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Abstract
In holographic data storage systems, the quality of the reconstructed data pattern is decisive and directly affects the system performance. However, noise from the optical component, electronic component and recording material deteriorates reconstruction quality. A high noise margin decoding method developed from compressed sensing technology was proposed to reduce the impact of noise in the decoding process. Compared with the conventional threshold decoding method, the proposed method is more robust to noise and more suitable for multilevel modulation. The decoding performance with five-level amplitude modulation was evaluated by both simulation and experimentation. For the combination of Gaussian noise, Rician noise and Rayleigh noise, the proposed decoding method reduces the BER of the threshold method to one-sixth with an SNR of -1 in the simulation. In the experiment, it behaves up to 8.3 times better than conventional threshold decoding.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With the development of information technology and mobile internet industry technology, the informationization pace of modern society has become faster than ever. A fair amount of data is generated by enterprises, organizations and individuals every second. The International Data Corporation predicts that the global data amount will grow from 33 Zettabytes in 2018 to 175 Zettabytes by 2025 [1]. As the amount of data increases, the demand for data storage devices with larger capacities and faster transfer rates has increased.
Page arranged data is used to store information by volumetric recording in holographic data storage (HDS) technology. Therefore, a high storage capacity and fast data transfer rate can be realized [2–7]. Depending on the number of modulation levels of each pixel, HDS is divided into binary modulation [8,9] and multilevel modulation [10,11]. Recent studies have focused on multilevel recording methods to further increase recording capacity and data transfer rate [12–14]. Although HDS technology can achieve high performance in theory, in practice, it is always deteriorated by noise from optical systems, materials and electrical components, especially for multilevel modulation. Therefore, research on improving the signal-to-noise ratio (SNR) of HDS systems is widely conducted [15,16]. However, despite the improvement in the SNR, noise still exists and is non-negligible.
Since removing the noise completely is almost impossible and HDS requires as low an error rate as possible, the task is focused on the encoding and decoding sections. It is obvious that data error will be reduced if the encoding or decoding method has a high noise margin. Some encoding methods have been proposed, such as applying error correcting code [17–19], but few decoding methods are available. Katano et al. proposed a demodulation method of multilevel HDS data with a convolutional neural network [20]. Their work shows the potential of decoding in HDS.
In 2004, an information processing technology named compressed sensing (CS) was proposed by Candès, Romberg, Tao and Donoho [21,22]. The sparsity of the signal is utilized to retrieve the original signal with a lower sampling rate than the Nyquist-Shannon sampling theorem requires. After being proposed, CS showed great potential in improving information retrieval quality [23,24], reducing the measurement requirement [25] and information denoising [26–28]. The data page used in HDS is a kind of special format image information; hence, it can also be handled with CS technology. In this manuscript, a high noise margin decoding method based on CS for multilevel modulation is proposed. The method is realized through improved K-SVD sparse dictionary training and orthogonal matching pursuit (OMP) retrieval algorithms. Simulation and experimental performances of the proposed decoding method are both evaluated. A comparison between the proposed decoding method and the conventional hard threshold decoding method is also performed.
2. Principle of collinear holographic data storage
2.1 Recording and reading
In a collinear holographic data storage system [4], user data is encoded and modulated into a two-dimensional information pattern called a data page. The data page is stored in the recording material by volumetric holography. A typical collinear holographic data storage system is shown in Fig. 1. In the recording process, both the data page and reference pattern are displayed on a spatial light modulator (SLM) to generate an information beam and reference beam. The information beam and reference beam are focused on holographic material, and the interference fringes are recorded holographically. In the reading process, only the reference pattern is displayed on the SLM to generate the reference beam. Irradiating the same spot of the recording material, the reference beam interacts with the interference fringe, and the information beam is reconstructed. The reconstructed information pattern is captured by a camera and decoded to the user data by an information processor.
