal and image processing. Analytical sparsifying transforms such as Wavelets and DCT have been widely used in compression standards. And，the problem formulations for learning sparsifying transformations from data have also been proposed^［25］. Moreover，the matrix Ψ=AΦ needs to satisfy the Restricted Isometry Property（RIP）criterion^［29］.

Under the above framework，a compressed sensing spectral measurement method based on AOTF is proposed in this paper.

The spectral measurement method based on AOTF can be configured to acquire a single narrowband spectrum，a composite multispectral image，or a broadband passband. This flexibility is achieved through the use of an AOTF driven by the programmable RF signal generator. The AOTF acts as an optical diffraction element and its output wavelengths are selected by RFs applied to it. The RF driver based on a high-speed DAC is capable of synthesizing arbitrary composite RF waveforms from a combination of sinusoidal signals.

By varying composite RF waveforms of the designed driver，the AOTF generates the spectral masks given by the designed matrix. Without changing the optical path setup of the conventional AOTF spectroscopy instrument，the compressive sampling process under the compressed sensing framework is realized. The optical signal is modulated in the spectral domain and the detector is used to record the modulated composite optical signal in sequence. Finally，the original spectrum is obtained with the compressive sensing reconstruction algorithm.

As shown in Fig. 1，a row of the sensing matrix A is inner-produced with the original spectral signal x to obtain an element of the measured value y. This is a compressive sampling process. The measurements obtained by compressive sampling represent the superimposed signal intensity of a particular combination of wavelengths in the spectral domain. This coding according to the sensing matrix A requires that the wavelength combinations of the actual diffracted light from the AOTF match those determined by the designed matrix H.

Based on this combination of wavelengths and the frequency table corresponding to the chosen AOTF device，the frequency combination required for each drive can be obtained. During the actual measurement，the drive subsystem needs to be able to drive the AOTF in a flexible and stable sequence according to a set series of frequency combinations，while the back-end detector subsystem records the measured values simultaneously. By measuring the spectrum of the light source and the diffracted light spectrum of the AOTF crystal，the overall equivalent spectral transmittance of the spectroscopic system can be obtained. Measuring the actual equivalent spectral transmittance allows verification that the actual modulation effect of the system is consistent with the designed measurement matrix.

For AOTF，the intrinsic momentum matching condition is highly selective in collimated light with fully collimated acoustic waves. However，due to the limited length of the transducer，the spectral resolution and bandpass shape are usually influenced by the diffraction of the acoustic field^［7］. This results in a sinc2 distribution of acoustic energy in the crystal in the plane of the optical axis，where sincx=sinπx/πx.

Defining the acoustic wave length Λ=Va/f，the spreading due to the momentum mismatch resulting from different directions of the acoustic wave vector at slightly different optical wavelengths，the full width at half maximum FWHMΔλFW of the passband can be approximated as Eq.（3）.

The bandpass shape of the homogeneous transducer and collimated input light is：

Eq.（4） describes the passband transmittance function of AOTF narrowband filtering at a specific wavelength. The transmittance functions at all wavelengths to be sampled are combined to form the matrix T.

In our approach，the broadened spectrum Tx is chosen instead of x as the target for reconstruction recovery.

In Fig. 2，we show the whole data flow process：

a. The light spectral distribution is modulated and multiplexed by AOTF to obtain the measured value y；

b. Using the broadened spectrum Tx sparsity feature，the sparse transformation matrix Φ is determined and combined with the designed matrix H to obtain the matrix Ψ，which is required to satisfy the RIP criterion.

Using y，Φ and A，solve Eq.（2） to obtain the sparse representation s.

c. A matrix calculation is performed to obtain the reconstructed target broadened spectrum.

In the next section，simulation experiments are performed using the measured data of selected AOTF devices，and the spectrum reconstruction from a semi-simulated experiment is evaluated for similarity to the target.

For the simulation experiments，the measurement spectral range was selected to be 600~855.5 nm with a wavelength interval of 0.5 nm and a total of 512 spectral bands.

The sensing matrix A is constructed from the selected measurement matrix H with elements 0 and 1 and the single wavelength filtered response passband matrix T of the AOTF at each wavelength point of the spectral signal to be measured，expressed as A=HT.

