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Abstract
A transmission matrix (TM) is a powerful tool for light focusing and imaging through scattering media. For measuring it, the normal way requires establishing a multiple-step phase-shifting interferometer, which makes the TM measurement not only complex and sensitive but also time-consuming. Imaging through scattering media using an intensity TM method can make the setup for TM measurement without the phase-shifting interferometer, thus it is much simple, more stable, and several times faster. Here, based upon a differential intensity TM method, we demonstrated it to do imaging through scattering media using different Hadamard orderings. To accelerate the TM measuring speed while degrading as little as possible of the imaging quality, a relatively reasonable strategy to plan Hadamard orderings for the TM measurement is designed since it can suggest us to preferentially measure the components in TM that are more important to the imaging quality. Thanks to the different Hadamard orderings, their influences on the imaging quality at different measuring ratios are investigated, thus an optimal measuring ordering for accelerating the TM measurement can be obtained, while only sacrificing as little as possible of the image fidelity. Simulations and experiments verify the effectiveness of the proposed method.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Scattering media widely exists in nature, such as turbid atmospheres and biological tissues, which scatter the propagation of light, leading to the failure of the traditional optical imaging modality. In the past, a lot of efforts have been done to overcome or utilize light scattering, such as adaptive optics technology [1], optical coherence tomography [2], confocal microscopy [3], and multiphoton microscopy [4], which focus on utilizing ballistic light. The methods harnessing the scattering light are mainly divided into four categories: (1) Scattering imaging using optical memory effect (OME) [5–9] reconstructs the hidden object by a phase retrieval algorithm using the autocorrelation of the recorded speckle pattern. However, its field of view is limited by the OME; (2) Optical phase conjugation techniques [10,11] utilize the reversibility of light propagation to obtain the phase of the scattering light field through interferometry, and then reversely input the conjugate phase of the scattering light field. However, the optical system is sensitive and complicated; (3) Wavefront shaping methods based on feedback optimization [12–17] mainly use the optimization algorithm to obtain the optimal wavefront corresponding to the target field through iterations, which makes it time-consuming; (4) Optical transmission matrix (TM) method, which can establish the relationship between the input and output light field of a scattering medium. Once the TM of the scattering medium is known, the target information can be retrieved from the speckle pattern by using it, which is not limited by the range of OME. Many researchers have demonstrated the utilization of the measured TM for imaging [18], focusing [19,20], and multispectral imaging [21–23] through scattering media. Popoff et al. [18] first proposed a method for measuring the complex-valued TM of a scattering imaging system using a common-path interferometer based on an SLM, in which the SLM was divided into the signal region and reference region, and a four-step phase-shifting method was used to measure the TM and then focusing and imaging through a scattering medium can be realized. However, in their setup, part of the pixels on SLM was used for generating the reference light in the experiment, which limited the number of measurable modes and thus reduced the spatial bandwidth. To overcome this problem, a self-reference method was proposed [24], which superimposes the reference light on the signal light and is suitable for almost all types of bases. Besides, there are some interference-free schemes [25–29] for complex-valued TM measurement, which use iterative algorithms to retrieve phase information of the light field using intensity information. However, this type of method is easy to converge to a local optimum. Recently, methods for TM measurement using deep learning algorithms were proposed [30,31]. Only the intensity information of the output light field was used for the image retrieval, a large number of input and output pairs was needed to train a neural network to recover the image from the output speckle pattern, which needs a time-consuming training process and lacks physical explanations. In brief, TM measurement only needs one calibration in theory and is not limited by the OME. However, it normally requires establishing a multiple-step phase-shifting interferometer, which makes the TM measurement not only complex and sensitive but also time-consuming.
In this work, a method for imaging through scattering media using a differential intensity TM with four different Hadamard orderings is demonstrated. First, only intensity TM is measured which avoids the phase-shifting interferometer, thus not only saving the measuring time and the storage-costs several folds but also making the setup for TM measurement simple and stable. Further, the optimal ordering gives priority to measuring the important components in TM, which can accelerate the TM measurement speed for saving time consumption, while only sacrificing as little as possible of the image fidelity.
2. Principle of differential intensity TM
The transmission of light through the scattering media can be represented by the matrix T with a size of $m \times n$, and the relationship between the input vector and output vector can be expressed as:
The TM calibration process can be expressed as:
Therefore, 2n binary Hadamard patterns are used as the input bases. In Eq. (2), subtract the lower formula from the upper one, the left can be represented by ${\textbf I}_{\textrm{out}}^{ + } - {\textbf I}_{\textrm{out}}^ {-}$. Besides, due to the symmetry of the Hadamard matrix, and its rows or columns are orthogonal to each other, which satisfies ${{\textbf H}^{ - 1}}{\textbf = }{{\textbf H}^\textrm{T}}/n$, the differential intensity TM (DITM) can be calculated as:
Fig. 2. The schematic of an image reconstructed by DITM. (a) Binary Hadamard patterns and the recorded speckle patterns. (b) The object behind scattering media (SC). (c) Results of ${\textbf I}_{\textrm{out}}^\textrm{ + } - {\textbf I}_{\textrm{out}}^ {-}$. (d) The calibrated DITM. (e) Image of the object reconstructed from the speckle pattern using the DITM.
