Abstract
Single-pixel cameras (SPCs) have been successfully used in different imaging applications during the last decade. In these techniques, the scene is illuminated with a sequence of microstructured light patterns codified onto a programmable spatial light modulator. The light coming from the scene is collected by a bucket detector, such as a photodiode. The image is recovered computationally from the photodiode electric signal. In this context, the signal quality is of capital value. One factor that degrades the signal quality is the noise, in particular, the photocurrent, the dark-current, and the thermal noise sources. In this chapter, we develop a numerical model of a SPC based on a photodiode, which considers the characteristics of the incident light, as well as the photodiode specifications. This model includes the abovementioned noise sources and infers the signal-to-noise ratio (SNR) of the SPCs in different contexts. In particular, we study the SNR as a function of the optical power of the incident light, the wavelength, and the photodiode temperature. The results of the model are compared with those obtained experimentally with a SPC.
Keywords
- single-pixel cameras
- structured illumination
- photodiodes
- signal-to-noise ratio
- photocurrent
- dark current
- noise in imaging systems
1. Introduction
Computational imaging with a single-pixel camera (SPC), or single-pixel imaging (SPI), is a remarkable alternative to conventional imaging [1]. SPCs are based on sampling the scene with a sequence of microstructured light patterns codified onto a programmable spatial light modulator (SLM), while the intensity of the light coming from the object is measured by a detector without spatial resolution. The image is computationally recovered from the fluctuations of the electric signal provided by the detector. Thus, the quality of this temporal signal is a key factor in order to recover a high quality image.
The simplicity of the detection stage in SPI is one of the main advantages of the technique. It can be exploited to use very sensitive light sensors in low light level applications [2, 3]. It is also useful in order to measure the spatial distribution of different parameters such as the spectral content [2, 4] or the polarization state [5] of the light coming from the objects. Besides, SPI has shown robustness to the presence of scattering media [6, 7]. Moreover, the SPC can be an interesting choice for imaging using light with a spectrum beyond the visible region, such as in the infrared (IR) and terahertz spectral regions. Furthermore, SPI techniques are very well suited for the application of compressive sampling (CS, also referred to as compressive sensing) methods, which noticeably reduce the measurement time, the bottleneck of this technique [8, 9].
Among the different possible detectors, photodiodes are the most common sensors in general single-pixel imaging applications. In this chapter, we develop a numerical model of a single-pixel camera based on a photodiode, which considers the characteristics of the incident light, as well as the photodiode specifications [10]. Our model takes into account the photocurrent, the dark current, the photocurrent shot noise, the dark-current shot noise, and the Johnson-Nyquist (thermal) noise. In particular, we study the signal-to-noise ratio (SNR) as a function of the optical power level and the wavelength of the incident light, as well as the photodiode temperature. We restrict our study to silicon (Si) and indium-gallium-arsenide (InGaAs) photodiodes.
In the following sections, first, we describe in detail the single-pixel cameras. Second, we review the properties of the electrical signal provided by photodiodes based on Si and InGaAs materials. Third, we present a numerical model of the single-pixel camera. Next, we apply this model to study the SNR of a single-pixel camera in different contexts. After that, we compare some of these numerical results with those experimentally obtained in the laboratory. Finally, we emphasize the main conclusions.
2. The single-pixel camera
The SPC is able to provide images of a scene with a bucket detector, such as a photodiode, by using light-structured illumination. A schematic representation of the optical system is shown in Figure 1. A set of microstructured light patterns is codified onto a programmable spatial light modulator (SLM) and sequentially projected onto the scene to be analyzed. The light reflected (or transmitted) by the scene is collected by a lens and focused onto a photodiode. The photodiode provides us with an electrical current proportional to the integrated light intensity, which is digitized by a data acquisition system (DAQ). The photodiode signal represents the dot product between each microstructured light pattern and the scene. The image is retrieved from the photodiode signal and the microstructured light patterns using computational algorithms.

