1. Introduction

Compressed sensing (CS) imaging [1–3] has shown great application value in recent years; compared with traditional imaging methods, CS imaging reconstructs the targets using a postprocessing algorithm with a subsampling strategy, such that the sampling process need not obey the Nyquist sampling theorem. Moreover, unlike the array detectors used in traditional imaging, CS imaging uses a single-point detector without spatial resolution to collect the overall measured results. Owing to its advantages in imaging applications, CS imaging has been widely used in many fields, such as spectral imaging [4–7], fluorescence imaging [8,9], and holography [10,11].

Natural targets are often analog variables, whereas digital images are discrete. Hence, the detectors used in imaging systems must be able to perform analog-to-digital conversion using a certain number of bits; therefore, quantization errors are often inevitable in imaging. In traditional imaging systems, the limited number of detector bits affects imaging quality. To obtain high-quality images, the number of detector bits must be sufficiently high to reduce distortions caused by quantization; however, this also results in problems involving large amounts of data and low transmission speeds. In CS imaging, high-flux measurements are used to increase the signal-to-noise ratio (SNR) of the detector; however, the demand on the number of detector bits is far greater than that of traditional imaging. Thus, quantization distortions in CS imaging are more serious than those in traditional imaging when using the same detector. Therefore, acquiring high-quality images with limited detector bits will be of significance. Thus far, several methods have been developed to reduce quantization errors when using fewer numbers of detector bits. One of the direct methods involves reducing the dynamic range of the measured signal. For example, instead of using the full dynamic range of the signal, researchers have considered measurements with an adaptive dynamic range to specifically increase the accuracy within the main distribution area of the signal [12]; however, this requires a priori knowledge of the dynamic range of the image and specialized hardware implementation. A balance detector has also been introduced to decrease the dynamic range during sampling [13], but this causes system limitations as the balance detector cannot be used at certain wavelengths or with high sensitivity. Moreover, with the development of 1-bit CS [14], numerous postprocessing algorithms have been reported to improve the recovered image quality from 1-bit quantizers [15–17]; however, these methods are not suitable for multiple-bit detectors.

Dither refers to a type of random noise added to the measured results before quantization and has been proven to be an effective method of reducing quantization errors [18,19]. After subtractive dither, a quantizer with a nonsubtractive dither was proposed in [20]. Li et al. proved that dithering can compensate for quantization distortions, even in binary measurements [21]. Compared with single dithering, researchers have proposed multiple dithering methods by averaging several measurements [22], but this incurs extra cost with respect to measurement time, which may reduce the imaging speed.

This work presents an improved compressive imaging method by combining sparse measurements and multiple dithers to realize high-quality imaging with limited detector bits. The relationship between the dynamic measurement range and reconstruction error is analyzed theoretically. Then, a sparse measurement matrix design and multiple dithering strategy are proposed to reduce quantization distortions. Numerical simulations and experiments are performed to show that both strategies are helpful in decreasing quantization distortions with limited detector bits; the combination of sparse measurements and multiple dithers in compressive imaging can achieve preliminary imaging even with 1-bit detector.

2. Compressed sensing imaging with sparse measurements and dithers

2.1 Dynamic range of compressed sensing imaging

Given an original image $x$ represented as an $N \times 1$ column vector, the process of compressive imaging with $M$ measurements, where $M < N$, can be written as

In imaging, the dynamic range refers to the maximum intensity of the image that can be captured using a given minimum resolvable intensity. Extensive studies have indicated that the imaging results will be distorted when measuring high-dynamic-range images when the number of detector bits is limited [12,21]. To resolve the distortions caused by high-flux measurements, we first discuss the relationship between the dynamic range and reconstruction errors in CS imaging mathematically. Here, the mid-riser model is used as the quantization model, and we assume that the saturation of the quantizers is not less than the dynamic range of the measurements in this study. The model can be expressed as follows:

In CS imaging, the relationship between the quantization bits and quantization steps can be expressed as

According to Eq. (5), for fixed quantization bits ${B_{cs}}$, the quantization step size $\Delta$ is proportional to the dynamic range to be measured, ${L_{cs}}$. For an image where the intensity of each pixel varies in the range of 0 to ${L_o}$, CS imaging collects the total intensity after sampling the measurement matrix; thus, for a given target, the full dynamic range ${L_{cs}} = N \cdot {L_o}$ is a large number in actual measurements. Therefore, we consider that compressive imaging has high-flux measurement characteristics, which results in a larger value of the quantization step for quantization of the same bits when compared with traditional imaging such that the sensors divide the grayscale range in a cruder manner. Hence, this drawback causes the reconstruction quality of compressive imaging to be more sensitive to the quantization bits.

