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Abstract
In this paper, we propose a new method for coding a full complex hologram with random phase. Since holograms with random phase have very unique spatial and frequency characteristics, a new compression method suitable for such holograms is required. We analyze the frequency characteristics of holograms with random phases and propose a new adaptive discrete wavelet transform (aDWT). Next, we propose a new modified zerotree alogrithm (mZTA) suitable for the subband configuration generated by the modified wavelet transform method. The results of the compression using the proposed method showed higher efficiency than the previous method, and the reconstructed images showed visually superior results.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Holography was first proposed by Gabor in 1948 [1], and research and development have been progressed in many fields because of its ability to record whole three-dimensional information. Analog holography uses three-dimensional information after recording it on a holographic film made of a special material, which is a rather limited technology for modern multimedia services [2]. Recently, studies on digital hologram technology have been widely conducted to fully utilize the capability of the hologram while overcoming the shortcomings of the analog method [3]. In order to use a digital hologram as electronic multimedia, digital hologram signal processing technology is required [4]. Hologram signal processing is largely classified into hologram rendering and compression. Hologram rendering may include hologram generation, editing, display, interpolation and enhancement. Hologram compression techniques include still hologram compression and moving hologram compression. JPEG Pleno is currently in the process of standardizing the compression of still holograms [5].
The studies of compressing the hologram may be classified into the method for the hologram itself and the method for the diffracted (reconstructed) domain after propagating the hologram at a certain distance [6–8]. The hologram has a very complicated feature unlike a 2D image, and it is very difficult to find a spatial correlation between holographic pixels. Therefore, even if a frequency transform is performed, it is difficult to show good energy concentration. In order to overcome this characteristic, the compression method in the reconstructed domain is to convert the hologram to a more spatially correlated form of data and then compress it. It can be understood that the approach in the reconstruction domain can achieve higher compression rates than compressing the hologram itself. However, since compressing data diffracted at a specific distance, errors may be concentrated in the data at a specific distance, and a difference may occur in the reconstruction quality of the hologram according to the distance.
There have also been many researches that compress the hologram itself, not the reconstruction domain. Initially, a method using a lossless compression algorithm [9] was developed. The study by Javidi et al. is representative. They researched lossless and lossy data compression algorithm for individual holographic frames [9]. Next, there is a hologram compression method using compressive sensing [10–12]. Javidi et al. proposed a new method to compress digital holograms (DHs) using a sparse matrix representation [10]. They applied an adaptive mask, based on a threshold filter, to the object wavefield by using digital holography to numerically manage complex wavefields. From there, tey stored the result of this filtering by sparse representation. They also provided an overview of the theoretical guidelines for application of CS in DH and demonstrate the benefits of compressive digital holographic sensing [11]. Poon et al. proposed a new lossy compression method which use row- and column-based uniform downsampling together with spline interpolation, whereas in the near-lossless compression method, and they used wavelet local modulus maxima and spline interpolation [12]. Cheremkhin et al. performed a research of compressing holograms using quantization. They considered common scalar and vector methods of digital holograms compression [13]. Wavelet transformation is a widely used tool in the field of compressing holograms [14,15], Darakis et al studied a compression method of phase-shifting digital holographic data [14]. They used standard baseline Joint Photographic Experts Group (JPEG) or standard JPEG2000 image compression techniques on the recorded interference patterns to reduce the amount of data to be stored. Cheremkhin et al. proposed a compression of off-axis digital hologram based on wavelet transform [15]. They used zero- and twin-order elimination and wavelet compression of the amplitude and phase components of the obtained Fourier spectrum, and they considered further additional compression of wavelet coefficients by thresholding and quantization. Compression of the method using the MPEG codec [16–19] and the method using the JPEG codec [20] was also researched. These include compression using MPEG1 [16], compression using H.264/AVC [17,18], and compression using MPEG-4 [19]. Most of these methods are focused on researching preprocessing techniques for using standard video codecs. Recently, research on the method using the latest standard codec has been actively conducted. Various studies have been conducted to compress holograms using standard codecs such as JPEG, JPEG2000, AVC, HEVC, and benchmark the compression efficiency between them [21,22]. Also, a color hologram compression technique based on matching pursuit was proposed [23]. In this paper, scalar and vector puruit were used to overcomplete Gabor’s dictionary and showed excellent compression efficiency.
Holograms can be represented in a variety of ways, in which case holograms are taken optically, intensity-based holograms are used, and in the case of phase-shifted holograms, the phase-shifted distances based representation is used for compression. Finally, the most common hologram is represented by a complex object wavefield. In the form of a complex wave, it can be expressed in real and imaginary numbers, or in amplitude and phase. Due to the various hologram representation methods, the efficiency of compression may vary even though the information about the same hologram is displayed.
