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Abstract
Focusing light through scattering media is essential for high-resolution optical imaging and deep penetration. Here, a two-step focusing method based on neural networks (NNs) and multi-pixel coding is proposed to achieve high-quality focusing with theoretical maximum enhancement. In the first step, a single-layer neural network (SLNN) is used to obtain the initial mask, which can be used to focus with a moderate enhancement. In the second step, we use multi-pixel coding to encode the initial mask. The coded masks and their corresponding speckle patterns are used to train another SLNN to get the final mask and achieve high-quality focusing. In this experiment, for a mask of 16 × 16 modulation units, in the case of using 8 pixels in a modulation unit, focus with the enhancement of 40.3 (only 0.44 less than the theoretical value) has been achieved with 3000 pictures (1000 pictures in the first step and 2000 pictures in the second step). Compared with the case of employing only the initial mask and the direct multi-pixel encoded mask, the enhancement is increased by 220% and 24%. The proposed method provides a new idea for improving the focusing effect through the scattering media using NNs.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Light passing through the inhomogeneous medium will be affected by scattering effects, which can disrupt the optical wavefront distribution [1]. With the progress of the spatial light modulator (SLM), the wavefront shaping technology based on SLM has become the mainstream and is widely used to achieve focusing through biological tissues [2–5], diffusers [6–8], optical fibers [9,10], and other substances. Currently, there are three common techniques for focusing light through scattering media. The first one is to employ the phase conjugation approach [4,11–14]. The outgoing light field of the reference point is recorded by interference. Due to the time inversion, the focus point can be obtained by the recorded light field. This technique has the advantage of short optimization time, but the process of recording light field requires high experimental accuracy. The second one is to use the transmission matrix method [7,15–19], which establishes the connection between the incident and the outgoing light fields. For obtaining the transmission matrix, one needs to measure the light phase, which also requires a technically more demanding interferometric approach. The third one is to utilize iterative wavefront optimization [6,20–29]. The stepwise sequential algorithm [6] and the continuous sequential algorithm [24] are used to modulate the phase or amplitude with SLM according to the feedback signal at the focus point. Therefore, the experimental setup using iterative wavefront optimization is relatively simple but requires much more time to get focus. With powerful computer capabilities, optimization algorithms such as genetic algorithm (GA) and particle swarm optimization (PSO) are widely used in iterative wavefront optimization to focus through the scattering media. GA and PSO have fast convergence speeds and high signal-to-noise ratio, but both of them are easy to fall into the local minimum [26–29]. Recently, machine learning has been widely studied in different areas [30–36]. Neural networks (NNs) as the classical method in machine learning have a stronger generalization ability. Hence, more and more scientists have used NNs to realize optical focusing through the scattering media.
Zuo et al. successfully realized focusing through the scattering media by using the BP network with a training time of 30 minutes [37]. Turpin et al. realized the fast focusing of light by using a single-layer neural network (SLNN) [38]. The focus point with the enhancement of around 17 is obtained by the SLNN trained by 9000 masks of 16 × 16 modulation units and their speckle patterns. Luo et al. combined DCNN with GA to propose GeneNN, which has a dataset of 10000 pairs of masks and speckle patterns [39]. Using GeneNN leads to higher enhancement and faster convergence speed than a single DCNN or GA. Liu et al. combined PSO with SLNN to optimize the convergence speed of PSO [40]. The enhancement obtained under the mask of 32 × 32 modulation units is 131.2, which is about 80% of the theoretical value. We could see from the previous reports, that the enhancements achieved by only using NNs are relatively low. Another step is always preferred to improve the enhancement. However, the reported second step is based on the technique of iterative wavefront optimization, to the best of our knowledge. Although the optimization algorithms are selected to get fast focusing, they still require re-optimization if another focus spot is needed.