3. Orderings
The Hadamard basis was used as an input basis, the Natural order Hadamard matrix can be expressed in the form of the matrix:
To measure the TM of a scattering imaging system, we need to input all the Hadamard patterns sequentially and collect the corresponding outputs. Each input Hadamard pattern, that is a column in the Hadamard matrix, can be considered as a spatial sampling of the imaging system, which corresponds to a Hadamard spectrum in the transform domain. As shown in Fig. 3, the Hadamard transforms with different orderings, i.e. Natural, Walsh, Cake-cutting, and ‘Russian dolls’, which means the different spatial sampling orderings of the system can be conducted although the reordered Hadamard patterns are input in the same sequential way for measuring the TM, or in other words, this means the different reordered Hadamard spectrums can be collected in a row-by-row way. Therefore, to reconstruct the image of an object behind scattering media by the TMs measured using the different orderings Hadamard patterns, if the reordered Hadamard spectrums are sparse and the important spectrums (normally the low-frequency components) are concentrated in the first few rows, one can reconstruct the object’s image using the TM measured at a low measuring ratio. Therefore, the motivation, that the speed of TM measurement is accelerated while sacrificing a little bit of the imaging fidelity, can be achieved.
4. Numerical simulations
In simulations, four different orderings, i.e. Natural, Walsh, Cake-cutting, and ‘Russian dolls’, were investigated to verify the effectiveness of the proposed method. Two images ‘Smile face’ and ‘Bear’ with a resolution of $32 \times 32$ pixels are utilized as the objects behind scattering media. The recorded speckle pattern consists of $500 \times 500$ pixels. The correlation coefficient (CC) [36] was used to evaluate the quality of the reconstructed results.
Figures 4 and 5 show the reconstructed images of the objects at the measuring ratios (R) of 0.1, 0.2, 0.4, 0.6, 0.8, and 1. The reconstruction quality of the Cake-cutting ordering and ‘Russian dolls’ ordering is better than that of the Natural ordering and Walsh ordering, especially at the low measuring ratios. For the reconstructed images using the Natural ordering shown in Figs. 4(a1)–(a6) and Figs. 5(a1)–(a6), when the measuring ratio is less than or equal to 0.6, the surroundings of the reconstructed images exist an aliasing effect, resulting in multiple copies in the image. However, this phenomenon does not exist in the other three orderings, they can reconstruct the image with better fidelity. By comparison, the Walsh ordering performs a better image quality than the Natural ordering does, as shown in Figs. 4(b1)–(b6) and Figs. 5(b1)–(b6), which can eliminate the above-mentioned image artifacts effectively. However, when the measuring ratio is less than or equal to 0.2, its reconstructed images show vertical blur, as can be seen in Figs. 4(b5)–(b6) and Figs. 5(b5)–(b6). For the other two orderings, the reconstructed images using the Cake-cutting ordering are almost identical in the overall structure to that of using the ‘Russian dolls’ ordering, as shown in Figs. 4(c)–(d) and Figs. 5(c)–(d). Both of them can effectively eliminate image artifacts and blur, even when the measuring ratio is 0.2, the details of the reconstructed images can still be recognized, thus verifying their good imaging performance. However, when the measuring ratio is 0.1, the imaging performances for both of them are degraded, and the quality of the reconstructed images for the ‘Russian dolls’ ordering is a little bit higher than that of the Cake-cutting ordering.
To observe the imaging performance of the four different orderings more intuitively, the CC curves as a function of the measuring ratio are given in Fig. 6, and the measuring ratio is from 0.1 to 1. As can be seen from the curves, with the measuring ratio increasing, all CC curves show an upward trend. At a measuring ratio of 0.7, the CC values of the image reconstructed by DITM can reach above 0.9. The differences in the CC values for the four orderings gradually increase with the decrease of the measuring ratio. The CC values of the Natural ordering are much lower than those of the other three orderings. For example, when the measuring ratio is 0.1, both of the CC values of Natural ordering are about 40% lower than other orderings, which suggests the Natural order Hadamard matrix is not suitable for TM measurement at a low measuring ratio. By contrast, the other three orderings have better imaging performance than the Natural ordering does, because they avoid artifacts that exist in Natural ordering as shown in Fig. 4 & 5. Compared to the Walsh ordering, the Cake-cutting ordering and the ‘Russian dolls’ ordering have better reconstruction performance, even at a measuring ratio of 0.2, they can reach up to 0.9. At a very low measuring ratio of less than or equal to 0.2, the imaging performance of the ‘Russian dolls’ ordering is best, followed by the Cake-cutting ordering and Walsh ordering.