In order to obtain a rational T，the bandwidth of the spectral response curve of the AOTF crystal corresponding to different diffraction center wavelengths was first measured. A HORIBA iHR 320 spectrometer was used to measure the diffracted light from a tungsten halogen lamp through the AOTF. Based on the relationship between the driving frequency of the AOTF and the diffraction wavelength，the frequency corresponding to one of the wavelengths to be measured was set. The diffracted light spectrum curve was then obtained using the spectrometer by fine-step scanning. The full width at half maximum of the curve was then used as the spectral resolution data for the AOTF at the wavelength. This procedure was repeated five times over the entire working spectral range to obtain all data.

Using the measured spectral resolution data of the selected AOTF device，as shown in Table 1，the full width at half maximum FWHMΔλFW of the passband at all wavelength-points in the operating spectral range is fitted according to the quadratic function of Eq.（3），and the spectral passband Tλ at each wavelength point required is generated by combining Eq.（4）. 512 corresponding narrow passbands were generated to construct the matrix T.

For AOTF spectroscopic instrumentation systems based on AOTF spectroscopy，the conventional measurement mode in which a single frequency drives the AOTF equally spaced non-ideal narrow-band filtering for target spectral sampling is equivalent to the discrete sampling of the AOTF measurement spectrum Tx. The coded measurements operate on spectral data，and each compressed sample can be equated to a linear combination of specific multiple single wavelength samples. The linear combination method is determined by the measurement matrix H. Considering the AOTF device and transducer energy threshold limits and the drive system output power，the maximum number of frequencies in a single drive needs to be strictly limited.

As mentioned above，Ψ=HΦ needs to satisfy the RIP criterion. The hierarchical condition for the RIP property is that the measurement matrix and the sparse transform base are uncorrelated，whereby the optimal measurement matrix can be selected and generated based on hardware metrics and spectral sparsity properties. Common random matrices include random Gaussian measurement matrices，random Bernoulli matrices，partially orthogonal matrices，sparse random matrices，etc. Considering the reconstruction recovery effect and hardware implementation feasibility，the Sparse Random Matrix（elements 0 and 1）is chosen as the measurement matrix H.

The RIP criterion is easily satisfied due to the Sparse Random Matrix with the determined sparse transformation matrix extremely incoherence. It has been demonstrated that given a signal s sparsity of K，the minimum number of measurements：

Since the sparsity of the target broadened spectrum Tx is unknown，we set the number of compressing measurements to 202 for this simulation experiment.

Therefore，if it can be reconstructed almost perfectly，the compression ratio is 0.39.

During a conventional raster spectral acquisition，only one channel of the AOTF is turned on and tuned in frequency to probe a single different wavelength at each acquisition（A_ci）. In sensing matrix A for acousto-optic coded sampling based on compressed sensing，several channels of the AOTF are turned on simultaneously in each acquisition. The additional passbands brought about an increase in optical throughput.

Many studies^{［15-16］［30］} have demonstrated that the multi-passband of an AOTF driven by multiple frequencies simultaneously is equivalent to the mathematical superposition of the narrow passbands when the AOTF is driven by each single frequency separately，under reasonable settings. Based on this assumption，a mathematically linear combination of the narrow passbands of different single-frequency driven AOTFs was also used in this paper to characterize the broad passbands when driven by multiple frequencies simultaneously.

The number of simultaneously driven frequencies determines the magnitude of the luminous flux in the passband. Since the sparse random matrix H chosen for the simulation，with the sparsity of the columns set to 2，the number of driving RFs per row，also called，each compressed measurement，is randomly distributed between 1 and 11，and the average number of simultaneous driving RFs for all 202 measurements is 5，so the system light throughput is considered to be increased by a factor of 5.

Fig. 3 shows the heat map of the normalized equivalent spectral transmittance（50 of the 202），which are denoted as A in the simulation，and the spectral response curves corresponding to the four measurements in it. The simulation uses a certain generated Sparse Random Matrix with the maximum number of wavelengths that need to be sampled simultaneously for a single sampling in all rows is 11.

As mentioned earlier，the maximum number of simultaneously driven RFs is limited by the RF power density that can be accepted by the transducer. The effective driving of 16 simultaneous passbands has been verified in previous studies^［30］. Therefore，it is easier to implement in hardware compared to other measurement matrices. For example，a partial Hadamard matrix，which requires the number of simultaneously turned-on passbands to be typically half of the original total number of samples，is difficult be implemented in hardware at this stage.

