Single-pixel Imaging （SPI）， which is different from traditional imaging， has its advantages. For example， in infrared and terahertz imaging， traditional array-based imaging sensors and systems are relatively bulky， complicated， and usually demand additional auxiliary modules which make them expensive. In contrast， SPI provides a better solution in which only a single point detector is needed， and the setup of the system is simple， which significantly reduces the costs. It is developed with the advent of the Compressed Sensing （CS） theory. In 2006， DONOHO D L ^［1］ and CANDÈS E J ^［2］ proposed the CS theory. In 2008， DUARTE M F ^［3］ realized the SPI based on CS， which suggested that CS ^［1-6］ had a broad prospect in the field of SPI. In addition to CS， Hadamard Transform （HT） ^［7-15］， Fast Fourier Transform （FFT） ^［14-21］， Wavelet transform ^{［15， 22］} and so on have been used in SPI to reconstruct the object， due to the low hardware requirements， short imaging time， and good imaging performance. Among them， HT has the merits of deterministic orthogonal basis patterns and can produce binary patterns that can be conveniently implemented by a Spatial Light Modulator （SLM） such as a digital Micro-Mirror Device （DMD）. Therefore， it is widely utilized in SPI.

However， with the increase of the imaging resolution， the measurements of SPI increase greatly， thus the imaging speed is limited. To solve this problem， current solutions are mainly to improve the sampling speed of the equipment or optimize the algorithm. At present， the generation of spatially structured light mainly depends on the SLM， but as its performance increases， its costs will increase. Therefore， XU Zihao et al.^［23］ proposed a computational ghost imaging scheme using LED-based high-speed lighting modules， which is a low-cost SPI scheme. However， it is only suitable for short-distance and low-resolution imaging， and its imaging resolution is limited. Reducing samplings through algorithm optimization is a cost-effective way to improve the imaging speed of SPI. The potential solution is to use the sparsity or compressibility of the signal to make a smaller number of measurements i.e. reducing the sampling rate of the SPI.

Here， based on the collected Hadamard spectrum， it is found that the distribution of the Hadamard spectrum has sparse characteristics， and the average value of the absolute value of the Hadamard coefficients in the oblique line shows a decreasing trend from the upper left to the lower right of the Hadamard spectrum. When it drops to a certain value， its implication on SPI is relatively weak. Taking this fact into consideration， an adaptive oblique zigzag sampling method is proposed and demonstrated to achieve a balance between the expected imaging fidelity and speed. It averages each oblique line of the absolute value of Hadamard coefficients along the oblique zigzag direction from the upper left corner to the lower right corner of the spectrum and compares the results with a threshold set by the user to automatically discard high-frequency signals and noises according to the expected imaging fidelity. Because only when the average values are larger than the set threshold， the Hadamard coefficients are collected， the sampling rate will be adaptively reduced to achieve a balance between imaging speed and the expected imaging fidelity.

Assuming that a digital image is represented as and its resolution is ， then its sparse representation in Hadamard domain can be expressed as

${\mathit{I}}_{\mathrm{H}}(u,v)=H\{\mathit{O}(x,y)\}=\frac{\mathrm{1}}{MN}{\displaystyle \sum _{x=\mathrm{0}}^{M-\mathrm{1}}}{\displaystyle \sum _{y=\mathrm{0}}^{N-\mathrm{1}}}\mathit{O}(x,y){(-\mathrm{1})}^{{\displaystyle \sum _{i=\mathrm{0}}^{N-\mathrm{1}}}\left[b_i(x)b_i(u)+b_i(y)b_i(v)\right]}$

inverse Hadamard transformation of the Hadamard coefficients matrix can be expressed as

$\mathit{O}(x,y)={\displaystyle \sum _{x=\mathrm{0}}^{M-\mathrm{1}}}{\displaystyle \sum _{y=\mathrm{0}}^{N-\mathrm{1}}}H\{\mathit{O}(x,y)\}{(-\mathrm{1})}^{{\displaystyle \sum _{i=\mathrm{0}}^{N-\mathrm{1}}}\left[b_i(x)b_i(u)+b_i(y)b_i(v)\right]}$

where are the coordinates in Hadamard domain. is the HT calculation. are the coordinates in spatial domain. is the kth bit of the binary representation of .

Based on the above-mentioned HT， the orthogonal Hadamard basis patterns can be generated using the inverse HT calculation， which can be set and expressed as

$\Theta \left(u,v\right)=\left\{\begin{array}{l}\mathrm{1}\hspace{1em}\hspace{1em}u=u_{\mathrm{0}},v=v_{\mathrm{0}}\\ \mathrm{0}\hspace{1em}\hspace{1em}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\right.$

The generated Hadamard basis patterns are normalized， and can be expressed as

${\varphi }_{\mathrm{H}}(x,y)=\frac{\mathrm{1}}{\mathrm{2}}[\mathrm{1}+H^{-\mathrm{1}}\{\Theta (u,v)\}]$

where is the inverse HT calculation， ， and each pattern corresponds to a unique 'Hadamard-frequency' component.

