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Abstract
A single-pixel camera (SPC) is a computational imaging system that obtains compressed signals of a target scene using a single-pixel detector. The compressed signals can be directly used for image classification, thereby bypassing image reconstruction, which is computationally intensive and requires a high measurement rate. Here, we extend this direct inference to phase object classification using single-pixel digital holography (SPDH). Our method obtains compressed measurements of target complex amplitudes using SPDH and trains a classifier using those measurements for phase object classification. Furthermore, we present a joint optimization of the sampling patterns used in SPDH and a classifier to improve classification accuracy. The proposed method successfully classified phase object images of handwritten digits from the MNIST database, which is challenging for SPCs that can only capture intensity images.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
A single-pixel camera (SPC) is a computational imaging system that obtains images using a single-pixel detector [1–3]. SPCs are particularly useful when image sensors are unfeasible owing to their cost, light sensitivity, or response timing. Their applications include terahertz imaging [4–6], short-wave infrared imaging [7–10], microscopy [11–13], imaging through scattering media [14–18], and three-dimensional sensing [19–21]. SPCs have a spatially multiplexed camera architecture that optically acquires linear projections of a target scene. Consequently, highly compressed signals of the target scene can be obtained, which enables image acquisition and reconstruction at sub-Nyquist rates.
Direct inference using measurements of SPCs, which eliminates the need for image reconstruction, has recently attracted significant attention. In several imaging applications, solving the inference tasks on images takes priority over high-quality image reconstruction. Moreover, direct inference from SPC measurements can bypass image reconstruction, which is computationally intensive and requires a high measurement rate. This type of inference in the compressed-signal domain has been investigated in the field of compressed sensing [22–28] and is sometimes called compressive classification [22], compressed learning [24–26], or compressive signal processing [23]. Recent studies have demonstrated image classification [29–35], face recognition [36–38], object tracking [39–41], and cell classification [42] using SPC measurements without image reconstruction.
However, most previous studies on SPC-based direct inference prioritized the optical intensity of a target object as the parameter for classification over its phase distribution. Phase distributions reflect the optical path length of a target scene, which is essential for analyzing optically transparent objects, such as biological cells or gases, and the three-dimensional structure of targets [43]. Several SPC-based phase imaging techniques have been proposed, such as those using Mach-Zehnder [44–48] or Michelson interferometers [49,50], common-path interferometers [13,51–54], coherent diffractive imaging with phase retrieval [55,56], the transport-of-intensity equation [57], wavefront sampling [58], or machine learning [59]. However, to the best of our knowledge, these techniques have rarely been used for compressive classification.
Here, we extend SPC-based direct inference to phase object classification using single-pixel digital holography (SPDH) [44–46,49,50]. Phase objects primarily induce phase shifts and minimally affect light intensity; therefore, their classification is difficult using SPCs that only capture intensity images. Therefore, we use SPDH, which is a complex-amplitude imaging technique that uses an SPC architecture and phase-shifting digital holography. Our method obtains compressed measurements of target complex amplitudes using SPDH and trains a classifier using those measurements for phase object classification. Furthermore, we present a joint optimization of the sampling patterns and classifier used in SPDH-based phase object classification to improve classification accuracy. We simulate and experimentally test our method on a phase object image dataset comprising images of handwritten digits from the MNIST database [60]. The results show that our method can classify phase objects using a simple neural network. Furthermore, they confirm that the proposed joint optimization improves the classification accuracy at low measurement rates and noise robustness.