Fig. 6. Curves of the CC values for (a) the ‘Smile face’ image and (b) the ‘Bear’ image evolute with the measuring ratio.
5. Experimental setup
To verify the effectiveness of the proposed method, experiments are carried out. The experimental setup is shown in Fig. 7, consisting of a laser with a wavelength of 532nm, a modulation part to illuminate the scattering media with binary patterns, and a detection part to measure the diffused intensity. The laser beam is expanded by L1 and L2 and steered to fully illuminate the DMD (DLP4500, $912 \times 1140$ pixels), which is used to modulate a set of binary patterns. Then the modulated beam illuminates on ground glass via L3. The transmitted light is collected by L4 and imaged by the sCMOS camera (ZYLA-5.5HI-USB3-2Y1).
6. Results and discussion
In the experiments, two images ‘Smile face’ and ‘Bear’ with a resolution of $32 \times 32$ pixels are utilized as the objects, and the recorded speckle pattern is with a resolution of $500 \times 500$ pixels. The reconstructed images of the objects, obtained by the RVITM at the measuring ratios of 0.1, 0.2, 0.4, 0.6, 0.8, and 1, are shown in Fig. 8 and Fig. 9.
From the comparisons between Figs. 8(a1)–(a6) and Figs. 8(b1)–(b6), the Natural ordering and Walsh ordering can obtain high-quality images at a measuring ratio of 0.8. However, the Walsh ordering performs better than the Natural ordering at a low measuring ratio, because with the decrease of the measuring ratio, the aliasing effect appears in the reconstructed images using the Natural ordering. By contrast, the Walsh ordering improves the image quality effectively. However, the orientation of the Hadamard matrix leads to image blur at a low measuring ratio, which is obvious at a measuring ratio of 0.1. This can also be noted from the comparisons between Figs. 9(a1)–(a6) and Figs. 9(b1)–(b6). Comparing the reconstructed results of the Cake-cutting ordering with these of the ‘Russian dolls’ orderings, as can be seen in Figs. 8(c)–(d) and Figs. 9(c)–(d), at a high measuring ratio, which has similar imaging performance. However, the ‘Russian dolls’ performs better than Cake-cutting at a measuring ratio of less than or equal to 0.2. For example, in the ‘Smile face’ image, compared with the Cake-cutting ordering (Fig. 8(c6)), the details of the image is better reconstructed in the ‘Russian dolls’ ordering (Fig. 8(d6)), the same phenomenon can be also seen in the ‘Bear’ image.
Comparing the experimental results with the simulations, they present a similar trend on the whole. However, the imaging quality of experimental results drops a little bit. This is reasonable since there is inevitable noise, which leads to the measuring accuracy decline of the TM and therefore degrades the imaging fidelity. Besides, due to different noise levels during the experiments, the results of full measuring for all orders should be different as can be noticed in the first column in Figs. 8 and 9. Despite all this, the overall experimental phenomena, i.e. (1) When the measuring ratio is less than or equal to 0.6, the Walsh, Cake-cutting, and ‘Russian dolls’ orderings perform well, and the aliasing effect exists in the Natural ordering; (2) The reconstructed images using the Walsh ordering show vertical blur at a low measuring ratio; (3) The ‘Russian dolls’ ordering has the best imaging performance at a low measuring ratio. All of them agree well with the simulations in Figs. 6 and 7, which suggest the effectiveness of the proposed method.
In brief, the Natural order Hadamard matrix is not suitable for TM measurement when the measuring ratio is very low. By contrast, the Walsh, Cake-cutting, and ‘Russian dolls’ orderings are more suitable for TM measurement at a low measuring ratio because their performances are better. And among them, the performance of the ‘Russian dolls’ ordering is the best. For example, at a measurement ratio of 0.2, the CC of the reconstruction image of the ‘Russian dolls’ ordering can reach up to over 0.9. This means the TM measurement speed can be increased by 5 folds while sacrificing 10% of the imaging quality. The ‘ Russian dolls’ ordering is the best choice for TM measurement at a low measuring ratio, it can accelerate the speed of TM measurement while sacrificing a little bit of the imaging fidelity.