2.2 Sparse measurements

In an imaging system, given a fixed number of bits, a smaller dynamic range of the measured signal means a finer quantization step, which decreases the loss of grayscale in quantization. Therefore, reducing the dynamic range of the measured signal can reduce the quantization errors and hence the reconstruction errors in CS imaging.

In the present study, we reduce the dynamic range of the measured signal by improving the design of the complementary measurement matrix. Based on this target, a sparse measurement matrix was proposed. Using the sparse matrix, the number of pixels of the information obtained from each sampling is reduced, such that the measured intensities can be decreased. We first construct an $M \times N$ sparse measurement matrix that consists of the elements −1, 0, and 1, where the total number of $\pm 1$ values in each row is $p$ with $p \ll N$, and the location distributions of the $\pm 1$ values in each row are completely random. Then, the matrix $A$ is divided into a pair of 0-1 complementary matrices, ${A_1}$ and ${A_2}$, for the actual modulations. This means that only the signals in $p/2$ pixels of the image are measured on average for each modulation, which is lower than half the pixels, as in traditional CS imaging. An example of the construction of the sparse complementary measurement matrices is presented in Fig. 1.

 

Fig. 1. Example of constructing sparse complementary measurement matrices, which have three $\pm 1$ in each row.

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According to Eq. (4), it is obvious that the range of quantization errors $\delta = Q\left ( y \right ) - y$ is

As the quantization error for each measurement satisfies the independent and identically distributed condition in the range of $\left ( { - \Delta /2,\Delta /2} \right ]$ with a zero mean value, the variance can be written as $\sigma _{y1}^2 = \sigma _{y2}^2 = {\Delta ^2}/12$. Thus, the variance of the complementary measurement result $y={y_1}-{y_2}$ is

As the modulation matrices ${A_1}$ and ${A_2}$ in the complementary measurements consist of elements 0 and 1, according to Eq. (1), the dynamic range ${L_{cs}}$ of the measured signal in one measurement during CS imaging can be expressed as

The reconstruction error $\varepsilon$ is then obtained by calculating the deviation between the original signal $x$ and reconstruction result $\hat x$ [24]:

It is obvious that the reconstruction errors in CS imaging are proportional to the sparsity $k$ of the measurement matrices. Thus, reducing the dynamic range of the measured signal by decreasing the sparsity of the measurement matrix is an effective method of alleviating quantization distortions. For an image consisting of $N$ pixels with the intensity of each pixel being ${L_o}$, the largest possible measured result with the classical 0-1 random matrix is ${NL_o}$, which is obtained when all the pixels in the image assume the maximum possible value ${L_o}$ and the modulation matrix has all entries as 1. However, the full dynamic range of the measured signal with a sparse matrix is ${pL_o}$, where $p$ is the maximum possible number of 1’s in a measurement. As $p \ll N$, the quantization error under the sparse measurement matrix is significantly lower compared with the imaging system using nonsparse matrices for a detector with the same number of bits. It should be emphasized that the sparsity cannot be arbitrarily reduced because under the premise of subsampling, a matrix sparsity that is too small will result in insufficient number of samples. In addition, the reduced detection SNR affects imaging quality.

2.3 Dither

The quantization error of a measurement is essentially determined by the relative levels of the dynamic range of the measured signal and that of detection. Thus, in addition to decreasing the dynamic range of the measured signal, as shown in the previous section, increasing the dynamic range of detection is an effective means to improve imaging quality. Adding the detector quantity is a direct method of increasing the total dynamic range of detection. However, simply using more detectors to measure the signals in CS imaging is invalid. According to Eq. (4), for a fixed input $\omega$, the quantization result $Q\left ( \omega \right )$ is also a constant; therefore, the fixed relationship results in the same quantization results for each detector.

Bennett [25] showed that the input–output function is smooth when the number of bits of the quantizer $B$ is large, meaning that the quantization step $\Delta$ is small. However, for fewer quantization bits, the input–output function becomes nonsmooth. It has been proven that adding an image with noise, which is called a dither, before quantization and subtracting it before reconstruction can break the fixed relationship between the input and output of the quantizer to allow smooth the function and decreasing the distortions caused by quantization [18]. The multiple dithering in traditional imaging is usually realized by multiple measurements. However, because CS imaging is based on point detection, spatial parallel dithering can be realized with a multi-point detector.