2. Methods
2.1 Single-pixel digital holography
SPDH is a complex-amplitude imaging method that combines a single-pixel camera and digital holography. Figure 1 shows a schematic of an SPDH setup. The laser beam is divided into two beams. A spatial light modulator (SLM) modulates one beam in the object arm to form a set of structured sampling patterns on the object plane. The light transmitted through the object interferes with the other beam in the reference arm, thereby generating an interference pattern. The single-pixel detector measures the spatial sum of the interference pattern through a lens. Suppose that the object, sampling patterns, and reference beam are pixelated into $D$ pixels. Let $\boldsymbol {x} \in \mathbb {C}^{D}$, $e^{j\phi }\boldsymbol {h}_k \in \mathbb {C}^{D} \,(k=1,\ldots,K)$, and $\boldsymbol {r} \in \mathbb {C}^{D}$ be the complex amplitude of the object, $k$th sampling pattern with $\phi$ phase shift on the object plane, and complex amplitude of the reference beam at the same distance from the single-pixel detector as the object, respectively. The intensity measured with the $k$th sampling pattern is expressed as:
where $h_{kd}$, $x_d$, and $r_d$ are the $d$th elements of $\boldsymbol {h}_k$, $\boldsymbol {x}$, and $\boldsymbol {r}$, respectively. The vector form of the measurements $\boldsymbol {y}^{\phi } = [y^{\phi }_1, \ldots, y^{\phi }_K]^{\top }$ can be expressed as:
Fig. 1. Schematic of single-pixel digital holography. The laser beam is divided into two beams by a beam splitter (BS). A spatial light modulator (SLM) modulates the beam in the object arm and forms sampling patterns on the object plane. A single-pixel detector measures the spatial sum of the interference pattern between the object and reference beams.
2.2 Compressive phase object classification
We introduce phase object classification using compressive SPDH measurements to eliminate the need for image reconstruction. SPC measurements are a linear projection of a target scene, and we can directly use them for image classification or object detection without image reconstruction. We extend this direct inference to phase object classification using SPDH. SPDH can obtain compressed measurements of target complex amplitudes and is sensitive to the phase distribution. Thus, it can be used for phase object classification, whereas SPCs can only capture the intensity distribution of an object.
Figure 2 shows a schematic of compressive phase object classification using SPDH. Phase objects are measured using SPDH, and the measurements are directly input to the classifier, which outputs a class label. The classifier should be trained on data comprising SPDH measurements so that it outputs the correct labels. This training set can be acquired using an SPDH optical setup, or it may be synthesized via simulations using phase object images and the SPDH measurement model (Eq. 2). In this study, we trained the classifier $\mathcal {N}_W$ with learnable parameters $W$ as follows:
Fig. 2. Schematic of compressive phase object classification using SPDH. The classifier outputs a class label from the SPDH measurements. To train the classifier, we can use data acquired using the optical setup or synthesized via simulations using phase object images and the SPDH measurement model. Training with simulated data can jointly optimize the sampling patterns with the classifier. The optimized patterns can be used in optical-setup measurements.
To improve the classification accuracy, we propose a joint optimization of the sampling patterns used in SPDH and a classifier. The proposed method follows the same scheme proposed in recent studies on SPCs [32–34,62] and compressed sensing [26]. The SPDH measurement process is related to dimensionality reduction, and the complex measurements are a linear projection onto the subspace, the basis of which is a set of sampling patterns. Thus, optimizing the sampling pattern realizes efficient dimensionality reduction and improves the classification accuracy, even if the number of measurements is small. The proposed method trains the patterns $\boldsymbol {H}$ and classifier $\mathcal {N}_{W}$ using a phase object image dataset as follows:
3. Results
3.1 Simulation
We evaluated the simulated compressive phase object classification using SPDH. Our training and test sets were created from the MNIST handwritten digit dataset [60] and Fashion-MNIST dataset [64], both of which comprise 60,000 training images and 10,000 test images from 10 categories. We generated the phase object image datasets by converting images $\boldsymbol {g}_i \, (i=1, \ldots, N)$ from the datasets to phase object images $\boldsymbol {x}_i = \exp [j (\pi \boldsymbol {g}_i + \psi _i \boldsymbol {1})]$, where $\exp (\cdot )$ is the element-wise exponential function, and $\psi _i \in [0, 2\pi )$ is a global random phase shift that differs for each image. This phase object image has a unit amplitude; therefore, we cannot classify it using SPCs that can only capture the target intensity. We used three types of sampling patterns: Hadamard patterns, random patterns, and patterns optimized using joint optimization for evaluation. The resolution of the sampling patterns was set as 32 $\times$ 32 pixels. We used a simple classifier comprising two fully-connected (FC) layers with a rectified linear unit (ReLU) between them (Fig. 2). The first FC layer outputs 100 features, and the second outputs the score of each class. The class label with the highest score is then selected. Cross-entropy loss was used as the loss function, and the model was trained using adaptive moment estimation (Adam) with 50 epochs. The classification accuracy was evaluated for the number of patterns $K=16$, 64, and 256.
