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Study of computational sensing using frequency-domain compression

Xiao Zhang, Haozhe Zhong, and Liangqi Cao

Open Access Open Access

Abstract

The computational sensing and imaging technique has been extended from spatial domain to temporal domain for capturing fast light signals with a slow photodetector. However, temporal computational sensing based on random source/modulation has to require a lot of measurements to reconstruct an object signal with acceptable SNR. In this paper, we study the frequency-domain acquisition technique for capturing a nanosecond temporal object with ten Hertz detection bandwidth. The frequency-domain acquisition technique offers a SNR gain of N, where N denotes the point number of Fourier spectrum. Because of the compressibility of data and the orthogonality and completeness of Fourier basis, it enables the reconstruction based on sub-Nyquist sampling. Because the slow detection only has low temporal resolution capability, the frequency-domain acquisition technique could provide robustness and is immune to the temporal distortion in experiments.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

As a category of computational imaging, single-pixel imaging, including single-pixel camera [1–3] and ghost imaging [4–6], allows capturing an object image with a single point detector. The object image is reconstructed by the correlation of the distribution of illumination/detected light field and the total light power from the object. Single-pixel imaging has considerable advantages over traditional approaches. Firstly, array detectors are unavailable or too expensive for some wavelength bands, due to the limitation of manufacture technology. Therefore, the operation wavelength range of traditional imaging is limited by the hardware of array detection. Single-pixel imaging could overcome this bottleneck in many bands, such as X-ray [7,8] and Terahertz [9,10], because it only requires a single point detector without spatial resolution. Secondly, compared with traditional imaging, single-pixel imaging is insensitive to distortion introduced by turbid [11], scattering [12,13], nonlinear [14,15] and dispersive media [16]. Thirdly, traditional imaging methods may cause the potential radiation damage to biological samples, due to the high cumulative photon requirement of array detection. Single-pixel imaging could address this issue, because the single-pixel detector only needs a low exposure dose for measurements [17]. Therefore, it is safer than traditional approaches, and ideal for medical imaging using high-frequency electromagnetic waves, such as X-ray [18].

The early single-pixel imaging used random distributions as the spatial light modulation patterns and reconstructed the object image through algorithms relying on ensemble average [4,5,19,20]. Then, compressed sensing [21,22] was introduced into single-pixel imaging, providing shorter data acquisition time and higher image quality even in the case of sub-Nyquist sampling [2,23,24]. Besides, basis scan single-pixel technologies, such as Hadamard single-pixel imaging [25], wavelet single-pixel imaging [26] and Fourier single-pixel imaging [27,28], applied orthogonal basis patterns to illuminate the object and therefore could retrieve the object image accurately. In particular, Fourier single-pixel imaging uses Fourier base patterns for spatial illumination, obtains the reflection/transmission light powers with a single-pixel detector, and then reconstructs the object image by the inverse Fourier transform of measurements. Till now, it has been developed for three-dimensional [29], color [30], fast [31] and microscopy imaging [32].

Recent years, as a special type of single-pixel imaging, ghost imaging has been extended from spatial domain to temporal domain, for capturing fast temporal objects or signals with a slow detector [16,33–37]. In addition, a number of techniques, such as time stretching [38] and wavelength conversion [39], have been applied in temporal ghost imaging to overcoming the limitations of time-domain resolution and operation wavelength range. However, temporal ghost imaging based on random source/modulation has to require a lot of measurements to reconstruct a target signal with acceptable signal-to-noise ratio (SNR).

In this paper, we study the Fourier single-pixel imaging in time domain. Two experimental configurations are demonstrated for obtaining the Fourier coefficients of temporal object respectively. Then the inverse Fourier transform is performed to the measured spectrum for reconstruction. In experiments, the measurement bandwidth of 250 MHz is achieved using a 10Hz-bandwidth detector. Additionally, the proposed technique provides a SNR gain of N, where N is the point number for discrete Fourier transform. Due to the sparsity of the measured Fourier spectrum, sub-Nyquist sampling is used to improve its measuring efficiency greatly. Furthermore, because the slow detection has few temporal resolution, the proposed technique offers robustness and is insensitive to temporal distortion experimentally.