Figure 10 gives the Hadamard spectrums of the reconstructed ‘Smile face’ and ‘Bear’ with different sizes by the TMs measured using the Natural, Walsh, Cake-cutting, and ‘Russian dolls’ orderings. The distribution of the Hadamard spectrums for the Natural ordering is not concentrated. The distribution of the Hadamard spectrums for the Walsh ordering seems to concentrate on the top left as a triangular shape, which is not fit for the TM measurement using the sequential way which means the Hadamard spectrums are collected in a row-by-row way. However, if one further designs the TM measurement using an oblique zigzag way, its performance could be improved. The distributions of the Hadamard spectrums for the Cake-cutting ordering and ‘Russian dolls’ ordering are concentrated in the first few rows, which are both well fit for the TM measurement using the sequential way which means the Hadamard spectrums are collected in a row-by-row way. Thus, at low measuring ratios, their performance should be better than that of the Natural ordering and Walsh ordering. As further comparing the ‘Russian dolls’ ordering with the Cake-cutting ordering, the concentration of the Hadamard spectrum for the former is a little bit better than that for the latter one, which illustrates why the performance of the ‘Russian dolls’ ordering is the best at a low measuring ratio in Figs. 8 and 9. In brief, thanks to the Cake-cutting and ‘Russian dolls’ orderings, one can reconstruct the object’s image using the TM measured at low measuring ratios, in other words, the speed of TM measurement can be accelerated while sacrificing as little as possible of the imaging fidelity.
Fig. 10. Hadamard spectrums of the experimentally reconstructed ‘Smile face’ and ‘Bear’ images with a resolution of (a)$32 \times 32$ pixels and (b)$64 \times 64$ pixels.
By comparing the spectrums of the reconstructed ‘Smile face’ image or ‘Bear’ image in different sizes, one can notice that the distributions of the Hadamard spectrums for the images in different sizes are overall similar, which suggests the image sizes have limited influence on the proposed method when the noise level of the TM measurement system is very low. For the same ‘Smile face’ image or ‘Bear’ image in a larger size, their Hadamard spectrum distributions are relatively more sparse and concentrative in the simulations without considering the noise introduced by the practical TM measurement process. However, in practice, there exists inevitable noise during the TM measurement, thus the reconstruction quality of the results should degrade gradually as the sizes of the TM increase and the degradation degree depends on the noise level of the TM measurement system.
As compared with the previous TM measurement methods [18,19,24–31], our method has its advantages. First, only the intensity information is exploited making the setup for TM measurement more simple and stable as well as without the need for the interferometer that is necessary for obtaining the phase information. However, in the TM measurement methods [18,19,24], a multiple-step phase-shifting interferometer is indispensable, making the setup complex and sensitive. Second, since the proposed method is without the need for the multiple-step phase-shifting interferometer, the speed of TM measurement for it is several times faster than that of the interference-based methods [18,19,24].
To further improve the speed of TM measurement, the influence of different ordering on TM measurement and image reconstruction is investigated. The results show that a rational ordering i.e.’ Russian dolls’ ordering can suggest us to preferentially measure the components in TM that make great contributions to the imaging quality. At a very low measuring ratio of less than or equal to 0.2, the imaging performance of the ‘Russian dolls’ ordering is best, followed by the Cake-cutting ordering, Walsh ordering, and the Natural ordering is the worst. An optimal ordering can accelerate the speed of TM measurement several folds while sacrificing a little imaging fidelity. Such a strategy could also be beneficial to methods for complex-valued TM measurement, in which the Hadamard measurement basis is applied.
Despite all this, this method also has the shortcoming that only can obtain intensity TM and realize intensity imaging. However, it could be further improved for complex-valued TM measurement, such as it could be improved using the iteration process similar to the interference-free schemes [25–29], or it could be combined with deep learning like the deep learning-based methods do [30,31].
7. Conclusion
In conclusion, a method for imaging through scattering media using DITM with four different Hadamard orderings is demonstrated, in which only the intensity information is exploited, making the TM measurement without the need for establishing the multiple-step phase-shifting interferometer, thus it is much simple and more stable as well as the speed of TM measurement for it is several times faster than that of the interference-based methods. We further investigate the influence of different orderings on imaging quality at different measuring ratios. Simulations and experiments indicate that optimal ordering i.e. ‘ Russian dolls’ ordering is the best choice for TM measurement at a low measuring ratio, it can accelerate the speed of TM measurement 5 folds while only sacrificing 10% of the imaging fidelity. Such a strategy could also be beneficial to methods for complex-valued TM measurement.
Funding
National Natural Science Foundation of China (62275188); International Scientific and Technological Cooperative Project in Shanxi Province (202104041101009); Research Project Supported by Shanxi Scholarship Council of China (2021-035); Research Project Supported by Natural Science Foundation of Shanxi Province (202103021223091, 20210302123169).
Acknowledgment
We thank Shuoning Zheng and Jiayu Zeng for their help to do the experiments. We thank the Editors and Reviewers a lot for their efforts to help us improve the manuscript during this difficult time due to COVID-19.
Disclosures
The authors declare no conflicts of interest.
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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