Figure 3 shows phase distributions of the Hadamard patterns, random patterns, and patterns optimized for the phase MNIST and Fashion-MNIST datasets with $K=16$. Although the optimized patterns appear random, they are more structured than random patterns.
Fig. 3. Phase distributions of Hadamard patterns (left), random patterns (center left), and patterns optimized for the phase MNIST (center right) and Fashion-MNIST (right) using joint optimization with $K=16$. The resolution of these patterns is 32 $\times$ 32 pixels.
Table 1 shows the classification accuracy based on the simulated data. We performed classification using images that were reconstructed from SPDH measurements with Hadamard patterns using the inverse Hadamard transform (IHT) to evaluate the effect of image reconstruction. The classification accuracies with and without image reconstruction are similar, and compressive classification can bypass image reconstruction without sacrificing classification accuracy. All the patterns successfully classified the phase object images with high accuracy, and the optimized pattern achieved the highest accuracy. Furthermore, the difference in the classification accuracy between the patterns was significant when the number of measurements was low.
Table 1. Classification accuracy of simulated compressive phase object classification using SPDH on the phase MNIST and Fashion-MNIST datasets. $K$ indicates the number of patterns. We evaluated the effect of image reconstruction via classification using images reconstructed from SPDH measurements with Hadamard patterns using the inverse Hadamard transform (IHT).
3.2 Experimental setup
Figure 4 shows the SPDH optical setup. The light source was a He-Ne laser with a wavelength of 632.8 nm. We formed sampling patterns using a DMD (digital micromirror device) module (V-7001, ViALUX, Germany) with 1,024 $\times$ 768 pixels, a 13.7 $\mathrm {\mu }$m pixel pitch, and a 22 kHz switching rate for the 1-bit binary patterns. The binary off-axis holograms were displayed using this setup, which is similar to that used in some previous studies [65,66]. The binary hologram encoding the slowly varying complex amplitude $A(x,y)\exp [j\phi (x,y)]$ was computed as follows:
Fig. 4. Experimental SPDH setup. The target object is a transmissive LC-SLM in phase modulation mode. The DMD displays binary amplitude holograms and forms complex amplitude patterns on the target plane. M1-2: Mirror, L1-4: Lens, BS1-2: Beam splitter, SF: Spatial filter, and PD: Photodetector.
We measured the phase target that displayed an MNIST image using the SPDH setup to demonstrate its phase imaging capability. For comparison, we measured the same phase target while blocking the reference beam. We used this configuration because it can only capture an intensity image, not a phase image. Thus, the configuration provides us with the requisite image for comparison. We refer to this configuration as an intensity-only SPC configuration. We used 16 $\times$ 16 Hadamard patterns ($K=256$) and reconstructed the images using IHT. Figure 5 shows the intensity image obtained with the intensity-only SPC configuration and the amplitude and phase images obtained with the SPDH setup. The LC-SLM primarily modulates the phase of the incoming light; therefore, identifying the image content in the amplitude and intensity images is difficult (Fig. 5 left and center). By contrast, the image of the digit 8 can be easily recognized as the phase image (Fig. 5 right). This result indicates that classifying target images in our setup without phase information is challenging.