2. Principle

We demonstrate the time-domain Fourier single-pixel imaging with two experimental configurations for detecting temporal object. Here the temporal object is defined as temporal signal or time-varying transmittance function of an optical element [16,33,39]. In configuration I, an unknown input light signal is modulated by an electro-optic modulator (EOM), driven by an arbitrary waveform generator (AWG), and then received by a light power meter with bandwidth of 10 Hz, which is used as a slow detector, as shown in Fig. 1(a). In configuration II, the CW light from light source is modulated by an AWG-driven EOM and then go through an unknown temporal object, i.e. an EOM driven by an unknown radio frequency (RF) signal, as shown in Fig. 1(b). Then the power of light signal is measured by the 10 Hz light power meter. In both two configurations, a computer generates and sends the waveform data of Fourier base, i.e. specific cosine signals with different initial phases and frequencies, to AWG via USB cable. In addition, the computer records the measured power values from the light power meter and further reconstructs the temporal object. Actually, these two configurations are temporal versions of single-pixel camera and computational ghost imaging respectively, which are proposed and studied frequently as two different imaging approaches [1,4,16]. For comparison, we demonstrate and discuss them in theory and experiment, although these two configurations work fundamentally on a similar principle.

 figure: Fig. 1.

Fig. 1. Two configurations of Fourier single-pixel imaging in time domain. (a) Configuration I for detecting temporal signal (b) Configuration II for detecting time-varying transmittance function of an optical element. EOM: electro-optic modulator, AWG: arbitrary waveform generator, RF: radio frequency.

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Let us consider the theoretical model of time-domain Fourier single-pixel imaging. The Fourier transform (FT) of signal f (t) is given by:

$$F(\omega )\textrm{ = }\mathrm{{\cal F}}[{f(t )} ]\textrm{ = }\int_{\textrm{ - }\infty }^\infty {f(t ){\textrm{e}^{\textrm{ - i}\omega t}}} \textrm{d}t,$$
where t and ω denote time and angular frequency, $\mathrm{{\cal F}}$ represents Fourier transform and i the imaginary unit. Conversely, f (t) can be reconstructed from F(ω) by the inverse Fourier transform (IFT):
$$f(t )= {\mathrm{{\cal F}}^{ - 1}}[{F(\omega )} ]\textrm{ = }{1 / {2\mathrm{\pi }}} \cdot \int_{ - \infty }^\infty {F(\omega ){\textrm{e}^{\textrm{i}\omega t}}} \textrm{d}\omega ,$$
where $\mathrm{{\cal F}}$-1 represents the inverse Fourier transform.

According to Euler's formula, Eq. (1) could be further written as:

$$F(\omega )\textrm{ = }\int_{\textrm{ - }\infty }^\infty {f(t )\cos ({\omega t} )} \textrm{d}t - \textrm{i}\int_{\textrm{ - }\infty }^\infty {f(t )\sin ({\omega t} )} \textrm{d}t.$$

The optical modulation function of EOM cannot be the cosine or sine wave, because it must be non-negative. Accordingly, differential measurement technique, i.e. four-step phase-shifting approach [27], has to been applied to obtain the Fourier spectrum. The normalized optical modulation function of EOM with initial phase ϕ is expressed as

$${E_{\phi ,\omega }}(t )= {{[{1 + \cos ({\omega t + \phi } )} ]} / 2}.$$

For configurations I and II, the signal waveform before the light power meter is f (t)Eϕ,ω(t), where f (t) represents the unknown temporal object. The light power meter’s bandwidth is far lower than that of f (t), therefore it could be actually regarded as a single-pixel detector in time domain. The corresponding photocurrent of the power sensor Iϕ(ω) is proportional to the average power of f (t)Eϕ,ω(t) over time:

$${I_\phi }(\omega )= {i_0}\int_{\textrm{ - }\infty }^\infty {f(t ){E_{\phi ,\omega }}(t )} \textrm{d}t,$$
where i0 is a constant. Then applying the equation below, we may get F(ω) by the measurements of Iϕ(ω):
$${I_0}(\omega )- {I_\mathrm{\pi }}(\omega )+ \textrm{i}[{{I_{{\mathrm{\pi } / 2}}}(\omega )- {I_{{{\mathrm{3\pi }} / 2}}}(\omega )} ]\textrm{ = }{i_0}\int_{\textrm{ - }\infty }^\infty {f(t ){\textrm{e}^{ - \textrm{i}\omega t}}} \textrm{d}t = {i_0}F(\omega ).$$

According to Eq. (6), the Fourier coefficient at frequency point ω, including real and imaginary parts, could be obtained by the differences of four photocurrent values, which are measured when EOM modulates at the same frequency point with initial phases of 0, π/2, π and 3π/2. Performing IFT on the measured F(ω), the input temporal object could be easily recovered.

It is worth noting that FT and IFT in theoretical model above have to be replaced by N-point discrete FT and IFT respectively, to realize time-domain Fourier single-pixel imaging in practical applications. The discrete version of Eq. (5) is written as

$${I_\phi }({\omega _k}) = {i^{\prime}_0}\sum\limits_{j = 0}^{N - 1} {f({t_j})} {E_{\phi ,{\omega _k}}}({t_j}),$$
where i’0 is a constant, and tj and ωk are discretized time and angular frequency, with j, k = 0, 1, … N-1. According to Eqs. (6) and (7), the measured discrete Fourier spectrum is given by
$$F^{\prime}({{\omega_k}} )\textrm{ = }{I_0}({\omega _k}) - {I_\mathrm{\pi }}({\omega _k}) + \textrm{i}[{I_{{\mathrm{\pi } / 2}}}({\omega _k}) - {I_{{{3\mathrm{\pi }} / 2}}}({\omega _k})] = {i^{\prime}_0}\sum\limits_{j = 0}^{N - 1} {f({t_j})} {\textrm{e}^{ - \textrm{i}{\omega _k}{t_j}}}.$$

Appling the discrete IFT, we get the reconstructed signal:

$$f^{\prime}({t_j})\textrm{ = }{\mathrm{{\cal F}}^{ - 1}}[F^{\prime}({\omega _k})] = \sum\limits_{k = 0}^{N - 1} {F^{\prime}({\omega _k})} {\textrm{e}^{\textrm{i}{\omega _k}{t_j}}}.$$

Let us consider the SNR of the reconstruction. Any signals could be regarded as a linear combination of scaled delta functions. Therefore, for simplicity we let f (tj) be a delta function:

$$f({t_j}) = \left\{ {\begin{array}{ll} {1,}&{{t_j} = 0}\\ {0,}&{{t_j} \ne 0} \end{array}} \right..$$

With Eqs. (8)–(10), the reconstructed signal is expressed as:

$$f^{\prime}({t_j}) = \left\{ {\begin{array}{ll} {N{{i^{\prime}}_0},}&{{t_j} = 0}\\ {0,}&{{t_j} \ne 0} \end{array}} \right..$$

In the shot noise limit, the noise at each frequency point is uncorrelated and its variance is independent of frequency, i.e. σ2(ωk)=σ02. Therefore, the noise variances add incoherently in the summation of discrete IFT:

$${\sigma ^2}({t_j}) = N\sigma _0^2.$$

Note that σ02 is determined by experimental parameters, such as mean detector photocurrent. According to Eqs. (11) and (12), the SNR of reconstructed signal is expressed as:

$$\textrm{SNR} = Ni^{\prime2}_{0}/\sigma _0^{2}.$$

According to Eqs. (8) and (10), (i’0)2/σ02 is actually the SNR of the measured Fourier spectrum. Therefore, for given experimental condition, the SNR of reconstructed signal is proportional to N. In other words, the proposed method could provide an improved SNR, which is up to N times as high as the SNR of the measured Fourier spectrum.