In this article, we propose a two-step focusing method based on neural networks and multi-pixel coding to achieve high-quality focusing. The first step is to train the SLNN to predict the initial mask, and the second step is to train another SLNN after encoding the initial mask. The multi-pixel coding breaks the limit of binary modulation. The existence of the initial mask guides the training of the second step to achieve better focusing. Finally, we realize the focus by using fewer datasets and its enhancement is close to the theoretical maximum value. Our method depends only on NNs can be used to get focused in different positions and multiple spots once the NNs are trained. Also, we realized scanning in a straight line.
2. Principle of the two-step focusing method based on multi-pixel coding and neural network
Figure 1 shows the basic flow of the two-step focusing method based on multi-pixel coding and neural network. In the first step, a single-layer neural network (SLNN) is trained by the binary masks loaded on the DMD and their corresponding speckle patterns. The SLNN consists of only one fully connected layer followed by the binary activation function bounding the output to 0 and 1, which can fit the mapping relationship between the masks and their speckle patterns. In principle, the binary activation function can be represented as the step function with a threshold value of 0. The process of forming scattering spots through the scattering media can be expressed by Y = F(X), where Y is the light field of the scattering spot, and X is the light field in front of the scattering media. Therefore, the incoming light field can be derived by using the scattering spot, and this process can be described as X = F-1(Y). Loading the randomly generated binary masks of 16 × 16 on the DMD results in different incident light fields. Here 16 × 16 represents the number of modulation units in a mask. In the first step, we do not use multi-pixel coding. Therefore, each modulation unit only contains one pixel. We load the mask in the range of 256 × 256 on the DMD so that each modulation unit has 16 × 16 micromirrors. Since there is a one-to-one correspondence between the mask and the incident light field, the mask loaded on the DMD can be derived by speckle pattern. As shown in Fig. 1, we generate a dataset, which includes all randomly generated masks and their corresponding speckle patterns, for training SLNN. In our experiment, we take the speckle pattern of 100 × 100 as the input and the mask of 16 × 16 as the output. The loss between the predicted value and the true value is calculated through the mean-square error. Therefore, the neural network can fit the mapping relationship between the mask and the speckle pattern. After that, the desired focused light field is input into the trained neural network to predict the initial mask.
Fig. 1. Diagram of two-step focusing method. In the first step, we use a single-layer neural network (SLNN) to learn the mapping relationship between the speckle patterns (input) and the masks (output). The predicted mask is used as the initial mask to focus the center point. In the second step, we encode the initial mask with multi-pixel coding and train another SLNN. Finally, the predicted mask is used to focus the center point with high enhancement.
Because the modulation unit in the initial mask only has two states (0 or 1), which results in the coarsely discretized modulation. Hence, in the second step, as shown in Fig. 1, we encode the modulation unit by the multi-pixel coding method to mimic a continuous modulation of the light amplitude. That is to say, each modulation unit no longer contains only one pixel, but consists of n pixels. Therefore, a mask with 16 × 16 modulation units will contain 16 × (16 × n) pixels. As in the first step, we also load the mask in the range of 256 × 256 on the DMD, so each modulation unit still has 16 × 16 micromirrors. However, since a modulation unit is composed of n pixels, the number of micromirrors contained in each pixel changes. Each pixel will contain 16 × (16/n) micromirrors. The encoded value x of the modulation unit is between 0 and 1. Therefore, the grey level of the ith (1 ≤ i ≤ n) pixel is calculated by Eq. (1) [41]:
3. Experiment
3.1 Experimental setup
The experimental scheme is shown in Fig. 2. A laser beam with a wavelength of 978 nm is incident on the DMD. The DMD we used is VIALUX V-650 L, which has a resolution of 1280 × 800, a single micromirror size of 10.8 µm, and can achieve high-speed amplitude modulation at a refresh rate of 10 kHz. After DMD, two lenses (f1 = 60 mm, f2 = 80 mm) are used to form a 4-f system, while an aperture (AP1) is added between the two lenses to filter out the higher-order modes of the DMD diffraction. A lens (f3 = 25 mm) is used to focus the light beam and irradiate it on the scattering media S (Daheng Optics GCL-201101, 220 mesh). After the scattering media S, the scattering light is collected by using a micro objective lens (Obj, 50X, 0.75 NA), and the speckle pattern is recorded by a CCD through the lens group (f4 = 80 mm). The CCD is MER-132-43U3M-L of Daheng Optics, which has a resolution of 1292 × 964, a single pixel side length of 3.75 µm, and a frame rate of 43 fps. The micro objective lens is mounted on the XYZ platform for easy alignment. We store the randomly generated binary masks in the memory of DMD to modulate the light’s amplitude. Finally, the speckle patterns are saved in the computer by the frame capture of a CCD. The computer is a Windows system with Intel CPU i7-12700, 64GB DDR4 RAM, and NVIDIA RTX 3070ti GPU.