Fig. 5. Reconstructed images of the phase target using the SPDH setup with 16 $\times$ 16 Hadamard patterns. From left to right: the intensity image obtained with the intensity-only SPC configuration and the amplitude and phase images obtained with SPDH.
3.3 Experimental results
We experimentally demonstrated compressive phase object classification using SPDH measurements. We used the SPDH optical setup shown in Fig. 4 and displayed the same MNIST phase images as those used in Section 3.2 on the LC-SLM to collect the real dataset. The same patterns (Hadamard, random, and optimized patterns) that were used for the simulation were also used for real data collection. The optimized patterns were trained using the simulated training set. To demonstrate the effectiveness of using phase information, we collected real data using the intensity-only SPC configuration with Hadamard patterns, as described in Section 3.2, which cannot capture the target phase information. The real dataset comprised 8,334 training images and 1,666 test images for each pattern. We trained the simulation classifier on the real training set and evaluated its classification accuracy on the real test set.
Table 2 shows the classification accuracy exhibited by the classifier when used on the real dataset. The LC-SLM marginally modulated the amplitude of the incoming light; therefore, the intensity-only SPC configuration could classify target images with low accuracy. However, the classification accuracy differed distinctly between the intensity-only SPC configuration and SPDH. Similar to the simulation, there was no significant difference in classification accuracy with and without image reconstruction via IHT, and classification using the optimized patterns achieved the highest accuracy. However, the accuracy of the random patterns was lower than that of the Hadamard patterns, which can be attributed to the inability of our setup to accurately replay random patterns. In addition, the classification accuracy using the real dataset was lower than that using the simulated dataset, which can be attributed to the severe corruption of the real dataset due to noise.
Table 2. Classification accuracy of compressive phase object classification experiment using SPDH on the phase MNIST dataset. $K$ is the number of patterns. Intensity-only refers to the intensity-only SPC configuration with Hadamard patterns. We evaluated the effect of image reconstruction via classification using images reconstructed from SPDH measurements with Hadamard patterns through the inverse Hadamard transform (IHT).
Figure 6 shows several phase images of class 8 reconstructed from SPDH measurements using the 16 $\times$ 16 Hadamard patterns. The background phase distributions on the phase images fluctuate, owing to the instability of our setup, which degrades the classification accuracy in the case of the real dataset. This degradation can be partially addressed by increasing the size of the real training set. In conclusion, the results indicate that the proposed joint optimization is effective, even when measurements are acquired in a severely noisy environment.
Fig. 6. Several phase images of class 8 reconstructed from SPDH measurements using the 16 $\times$ 16 Hadamard patterns. The phase images are severely noisy because the background phase distributions fluctuate owing to the instability of the experimental setup.
4. Conclusion
We demonstrated phase object classification using compressed measurements acquired using SPDH. This method enables phase object classification while bypassing image reconstruction, which requires high computational costs and a high measurement rate. The simulation and experimental results showed that we could successfully classify phase object images constructed from the MNIST dataset using SPDH measurements. Furthermore, we proposed a joint optimization of sampling patterns and a classifier to improve the classification accuracy. The results indicate that the proposed optimization is particularly effective when the number of measurements is small and the measurement environment is noisy. Phase distribution provides several advantages over light intensity, particularly for transparent objects, such as biological cells or gases. Therefore, our proposed compressive classification provides a new analysis technique for life sciences and remote sensing.
Although this study proves that phase object images can be successfully classified using compressed SPDH measurements, the current form of the proposed method is limited. The classification accuracy of the experiment was lower than that of the simulation owing to the instability of our experimental setup. Therefore, further investigation is required to develop a stable SPDH setup and to improve the classification accuracy on real datasets, which will be the focus of our future work.
Funding
Japan Society for the Promotion of Science (JP22K17908); The Mazda Foundation (19KK-271); Kayamori Foundation of Informational Science Advancement (K31-XXIV-543); Hokuriku Bank Research Grant for Young Scientists.
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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