2.2 Encoding and multilevel modulation of data page
The binary modulation method is widely used in generating data pages in holographic data storage systems. The 8:16 two-dimensional constant-weight sparse modulation code is popular in binary modulation due to its balance between the code rate and raw bit error rate (BER) [29]. However, the 8:16 modulation code has a crucial flaw, i.e., a low code rate of 0.5. In order to develop this code principle, the modulation level of each pixel is improved from binary to five-level. The 15-bit information is represented in 16 pixels by applying the new 15:16 modulation code principle. The code rate of the 15:16 modulation code is 0.9375, which is 1.875 times that of the 8:16 modulation code. In each 16-pixel data symbol, data information is represented by three bright pixels with four different intensities, and the others are regarded as dark pixels. The total number of data symbol varieties is
Of these symbols, a total of 32768 (215) symbols are selected to encode the data page with five-level modulation. In addition, the sync mark is set to the brightest intensity level, and this intensity is used to normalize the whole data page in the decoding process. Some examples of the 15:16 modulation code are shown in Fig. 2.Hard threshold decisions are usually used as decoding methods in HDS systems. For the binary modulation system, the three brightest pixels are regarded as modulated pixels, and the others are regarded as dark pixels. However, for a multilevel modulation system, a modulated pixel must be classified to the corresponding level. The level of the three modulated pixels is decided by using threshold values between each adjacent modulation level. However, the threshold decoding method does not work well with multilevel modulation, and misdecoding of the adjacent modulation level occurs frequently due to the reduced level interval. For multilevel modulation, there should be a more suitable decoding method.
2.3 Noise type in holographic data storage system
Noise exists in every process in the HDS system and deteriorates the reconstructed data page directly. The main components of the HDS system, i.e., the SLM, optical element, material and camera, are all noise sources. However, their noise types are different. Noise from electrical elements such as thermal noise have been modeled as Gaussian noise. Noise from optical elements such as crosstalk noise and scattering noise have been modeled as Rician statistics [30,31]. The amplitude distribution of speckle noise caused by the uneven surface of the material has been modeled as Rayleigh statistical noise [32,33]. These three noise types are considered in this manuscript. They are also used in the evaluation of the proposed decoding method.
3. Principle of compressed sensing
Compressed sensing goes against conventional wisdom in data acquisition, showing that signals can be recovered from fewer samples or measurements than the Nyquist-Shannon sampling theorem requires. Compressed sensing relies on two principles: sparsity, which relates to the signal, and incoherence, which relates to the measurement. In the following, a simple introduction of compressed sensing theory with a one-dimensional signal as an example is given; a more detailed explanation and theoretical derivation can be found in proof papers [21,22] and some review papers [34].
The general signal acquisition process can be expressed by Eq. (2). Here, the one-dimensional N length signal x is used as an example. The measurement of signal x is represented by the $M \times N$ measure matrix $\Phi $. Thus, the $M$-length observed signal y is obtained.
In traditional information process technology, if the measurement of signal x is insufficient, in other words, M is less than N, then Eq. (2) becomes an underdetermined system with infinite solutions. The original signal x cannot be retrieved successfully from the observed signal y. However, CS technology can realize retrieval of the original signal x with high likelihood when M is less than N as long as it meets the two principles mentioned earlier. The first principle is the sparsity requirement: the signal x is compressible. A sparse domain of x exists, and the transformation from the original signal domain to the sparse domain is expressed by $N \times N$ sparse trans-matrix $\Psi $. The sparse expression of the original signal is the $N$-length sparse vector $\alpha $. Their relationship can be expressed as follows: Combining Eqs. (2) and (3), the relationship between y and $\alpha $ is as shown in Eq. (4). The second principle is the incoherence requirement. The measurement matrix $\Phi $ and sparse transformation matrix $\Psi $ must meet the restricted isometry property (RIP) shown in Eq. (5). ${|{} |_{{l_2}}}$ represents the ${l_2}$ norm.Meeting the two principles of CS, the original signal can be retrieved with high likelihood by solving the least squares problem in ${l_1}$ regularization or using an approximate estimation algorithm such as OMP.
4. High noise margin decoding method in holographic storage
4.1 Noise depression of compressed sensing
It is precisely because of the strict requirements of the sparsity that CS offers the ability of denoising. The great majority of noise is randomly distributed in an arbitrary domain and thus incompressible. In other words, the noise does not meet the sparsity principle; hence, the noise cannot truly be retrieved with CS.
The noise depression ability of CS can be explained by Fig. 3. The compressible signal is polluted by noise.
