^{ − 1}with increasing compression ratios using adaptive basis scan wavelet prediction strategy. In summary, our findings indicate that the extracted optical properties remained usable up to 99% of compression rate at a spatial frequency of 0.2 mm

^{ − 1}with errors of 5% in reduced scattering and 10% in absorption.

## 1.

## Introduction

Over the last decade, extensive progress has been made in diffuse optics toward developing quantitative, wide field, rapid, and inexpensive methods to visualize the physiological properties of living tissues. Among the numerous approaches proposed in this direction, the most recent ones rely on the spatial modulation of the light and processing in the frequency domain.^{1}^{–}^{3} Spatial frequency-domain imaging (SFDI) techniques allow the measurement of tissues absorption and reduced scattering optical properties over an entire image at once, which are used to retrieve physiological tissues parameters.^{4}^{–}^{10} However, major limitations of such imaging system include the necessity to project several patterns of light and to illuminate sequentially at several wavelengths to extract physiological parameters, impeding the use of such technology for real-time imaging.

To increase the speed of acquisition, some approaches have proposed to reduce the number of projected patterns.^{11}^{–}^{14} Methods, such as single-snapshot imaging of optical properties (SSOP), make use of a single pattern of light,^{11}^{,}^{12} and the combination of this method with temporal encoding of the wavelengths has been proposed to achieve rapid multispectral acquisitions.^{15}^{,}^{16} However, these approaches still make use of standard camera-based technology such as complementary metal–oxide–semiconductor (CMOS) or charge-coupled device (CCD) sensors that are generally expensive, monochromatic, limited in dynamic range, and importantly are limiting for multi- and hyperspectral imaging configurations.

In front of these limitations, compressive optics is a promising alternative to conventional imaging for diffuse optics. In particular, single-pixel cameras (SPCs) exhibit increased sensitivity and higher dynamic range due to efficient electronic designed for high-end photodiode or single-photon detector,^{17} as well as low-cost multispectral imaging capabilities.^{18}^{–}^{21}

In this paper, we explore numerically, using real SSOP data, the possibilities and limitations offered by compressive optics for real-time implementation using SSOP associated to an SPC (Fig. 1). In particular, we compressed existing SSOP data using compressive optics algorithms to evaluate the accuracy in extracting optical properties and the theoretical acquisition time required as a function of the compression rate of the SPC acquisition.

## 2.

## Materials and Methods

## 2.1.

### Single-Snapshot Imaging of Optical Properties

The standard SFDI method requires to acquire several images of the scene sequentially at two different spatial frequencies (e.g., ${f}_{x}=0$ and $0.2\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$) and three different phases (0 deg, 120 deg, and 240 deg) to obtain the modulation amplitude of each spatial frequency.^{3} In comparison, the SSOP method requires the acquisition of a single image and relies on filtering in the Fourier domain to extract the amplitude modulation at two spatial frequencies: ${f}_{x}=0\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$ and the projected spatial frequency (e.g., ${f}_{x}=0.2\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$). From these measurements, diffuse reflectances at these spatial frequencies are obtained by calibration using a phantom with known optical properties.^{3} The measured diffuse reflectances allow to extract the absorption as well as the reduced scattering properties from a precomputed look-up table.^{24}^{,}^{25}

## 2.2.

### Compressed Sensing

An SPC is an experimental setup that can measure the dot product of a scene and some user-defined pattern. See Ref. 26 for a review of this approach. The key idea is to acquire by hardware a compressed version of the scene by measuring

where $\mathbf{h}\in {\mathbb{R}}^{{N}^{2}}$ represents the image of the scene, $\mathbf{p}\in {\mathbb{R}}^{{N}^{2}}$ represents the pattern that is loaded on the spatial light modulator, and ${N}^{2}$ denotes the number of pixels of the spatial light modulator.The image $\mathbf{h}$ can be recovered by postprocessing a collection of measurements $\{{m}_{i}\}$, $1\le i\le I$, obtained for different patterns $\{{p}_{\mathrm{i}}\}$, where $I$ represents the number of patterns. Let $\mathbf{m}={[{m}_{1},\cdots ,{m}_{I}]}^{T}$ be the measurement vector and $\mathbf{P}={[{\mathbf{p}}_{1},\cdots ,{\mathbf{p}}_{I}]}^{T}$ be the measured patterns. Assuming that the measurement vector is a compressed version of the scene, we estimate the scene by the least-squares method:

for which fast implementations are available.^{27}In this paper, we choose the patterns as Daubechie wavelets with four vanishing moments, which is a common choice in image compression.