Herein, we add a random dither $\nu$ to the input $\omega$ before quantization, and the quantization input $y$ is expressed as

For a complementary measurement, the quantization input signals are ${\omega _1}$ and ${\omega _2}$. We stipulate that the dithers added to a pair of measurements are the same. In addition, to reduce the system complexity, the $t$ dithers between different pairs of measurements are identical. The final quantization result $Q\left ( Y \right )$ for the complementary measurements can be expressed as the difference between the quantization result $Q\left ( Y_1 \right )$ from ${\omega _1}$ with $t$ added dithers and $Q\left ( Y_1 \right )$ from ${\omega _2}$ with the same $t$ added dithers:

From Eq. (14), the quantization result $Q\left ( Y \right )$ is a random value with different ${\nu _i}$. The fixed relationship between the quantization input ${\omega }$ and quantization result $Q\left ( Y \right )$ is thus eliminated, so that the quantization input–output function can be smoothed, which means that the quantization error is reduced.

The effects of dithers in measurement quantization are proven via simulations as follows. Referring to the complementary measurements used in the CS imaging process, we use 2000 random values that obey a Gaussian distribution in the range of $\left [ {0,{L}=1} \right ]$ as ${y _1}$, and another 2000 random numbers with the same distribution are introduced as complementary data ${y _2}$. Let ${y _1}$ and ${y _2}$ pass through the quantization system with $t$ pixel dithers of a uniform distribution individually, so that we can calculate the root mean-squared errors (RMSEs) between the original complementary measured result $y={y_1}-{y_2}$ and that after quantization $Q\left ( y \right ) = Q\left ( {{y_1}} \right ) - Q\left ( {{y_2}} \right )$. The RMSE is defined as follows:

The curves for the relationships between the dithers and quantization RMSEs are shown in Fig. 2; Fig. 2(a) indicates the influence of the dithering quantity on the reconstruction errors, and Fig. 2(b) reveals the effects of the dithering intensity, which is the maximum dithering value. The dithering intensities used in Fig. 2(a) is $\Delta$, and the number of dithers in Fig. 2(b) is 16. It is worth noting that the zero points on the leftmost side of the two pictures indicate the situations without dithering. Figure 2(a) shows that the RMSEs decrease with increasing dithering quantities. Because the increase in the number of dithers means increasing the number of detectors and dithers break up the fixed relationships between the input and output, the dynamic range of the total measurement system increases, and the detectors have the ability to quantize the signal more finely. From Fig. 2(b), we observe that the requirements regarding the intensities of the dithers used for complementary measurements are not stringent. The optimal intensity for the dithers in complementary quantization is approximately $\Delta$ from the image. Dithers with extremely low intensities almost cannot decrease quantization distortions. When the intensity exceeds the optimal value, the quantization error increases gradually; however, the distortion is still better than not using dithers. The drawback of high-intensity dithers is not obvious for increasing the number of quantizer bits. For example, for a 6-bit quantizer, a $10\Delta$ intensity achieves a similar RMSE as dithering with an intensity of $\Delta$.

 

Fig. 2. Relationships between quantization RMSE and dithering for different numbers of detector bits (a) quantization results with different dithering quantities when the intensity of the dithers is $\Delta$; (b) quantization results with different dithering intensities while the dithering quantity is 16.

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The relationships between the dithering intensities and RMSEs can be further explained as follows. When dithers with low intensities that are far lower than the quantization step $\Delta$ are used, a large number of quantization results cannot be changed after dithering; thus, the ability of dithering to smooth the input–output function is limited. However, adding dithers with high intensities to the quantization system not only improves the dynamic range but also causes strong noise in the signal. However, in a complementary system, there are subtraction operations after quantization between the complementary signals ${y _1}$ and ${y _2}$. As the dithers for the complementary signals are the same, the actual dithering effect for the complementary result $y$ is no more than $\Delta$, even when dithers with high intensities are added.