Fig. 2. Experimental setup (f1: the lens of 60 mm, AP1: the aperture, f2: the lens of 80 mm, f3: the lens of 25 mm, S: the scattering media, Obj: the micro objective lens, f4: the lens group of 80 mm).
3.2 Experimental results
In the first step, we randomly generate the binary masks. A single mask has 16 × 16 modulation units, where each modulation unit includes 16 × 16 micromirrors. The pattern of a resolution of 256 × 256 micromirrors is loaded on the DMD. The speckle patterns corresponding to different masks are recorded by a CCD and cut to 100 × 100 pixels. Due to the limitation of the CCD frame rate, too short a sampling interval will lead to a mismatch between the speed of collecting pictures by CCD and the speed of loading masks by DMD. Therefore, we set 250 ms to collect a picture. It takes us 20 minutes and 50 seconds to collect 5000 pictures. In order to test the stability of the system, we tested the correlation curve of the speckle pattern by using the first picture collected at the beginning as the benchmark to calculate the correlation coefficient between the subsequently collected pattern and it. The result is shown in Fig. 3. It can be seen that the stability of the system is good in the process of collecting 10000 pictures within 2500 s. We can better collect speckle patterns that are conducive to neural network training.
After training, the required light field is put into the neural network to obtain the mask and focus through the scattering media. The effect of focusing through the scattering media is usually described by the enhancement, which could be calculated by Eq. (2):
where Ifocus is the intensity of the generated foci and < Ispeckle > is the reference intensity, which can be expressed as the mean value of the background speckle. In order to explore the impact of the size (N) of the dataset which is used to train the neural network and the training epochs on the focusing effect, we train the neural network by 1000 to 5000 speckle patterns respectively and record the enhancement obtained under different training epochs. The results are shown in Fig. 4(a). It can be found that with the increase of the number of training epochs, the enhancement has not been greatly improved but will decline. The maximum enhancement we achieved here is about half lower than the theoretical value of 16 × 16 × (1/2π) ≈ 40.74. In order to find the reason, the loss value of the NN with different sizes of a dataset is shown in Fig. 4(b). It is clear that the loss value drops rapidly in the first 20 epochs and after that the drop rate decreases. After 40 epochs, with the increase of the training epochs, the loss of the neural network basically remains stable. At this point, the fitting of the corresponding relationship between the speckle patterns and the masks by the neural network is close to the limit. Therefore, with the increase of training epochs, the performance of the neural network will fluctuate, leading to the decline of the enhancement. Therefore, it is difficult to improve the focus quality simply by increasing the number of training epochs. From Fig. 4(a), we can also see that with the increase of the dataset size, the enhancement is not significantly improved. The enhancement achieved with the dataset size of 4000 is higher than 5000. As shown in Fig. 4(b), the loss value of the neural network is very close when 4000 and 5000 data sets are used. The enhancement tends to saturate beyond a certain dataset size. At the same time, as mentioned earlier, with the increase of training epochs, the performance of the neural network will begin to fluctuate. Therefore, the enhancement trained from 5000 datasets is close to that trained from 