3. Experimental results

The bandwidth of EOMs used in experiments is 10 GHz. The sampling rate of AWG is 500MSa/s, therefore its maximum bandwidth is theoretically 250 MHz, according to Nyquist-Shannon sampling theorem. Using two proposed configurations, N Fourier coefficients at evenly spaced frequency points between -250 MHz and 250 MHz are achieved, as shown in Fig. 2(a) and (c). Obviously, these Fourier spectra, measured under N = 32, 64, 128 and 256, show a similar trend. Note that the frequency-domain sampling spacing Δf equals 2B/N, where B = 250 MHz is the bandwidth of AWG. Thus, the period of reconstruction is 1/Δf = N/2B. The sampling interval for reconstruction in time domain is 1/2B. In theory, bigger N means shorter frequency point spacing and longer period of reconstruction. The waveform of input light signal/temporal object for configuration I/II is a 32-bit binary sequence of nanosecond pulses. For a comparison, we also measure the temporal object directly using a fast detector and oscilloscope. For direct detection, the AWG signal is set to allow the light go through the modulator directly. Under different N, the reconstructed signals of the proposed algorithm are entirely consistent with the waveform data of binary sequence and the results of direct detection, as shown in Fig. 2(b) and (d). It means that the proposed technique could break the temporal resolution limitation of detection, thus allowing the exact and effective measurement of nanosecond pulses using a 10 Hz bandwidth detector.

 figure: Fig. 2.

Fig. 2. (a)(c) The amplitudes of spectrum F(ω) measured by the proposed technique using configurations I and II respectively, when N = 32, 64, 128 and 256 (blue, red, yellow and purple). (b)(d) The binary waveform data of input temporal object used in configuration I/II (olive drab). The signals measured by direct detection using a fast detector (green). Reconstructed results of the proposed technique with different N (blue, red, yellow and purple). The fixed threshold to distinguish 1 from 0 for binary bit (black dash line).

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When point number N = 32, 64, 128 and 256, the experimental SNR values of the traditional method, i.e. direct detection, and the proposed technique are compared in Fig. 3. For a given sampling rate, the SNR of direct detection without averaging is only determined by input light power and technical parameters of detector and oscilloscope, and therefore independent of sampling length i.e. sampling point number N. Note that the SNR is defined as the ratio between the signal power and the power of background noise. As discussed in Section 2, the SNR of the proposed technique is proportional to N. Interestingly, when N doubles, the experimental SNR of the proposed technique gains ∼3 dB and therefore agrees well with the theoretical prediction. The reason is that the power of reconstructed signal increases by N2 times, while the noise variances only increases by N times, as illustrated in Eqs. (11) and (12). As a result, not only sampling period but also SNR are improved by a factor of N for time-domain Fourier single-pixel imaging.

 figure: Fig. 3.

Fig. 3. Comparison of the experimental SNR values of signals obtained by configuration I (blue square), configuration II (red diamond) and direct detection (black triangle). The SNR values for configuration I are 22.8, 27.0, 29.6 and 31.9 dB respectively. The SNR values for configuration II are 23.2, 26.1, 29.1 and 31.1 dB respectively. The SNR values for direct detection are 23.7, 23.9, 23.7 and 23.6 dB respectively.

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Generally, the main component of the reconstructed signals is medium-frequency and low-frequency information, which forms the basic binary codes [2,27]. In other words, the Fourier spectrum of binary sequence is sparse and compressible in most cases. Therefore, the high-frequency component of signal does not have to be measured with the proposed method. To evaluate the sparsity of F(ω), we define a parameter called compression ratio (CR), i.e. the percentage of measured frequency points in total frequency points. The binary waveform of signal to be measured is shown in Fig. 4(a). When CR = 100%, 50% and 30%, the binary code could still be extracted exactly with the same threshold, even if the rising and falling edges of reconstructed signals are smoothed as the CR decreases, as shown in Fig. 4(b)-(d). For a fair comparison, the lengths of measured Fourier spectrum are extended to N by zero padding before applying reconstruction algorithm, when CR = 50% and 30%. The corresponding F(ω) for reconstruction are shown in the insets. Thus the reconstructed signals have the same length, regardless of CR. These experimental results indicate that quite a number of measurements for time-domain Fourier single-pixel imaging could be saved, especially in digital communication applications.

 figure: Fig. 4.