Figure 1.
Schematic representation of a single-pixel camera.
The ideas on which SPCs are based were proposed by Golay in 1949, for spectroscopy applications [11], and by Decker in 1970, for imaging [12]. However, the first efficient SPC was created in 2006 [13], by using a programmable SLM. The most common types of SLMs are the liquid crystal spatial light modulator (LC-SLM) and the digital micromirror device (DMD) [14]. In general, DMDs are more used than LC displays, in SPI applications, except when phase modulation is required. A DMD is a microelectromechanical system device that contains a pixelated display composed by millions of tiny switchable mirrors. Each mirror is able to switch to either
As a result, the SPCs have been successfully applied in many different imaging areas during the last decade. Among them, we can mention infrared imaging [17, 18], terahertz imaging [19], ultrasonic imaging [20], 3D computational imaging [21, 22], imaging through scattering media [23, 24, 25], 3D and photon counting light detection and ranging (LIDAR) imaging systems [3, 26, 27], stereoscopic imaging [28], microscopy [29, 30], holography [31, 32], and ophthalmoscope imaging [33].
The set of microstructured light patterns is also an important element of the illumination stage in SPI. The light patterns commonly used for illumination are speckle patterns, binary random distributions, and functions of different basis such as noiselets [34], wavelets [35], Fourier, and Walsh-Hadamard (WH) functions. In fact, the measurement time and the resolution of the image are directly related with the properties of the light patterns. Particularly, the WH functions are easily coded on the DMD display due to the binary modulation nature of the DMD.
3. The photodiode signal in a single-pixel camera
In SPCs, the image is computationally reconstructed from the electric signal provided by the photodensor; in our case, a photodiode. In this context, the photodiode signal represents the inner product of the set of microstructured light patterns with the scene. Therefore, to analyze the quality of the image, it is convenient to study the properties of the electrical signal provided by the photodiode and its noise sources.
By definition, a photodiode is a semiconductor device that converts the optical signal into a current signal by electronic processes [36]. The electrical current of the photodiode is composed by two terms, the photocurrent
where P is the optical power level of the light source and
where
Assuming that the temperature is lower or close to room temperature, the first term in Eq. (3) can be considered negligible [36]. Taking this into account and substituting this expression in Eq. (2), the dark current is given by
where
and by the Sajal Paul relation for the In
where
Si | |||
GaAs | |||
InAs |
Table 1.
Values of material parameters
In the single-pixel camera process, the photocurrent and the dark current signals have an associated error, due to the discrete nature of the electrical charge [45]. The noise of the former one is known as the photocurrent shot noise
where
where
For the sake of completeness, we will consider the Johnson-Nyquist (or thermal) noise
where
Consequently, the signal-to-noise ratio (SNR) of the electrical current signal in units of decibels is defined as [37].
4. Numerical model of the single-pixel camera
In this section, a numerical model of the SPC is described. The camera model takes into account the optical power level
In this model, the microstructured light patterns are 2-D functions
In absence of noise, the mathematical properties of
The numerical process developed to simulate the SPC, from the moment in which the light source illuminates the DMD up to the image reconstruction (see Figure 1), is as follows:
Step 1: Obtain the number of photons per second
where
Step 2: Distribute the
Step 3: Multiply the photon matrix
Step 4: Calculate the number of photons per second
Step 5: Obtain the optical power level (
Step 6: Calculate the total current as the sum of the photocurrent and the dark current according to Eq. (7),
Step 7: Obtain the noisy current
Step 8: Obtain the normalized photodetector signal related to
where
Step 9: Calculate the image
5. Results
5.1. Numerical results
The numerical model described in the previous section was used to analyze the performance of a SPC formed by photodiode detectors under different circumstances. Three different studies were developed analyzing the image quality when: (1) the optical power level of the light source changes; (2) we use light sources with different wavelengths; and (3) the photodiode temperature varies. The simulations were performed for two commercial photodiodes, DET36A Thorlabs and DET10C Thorlabs, whose specifications are shown in Table 2. Moreover, the dark current
Parameter | Symbol | Silicon biased detector | In |
---|---|---|---|
Detector name | DET36A Thorlabs | DET10C Thorlabs | |
Photodiode active area | |||
Wavelength range | 350–1100 nm | 900–1700 nm | |
Band gap energy at 298 K | 1.1114 eV | 0.7379 eV | |
Rise time response | 14.0 ns | 10.0 ns | |
Noise bandwidth | 0.025 nHz | 0.035 nHz | |
Bias voltage | 10.0 V | 5.0 V | |
Saturation current at 298 K | 0.35 nA | 1.0 nA | |
Shunt resistance | 1.0 G | 10.0 G | |
NEP at |
Table 2.
Photodiode parameters data.

Figure 2.
Simulated dark current, dark-current shot noise and thermal noise as a function of the temperature for (a) Si biased detector (DET36A Thorlabs) and (b) InGaAs biased detector (DET10C Thorlabs).
Firstly, we analyzed the image quality as a function of the optical power level of the light source. We fixed the wavelength of the light source to 520 nm and the photodiode temperature to 298 K. Figure 3 shows the photocurrent, the dark current, and the total current associated to the single-pixel camera for two different optical power levels; (a)

Figure 3.
Photocurrent, dark current, and total current with their associated noise values as a function of the WH pattern index. Two different optical power values were considered (a) 42.49 μ W and (b) 0.0085 μ W . The wavelength of the light source was fixed at 520 nm and the photodiode temperature at 298 K. Images computed from these electric signals are shown on the right. In both cases, the resolution of the WH patterns is 64 × 64 pixels (Reprinted from [10]).
On the other hand, we numerically evaluated the image quality using the SNR metric defined by [48]. The reference image is obtained by SPI techniques but using only the photocurrent values without considering the noise sources. Afterward, we plot in Figure 4 the SNR as a function of the optical power of the light source. As expected, the image quality obtained by the SPC improves when the optical power increases. In the same figure, we also plot the curve of the SNR of the photodiode signal as a function of the optical power level. The reference signal is again the photocurrent signal without noise values. Of course, the SNR is the same in both cases. Therefore, we will use the SNR applied to the images from now onwards.