4000 datasets, and will fluctuate with the performance of the neural network, i.e., the enhancement trained from 5000 data sets training may be worse than it from 4000 datasets. Hence it is difficult for the trained model to improve greatly. Therefore, increasing the size of dataset as well as the training epoch can not bring a positive impact on the improvement of the enhancement. In order to take into account the time of recording speckle patterns and training as well as the focusing effect, we record the intensity distribution by the CCD in the case of 40 epochs of training with 1000 to 5000 masks and their speckle patterns. The results are shown in Fig. 4(c). The focusing effect is not greatly improved with the increase of the dataset. Meanwhile, we record the intensity distribution by the CCD in the case of training SLNN with 1000 masks and their speckle patterns after 20 to 100 epochs of training. The results are shown in Fig. 4(d). The focusing effect is also not greatly improved with the increase of the training epochs. Therefore, in the following experiments, we choose the initial mask predicted by the neural network after 40 epochs of training with 1000 masks and their speckle patterns. In this case, the training time of the neural network is 12 s, and the prediction time is 0.7 s. Finally, we get the focus at the central point with an enhancement factor of 12.59.Fig. 4. (a) Relations between the enhancements and the epoch of training, if the size N of the dataset are 1000, 2000, 3000, 4000, and 5000. (b) Relations between the loss of the test set and the epoch of training, if the size N of the dataset are 1000, 2000, 3000, 4000, and 5000. (c) The intensity distribution of the images captured by the CCD after 40 epochs of training, if the size N of the dataset are 1000, 2000, 3000, 4000, and 5000. (d) The intensity distribution of the images captured by the CCD after 20, 40, 60, 80, and 100 epochs of training, if the size N of the dataset is 1000.
In the second step, we encode the initial mask that can focus at the center point. The initial mask includes 16 × 16 modulation units. Each modulation unit is composed of one pixel. If the gray scale of the pixel included in the modulation unit is 255, we randomly select a number from 0.5 to 1 as the encoded value. The pixel arrangement corresponding to the encoded value is determined by Eq. (1). If the gray scale of the pixel included in the modulation unit is 0, we randomly select a number from 0 to 0.5 as the encoded value. As described above, each modulation unit is coded in turn. Since the random coding value is adopted for the same modulation unit in the initial mask, different binary matrices can be formed as new masks. In order to check the effect of coding, a single modulation unit of 16 × 16 micromirrors on the DMD with different pixel distributions is tested. We encode the modulation unit with multi-pixel coding and collect the speckle patterns corresponding to different encoding conditions with CCD. The results are shown in Fig. 5. We can see that as the coding pattern changes (i.e., the number of pixels with a gray scale of 255 decreases), the distribution of speckles remains unchanged but the light intensity gradually decreases. Therefore, the amplitude modulation can be manipulated more continuously by using multi-pixel coding. And the larger the n is, the more precisely the amplitude can be adjusted.
Fig. 5. Speckle patterns corresponding to different pixel distributions. Only one modulation unit is loaded in the range of 16 × 16 on the DMD, which is coded with different pixel distributions. The first row is pixel distributions (n = 4), and the second row is Speckle patterns.