The sparse domain of compressible signal x and the corresponding measurement are applied to polluted signal $({x + z} )$. As mentioned in the Section 3, only the compressible signal can truly be retrieved with CS in the case of undersampling. After the CS process, the retrieved signal $x^{\prime}$ is approximately equal to the original signal. Meanwhile, the retrieved noise $z^{\prime}$ is not approximately equal to the original noise owing to the fact that z is dissatisfied with the sparsity principle of CS. Moreover, the power of $z^{\prime}$ is much lower than z due to undersampling. Therefore, the noise is depressed after the CS process compared with the signal. The feature of noise depression is widely applied to signal and image denoising [26–28]. In this manuscript, this noise depression feature is applied in the decoding process of an HDS system to enhance the anti-noise ability of decoding and achieve a low decoding error.
4.2 Decoding method
The data page in the HDS system can be regarded as a kind of special image. The CS noise depression feature of images is applied to the reconstructed data page in the decoding process for better performance. As mentioned before, the noise depression ability of CS relies on sparsity. Hence, the signal retrieval quality and the noise depression level are determined by the choice of sparse domain. Fortunately, compared with the natural image full of details, the data page used in HDS is designed artificially following particular regulations with fewer details. Therefore, a more targeted and suitable sparse domain of the data page can be found more easily. To find a universal sparse domain of the data page, the basic component unit of the data page, i.e., a symbol, is fully considered. The sparse dictionary training method named K-SVD [36] is applied to these symbols to find the specific domain, also called the dictionary here, as shown in Fig. 4.
In dictionary training, a fixed number of coefficients of sparse atoms are used for the representation of each symbol. The OMP algorithm used in training is also modified more specifically: the judgment principle in OMP is changed to the cosine similarity from the Euclidean distance, which focuses more on the location distribution and relative intensity of data points in a symbol. A flow chart of the sparse dictionary training is shown in Fig. 5.
After finishing the sparse dictionary training, the reconstructed data page of the HDS in the reading process can be decoded through CS technology with the trained sparse dictionary and OMP retrieval algorithm. By searching the most likely sparse atom of the dictionary and the corresponding large sparse coefficient, every symbol block of the reconstructed data page is decoded correctly to data information with high probability. The decoding process can be simply represented by Fig. 6. In the next section, the decoding performance is evaluated in both the simulation and experiment.
5. Simulation and experiment
5.1 Simulation
The simulation model is based on collinear transmissive holographic data storage with a 4 f optical system. The calculation is according to coupled wave theory [37] and in k vector form [38].
Considering the reduction in the complexity of the codebook and calculation amount in this study, 256 randomly generated symbols are used, and 8-bit information is represented by one symbol. The data page used in both the simulation and experiment consists of 384 symbols that represent 3072 bits of information in total. As mentioned in Section 2.2, for complete five-level modulation, there should be 35840 kinds of symbols, and 15-bit information is represented by one symbol. The schematic diagram of the simulation and the simulation pattern are shown in Fig. 7, and the parameters are listed in Table 1.
Fig. 7. (a) Simulation model with 4 f optical system; (b) simulation pattern, where the inner pattern is a 5-level data pattern and the radiation is the reference pattern.
Table 1. Simulation parameters
The simulated reconstructed data page without any extra noise and its gray statistical histogram are shown in Figs. 8(a) and 8(b). The reconstructed data page is normalized by using the sync mark intensity as the maximum value. Different Gaussian-type noise, Rayleigh-type noise, Rician-type noise and combined noise are artificially added to the reconstructed data page to evaluate the anti-noise ability of the proposed decoding method. Different types of noise and their combined noise are shown as examples in Figs. 8(c)–8(k). The noise level is controlled by changing the standard deviation of the generated noise distribution. To match the experimental situation, the ratio of the combination of Gaussian noise to Rician noise to Rayleigh noise is 1:1.2:1. This ratio is obtained by statistical analysis.
Fig. 8. Simulation results. Reconstructed data pattern: (a) without extra noise and its statistical histogram (b); (c) with Gaussian noise from camera and its statistical histogram (g); (d) with Rayleigh-type noise and its statistical histogram (h); (e) with Rician-type noise and its statistical histogram (j); and (f) with combined noise and its statistical histogram (k).
The proposed decoding method is applied to a simulation reconstruction data page with different noise types and levels. The BER curve of the simulation results for different types of noise is shown in Fig. 9.