The target spectrum is selected as the broadened spectrum of the Hg-Ne-Ar spectral calibration source of the IntelliCal series from Princeton Instruments（PI），USA，after the conventional sampling by AOTF.

The red line spectrum shown in Fig. 4（a）is the original emission spectrum of the source，and the blue spectral curve is the conventional measured spectrum of AOTF，which is represented as x and Tx，respectively，in the simulation. The measured values obtained from 202 compressive measurements are shown in Fig. 4（b）which are denoted as y in the simulation.

Our approach，choosing Tx as the compressed sensing reconstruction target instead of x，brings multifaceted benefits. Including，the broadened spectrum is naturally sparse，the measurement matrix can be chosen randomly without introducing fixed T，and the spectral resolution can be unambiguous（remain unchanged with the one by conventional methods）.

For some application scenarios where the target data to be detected does not require higher spectral resolution，our method can be designed with a random measurement matrix for a given spectral resolution so that performance close to the theoretical minimum number of measurements can be achieved for a given sparse transformation. This minimizes sampling costs.

As mentioned before，the size of the sparsity after the sparse transformation determines the minimum number of measurements. Hence，the most suitable sparse transform for the measured signal is worth to be explored. Without loss of generality，one of the sparse transforms is selected in this paper for simulation and verification. If a better sparse transform is replaced in the future，it will be possible to further reduce the number of measurements while achieving a relatively perfect recovery.

In this paper，the sparsifying transform used for reconstructing the broadened spectra Tx is from the family of orthogonal Daubechies wavelet. The Discrete Wavelet Transform（DWT）matrix W is generated by performing a level 3 decomposition of the signal using the db6 wavelet. The sparse transformation matrix Φ is obtained by solving the inverse matrix for W.

At this point，we obtained A，y and Φ. The next step is the signal reconstruction simulation.

To solve the optimization problem as in Eq.（2），reconstruction simulations were performed using the L1-MAGIC toolbox^［26］ and the l1_ls solver ^［27］，respectively. The Orthogonal Matching Pursuit（OMP）algorithm^［28］ was also applied for reconstruction recovery.

The iterative optimization yielded s，and then the matrix multiplication calculation with the sparse transformation matrix Φ was performed to obtain the reconstructed spectral signal Tx̃.

Fig. 5（b）visualizes the comparison of the reference spectral curve Tx and the three reconstructed spectral curves Tx̃ generated by L1-MAGIC，l1_ls，and OMP. A randomly selected part of the area is enlarged to show that the curves match relatively well. Almost perfect recovery was achieved，both in terms of the overall curve trend and the shape，area，and position of the spectral emission peaks. All three algorithmic tools availably completed the refactoring task.

Since the sparse inverse transformation from the wavelet domain sparse signal to the original spectral intensity signal is not a perfect recovery. We also examined the difference between the reconstructed sparse signal before the sparse inverse transformation by matrix multiplication and the actual sparse transformation of the broadened spectrum.

The actual sparse signal in the wavelet domain of the AOTF broadened spectrum Tx is denoted as s0.

Fig. 5（a）shows the comparison of the reference sparse signal s0 in the wavelet transform domain and the sparse representation s obtained by the three algorithms. A randomly selected part of the area is also enlarged to show that the curves match relatively well.

Quantitative evaluation of reconstruction accuracy^［21-22］ can be done using the peak signal-to-noise ratio（PSNR）.^{PSNR is a scientific parameter for the measurement of the similarity of two signals，which is calculated as follows：}

where MAXI is the maximum intensity value of the signals and MSE is Mean Squared Error between the reference and reconstruction signal.

The peak signal-to-noise ratios of the three different reconstruction algorithms were calculated in the sparse and spectral domains，respectively. As shown in Table 2. We can see that the l1_ls solver obtains the highest PSNR，which means the highest reconstruction accuracy with the least distortion. In general，a PSNR above 40 dB means that the signal quality is pretty good（i.e.，very close to the original signal）. And，PSNR between 30-40dB usually means that the signal quality is good（i.e. distortion is detectable but acceptable）. Thus，the recovery of L1-MAGIC and l1_ls is almost perfect，while the recovery of OMP is also acceptable.

