## Abstract

A coupled deep learning approach for coded aperture design and single-pixel measurements classification is proposed. A whole neural network is trained to simultaneously optimize the binary sensing matrix of a single-pixel camera (SPC) and the parameters of a classification network, considering the constraints imposed by the compressive architecture. Then, new single-pixel measurements can be acquired and classified with the learned parameters. This method avoids the reconstruction process while maintaining classification reliability. In particular, two network architectures were proposed, one learns re-projected measurements to the image size, and the other extracts small features directly from the compressive measurements. They were simulated using two image data sets and a test-bed implementation. The first network beats in around 10% the accuracy reached by the state-of-the-art methods. A 2x increase in computing time is achieved with the second proposed net.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Compressed sensing (CS) has emerged as a sensing paradigm that instead of acquiring $N$ samples of a given signal $\mathbf {x} \in \mathbb {R}^{N}$, captures $M \ll N$ linear projected measurements ($\mathbf {y}=\mathbf {\Phi }\mathbf {x} \in \mathbb {R}^{M}$), resulting in hardware compression [1]. The single-pixel camera (SPC) has been lately exploited as part of research advances on the CS theory [2]. This low-cost camera acquires a multiplexed version of a scene with a single-pixel detector computing random linear measurements of the scene using a binary coded aperture [2]. This architecture is especially useful in applications where multiple-pixel sensors are expensive or infeasible, such as at shortwave-infrared and terahertz wavelengths [3]. SPC is an efficient sensing system at the expense of a slow computational recovery, that leads with the expensive task of finding a solution of an under-determined system [1]. To this end, CS provides theoretical guarantees for image recovery, assuming that the underlying image is sparse in some basis [4]. More recently, deep learning approaches have been proposed for CS recovery, addressing the speed and sparsity limitations [5–7]. However, high computation workloads come as a trade-off for the fast signal recovery from its compressed measurements under these approaches [6,8].

From a hardware point of view, recent works have focused on designing the binary coded aperture in order to improve the storage space, speed, and accuracy of the image reconstruction [3,9–11]. However, when the objective is an inference task such as segmentation, detection, or classification, the two-step process of reconstruction and task-solving is suboptimal in terms of efficiency; indeed, some compressed learning (CL) approaches have shown that it is possible to perform inference tasks directly in the compressive domain without the need to restore the scene [12,13]. Specifically in [12], theoretical and simulation results have shown that it is possible to learn features from the CS measurements, and these can be used in classifiers such as support vector machines [14], and sparse subspace clustering [15]. Also, in [16] a convolutional neural network (CNN) for compressed image classification using random sensing matrices was developed. In particular, the input of the CNN is a re-projected measurement vector $\mathbf {\hat {x}=\boldsymbol {\Phi }}^T\mathbf {y}$, where $\mathbf {\hat {x}}$ has the same size of the target image. More recently, [17] used a deep learning approach to simultaneously learn the linear projections and the non-linear classification net. Different from [16], in [17] the re-projection to the image size ($\boldsymbol {\Phi }^{T}$) is learned in the second fully connected layer of the network. Although [17] learned the linear projection operator, it cannot provide linear projections that fit with specific structures and properties of implementable CS systems. For instance, in optical imaging, elements such as spatial light modulators (SLM) or digital micro-mirror devices (DMD) are used in the acquisition as coded apertures, whose patterns are mathematically modeled by a binary sensing matrix with a specific structure determined by the optical configuration [18].

In contrast to state-of-the-art methods, this work proposes a coupled deep learning approach for the sensing matrix design, accounting for real/implementable SPC, and image classification directly from the compressed measurements. In the proposed approach, a neural network (NN) is trained to simultaneously learn the linear sensing matrix and the parameters of the non-linear classification network, considering the constraints imposed by the SPC. The first layer learns the binary sensing matrix, and the subsequent layers learn the classification parameters. After that, the optimized sensing matrix is used in a real SPC to acquire the projected data. These measurements are classified by using the already trained subsequent part of the network. In particular, for the classification task, two different NN architectures are proposed. The first, similar to [17], learns a fully connected layer to re-project the measurements to the image size so that any known CNN classifier can be applied. The second NN architecture proposes to extract small features directly from the compressive measurements without learning an image size re-projection operator. The performance of the proposed approach is demonstrated using MNIST and CIFAR-10 data sets, for which it provided better average classification accuracy for different sensing ratios compared to the works in [16] and [17]; and in some cases, it obtained similar results compared to non-implementable (non-binary) learned sensing matrices [17] for sensing ratios below $0.05$. Additionally, an optical setup was built to validate the classification results from SPC compressed measurements using the learned coded apertures.