^{27}The wavelet patterns to acquire are determined iteratively during acquisition according to the adaptive basis scan wavelet prediction strategy described in Ref. 28.

The number of pattern $I$ is an acquisition parameter. Large numbers of patterns lead to high image quality but long acquisitions. We define the acquisition compression ratio (in %) as:

## 2.3.

### Data Processing and Results Analysis

To evaluate the performances of an SSOP-SPC system, we acquired a set of hand images at multiple spatial frequencies (${f}_{x}=0$, 0.1, 0.2, 0.3, and $0.4\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$) at a wavelength of 665 nm and three evenly distributed phases (SFDI acquisition).^{29} This range of spatial frequencies was chosen for investigating the trade-off between compression ratio and image quality, in the context of SSOP-SPC.^{30} Then, we compressed a single-phase image from the set of images acquired with SFDI to simulate the acquisition of a single SSOP image by an SPC, with increasing compression ratios of 0%, 50%, 90%, 98%, 99%, and 99.5% using the adaptive basic scan described above with $N=1024$. The resulting images are shown in Fig. 2. Each of the simulated SSOP-SPC images was then processed using the SSOP method to extract the absorption and reduced scattering maps.

The SSOP-SPC approach was evaluated by comparing the absorption and reduced scattering maps recovered from the original SFDI images and from the SSOP-SPC images. This comparison was done by measuring the mean percentage error in absorption and reduced scattering over a $400\times 350\text{\hspace{0.17em}\hspace{0.17em}}\text{pixels}$ region of interest (see Fig. 3) given by the following equation:

## Eq. (4)

$$\text{Error}\text{\hspace{0.17em}}\%=\frac{1}{N\times M}\sum _{n=0}^{N-1}\sum _{m=0}^{M-1}|100\times \frac{({\mu}_{n,m}^{\mathrm{SFDI}}-{\mu}_{n,m}^{\mathrm{SSOP}})}{{\mu}_{n,m}^{\mathrm{SSOP}}}|,$$## Eq. (5)

$$\text{Time\hspace{0.17em}}(\text{in}\text{\hspace{0.17em}}\mathrm{ms})=\frac{N\times M}{{f}_{\mathrm{DMD}}}\times (1-\frac{\mathrm{cr}}{100}),$$## 3.

## Results

The results of the absorption and reduced scattering coefficient recovered by the SFDI method and by the SSOP-SPC method at the different compression rates are shown in Figs. 3 and 4, respectively. These figures make it possible to visually assess the qualitative degradation of the images extracted based on different compression rates and the change in reduced scattering due to the more superficial measurement when the spatial frequency is increasing.

Results quantifying the mean and standard deviation percentage error in absorption and reduced scattering are presented in Tables 1 and 2, respectively. These tables show the average percent error based on the compression rate at each spatial frequency projected. Bold values are the results similar to the SFDI reference measurement. Overall, good results (i.e., without major difference in errors compared to the 0% compression rate results) can be obtained with 99.5% compression rate at $0.1\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$, 99% compression rate at $0.2\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$, and 90% compression rate at $0.3\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$. The method is not usable without significant inaccuracies for $0.4\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$.

## Table 1

Mean and standard deviation percentage error in absorption.

Compression rate | |||||||
---|---|---|---|---|---|---|---|

0% | 50% | 90% | 98% | 99% | 99.5% | ||

Spatial frequency | $0.1\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$ | $\mathbf{16.2}\pm \mathbf{12.9}$ | $\mathbf{16.2}\pm \mathbf{12.9}$ | $\mathbf{16.2}\pm \mathbf{12.9}$ | $\mathbf{16.2}\pm \mathbf{12.8}$ | $\mathbf{16.1}\pm \mathbf{12.8}$ | $\mathbf{15.8}\pm \mathbf{12.7}$ |

$0.2\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$ | $\mathbf{9.6}\pm \mathbf{7.5}$ | $\mathbf{9.6}\pm \mathbf{7.6}$ | $\mathbf{9.6}\pm \mathbf{7.6}$ | $\mathbf{9.7}\pm \mathbf{7.6}$ | $\mathbf{10.1}\pm \mathbf{8.1}$ | $\mathrm{34.1}\pm \mathrm{28.7}$ | |