To further improve the imaging fidelity， we adopt a differential algorithm^［9］ to suppress noise and improve the system of signal-to-noise. The basis pattern generated by Eq.（4） is projected onto the object， and the reflected light signal is detected. Then， reversing the basic pattern ， the reflected light signal is detected. The absolute value of Hadamard coefficients matrix can be obtained by

$\mathit{S}(u,v)={\mathit{D}}_{+\mathrm{1}}\left(u,v\right)-{\mathit{D}}_{-\mathrm{1}}\left(u,v\right)$

therefore， the object can be reconstructed by calculating the inverse HT of it.

The proposed method makes full use of the fact that the low frequencies of Hadamard single-pixel imaging are concentrated on the upper left corner as shown in Fig. 1（b） and （c）. Fig. 1（a） is the target image to be restored. Fig. 1（b） is the sequential sampling. After carefully investigating the Hadamard spectrum in Fig. 1（c）， we found that the average value of the absolute value of the Hadamard coefficients for each oblique line has the characteristic that it decreases rapidly as shown in Fig. 1（d）， which inspired us to propose the adaptive oblique zigzag sampling method to achieve a balance between the imaging fidelity and speed.

Most information on a natural object normally concentrates on the low-frequency components， and the subtle details and noise concentrate on the high-frequency components ^［14］. Therefore， effectively collecting the low-frequency components in the upper left corner and ignoring a few high-frequency components can reduce the sampling rate， and improve the imaging speed of the single-pixel camera.

As shown in Fig. 1（c）， the low-frequency components are concentrated on the upper left corner of the Hadamard spectrum. We found that in the oblique zigzag direction from the upper left to the lower right of the spectrum， the average value of the absolute value of the Hadamard coefficients for each oblique line can be calculated as

$C_i=\frac{\sum _{u+v=i-\mathrm{1}}|\mathit{S}(u,v)|}{i}$

in which the oblique line ， is the absolute value of the Hadamard coefficients. The evolution curve is plotted in Fig. 1（d）， which shows that the rapidly decreases with the increase of the oblique line . Therefore， sampling is performed in an oblique zigzag direction from the upper left corner to the lower right corner of the spectrum. A threshold can be set to perform the adaptively sampling according to the desired imaging fidelity. Only when the average values are larger than the set threshold， the Hadamard coefficients are collected， the sampling rate will be adaptively reduced to achieve a balance between imaging speed and the expected imaging fidelity.

As shown in Fig. 2， light emitted from an ultra-high-pressure mercury lamp incidents on the DMD coded by the basis of Hadamard patterns， which projects the coded illuminations onto the object. The light reflected from the illuminated object is detected by a photodetector （PD KG-PR-200K-A-FS）， which converts the optical signal into an electrical signal. The electrical signal is collected by a digital acquisition card （NI USB-6216） to convert it to be a digital signal for computer processing. Finally， the image of the object can be reconstructed computationally.

We adopt the differential algorithm to improve the performance of Hadamard SPI and enhance the anti-interference ability to the system noise. Fig. 3（a） shows the object to be reconstructed. Fig. 3（b） is the image reconstructed without the differential algorithm. Due to the influence of the noise， the edge of the image is blurry. Fig. 3（c） is the reconstructed image after noise reduction by the difference algorithm， thus the contrast of it is better as compared to the reconstruction shown in Fig. 3（b）. Besides， the reconstructed image has a better fidelity， which shows that the differential algorithm has a strong anti-interference ability. In the experiments， we adopted the differential algorithm.

To verify the effectiveness of the proposed oblique zigzag adaptive sampling method， the same object as shown in Fig. 3（a） was reconstructed by using the method in which the threshold was set to be 0， 200， 300， and 400. The corresponding experimental results are given in Fig. 4， in which the horizontal coordinate is the oblique line， and the vertical coordinate calculated by Eq.（6）， is the average value of the absolute value of the Hadamard coefficients for each oblique line. Besides， there are two inserts， the left one is a zoom-in of the part indicated by the dashed ellipse and the right one is the reconstructed image of the object with a resolution of 128×128 pixels. As can be seen in Fig.4， the evolution curves decrease with the increase of the oblique line ， and there is a sharp decrease within which suggests that the low-frequency components are concentrated within a small area （） at the upper left corner of the Hadamard spectrum matrix as predicted and shown in Fig. 1（c）. It can be seen from the comparison of the evolution curves shown in Fig. 4（a）~（d） that， for the chosen threshold equals to 0， 200， 300 and 400， the sampling rates are respectively 100%， 46.17%， 36.92% and 22.04%， and the object can be successfully reconstructed， which suggests that， by setting different thresholds according to the expected imaging fidelity， the proposed method can adaptively sample along the oblique zigzag path in Hadamard spectrum domain. Therefore， without the need for CS algorithms， it can efficiently collect the low-frequency components in the upper left corner and ignore a few high-frequency components to adaptively reduce the sampling rate for improving the imaging speed of the single-pixel camera.