## 2. Single-pixel camera model

The single-pixel camera (SPC) spatially encodes the full image before a single-pixel detector acquires projections of the image. The optical architecture consists of an objective lens, a coded aperture, a collimator lens, and a single-pixel sensor, as illustrated in Fig. 1. Precisely, the scene $f(x,y),$ where $(x,y)$ index the spatial dimensions of the scene, is modulated by a binary coded aperture $\phi ^{k}(x,y)$, for $k=1,\ldots ,K$ different number of snapshots. The coded aperture in Fig. 1, blocks-unblocks some pixels of the scene. Then, the encoded scene passes through the collimator lens, which concentrates the light into a single spatial point. This operation is expressed as

where a single-pixel detector with a pixel size $\Delta _g$ captures the incoming light intensity. In particular, the single discrete measurement is written as## 3. Coded aperture design and classification: a compressed learning framework

The proposed approach aims to design a binary sensing matrix $(\boldsymbol {\Phi })$ accounting for real and implementable SPC coding patterns such that the obtained measurements can be effectively used for classification employing a deep learning (DL) scheme. In particular, a coupled neural network approach which simultaneously learns the sensing matrix and the parameters of the classification network is summarized in the top of Fig. 2. Once the sensing matrix is optimized, new SPC measurements can be acquired, and the trained inference network, can be directly applied over the compressive measurements to obtain the classification results as shown at the bottom of Fig. 2.

#### 3.1 Training stage

The training step consists of two main blocks: the first one related to the binary sensing matrix optimization, and the second related to the non-linear classification operator $(\mathcal {M}_{\boldsymbol {\theta }})$ as summarized in the top of Fig. 2. Specifically, with a set of $L$ images $\{\mathbf {x}_{\ell }\}_{\ell = 1}^{L}$ and its respective labels $\{\mathbf {d}_\ell \}_{\ell = 1}^{L}$, the joint learning problem can be formulated as follows

### 3.1.1 Structure of the classification operator

The inference task considered in this paper is classification; however, any other inference task can be solved using the proposed approach. In particular, defining the structure of the inference operator as a neural network, the classification loss function is usually defined as

The second approach is a direct way to extract the features from the compressive measurements, without the need to return to the image size. This can be achieved with smaller fully connected layers followed by an element-wise non-linear function. This new approach has advantages because the re-projection to the image size implies a higher number of parameters to train which therefore, is prone to overfitting and requires more training time as it will be shown in the results section [25].

#### 3.2 Testing stage

The testing stage splits in hardware and software sub-stages. In terms of hardware, once the training stage is complete, each resulting row of $\boldsymbol {\Phi }$ represents a distribution of a coded aperture, where the number of rows represent the number of shots. Thus, the trained $\boldsymbol {\Phi }$ provides the physical pattern used to acquire new compressed measurements $\mathbf {g}$ employing the SPC through the digital micromirror devices (DMD) as explained in section 2. Additionally, in the software counterpart, the classification operator $(\mathcal {M}_{\boldsymbol {\theta }})$ and its learned parameters, are used as inference operator, which can be applied directly over the compressed measurements as

## 4. Simulations and results

This section evaluates the performance of the proposed coupled SPC binary sensing matrix design for compressive image classification. Specifically, the first approach, which preserves the image size when the classification stage starts, is denoted as Binary-Pres-Net. The second approach does not require to preserve the image size, and will be referred as Binary-NoP-Net. The proposed methods are compared with two state-of-the-art methods that perform classification over compressed measurements: Random + CNN [16] and End-to-End [17]. It should be noted that the sensing matrix used in Rand + CNN can be directly implemented as a coded aperture in the SPC architecture, since it can use binary sensing matrices, while the opposite happens with the End-to-End method, since it provides a real sensing matrix; however, End-to-End is considered in this paper for comparison purposes. The four methods were evaluated over two different data sets, whose images were divided into training and testing subsets; details of the size for each database are presented in the following subsection. The training data was used to simultaneously train the binary sensing matrix $\boldsymbol {\Phi }$ and the parameters of the classification network $\boldsymbol {\theta }$. After that, the designed sensing matrix was used to acquire compressed SPC measurements from the testing data set. These measurements were contaminated with Gaussian Noise of $30$ dB signal-to-noise ratio (SNR). Then, the resulting SPC measurements were used as input data in the trained network to classify them and to obtain the test results. It is worth mentioning that the noise is only applied in the testing step. All the methods were trained with the Adam algorithm [23], using a learning rate of $0.001$, over $100$ epochs. For the proposed method, the hyper-parameter $\mu$, which promotes binary weights, was fixed as $0.01$. These values were determined using a cross-validation strategy such that each simulation uses the value that results in the best classification accuracy. All simulations were implemented in Matlab 2018a on an Intel Xeon E5-2697 2.6GHz CPU with 192GB RAM, coupled with a Nvidia Quadro K6000 12GB GPU.

