Abstract
Acousto-Optic Tunable Filter （AOTF） based spectroscopy instruments have been widely applied in biomedical， agricultural， aerospace， and other fields. However， the conventional AOTF spectrometers struggle to achieve increased system luminous flux while maintaining spectral resolution and reducing the number of samples. To address the above problems， this paper proposes an AOTF spectral measurement method based on the compressed sensing （CS） theory. Sparse randomly coded composite optical signal modulation in the spectral domain using the multi-frequency acousto-optic diffraction of AOTF. A modulated composite optical signal is obtained in the spectral domain and recorded sequentially using a single-element detector or a focal-plane detector array. The original spectrum or spectral image data cube is then obtained by using compressed sensing reconstruction algorithms. In order to verify the effectiveness of the present method， we constructed a sensing matrix using the actual measured AOTF spectral response bandwidth data and simulated the effect of compressed sampling and target data reconstruction with the spreading spectrum as the recovery target. The simulation results show that the method can reconstruct the spectral data of 512 wavelength points with 202 compressed samples， and the spectral data sampling rate and compression ratio is 0.39. Under this sampling rate， the method can recover the spectral curve with high accuracy， and the PSNR index reaches 41.75 dB， and the SAM and GSAM indexes are 0.9998 and 0.9754. With the simultaneous multi-frequency drive， the system optical throughput is improved by a factor of 5 on average. Compared with the traditional wavelength-by-wavelength point scan sampling method， this method can reduce the total number of samples and improve the luminous flux of the system while maintaining the original spectral resolution， and also compressing the spectral data， which is of great importance in the fields of weak signal detection， rapid identification of substances， and spectral data transmission and storage.
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 exploratio
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 larg
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 transduce
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 swee
Since the compressed sensing （CS） theory was put forwar
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 of the light source and the modulated signal collected by the detector can be expressed as：
| , | (1) |
where 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 . 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 of a wavelength point . At this point， the row is generally equivalent to the spectral transmittance curve of a narrow-band filter with a central wavelength . The number of spectral measurements is also the number of spectral passbands， i.e.， . Thus， the sensing matrix for conventional measurements can be expressed as： ， where 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 rati
The most distinctive difference between computational and traditional spectroscopic instruments is the sensing matrix. Based on compressed sensing theory， the sensing matrix ， implying fewer measurements， is the transmittance matrix of the multi-passband optical devices. By applying multiplexing and computational reconstruction method
To solve CS problems， several algorithms were develope
| , | (2) |
where is a regularization parameter and the norm.
The reconstructed original spectra can be obtained from the sparse inverse transform：
| . |
where is the sparse inverse transform matrix.
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 propose
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 Compressive sampling based on AOTF
图1 声光编码压缩采样
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 fiel
Defining the acoustic wave length ， 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 of the passband can be approximated as
| . | (3) |
The bandpass shape of the homogeneous transducer and collimated input light is：
| . | (4) |
In our approach， the broadened spectrum is chosen instead of as the target for reconstruction recovery.
In
Fig. 2 Sparse-reconstruction Compressed Sensing System model
图 2 稀疏重构压缩感知系统模型
a. The light spectral distribution is modulated and multiplexed by AOTF to obtain the measured value ；
| . |
b. Using the broadened spectrum sparsity feature， the sparse transformation matrix is determined and combined with the designed matrix to obtain the matrix ， which is required to satisfy the RIP criterion.
| . |
Using ， and ， solve
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 is constructed from the selected measurement matrix with elements 0 and 1 and the single wavelength filtered response passband matrix of the AOTF at each wavelength point of the spectral signal to be measured， expressed as .
In order to obtain a rational ， 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
| Wavelength/nm | Driver frequency/MHz | Diffracted efficiency | FWHM/nm |
|---|---|---|---|
| 600 | 121.8 | 0.735 | 5.401 |
| 650 | 110.5 | 0.935 | 6.2 |
| 700 | 101.1 | 0.882 | 7.8 |
| 750 | 93.3 | 0.727 | 9.4 |
| 800 | 86.7 | 0.522 | 10.999 |
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 . 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 . 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， 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 .