2.4 Simulation

Based on the above conclusions, the reconstruction quality can be improved using sparse measurement matrices and by adding dithers with the same quantizations. The combination method of these two operations can further improve the dynamic range of compressive imaging, the flow chart of which is shown in Fig. 3. The imaging target is modulated by sparse complementary matrices, and the collected uniform facula is then superimposed with the $t$ pixel dithering image before measured by a multi-point detector in parallel. The proposed CS imaging method reduces the requirement for the dynamic range of the detector and improves imaging quality when the number of detector bits is limited, as proved by simulations herein.

 

Fig. 3. Flow chart of the combination method of sparse measurement and parallel dithers.

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In the simulation, the target is a natural image of a house sized $64 \times 64$, meaning $N=4096$, and the intensity of each target pixel is an integer in the range of 0 to 255 which means ${L_{o}}=255$. In addition, for the sparse measurements, the measurement matrix $A$ has forty $\pm 1$ values in each row, such that the sparsity of the matrix is ${k_s} = 40/4096 \approx 0.01$; contrarily, in classical CS imaging, the sparsity of the complementary measurement matrix is ${k_c} = 4096/4096 = 1$. Therefore, the dynamic range of the signal in the classical measurement is the maximum value ${L_{cs1}} = {k_c}N{L_o} = 1.04 \times {10^6}$ and that in the sparse measurement is ${L_{cs2}} = {k_s}N{L_o} = 1.02 \times {10^4}$. We selected $t=16$, which means that 16 dithers were added to each measurement. The image was reconstructed using the TVAL3 algorithm [26] with 2000 measurements.

The images recovered using the classical CS imaging method are shown in Fig. 4(a); the reconstructed images measured using the sparse random binary matrices are shown in Fig. 4(b); the images reconstructed with the classical measurement method and 16 dithers before quantization are displayed in Fig. 4(c); lastly, Fig. 4(d) shows the images recovered using sparse measurements by combining the method with 16 dithers. According to Fig. 4(a), in the traditional CS imaging method, if the number of quantizer bits is less than 4, then the reconstructed results are completely black. This is mainly attributed to the difference between the maximum and minimum measured results being less than the quantization step $\Delta$ for a small number of detector bits, and all the results are quantified to the same value according to Eq. (4). Consequently, the results of complementary measurements after quantization all become zero, such that there is no effective information in the reconstructed image. However, as shown in Fig. 4(b) and Fig. 4(c), both sparse measurements and dithering can decrease the number of detector bits required. Signals are observed with 2 bits, and the image can be roughly obtained with a 2- or 3-bit detector. From Fig. 4(d), for sparse measurements combining dithers, it is surprising that the imaging system can recover a distinguishable image even with only a binary 1-bit detector, and a clear reconstructed image can be acquired using a 3-bit detector, thereby greatly reducing the demand for the number of detector bits.

 

Fig. 4. Images recovered with different numbers of bits: (a) images reconstructed using the classical CS method; (b) images reconstructed using sparse random binary measurement matrices; (c) images reconstructed using 16 dithers before detection; (d) images reconstructed using sparse measurements combined with 16 dithers.

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The reconstruction errors are expressed in terms of the RMSEs between the reconstructed and original images, and the data of the RMSEs for Fig. 4 are presented in Table 1. The RMSE for the classical CS imaging method is 11.94% with a 6-bit detector, and the sparse matrices and dithering methods reduce this RMSE to 3.30% and 6.26%, respectively, for the same number of bits, whereas the RMSE for sparse measurements combined with dithering is 2.57%. In addition, the reconstruction error for the sparse measurements combined with dithers is close to convergence (2.57%) for a 6-bit quantizer; however, the classical CS imaging method requires a 9-bit quantizer to achieve the reconstruction error of 4.56%. We also evaluate the imaging quality with feature similarity (FSIM) [27], the data of which are shown in Table 2. The variety regularization is similar to RMSE. With the detector of same bit, the FSIM with sparse matrices or dithering is larger than classical CS imaging, and the combination of the two methods can make the FSIM be even larger. Both RMSE and FSIM data clearly show that the sparse measurement matrices and dithers in quantization can both decrease reconstruction errors, while the combination of these two methods can help further decrease the RMSEs.

Table 1. RMSEs of CS imaging with different imaging methods and detector bits.

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Table 2. FSIMs of CS imaging with different imaging methods and detector bits.