Figure 4.
SNR of the signal and the recovered images as a function of the optical power P inp . The recovered images on the right part are a sample of the SNR red data points (reprinted from [10]).
Second, we analyzed how the wavelength of the light source influences the performance of the SPC. The optical power of the light source was set to

Figure 5.
(a) SNR as a function of the wavelength of the light source; (b) responsivity data of both photodiodes [49, 50]; and (c) recovered images for different wavelengths.
Finally, we analyzed the dependence of the image quality with the photodiode temperature. The wavelength of the light source was set to 520 and 1600 nm for the DET35A and the DET10C detectors, respectively. For each detector, three curves of the SNR as a function of the photodiode temperature are plotted for constant values of the optical power (

Figure 6.
SNR dependence with the photodiode temperature for three optical power levels: 42.49 μ W , 8.49 μ W , and 0.21 μ W for (a) the Si biased detector (DET36A Thorlabs), and (b) the InGaAs biased detector (DET10C Thorlabs). The recovered images obtained for those optical power levels are shown as well. For those images, the temperature range starts at 273 K and ends at 373 K in 25 K steps.
5.2. Experimental results
A scheme of the experimental setup by using transillumination is depicted in Figure 7. In this case, a DMD (DLP Discovery 4100, Texas Instrument) was illuminated with a collimated light beam generated with an incoherent white-light source. A narrow band pass filter (P10-515-S 93819, Corion) centered at a wavelength of 520 nm with a bandwidth of 20 nm was used to avoid spectral artifacts. In order to apply SPI techniques, microstructured light patterns corresponding to 2-D functions of the orthonormal WH basis with 64

Figure 7.
Experimental setup of the single-pixel camera.
Figure 8(a) shows numerically and experimentally recovered images with different levels of the optical power. We can see that, in both cases, the noise level decreases with the optical power. This is corroborated by the results in Figure 8(b), which show that the SNR curve corresponding to images obtained with the simulated and the experimental systems have a similar dependence with the optical power. This fact confirms the validity of our numerical model.

Figure 8.
(a) Numerically and experimentally recovered images for different optical power levels P inp ; (b) SNR dependence with P inp for the experimental and numerical images.
However, even though the model has been developed taking into account the most important noise factors during the imaging process, there is still a discrepancy in the values of the SNR for the experimental and the simulated images. This difference is produced by several other noise sources not included in the model. First, we have considered that both the DMD reflectance and the object transmittance are ideal binary functions, which is not true in practice. Second, we did not introduce background, or ambient, light into the numerical model, with the unavoidable associated noise. Finally, we did not consider the current-to-voltage and the analog-to-digital conversion processes, which produce certain amount of noise. A clear example in the last case is the quantization noise.
6. Conclusions
In this chapter, a numerical model of a single-pixel camera (SPC) has been developed, considering the characteristics of the incident light and the physical properties, as well as the specifications of the photodiode. We have accomplished a careful and rigorous mathematical review of the electrical behavior of Si and InGaAs detectors. Our model takes into account the photocurrent, the dark current, the photocurrent shot noise, the dark-current shot noise, and the Johnson-Nyquist (thermal) noise of two commercial photodiodes, a Si and an InGaAs photodetector.
Numerical simulations with our model have allowed us to analyze the behavior of the single-pixel imaging (SPI) technique in different contexts. In particular, we have studied the quality of the final image as a function of the power of the light source. We have corroborated the reduction of the SNR for low light levels. We have also observed the clear link between the quality of the photocurrent signal and the quality of the reconstructed image. These results can be useful to predict the behavior of imaging systems working in low light level conditions.
We have also studied the dependence of the SNR with the wavelength of the light source. In this case, we conclude that the influence of the wavelength arises from the variation of the quantum efficiency and the responsivity of the photodetector. Such analysis could be the first step in the application of SPI techniques to multispectral imaging.
Finally, we have analyzed the quality of the images provided by the SPC as a function of the photodiode temperature. The study is performed for both a Si biased and an InGaAs biased detector. The main conclusion in this case is that the SNR of the reconstructed images changes only slightly with the temperature for high values of the light power. However, the reduction is clearly significant for low light levels. Therefore, cooling the detector can play an important role in photon counting or low light level applications.
An experimental SPC has been developed in order to validate the results provided by our model. The quality of the images obtained experimentally does not match perfectly with that predicted by the model. The discrepancy is due to several unaccounted sources of uncertainty such as nonuniformities in the mirrors of the DMD or in the object substrate, as well as noise introduced in the signal amplification process or the analog to digital conversion procedure. However, we have shown that the quality of the final image, in terms of the SNR, changes in a similar way with the light power. This allows us to confirm that the model can be useful to predict the behavior of SPI systems based on photodiodes under different circumstances.
Funding information
We acknowledge financial support from MINECO (FIS2016-75618-R and FIS2015-72872-EXP), Generalitat Valenciana (PROMETEO/2016/079), and Universitat Jaume I (P1-1B2015-35). Yessenia Jauregui-Sánchez acknowledges the Santiago Grisolía support from Generalitat Valenciana (GRISOLIA/2015/037).