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 sparsity of ， the minimum number of measurements：
| or . | (5) |
Since the sparsity of the target broadened spectrum 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 （Aci）. 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 studie
The number of simultaneously driven frequencies determines the magnitude of the luminous flux in the passband. Since the sparse random matrix 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 Sampling matrix visualization （a） sensing matrix Ac for conventional wavelength-by-wavelength point sampling， each row represents the spectral transmittance data of AOTF driven by a single frequency， （b） four of the spectral transmittance curves， （c） sensing matrix A （50 of 202） for acousto-optic coded sampling based on compressed sensing； each row represents the spectral transmittance of the system at a given state of AOTF， （d） four of the equivalent spectral transmittance curves
图3 采样矩阵的可视化 （a） 传统逐波长点采样的传感矩阵Ac，每一行代表单频驱动下的AOTF的光谱透过率数据， （b） 其中四条光谱透过率曲线， （c） 压缩感知声光编码采样的传感矩阵A（202中的50个），每一行代表系统在AOTF给定状态下的光谱透过率， （d） 其中四条等效光谱透过率曲线
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 studie
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 Comparison of the measured values obtained from the two methods
图4 两种方式直接得到的测量值对比
Our approach， choosing as the compressed sensing reconstruction target instead of ， brings multifaceted benefits. Including， the broadened spectrum is naturally sparse， the measurement matrix can be chosen randomly without introducing fixed ， 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 is from the family of orthogonal Daubechies wavelet. The Discrete Wavelet Transform （DWT） matrix is generated by performing a level 3 decomposition of the signal using the db6 wavelet. The sparse transformation matrix is obtained by solving the inverse matrix for .
At this point， we obtained ， and . The next step is the signal reconstruction simulation.
To solve the optimization problem as in
The iterative optimization yielded ， and then the matrix multiplication calculation with the sparse transformation matrix was performed to obtain the reconstructed spectral signal .
Fig.5 Spectral reconstruction results of L1-MAGIC， l1_ls， and OMP， from 202 measurements of 512 spectral bands （600~855.5 nm）， （a） reference signal vs reconstruction signals in the wavelet transform domain， （b） reference signal vs reconstruction signals in the spectral domain
图5 三种算法 L1-MAGIC、l1_ls和OMP使用202次测量值重建512个波长点数据（600~855.5 nm）的结果，（a） 小波变换域中的参考信号与重建信号对比，（b） 光谱域中的参考信号与重建信号的对比
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 is denoted as .
Quantitative evaluation of reconstruction accurac
| , | (6) |
where is the maximum intensity value of the signals and 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
| Solvers | PSNR in the spectral domain | PSNR in the sparse domain |
|---|---|---|
| L1-MAGIC | 41.4802 | 50.2913 |
| l1_ls | 41.7517 | 50.5628 |
| OMP | 36.8563 | 45.6674 |
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 Pape
| Solvers | SAM | GSAM |
|---|---|---|
| L1-MAGIC | 0.9997 | 0.9731 |
| l1_ls | 0.9998 | 0.9754 |
| OMP | 0.9561 | 0.8868 |
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 matrix of the AOTF spectral instrument can be used to broaden the real spectrum to obtain the measured spectrum . Then the compression sampling simulation and reconstruction recovery simulation were completed by combining the measurement matrix and the sparse transformation matrix . 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 the 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 . As can be seen from the previous simulation， the matrix is only involved in the process of obtaining . With the known ， the compressed sampling simulation can be completed by setting only. In the evaluation of reconstruction recovery results， was also directly used as the target reference spectrum without explicit matrix. Therefore， the spectral library data were simulated throughout with no 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 as in the previous section， with M = 202 and N = 512， the direct simulation results are presented below.
Fig.6 Reconstruction results of spectrum data of 75 materials in the spectral library. (a) comparison of SAM metrics for the recovery results of the two reconstruction algorithms, (b) comparison of GSAM metrics for the recovery results of the two reconstruction algorithms, (c) comparison of PSNR metrics for the recovery results of the two reconstruction algorithms. (d) spectral reflectance curves of 75 materials (0.714~2.5 μm)
图6 光谱库中75种材料的光谱数据的重构结果,(a)两种重构算法恢复结果的SAM指标对比,(b) 两种重构算法恢复结果的GSAM指标对比,(c) 两种重构算法恢复结果的PSNR指标对比,(d) 75种材料的光谱反射率曲线（0.714~2.5 μm）
A statistical analysis of the spectral fidelity metrics was performed and is displayed in
| Solvers | PSNR | SAM | GSAM |
|---|---|---|---|
| L1-MAGIC | 52.5477 | 1.0000 | 0.9091 |
| l1_ls | 50.1489 | 1.0000 | 0.9006 |
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.
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