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The relationship between the number of detector bits and reconstruction error is depicted in Fig. 5. The red line with circles shows the relationship for traditional CS imaging, while the blue line with stars is for CS imaging using a sparse matrix combined with dithers. For RMSE=4.00% or FSIM=0.90, approximately 9-bit quantization is required in classical CS imaging, whereas the proposed sparse measurements combined with dithers requires only a 4-bit detector. This implies that the introduction of sparse matrices and dithers can reduce the demand for the number of quantization bits, so that the imaging system can recover distinguishable images even with a 1-bit detector and achieve high-quality imaging with limited numbers of quantization bits.

 

Fig. 5. Relationship between number of detector bits and reconstruction error. The red line with circles is for classical CS imaging, and the blue line with stars represents CS imaging using sparse matrices combined with dithers.

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We test the effect of the proposed method under different sampling rate, which is revealed in Fig. 6. The sampling rate of imaging system changes from 30% to 70%, with the detector bit of 6. It can be seen that the proposed method is effective at any sampling rate, as the sparse matrices and dithers can always increase the reconstruction quality whatever they are used individually or in combination. From another perspective, the proposed method reduces the demand of sampling rate for the same imaging quality. For example, to achieve RMSE of 5.00%, the classical CS imaging method requires a sampling rate greater than 70%, while sparse measurement or dithers methods only needs about 50% sampling, and the combination method further reduces the demand to approximately 30%. Since the detection time remains the same as classical CS imaging by parallel dithering method, the overall imaging speed is mainly determined by the sampling rate. Therefore, the proposed method can also be applied to improve the imaging speed on the precondition of the same imaging quality.

 

Fig. 6. Relationship between sampling rate and reconstruction error.

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3. Experimental results and discussion

3.1 Experiment and results

We built an imaging setup based on digital micromirror devices (DMDs) and a multi-point detector to confirm the effectiveness of the proposed method, as shown in Fig. 7. The light source1 is a halogen lamp that illuminates DMD1 directly. The DMD1 consists of $1024 \times 768$ micromirrors with each micromirror having dimensions of $13.68 \times 13.68\mu m$. Each micromirror in the DMD1 flips the angle separately based on the loaded measurement matrices to realize spatial modulation. A digital object is used as the imaging target, whose original image can be obtained by point scanning the DMD1 pixels. Thus, the actual modulations for DMD1 are the results of the dot product of the digital image and measurement matrices. The modulated light signal is then collected using a lens L1 and fiber collimator into the fiber, whose function is to produce a uniform facula. Then, the facula is transmitted to a charge-coupled device (CCD) used as the multi-point detector after passing through a beam splitter (BS). The dithers are provided by a light beam modulated with a grayscale DMD2. For the DMD2, $6 \times 6$ micromirrors are used as the superpixels, and the modulated random speckle forms dithers that obey a uniform distribution. The dithering signals are collected and imaged using lens L3 on the multi-point detector. For the 16 dithers, $4 \times 4$ fixed pixels are chosen in the detector, and the speckle on the $4 \times 4$ pixels are ensured to correspond to different superpixels of the DMD2. The final sampling result is obtained by summing the total light intensities of all 16 pixels, from which the target image is reconstructed by the CS algorithm in combination with the measurement matrices of DMD1.

 

Fig. 7. Experimental setup for the proposed CS imaging method.

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To set the appropriate intensities of the dithers, we first obtain the facula intensity of one pixel in the detector without the dither and calculate the quantization step $\Delta$ according to Eq. (5), wherein the dynamic range ${L_{cs}}$ is set to the maximum intensity value captured by the detector over all measurements. As the dither intensity in the complementary measurement is not strictly restricted, it can reduce quantization distortions even at higher intensities. Thus, we simply set the dithers in the range of 0–$\Delta$ with the quantization step for a 1-bit detector. Further, we use the same dithers for the detectors with different numbers of bits.

Without losing generality, in the experiment we use a 0-1 digital object as the target as it can be easily generated by a binary modulated DMD. The digital object used is a Tai Chi image with $64 \times 64$ pixels with 2000 measurements. For the sparse measurement matrix, there are forty $\pm 1$ in each row which means $k=40/4096\approx 0.01$. The measured results are quantified using different numbers of bits in the range of 1 to 6. The reconstructed images obtained using the classical CS imaging method are shown in Fig. 8(a), recovered images measured using sparse complementary matrices are shown in Fig. 8(b), images with traditional matrices combined with 16 dithers are shown in Fig. 8(c), and images with sparse measurements combined with 16 dithers are shown in Fig. 8(d). For objective evaluations, the RMSEs and FSIMs for the reconstructed images are displayed in Table 3 and Table 4, respectively.