#### 4.1 MNIST data set

The MNIST data set (available at http://yann.lecun.com/exdb/mnist/) contains 60,000 hand written images of the numbers from $0$ to $9$, each with $28 \times 28$ pixels. All results for this database are the average of $5$ trial runs, where $50,000$ and $10,000$ images were randomly selected for the training and testing sets, respectively. For the MNIST data set, the Binary-Pres-Net is a modification or extension of the LeNet-5 model [26], i.e., once the second layer output is reshaped as an image, the LeNet-5 model is concatenated to the first two layers, as described in Section 3.1.1. Figure 3(a) summarizes the layer description of the Binary-Pres-Net. Similarly, End-to-End and Random+CNN employed the same network configuration after the re-projection layer. For this data set, the proposed Binary-NoP-Net uses the binary layer followed by two fully connected layers with a ReLU as a non-linear operator and a 10-class softmax classifer as it is summarized in Fig. 3(b). Notice the difference in the number of layers used for each net, this difference is remarkable in terms of computing time, as it will be reported in the results tables.

Table 1 presents a comparison of the classification accuracy results for the two proposed approaches (Binary-Pres-Net, Binary-NoP-Net), the End-to-End, and the Random+CNN methods, for different sensing ratios. Boldface indicates the best result for each case, and the second best result is underlined. It is worth noting that each sensing ratio entails a different sensing matrix, which requires weights training into the networks for each case. The evaluated sensing ratios include $\gamma = \{0.01, 0.05, 0.01, 0.25\}$ which are equivalent to $K=\{8,39,78,196\}$ SPC shots calculated as in Eq. (6), respectively.

From Table 1, It can be observed that the proposed methods obtain better results as the sensing ratio increases. Specifically, for more than $39$ shots, the Binary-Pres-Net performs similar to the End-To-End approach. Moreover, the performance decreases for very low sensing ratios, i.e., $1\%$ and $5\%$ of the data, but these results still outperform those from the Random+CNN, which is the most comparable method because of its implementation. Notice that the simple configuration chosen for the Binary-NoP-Net needs less number of parameters to train and shows that it is possible to extract features directly in the compressed domain, nevertheless, it obtains results comparable with those achieved by the Random + CNN. Additionally, Table 2 shows, the average training time (in seconds) for each epoch using a mini-batch size of 256. Notice that the Binary-NoP-Net needs less training time as expected, since it has fewer parameters compared with the other approaches.

#### 4.2 CIFAR-10 data set

The CIFAR-10 data set (available at https://www.cs.toronto.edu/~kriz/cifar.html) [27] consists of $60,000$ RGB images with $32\times 32$ pixels, where each image belongs to one of $10$ different classes. The data set is divided into $50,000$ training images and $10,000$ test images. As in the previous experiment, all the results are the average of $5$ trial runs. For the Binary-Pres-Net, End-to-End, and Random+CNN methods, the classification block is a variant of AlexNet [28] as it is shown in Fig. 4(a). Due to the complexity of the network and the database, the weights of the layers $3$ to $11$ were initialized independently of the first 2 layers with the weights learned from a training process, with $50$ epochs and with the same hyper-parameters used with the MNIST data set. For the Binary-NoP-Net, we extract features from the compressed measurements using a small fully connected layer and then we reshape the outputs of this layer into a 3D structure, then, a convolutional layer is employed as shown in Fig. 4(b). The same sensing ratios of the previous experiment were used for the CIFAR-10 data set. The obtained average classification accuracy results are summarized in Table 3. It can be observed that for this deeper network, the relative performance of the proposed Binary-Pres-Net method for lower ratios is better than for the previous experiment with the MNIST data set. Specifically, it provides the best test results for sensing ratios as low as $0.05$. Even though End-to-End provides the best accuracy for $0.01$, it yields to non-binary sensing matrices. Finally, for this data set, the training time for each epoch is summarized in Table 4. Note that Binary-NoP-Net needs less training time compared to the other methods and also has comparable accuracy as observed in Table 3.

## 5. Experimental setup

To evaluate the effectiveness of the designed sensing matrices and the proposed CCNs to classify single-pixel measurements, a SPC testbed was implemented to acquire real measurements. The experimental setup is shown in Fig. 5. It is composed of a 100-mn objective lens, a high-speed digital micro-mirror device (DMD), Texas Instruments, DLi4130 .7" VIS XGA, with a pixel size of $13,6 \mu m$ placed at the image plane; a 100-mm relay lens; a Thorlabs F220SMA-A as a condenser lens that projects the scene at a single point, where the incoming light passes through the fiber optic; and an Ocean Optics Flame S-VIS-NIR-ES spectrometer used as a detector.