Fig. 5 gives the results reconstructed using the proposed method and the sequential sampling method with the same sampling rates which are respectively 100%， 46.17%， 36.92%，and 22.04%. SSIM and PSNR ^［14］ are used for assessing the reconstructed results in Fig. 5. The corresponding results are presented in Fig. 6（a） and （b）. The red triangle and blue square lines respectively correspond to the proposed method and the sequential sampling method. As can be observed in Fig. 5， as the threshold increases from 0， 200， 300 to 400， both PSNR and SSIM parameters show a downward trend. As predicted in Fig. 5（a）， when the threshold is 0， the corresponding sampling rate is 100%， and the quality of the reconstructed images is the best. The PSNR is close to positive infinity， and the SSIM is close to 1. As the threshold increases， PSNR and SSIM show a decreasing tendency. Under the same threshold， the red triangle line is always higher than the blue square line. Therefore， under the same sampling rate or threshold， the adaptive oblique zigzag sampling method is better than the sequential sampling method. Besides， the red triangle line decreases more slowly than the blue square line. As the threshold increases， the gap between the PSNR and SSIM parameters of the two sampling methods gradually increases. Therefore， by increasing the threshold （i.e. reducing the sampling rate）， the image quality reconstructed by the adaptive zigzag sampling method is better.

To further investigate the influence of the dark noise of the PD on SPI using the proposed method. We measured the dark noise signals of two different PDs as shown in Fig. 7（a）. The tortoise image with the resolution of 128×128 is reconstructed using the two PDs. The results are shown in Fig. 7（b）. The smaller the dark noise of the PD， the better the quality of the reconstructed image， which suggests that the much of dark noise of the detection system can not be suppressed using the differential algorithm ^［14］. In this situation， during the experiments， it can be predicted from Fig. 7 that as the dark noise of the detector increases， the imaging quality will gradually deteriorate. When a PD with high dark noise， such as PD1， is used， the high frequencies of the object to be reconstructed will be drowned in the high-frequency noise， which means SPI using the fully sequential or zigzag sampling will not give a satisfactory result， this can be seen in Fig. 8（a） in which more samplings give poorer results due to the high-frequency dark noise. In this case， SPI using the proposed adaptive zigzag sampling is necessary and can automatically get rid of the high-frequency noise by setting an appropriate threshold to get better reconstructions as shown in Fig. 8（b） and （c）， at the same time a balance between imaging quality and speed can be achieved. As shown in Fig. 8， the three restored pictures are marked with a red dashed frame in the same place. As the threshold increases， the dark noise gradually decreases.

A Hadamard SPI using the proposed adaptive oblique zigzag sampling method has been proposed and demonstrated. Hadamard transform is one of the orthogonal transforms， therefore， the proposed method not only can apply to other orthogonal transform based SPIs， but also inherits the advantages of orthogonal transform based SPIs.

As indicated in Ref.［21］， compared with the traditional compressed sensing method， orthogonal transform based SPIs’ advantages are as follows： First， it can reconstruct the object image without distortion. Second， it can also make better use of the sparsity of natural images in the transform domain to reconstruct high-quality object images by only obtaining larger coefficients in the transform domain， while the compressed sensing method reconstructs object images by dealing with coefficients corresponding to the random patterns without distinction. Third， it is only a simple two-dimensional inverse transformation， so the operation time can be ignored. By contrast， a compressed sensing algorithm takes more time to reconstruct the image.

Besides， our method has the following advantages： First， a threshold can be adaptively set according to the expected imaging fidelity. The method can achieve a balance between imaging speed and the expected imaging fidelity. Second， it can adaptively reduce the sampling rate， thus it can adaptively improve the imaging speed of the SPI significantly. Third， we can further investigate the influence of the dark noise of the PD on SPI. Experimental results show that the proposed adaptive zigzag sampling is capable of automatically reducing the influence of the high-frequency dark noise on the reconstruction quality by setting an appropriate threshold to get better reconstructions.

We proposed and demonstrated the Hadamard SPI using the proposed adaptive oblique zigzag sampling method. Experimental results show that it can adaptively reduce the sampling rate while meeting the fidelity required by users by setting a threshold. In other words， it can adaptively achieve a balance between imaging quality and speed. When the dark noise of the PD is high， the advantage of the proposed method is highlighted. Since when the dark noise of the imaging system is high and the high-frequencies of the object to be reconstructed are drowned in the noise， SPI using the fully sequential or zigzag sampling will not give a satisfactory result. However， SPI using the proposed method can automatically discard the high-frequency noise by setting an appropriate threshold to get a better reconstruction， at the same time a balance between imaging quality and speed can be achieved.