 

Fig. 8. Experimental results with different imaging methods: (a) images reconstructed using the classical method; (b) images reconstructed using measured sparse random binary matrices; (c) images reconstructed using the classical method with 16 dithers before quantization; (d) images reconstructed using the combined method.

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Table 3. RMSEs of the imaging results in Fig. 8.

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Table 4. FSIMs of the imaging results in Fig. 8.

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From the experimental results, it is observed that identifiable reconstructed images with the classical CS method require at least a 3-bit detector while those for sparse measurements combined with dithers requires only a 1-bit detector. When measuring with a 3-bit detector, both sparse measurements and dithers can increase the imaging quality compared with the classical method, and the corresponding RMSEs are 13.28% and 8.77%, respectively; the image quality of the combined method is the best (4.85%) amongst all methods. Similarly, the FSIM for classical method with a 3-bit detector is 0.6000, and the introduction of sparse measurement and dithers can increase it to 0.6254 and 0.7801 respectively. Furthermore, the combination method has the largest FSIM of 0.8662. These results indicate that the sparse measurements combined with dithers not only help decrease the reconstruction errors for the same number of sensor bits but also reduce the detector requirements for recovering distinguishable images.

We note that there are some slight differences between the experimental and simulation results, which are mainly because of the system noise in the real experiment. In the experiment the sparse measurement matrix is not always beneficial for all detector bits. The reduction in signal intensity caused by the sparse modulation makes the imaging system sensitive to environmental noise and degrades performance than with the classical measurement matrix when using 5- or 6-bit detectors. Although the target images are different in the simulation and experiment, we can still find that the effect of the proposed method in improving the imaging quality is slightly affected by the system noise. Whereas the experimental noise also exists in the classical imaging method, it plays the role of dithering and makes the imaging quality even better than ideal simulation, which will be discussed in detail in the next subsection.

3.2 Dithering effect of detector noise

To confirm that the improvement in reducing the recovered distortions is mainly because of the effectiveness of the designed dithers rather than the multi-point configuration of the detector, we further compared the reconstruction errors between the different methods. We selected the same multi-point for imaging with classical and sparse matrices. The only difference between these experiments and that in the previous subsection is the absence of the dithers. The recovered images are shown in Fig. 9, and the corresponding reconstruction errors are listed in Table 5. Due to the consistent change rule of RMSE and FSIM, we only show the result of RMSE below.

 

Fig. 9. Experimental results for multi-point detection without dithers: (a) images reconstructed using classical random binary matrices; (b) images reconstructed using measured sparse random binary matrices.

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Table 5. Imaging results for multi-point detection without (the upper two rows) and with (the bottom two rows) dithers.

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Although the use of multiple pixels has a positive effect on decreasing the reconstruction errors, the impact is limited, especially with detectors with lower numbers of bits. Comparing Tables 3 and 5, for sparse measurements with 2-bit sensors, the RMSE for single-pixel detection is 16.75% and that for multi-point detection without dithers reduces to 12.36%. However, the introduction of dithers decreases the RMSE to 10.49%. The regularity is similar to that of the classical measurement matrix. For finer sensors with 5- or 6-bit resolutions, the introduction of dithers has little effect. In fact, owing to the existence of dark noise at the detector, the multi-point detection system itself can be seen as having random dithers with small intensities. Thus, for detectors with higher numbers of bits, the dark noise is close to the quantization step $\Delta$ and is sufficient to be seen as the required dithers; however, for fewer detector bits, the intensity of the dark noise is much smaller than the quantization step $\Delta$, and extra dithers are required for good reconstruction results. It should be noted that for detectors above 5-bit resolutions, the reconstruction error with extra added dithers (4.07%) is slightly higher than that using the dithers from the dark noise of the detector (3.52%). This is mainly because the intensity of the added dithers is designed according to the $\Delta$ of the 1-bit detector, which is larger than the quantization step of a 5- or 6-bit detector. Thus, the effect of decreasing the quantization error in the measurements is gradually weakened as shown in Fig. 2(b), and the imaging quality converges to a fixed level which is determined by the specific CS algorithm. On the contrast, the smaller system noise may continue to work at higher detector bits and makes the RMSE of classical method with multi-point detector converge to a lower level. It should be emphasized that if we use dithers with appropriate intensities according to the number of detector bits, the effectiveness of decreasing reconstruction errors will be more significant. Further, if a high-quality multi-point detector with very low noise is used, the introduction of extra dithering is necessary for improving imaging quality even with a large number of detector bits.