Moreover，as expected，the accuracy of reconstruction recovery in the sparse domain is much higher before undergoing sparse inverse transformation. PSNR in the sparse domain even exceeds 50dB.

In Paper^［31］，it is indicated that the fidelity of spectral restoration can be evaluated using the Spectral Angle Model（SAM）and Gradient Spectral Angle Model（GSAM）metrics. The SAM and GSAM were calculated for the three reconstruction algorithms，as shown in Table 3.

The simulation results show that with the above settings，only 202 measurements successfully reconstructed the target data of 512 points almost perfectly by the method proposed in this paper.

In the above Hg-Ne-Ar spectral curve simulation，since the real spectrum of the Hg-Ne-Ar source is known，the response T matrix of the AOTF spectral instrument can be used to broaden the real spectrum to obtain the measured spectrum Tx. Then the compression sampling simulation and reconstruction recovery simulation were completed by combining the measurement matrix H and the sparse transformation matrix Φ. To further validate the method in this paper，simulations were next performed using data from a spectral database with richer features.

When using the data from the spectral library for simulation，since the data from the spectral library obtained from the actual acquisition of the typical instruments，the broadening characteristics of the instruments such as the wavelength position data of each channel and theFWHM of each channel were recorded in addition to the detailed reflectance spectral curve data of different geographical targets.

In the simulation，it was assumed that the spectral radiation energy of the illumination source of the feature target is flat in the working spectral range，at which time the reflectance spectral curve in the spectral library can be regarded as the spectral curve of the reflected light entering the instrument in the simulation. Therefore，the spectral library data can be considered as Tx. As can be seen from the previous simulation，the T matrix is only involved in the process of obtaining Tx. With the known Tx，the compressed sampling simulation can be completed by setting H only. In the evaluation of reconstruction recovery results，Tx was also directly used as the target reference spectrum without explicit T matrix. Therefore，the spectral library data were simulated throughout with no T involved. Since the OMP recovery results in the previous simulations were relatively poor，while the l1-MAGIC and l1-ls algorithms worked better，the following simulations were reconstructed using these two algorithmic tools.

The reflectance spectrum data for the 75 materials used in the simulations are located in the USGS_Spectral_Library_Version_7 spectral library under the veg_1dry and veg_2grn categories. For each spectrum data，a total of 512 wavelength points in the spectral range of 0.714-2.5 μm were intercepted as the recovery target. Using the same measurement matrix H as in the previous section，with M = 202 and N = 512，the direct simulation results are presented below.

A statistical analysis of the spectral fidelity metrics was performed and is displayed in Table 4. The results show that the average PSNR can reach 52.54，and the average SAM and average GSAM are 1.000 and 0.9091，respectively，when the compressed sampling rate is no higher than 0.4.

This method chooses the broadened spectrum by conventional measured as the sparse transform target so that the final spectral data obtained by the method is consistent with the spectral resolution characteristics of the spectral data obtained by the conventional AOTF spectrometer. The compressive sampling process reflects the system's luminous flux advantage and reduced sampling points. At the same time，the new form of transmittance coding model based on AOTF does not change the original optical path setup and does not need additional coding optical components.

It can be seen from the results that the scheme can achieve sufficient recovery accuracy with a reasonable computational overhead. The simulation experimental results verify that the selection of the measurement matrix and the overall data flow design of the system in this paper can reach the goal of coding for compressed sensory spectral measurements. This simulation experiment did not introduce noise，and further studies based on this can take noise into account.

In this paper，we propose an AOTF spectral measurement method based on compressive sensing，and conduct simulations to verify the feasibility of computational spectral measurement through AOTF. Simulation results show that the method proposed in this paper can achieve fast acquisition of target spectral data with fewer measurements and higher luminous flux for signal sampling，and simultaneous data compression in the spectral domain to accurately reconstruct signals at 39% or even fewer measurements. Based on the method proposed in this paper，the acousto-optic multi-RF drive source and acousto-optic encoded spectral measurement system can be developed，and it is expected that the number of measurements can be further reduced，the reconstruction accuracy and the reconstruction calculation speed can be improved by selecting a better sparse strategy and a reconstruction algorithm with better performance. This method will provide a reference for the acousto-optic encoded spectral detection system based on compressive sensing and broaden the application of compressive sensing theory in the field of spectral detection.