Fig. 9. BER curves of the proposed decoding method and conventional threshold decoding for different single noise types.
The minimum value of the SNR of the adjacent modulation level is used to represent the noise level; the lower the SNR is, the higher the noise level. The SNR is calculated according to Eq. (6), where A and B represent adjacent levels.
Fig. 10. BER curves of the proposed decoding method and conventional threshold decoding for combined noise.
5.2 Experiment
To verify the anti-noise ability of the proposed decoding method in a real optical system, a proof of concept experiment is also carried out. The experimental setup is built according to an amplitude-modulated collinear holographic data storage system with a transmissive-type medium.
Four data pages generated from different random information are used for the evaluation. The symbols of the data page follow the same generating rule in the simulation. The laser source employed in the experiment is a Nichia tunable laser NUV603E with a 405-nm wavelength. The spatial light modulator (SLM) is X10468-05 LCOS-SLM, produced by Hamamatsu, and its pixel pitch is 20 $\mu m$. A DVC-1500M CCD with a 6.45-$\mu m$ pixel pitch is chosen as the camera of the reconstructed data page. The holographic data storage medium is provided by Kyoeisha Corporation, a nano-gel photopolymer with a thickness of 0.4 mm. The experimental setup is shown in Fig. 11.
Fig. 11. Experimental setup. HWP: half wave plate, PBS: polarization beam splitter. The focus lengths of lenses ${L_1}$, ${L_2}$, ${L_3}$, and ${L_4}$ are 450 mm, 200 mm, 200 mm, and 300 mm, respectively. The two objective lenses are the same, with 20X and 0.4 NA.
The noise level of the experimental results is evaluated by the same SNR defined in Eq. (6). The reconstructed data page in the experiment is shown in Fig. 12. Its SNR is calculated as -0.3168. For this reconstructed data page, the BER of the proposed decoding method is 4.92%, and this BER can reach zero with the assistance of code correction. More reconstructed results and their corresponding decoding BERs are shown in Fig. 13. The conventional threshold decoding method is also applied with an optimal threshold value. The partial BER curves of the combined noise simulation are also shown in Fig. 13 as dotted lines.
The same trend of the combined noise in the simulation occurs in the experimental results. Adjacent modulation levels cross each other, making it difficult to distinguish them using a threshold value. The experimental result indicates that decoding by the conventional threshold decoding method retains a high BER level between 40% and 50%. It should be noted that the 50% BER is actually the highest value for randomly generated binary information. Compared with the conventional threshold decoding method, the proposed decoding method behaves much better. It has a strong decoding ability when the SNR is below zero, which means a high noise margin. The decoding results with 4.92% and 5.53% BERs can be made error free with a code correction; however, this is impossible for the corresponding decoded results of the threshold decoding method. In these experimental results, the proposed decoding method behaves 8.3 times better than the threshold decoding method in terms of the BER at most.
6. Summary
Holographic data storage suffers from noise, which causes errors, especially for multilevel modulation. To alleviate this situation, a decoding method based on compressed sensing is proposed. In addition, the noise depression character of compressed sensing is utilized. The proposed decoding method is realized by using improved K-SVD dictionary training and the OMP algorithm. The performance of the proposed decoding method is evaluated by both a simulation and an experiment with five-level modulation, which improves the code rate by 1.875 times compared to binary modulation. Compared with the conventional threshold decoding method, the proposed decoding method is superior. In the simulation, the proposed method can realize a quarter of the error rate of the threshold method for Gaussian and Rician noise and a twentieth of the error rate for Rayleigh noise. For combined noise, the proposed decoding method reduces the BER of the threshold method to one-sixth in the simulation with an SNR of -1. The experiment shows the same tendency: the proposed method behaves up to 8.3 times better than threshold decoding. However, there are some limitations of the proposed decoding method, e.g., it requires a data page consisting of symbols, and the modulation should impact only the amplitude. Although more features remain to be studied, such as more suitable and faster training and retrieval methods, the proposed decoding method is more robust than threshold decoding, especially for multilevel modulation.
Funding
Key Technologies Research and Development Program (2018YFA0701800); China Scholarship Council (201806030164).
Disclosures
The authors declare no conflicts of interest.
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