$0.3\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$ | $\mathbf{7.3}\pm \mathbf{5.8}$ | $\mathbf{7.4}\pm \mathbf{6.1}$ | $\mathbf{7.8}\pm \mathbf{7.5}$ | $\mathrm{10.8}\pm \mathrm{11.6}$ | $\mathrm{16.8}\pm \mathrm{14.9}$ | $\mathrm{58.7}\pm \mathrm{30.6}$ | |

$0.4\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$ | $\mathbf{8.0}\pm \mathbf{6.4}$ | $\mathrm{9.6}\pm \mathrm{10.2}$ | $\mathrm{12.2}\pm \mathrm{13.5}$ | $\mathrm{36.9}\pm \mathrm{29.2}$ | $\mathrm{54.3}\pm \mathrm{32.7}$ | $\mathrm{70.3}\pm \mathrm{31.4}$ |

## Table 2

Mean and standard deviation percentage error in reduced scattering.

Compression rate | |||||||
---|---|---|---|---|---|---|---|

0% | 50% | 90% | 98% | 99% | 99.5% | ||

Spatial frequency | $0.1\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$ | $\mathbf{5.8}\pm \mathbf{4.3}$ | $\mathbf{5.8}\pm \mathbf{4.3}$ | $\mathbf{5.8}\pm \mathbf{4.3}$ | $\mathbf{5.7}\pm \mathbf{4.3}$ | $\mathbf{5.7}\pm \mathbf{4.3}$ | $\mathbf{6.0}\pm \mathbf{4.6}$ |

$0.2\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$ | $\mathbf{3.9}\pm \mathbf{3.2}$ | $\mathbf{4.0}\pm \mathbf{3.7}$ | $\mathbf{4.0}\pm \mathbf{3.7}$ | $\mathbf{4.1}\pm \mathbf{3.7}$ | $\mathbf{4.9}\pm \mathbf{5.3}$ | $\mathrm{33.1}\pm \mathrm{30.3}$ | |

$0.3\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$ | $\mathbf{4.2}\pm \mathbf{3.4}$ | $\mathbf{4.3}\pm \mathbf{4.0}$ | $\mathbf{4.8}\pm \mathbf{6.6}$ | $\mathrm{9.2}\pm \mathrm{11.5}$ | $\mathrm{16.7}\pm \mathrm{15.5}$ | $\mathrm{58.1}\pm \mathrm{31.2}$ | |

$0.4\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$ | $\mathbf{6.0}\pm \mathbf{4.8}$ | $\mathrm{8.2}\pm \mathrm{10.4}$ | $\mathrm{10.9}\pm \mathrm{14.3}$ | $\mathrm{38.4}\pm \mathrm{30.1}$ | $\mathrm{55.9}\pm \mathrm{33.1}$ | $\mathrm{71.7}\pm \mathrm{31.3}$ |

These tables also shows that best overall performances indicated by the lowest mean percentage error are obtained with a spatial frequency of $0.3\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$. Overall, the bold values show almost constant mean percentage error for one given spatial frequency that could be used to perform SSOP imaging with an SPC.

Results of minimum theoretical acquisition time are reported in Table 3 using a 20-kHz DMD with a single photodetector synchronized with the DMD refresh rate (i.e., acquisition time of $50\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{s}$). Logically, acquisition time decreases with compression rate and is not dependent on the spatial frequency of patterns projected on the scene. In summary, best overall performance in both acquisition time and accuracy, shown in bold fonts, are obtained at 99% compression rate with $0.2\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$ and a resulting acquisition time 520 ms.

## Table 3

Minimum theoretical acquisition time in second.

Compression rate | |||||||
---|---|---|---|---|---|---|---|

0% | 50% | 90% | 98% | 99% | 99.5% | ||

Spatial frequency | $0.1\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$ | 52.43 | 26.21 | 5.24 | 1.05 | 0.52 | 0.26 |

$0.2\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$ | 52.43 | 26.21 | 5.24 | 1.05 | 0.52 | 0.26 | |

$0.3\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$ | 52.43 | 26.21 | 5.24 | 1.05 | 0.52 | 0.26 | |

$0.4\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$ | 52.43 | 26.21 | 5.24 | 1.05 | 0.52 | 0.26 |

## 4.

## Discussion

In this study, we evaluated the potential of using an SPC in an SSOP system. To this end, images were acquired on an experimental SFDI system at different spatial frequencies and compressed at different compression rates according to the adaptive basis scan wavelet prediction strategy. These images were then processed using the SSOP processing method to extract their optical properties. Finally, these results were compared to the SFDI processing results on the original images.