For the experiments, four randomly selected images per digit from the MNIST testing data set were printed with a size of $25 \times 25 mm$ and used as a target in the test-bed implementation. Figure 6 shows the printed numbers, they were illuminated by a lamp for the visible spectrum as illustrated in Fig. 5. As the scenes are gray-scale images, and the proposed coded apertures were trained for a single band, the sum of the spectral bands captured by the spectrometer in the range of 470 and 620 mn was used as the SPC measurement for each shot to emulate a photodiode.

Similarly, as in section 4.1, the designed coded apertures were trained using the MNIST training data set, generating binary masks of $28\times 28$ pixels that were implemented into a DMD with pixel size of $13.5 \mu m$, and using a ratio of $1$ to $25$ as shown in Visualization 1, i.e., each binary pixel has a size of $340 \mu m$. To guarantee the quality of the measurements, a black scene was subtracted to each measurement shot. For the Binary-Pres-Net and the Binary-NoP-Net methods, different number of shots were evaluated, specifically 5, 10, 30, 50 and 100 shots, which are equivalent to a compression factor of 0.64, 1.27, 3.83, 6.38 and 12.76%, respectively. Figure 7 shows the distribution of the coded apertures for the set of 5 and 10 shots, obtained with the proposed design schemes. Additionally, the two proposed network configurations were evaluated using random coded apertures (which is the equivalent to the Random+CNN methodology), and they were denoted as Rand+CNN(Pres-Net) and Rand+CNN(NoP-Net).

Table 5 presents the overall accuracy obtained for the experiments. It can be observed that for low sensing ratios, the proposed method that preserves the image size achieves better results, and the behavior is similar for large sensing ratios in Table 5; nevertheless, for the $0.0383$ and $0.0638$ sensing ratios, the proposed methods have similar results. For both cases, the use of the designed coded apertures shows better performance compared with random coded apertures, and it can be appreciated that as more shots are acquired, the quality of the classification increases.

To analyze in more detail the results, the confusion matrix for the set of 50 shots (6.38%) with the two proposed schemes is presented in Fig. 8. These matrices show the behavior per class, where the rows and columns correspond to the predicted and correct class, respectively. The values and respective percentages in the diagonal correspond to the observations that were correctly classified, and the values in the off-diagonal correspond to incorrectly classified observations. Notice that both classifiers correctly matched in all cases for the numbers 0,2,4, and 6. Additionally, the last column of each table shows the precision achieved by the classification, and the false discovery rate. The precision for both classifiers is 100% for six of the ten digits. In the bottom row, the recall and false negative rates are shown, the recall metric is higher in most of the digit cases and for both classifiers schemes. Finally, the overall accuracy reported in the cell in the bottom right of the matrix is for both classifiers 82.5%.

## 6. Conclusions

Two coupled deep learning approaches to simultaneously learn the binary SPC sensing operator and extract non-linear features directly from SPC measurements have been proposed. To demonstrate the capabilities of the approaches, they were successfully applied in a classification task. After the training stage, the trained sensing matrix is employed to acquire the SPC measurements, and the trained classification network is used as an inference operator applied directly to these measurements. In particular, the effectiveness of the proposed approaches has been demonstrated for two well-known data sets: the MNIST and CIFAR-10. For lower sensing ratios, the proposed Binary-Pres-Net provides comparable results to the End-to-End method, however the latter is not able to provide binary sensing matrices. On the contrary, when the sensing ratios increases, the Binary-Pres-Net provides the higher classification accuracy. In terms of computing time, the proposed Binary-NoP-Net method, which is a simpler configuration, outperforms all the other compared methods.

## Appendix

Notice that the second term of (8), induces the sensing matrix to have $\{-1,1\}$ values; however, the DMD can only model $\{0,1\}$ values. An intuitive post-process to get negative values is to acquire two sets of complementary measurements and subtract them [3]. However, this means taking twice the number of shots. On the other hand, a more efficient process is to first acquire a set of measurements $g_0 = \mathbf {d}^T\mathbf {f}+ \omega _0$, where $\mathbf {d} \in \{1\}^{MN}$ represents a sensing matrix with all the elements in on, i.e, all the information of the scene passes in this measurement; then, the coded measurements obtained with $\{1,-1\}$ are calculated using these measurements as follows

## Acknowledgments

Universidad Industrial de Santander under VIE-project 2467 and Colciencias-grant No. 811-2018 titled: "Agricultura de precisión a través de la fusión de imágenes multiespectrales e hiperespectrales adquiridas bajo un sistema de muesttreo compresivo, empleando sensores de bajo costo para ser utilizado en un sistema de detección y clasificación de plagas y enfermedades en cítricos y análisis de los requerimientos mínimos para su aplicación en los procesos agrícolas colombianos”.

## Disclosures

The authors declare no conflicts of interest.

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