Fig. 4. (a) The binary waveform of signal to be measured. (b)-(d) The reconstructed signals with CR = 100%, 50% and 30% when N = 32, 64, 128 and 256 (blue, red, yellow and purple) and the fixed threshold to distinguish 1 from 0 for binary bit (black dash line). The corresponding F(ω) are shown in the insets.

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It has been reported that an inherent advantage of single-pixel imaging is the immunity against the distortion before detector [11,12,16]. For the proposed method, it is confirmed again by experiment. We place a distortion unit in front of the detector in configurations I and II. The distortion unit is actually an optical ring cavity based on a 50:50 coupler with cavity fiber length of ∼0.4 m, as shown in Fig. 5(a). In this distortion unit, the copy of signal is delayed for ∼2 ns per round and overlaps other copies at the output. Therefore, the signal obtained by direct detection is broadened remarkably, as shown in Fig. 5(b) and (c). However, the reconstructions of the proposed method are unaffected by distortion and consistent with the original waveform data of binary sequence. Because the very low-bandwidth detection used in the proposed method is insensitive to temporal distortion.

 figure: Fig. 5.

Fig. 5. (a) Distortion unit. (b)(c) The comparison of signals obtained by direct detection (green), configurations I and II when N = 32, 64, 128 and 256 (blue, red, yellow and purple) in the presence of temporal distortion.

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4. Discussion and conclusion

In proof-of-principle experiment, time-domain Fourier single-pixel imaging is demonstrated to retrieve a periodic temporal object. Actually, using parallel single-pixel technology such as spatially multiplexed measurement [33] and wavelength division multiplexing technology [40], the proposed technique also allows the capture of a single non-reproducible temporal object. But limited by the speed of parallel modulation and/or parallel detection, the typical temporal resolution of parallel single-pixel technology for single-shot measurement is only at sub-second level [33,40]. In addition, wavelength division multiplexing technology could only be applied for the temporal objective with flat spectral response. In a word, the serial measurement method demonstrated here can hardly be replaced by parallel single-pixel technology, because it has high temporal resolution and no requirement for spectral response of temporal objective.

In principle, the temporal resolution of the proposed technique is basically independent of detection bandwidth but limited by modulation bandwidth. With dispersive Fourier transform technology [38], the proposed technique could overcome this limitation and achieve super-resolution beyond hardware capability. Furthermore, using wavelength conversion [39], it can even be applied in the wavelength range where fast modulators are unavailable.

Compared with other basis scan single-pixel technologies, the proposed method could obtain Fourier spectrum of the temporal object directly, which is very important and useful in signal analysis and signal processing. Therefore, the information extraction and enhancement based on frequency domain may be integrated into the proposed method conveniently in a reconstruction-free way [41].

In conclusion, we study Fourier single-pixel imaging in time domain for capturing temporal object. Using the proposed technique, binary sequence of nanosecond pulses is obtained experimentally with ten Hertz detection bandwidth. Furthermore, the theoretical and experimental results show that the proposed technique has a SNR gain of N, where N is the point number for discrete Fourier transform, similar to spatial Fourier single-pixel imaging [42]. In experiments, due to the sparsity of Fourier spectrum, the proposed technique allows the reconstruction based on sub-Nyquist sampling, reducing the number of measurements by up to 70%. Finally, it proves immune to the temporal distortion before detector. We believe this study will lead the development of temporal sensing technology and may have important applications in communications, high-speed sensing and distortion-free detection, especially for wavebands where fast photodiode is unavailable, such as far infrared light, THz and X-ray.