In summary, this study shows that at spatial frequencies of 0.1 and $0.2\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$ optical properties can be extracted accurately with a 99% compression rate resulting in a 520-ms theoretical acquisition time. When adding the constraint for a good quality image, SSOP-SPC imaging can be performed optimally at $0.2\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$ with a compression rate of 99%. At higher spatial frequencies, while better precision in extracting optical properties and improved image quality can be obtained at low compression rates, degradations at higher compression rates significantly impacts the precision of the extracted optical properties, limiting its potential for fast imaging.

This study highlights the trade-off that exists between the choice of the spatial frequency used for acquisition and the capacity for the SSOP-SPC method to perform in real time. As evidenced through the results, the choice of the spatial frequency has naturally a direct consequence on the image quality when using SSOP. Decreasing spatial frequency in SSOP-SPC allows faster acquisition but results in image degradation.^{30}

This study reveals that more technological progress is necessary for using SSOP with an SPC in real time. Using the current technology (20-kHz DMD), 520-ms acquisition time is obtained at 99% compression rate when a scientific CMOS camera can reach up to 200 frames per second (i.e., 5-ms acquisition time), nearly a 100-fold difference. In addition, at a spatial frequency of $0.3\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$, where best precision is obtained, it would take 5.24 s of acquisition time at 90% of compression rate. It is clear that faster spatial light modulator technology is required to be able to use the SSOP-SPC method in real time.

Despite this limitation, it is important to remember a few points. First, an SPC is potentially more sensitive than the conventional CMOS or CCD cameras in the near-infrared, a wavelength band that is regularly used in diffuse optics. It will, therefore, not be necessary to increase the exposure time as much as with a conventional camera. Second, the main interest in using a single-pixel architecture is the potential for multi- or hyperspectral measurements. Current SFDI hyperspectral measurements typically require tunable filters, filter wheel, and sequential acquisition on a single camera or a hyperspectral camera.^{10}^{,}^{31}^{–}^{34} None of these technologies is either cost-effective or allows real-time measurement. A single-pixel architecture can easily scale to several wavelengths by replacing the photosensor in SPC with line array of photosensors.

## 5.

## Conclusion

In this paper, we explore the possibilities and limitations offered by compressive optics for real-time implementation using SSOP associated to an SPC. In particular, we evaluate the accuracy in extracting optical properties and the theoretical acquisition time required as a function of the compression rate of the SPC acquisition. The results show that usable optical properties with errors $<10\%$ in absorption and 5% in reduced scattering can be extracted at a spatial frequency of $0.2\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{mm}}^{-1}$ with a compression rate of 99% resulting in a 520-ms acquisition time. This work contributes to the exploration of the benefits of SPC in the development of diffuse optical imaging.

## Acknowledgments

Funding for this research was provided by the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Program under Grant Agreement No. 715737 (QuantSURG); France Life Imaging WP3, the French program “Investissement d’Avenir” run by the “Agence Nationale de la Recherche” under Grant Agreement No. ANR-11-INBS-006; the University of Strasbourg IdEx; and the ICube Laboratory.

## References

## Biography

**Foudil Dadouche** received his engineering degree from the University of Sciences and Technology of Oran, Algeria, and his electronic master’s and PhD degrees from the University of Pierre and Marie Curie (Paris VI). He joined the University of Strasbourg as an associate professor in 2010. He is member of the ICube Laboratory. His main research interests include imaging systems, digital electronics, and photovoltaics.

**Nicolas Ducros** received his MSc in electrical engineering and MSc in biomedical engineering, both from the University of Strasbourg, France, in 2006. He obtained his PhD in electrical engineering from the University of Lyon, in 2009. After a postdoctoral fellowship at Politecnico Milan, he became an associate professor at the University of Lyon with the Biomedical Imaging Laboratory CREATIS. His research interests include image processing and inverse problems.