We generate the encoded masks and load them on the DMD. Then we record speckle patterns corresponding to different masks with a CCD and train another SLNN. The image collection rate, the size of the input of the neural network, the activation function, and the loss function are the same as those in the first step. The size of the output of the neural network will change with the number n of pixels included in each modulation unit. In order to explore the impact of the size of the dataset on the focusing effect, the SLNN is trained by 1000 to 5000 encoded masks and their speckle patterns. After 40 epochs of training, the results are shown in Fig. 6(a). It can be found that the enhancement increased significantly before N = 2000, but stabilized after it. Therefore, using 2000 encoded masks and their speckle patterns for training can keep the dataset size as small as possible and have a good focusing effect. We use different numbers of pixels in a modulation unit to generate 2000 coded masks and their speckle patterns for training to explore the impact of the number of coded pixels on the focusing effect. The results after 40 epochs of training are shown in Fig. 6(b). When n = 2, 4, and 8, the enhancement increases. But when n = 16, the enhancement decreases. The fineness of modulation increases with the increase of n. The more the number of pixels encoded in a modulation unit, the finer the amplitude modulation of the input beam is. That is, the fineness of modulation represents the continuity of amplitude modulation. However, as shown in Fig. 6(c), with the increase of n, the minimum threshold value of the loss increases too. In the second step, the size of the output of the neural network is 16 × (16 × n). Therefore, it can be seen that a larger n increases the complexity of the training of the neural network. Therefore, if the advantage of high modulation fineness brought by the increase of n cannot compensate for the disadvantage brought by the increase of the complexity of the training, the enhancement will decrease. We also explored the guiding role of the initial mask for the second step focusing. As shown in Fig. 6(b), 2000 masks of different numbers n of pixels as a modulation unit are randomly generated for 40 epochs of training. It is found that the enhancement of the second step focusing increased at n = 4, 8, and 16 compared with that of the first step. As shown in Fig. 6(d), the minimum threshold of the loss after encoding the initial mask is smaller than that of the direct multi-pixel encoded mask. It indicates that the initial mask has a guiding role for the two-step focusing and the model can have a better prediction effect. When n = 2, the enhancement of the two-step focusing method is smaller than that of the one-step focusing method. In the case of an initial mask, the characteristics of the speckle pattern are not obvious with the decrease of n. Therefore, when n is small, the generalization of the neural network model is reduced and the enhancement of the two-step focusing method will be less than that of the one-step focusing method (focusing by training the SLNN with the direct multi-pixel encoded mask). As shown in Fig. 6(b), when n = 8, the enhancement of 40.3 is the highest and only 0.44 less than the theoretical value of 40.74.
Fig. 6. (a) Relations between the enhancement and the size N of the dataset, if the number of the encoded pixels n is 4. (b) Relations between the enhancement and the number n of the encoded pixels, if we use one-step focusing method (focusing by training the SLNN with the direct multi-pixel encoded mask) and two-step focusing method. (c) Relations between the loss of the test set and the epoch of training, if the number n of the encoded pixels are 2, 4, 8, and 16. (d) Relations between the loss of the test set and the epoch of training, if we use one-step focusing method and two-step focusing method.
Compared with the case of employing only the initial mask or the direct multi-pixel encoded mask, the enhancement is increased by 220% and 24%. In this case, it takes us 17 s for encoding, 27 s for training, and 1 s for prediction. If better hardware is used, training will be faster. The focused image is shown in Fig. 7(a).
Fig. 7. Intensity distribution of focusing in different positions captured by the CCD, if the size of the dataset N is 2000, the number n of the encoded pixels is 8 and the number of epochs is 40. (a) Middle. (b) Top right. (c) Bottom right. (d) Top left. (e) Bottom left.
In order to test the generalization characteristics of neural networks, after 40 epochs of training with N = 2000, n = 8 in the second step, we achieve focusing in different positions without extra training process. The results are shown in Fig. 7(b)-(e). The enhancement factors are 40.3 (Middle), 23.7 (Top right), 33.4 (Bottom right), 19.2 (Top left), and 25.9 (Bottom left). The reason why different enhancement is obtained is that we only code the mask for center point focusing in the second step. It can be considered that there is an emphasis on focusing on the middle spot, but focusing at other positions is also realized and relatively enhanced by using the generalization characteristics of neural networks. Therefore, the enhancement of the middle point focus is higher than that of other positions. We also carried out the experiment of simultaneous focusing of multiple spots, and the results are shown in Fig. 8. In Fig. 8(a), we achieved three spots focusing, and in Fig. 8(b), we achieved five spots focusing. In addition, we tested the scanning ability of the focus point generated by the two-step focusing method, which is shown in the supplementary part (see Visualization 1). The scanning rate is limited by the frame rate of the CCD, which is 43 Hz in the present setup. Definitely, we can achieve kHz scanning if a high frame rate CCD is used.
Fig. 8. Intensity distribution of focusing at multiple spots captured by the CCD. (The size of the dataset N is 2000, the number n of the encoded pixels is 8 and the number of epochs is 40 in the second step) (a) Three spots. (b) Five spots.