Funding

National Natural Science Foundation of China (62275018).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (5)
Fig. 1.
Fig. 1. Two configurations of Fourier single-pixel imaging in time domain. (a) Configuration I for detecting temporal signal (b) Configuration II for detecting time-varying transmittance function of an optical element. EOM: electro-optic modulator, AWG: arbitrary waveform generator, RF: radio frequency.

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Fig. 2.
Fig. 2. (a)(c) The amplitudes of spectrum F(ω) measured by the proposed technique using configurations I and II respectively, when N = 32, 64, 128 and 256 (blue, red, yellow and purple). (b)(d) The binary waveform data of input temporal object used in configuration I/II (olive drab). The signals measured by direct detection using a fast detector (green). Reconstructed results of the proposed technique with different N (blue, red, yellow and purple). The fixed threshold to distinguish 1 from 0 for binary bit (black dash line).

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Fig. 3.
Fig. 3. Comparison of the experimental SNR values of signals obtained by configuration I (blue square), configuration II (red diamond) and direct detection (black triangle). The SNR values for configuration I are 22.8, 27.0, 29.6 and 31.9 dB respectively. The SNR values for configuration II are 23.2, 26.1, 29.1 and 31.1 dB respectively. The SNR values for direct detection are 23.7, 23.9, 23.7 and 23.6 dB respectively.

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Fig. 4.
Fig. 4. (a) The binary waveform of signal to be measured. (b)-(d) The reconstructed signals with CR = 100%, 50% and 30% when N = 32, 64, 128 and 256 (blue, red, yellow and purple) and the fixed threshold to distinguish 1 from 0 for binary bit (black dash line). The corresponding F(ω) are shown in the insets.

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Fig. 5.
Fig. 5. (a) Distortion unit. (b)(c) The comparison of signals obtained by direct detection (green), configurations I and II when N = 32, 64, 128 and 256 (blue, red, yellow and purple) in the presence of temporal distortion.

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Equations (13)

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F ( ω )  =  F [ f ( t ) ]  =  ∫  -  ∞ ∞ f ( t ) e  - i ω t d t ,
f ( t ) = F − 1 [ F ( ω ) ]  =  1 / 2 π ⋅ ∫ − ∞ ∞ F ( ω ) e i ω t d ω ,
F ( ω )  =  ∫  -  ∞ ∞ f ( t ) cos ⁡ ( ω t ) d t − i ∫  -  ∞ ∞ f ( t ) sin ⁡ ( ω t ) d t .
E ϕ , ω ( t ) = [ 1 + cos ⁡ ( ω t + ϕ ) ] / 2 .
I ϕ ( ω ) = i 0 ∫  -  ∞ ∞ f ( t ) E ϕ , ω ( t ) d t ,
I 0 ( ω ) − I π ( ω ) + i [ I π / 2 ( ω ) − I 3 π / 2 ( ω ) ]  =  i 0 ∫  -  ∞ ∞ f ( t ) e − i ω t d t = i 0 F ( ω ) .
I ϕ ( ω k ) = i 0 ′ ∑ j = 0 N − 1 f ( t j ) E ϕ , ω k ( t j ) ,
F ′ ( ω k )  =  I 0 ( ω k ) − I π ( ω k ) + i [ I π / 2 ( ω k ) − I 3 π / 2 ( ω k ) ] = i 0 ′ ∑ j = 0 N − 1 f ( t j ) e − i ω k t j .
f ′ ( t j )  =  F − 1 [ F ′ ( ω k ) ] = ∑ k = 0 N − 1 F ′ ( ω k ) e i ω k t j .
f ( t j ) = { 1 , t j = 0 0 , t j ≠ 0 .
f ′ ( t j ) = { N i ′ 0 , t j = 0 0 , t j ≠ 0 .
σ 2 ( t j ) = N σ 0 2 .
SNR = N i 0 ′ 2 / σ 0 2 .
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