3.3 Parameter effects on imaging quality

We compare the reconstruction errors for various quantities of dithering from the RMSE data presented in Table 6. The results indicate that with increasing dithering amounts, the distinguishing ability of the detection system increases and reconstruction RMSEs decrease. For instance, for a 1-bit detector, 9 or 16 dithers added to one set of measurements cannot help reconstruct the image, whereas the general information of the object can be recovered if we increase this to 100 times. Moreover, the RMSE for 9 dithers is 15.40% when using a 2-bit detector, while that for 400 times decreased to only 3.88%.

Table 6. Reconstruction errors for different quantities of dithering.

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The effects of sparsity of the measurement matrix are discussed next. According to Eq. (11), the sparsity of the measurement matrices affect the measured dynamic range; thus, we test the reconstruction RMSEs for imaging systems with different sparsities of the measurement matrices, as shown in Table 7. The results show that for sensors with the same numbers of bits, as the sparsities of the measurement matrices decrease, the recovery errors decrease significantly. For example, the classical measurement matrix $A$ with the number of $\pm 1$ values being 4096 cannot recover an object from a 1-bit detector, while a matrix containing only 100 $\pm 1$ values can help reconstruct the object roughly with an RMSE of 20.72%. Nevertheless, with the increase in the number of sensor bits, the best imaging quality may not be obtained with the lowest matrix sparsity. This is mainly because of the low SNR in the sparsity measurements. For sensors with higher numbers of bits, the quantization step $\Delta$ decreases to a small value, which results in the imaging system being more sensitive to background noise. Thus, imaging with an extremely low sparsity matrix shows worse antinoise ability than that with a relatively higher sparsity matrix. In real applications, the choice of sparsity of the measurement matrix will be a tradeoff between the quantization error and SNR of the measurements.

Table 7. Reconstruction errors for different sparsities of measurement matrix.

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According to the conclusions above, the effect in improving the imaging dynamic range is depended on dithering intensity, dithering quantity and sparsity of measurement matrix. For the dithering, the optimal dithering intensity is quantization step $\Delta$, and the increase of dithering quantity is continuously conducive to the improvement of imaging quality. However, in practice the dithering quantity is limited by the pixel number of the detector, especially considering that CS imaging is usually applied in the case of insufficient detector resolution. For the sparse measurement, it has been proved the reconstruction errors are proportional to the sparsity of matrix from Eq. (11), whereas in real applications, an extremely sparse matrix will cause low detection SNR and pixels not sampled. Therefore, an appropriate low sparsity should be chosen by comprehensive considering these factors.

4. Conclusion

In this work, we realize high-quality CS imaging using a low number of detector bits with improved modulation and detection design. The high-flux measurements in CS imaging can cause serious distortions with limited detector bits than in traditional imaging, thereby decreasing the imaging quality. To reduce the quantization errors, a sparse measurement matrix design is introduced to reduce the dynamic range of the measured signal. Moreover, we propose a method of multiple random dithers before detection to eliminate the fixed relationship between the quantization input and output, thus decreasing quantization distortions and improving reconstruction quality. By combining the above two operations, we propose an improved imaging and quantization method, wherein the imaging object is first measured with sparse matrices and then supplied with multiple random dithers before quantization. The simulated and experimental results show that the proposed method improves reconstruction quality effectively, thus achieving preliminary imaging even with binary 1-bit detectors.

The proposed method can be used to recover valid information when the number of detector bits is limited, where the classical CS imaging method is inadequate. In practice, data acquisition with low-bit detectors can reduce the memory and transmission requirements and therefore have advantages when there are large numbers of image pixels. Moreover, low-bit detection is insensitive to small amounts of noise and increases the robustness of the imaging system. Although a multi-point detector is used instead of the single-point detector in traditional CS imaging, there is no imaging relationship between the detector and the modulated target. The system maintains the bucket detection advantage of traditional CS imaging, which is important in the imaging through turbulence or scattering media. Therefore, the proposed method is applicable to many fields such as aerospace imaging, remote sensing and high sensitivity fluorescence detection. We expect that these observations will further inspire the application of CS imaging.