Our method can improve the enhancement in the case of using the mask with 16 × 16 modulation units. In order to explore the effect of this method on 32 × 32 modulation units, we modified the size of the output of SLNN. The size of the output is 32 × 32 in the first step and 32 × (32 × n) in the second step. We load the mask to 256 × 256 micromirrors on the DMD. The first row of Fig. 9(a) shows the focusing spot achieved from the first step of training in the case of N = 1000, and epoch = 40. Its enhancement is 17.96. The second row shows the focusing spot after the second step of training in the case of N = 2000, n = 8 and epoch = 40. Its enhancement is 49.69. The enhancement is increased by 180%. Then, we expanded the mask to 512 × 512 micromirrors, and the focused spots are shown in Fig. 9(b). The first row of Fig. 9(b) shows the focusing spot achieved from the first step of training in the case of N = 1000 and epoch = 40. Its enhancement is 14.11. The second row shows the focusing spot after the second step of training in the case of N = 2000, n = 8, and epoch = 40. Its enhancement is 83.09. The enhancement is increased by 488%. We could see from these figures that with increasing modulated beam size (i.e., the range of mask loading on DMD), the enhancement increases. But due to instruments limitation in the present experiment setup, the largest incident beam we have can only be incident in the range of 512 × 512 micromirrors on the DMD. We think if a larger incident beam size is achieved, enhancement close to the theoretical value with 32 × 32 or even larger modulation units should be expected.
Fig. 9. Intensity distribution of focusing at the center point captured by the CCD. (a) The mask of 32 × 32 is placed in the 256 × 256 area on the DMD. The first row is the focus obtained in the first step, if the size of the dataset N is 1000 and the number of epochs is 40. The second row is the focus obtained in the second step, if the size of the dataset N is 2000 and the number n of the encoded pixels is 8 and the number of epochs is 40. (b) The mask of 32 × 32 is placed in the 512 × 512 area on the DMD. The first row is the focus obtained in the first step, if the size of the dataset N is 1000 and the number of epochs is 40. The second row is the focus obtained in the second step, if the size of the dataset N is 2000 and the number n of the encoded pixels is 8 and the number of epochs is 40.
The reason why the second step can improve the focusing effect is that the second step of multi-pixel coding breaks the bottleneck that DMD can only perform binary amplitude modulation. Therefore, using the two-step focusing method in the spatial light modulator (SLM) also needs to break the bottleneck of SLM by coding in the second step. The change of complex amplitude of light passing through a scattering medium includes two parts: phase and amplitude. Both DMD or SLM modulates only one part. Therefore, our method can be extended to SLM.
4. Conclusion
We propose a two-step focusing method based on neural networks and multi-pixel coding. After fitting the mapping relationship between the mask loaded on the DMD and its speckle patterns by using a single-layer neural network, the mask required for focusing is predicted as the initial mask. Through exploration, it is found that increasing the number of training sheets could not always improve the focusing effect. We encode the initial mask, and then another neural network is trained again to establish the mapping relationship between the encoded masks and their corresponding speckle patterns in order to achieve high-quality focusing. For the mask of 16 × 16 modulation units, in the experiment, 1000 pictures are used in the first step and 2000 pictures are used in the second step, i.e., a total of 3000 pictures are set as the training set. In the case of n = 8, the enhancement can reach 40.3, which is only 0.44 less than the theoretical value. Also, the enhancement is 220% higher than that of focusing by the initial mask. We proved the guiding role of the initial mask for training. In the case of 8 pixels in a modulation unit, the enhancement of the two-step focusing method is 24% higher than that of focusing by the direct multi-pixel encoded mask. At the same time, we achieve focusing in different positions. The proposed method can provide a solution to enhance the focusing effect through a scattering medium to the theoretical value by using the neural network only, which will bring us a simpler experimental setup.
Funding
National Natural Science Foundation of China (61905168, 61975137).
Disclosures
The author declares 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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