In previous studies, it has been experimentally known that compressing a full complex digital hologram, usually in the form of real and imaginary parts, is more efficient. Complex data can typically be represented by real and imaginary pairs or amplitude and phase pairs. The amplitude is easy to compress because it has a shape similar to the original object. However, the phase component has very low compression efficiency because there is no correlation in the pixel unit. Therefore, even if the compression efficiency for amplitude is high, the phase component is not so compression efficient, so compression in amplitude and phase pairs results in lower compression efficiency than compression in real and imaginary pairs. If the phase can be efficiently compressed, a fully complex digital hologram of amplitude and phase pairs will be able to compress at high compression rates.
In this paper, we propose a compression scheme that applies a zerotree-based algorithm to various subbands after selecting a filter based on the characteristics of holograms with random phase and constructing subbands. In case of a hologram with random phase, the frequency characteristic tends to be very unique. Therefore, if frequency decomposition is taken into consideration of these characteristics, efficient energy compaction for compression can be obtained. Also, high compression efficiency can be obtained by constructing a zerotree by analyzing various subband structures.
This paper is organized as follows. Section 2 presents the results of the analysis of the full complex hologram with random phase, and Section 3 describes the proposed hologram compression codec. Section 4 describes the compression results and concludes in Section 5.
2. Analysis of hologram
In this section, we introduce the hologram data set released by JPEG Pleno to establish the hologram compression standard, and analyze the frequency characteristics of the hologram using wavelet transform. Based on this, we intend to show that the wavelet transform method commonly used in 2D images is not suitable for holograms. JPEG Pleno provides data from ERC Interfere and B-com and UBI EmergIMG for holographic compression. The resolution of Dices1080p in Fig. 1 is $1,920\times 1,080$. The pixel size of SLM is 6.4um, and the restoration distance is 0 0.655cm. The wavelength of the reference wave for restoration is 640nm, and the precision of the provided data is 32 bits [5]. As such, JPEG Pleno provides various information about holograms along with various holograms for compression. Since JPEG Pleno’s hologram data set contains holograms of various types and characteristics, the need for an adaptive hologram compression tool suitable for these situations can be considered.
Fig. 1. Hologram example, Dices1080p: (a) real and (b) imaginary hologram; (c) amplitude of the reconstruction hologram.
In order to identify the frequency characteristics of the hologram according to the use of various wavelet filters, the hologram is transformed using another wavelet filter, and the frequency characteristics of the subband are analyzed. One of the results of this experiment is shown in Fig. 2, and in this figure, it can be seen that the frequency distribution differs according to the type of filter. When a bi-orthogonal filter is used for a Dicces1080p hologram, the energy distribution of the frequency for the hologram spreads very widely. However, when using a reverse Bi-orthogonal filter, it can be seen that energy is concentrated in the high frequency subband. Through these experimental results, we can confirm that the compression efficiency can be improved if the wavelet filter is used by reflecting the characteristics of the hologram.
Fig. 2. Frequency distribution of wavelet subband using the bi-orthogonal filter (a) real and (b) imaginary holograms and using the reverse bi-orthogonal filter (c) real and (e) imaginary holograms.
3. Hologram compression
This section describes the structure of the proposed codec and the detailed techniques used.
3.1 Codec structure
The proposed codec has a structure that uses a wavelet transform and a zerotree algorithm. Figure 3 shows the structure of the proposed codec. The real and imaginary parts of the complex hologram are compressed independently. That is, real and imaginary holograms are separately input to the proposed codec in Fig. 3, and the compression process is performed. Since real and imaginary holograms are data with a wide variety of dynamic range, they are first normalized to the data on a specific scale. We normalized the data with a distribution of -128 to 127, but we can determine the normalized distribution according to the characteristics of the hologram. Next, a 1-level forward DWT and a next-level forward DWT process are performed, and this process is linked with rate distortion optimization. This process is called adaptive discrete wavelet transform (aDWT). The resulting DWT subbands are compressed using a modified zerotree algorithm (mZTA) based on SPIHT [24]. Distortion optimization includes energy distribution optimization and energy compaction analysis. Compressed data is decoded in the reverse process.
3.2 Adaptive discrete wavelet transform
This section introduces the adaptive wavelet transform algorithm. First, one of the filters prepared in advance is selected. Using this filter, normalized data is subjected to 1-level DWT, and 4 subbands are generated. Next, after the DWT is performed on the four subbands, the energy distribution of each subband is analyzed to perform the next level DWT on the subband having the highest energy. After performing the DWT up to the target level limit in this way, the DWT for the first filter is completed. Next, the above process is repeated using the second filter, and the entire process is repeated until all of the previously prepared filters are used. After completing the transform for all filters, the filter and subband structure with the highest compaction among the energy distributions are selected through the filter and suband selection process. The aDWT is shown in Fig. 4. The energy distribution and campaction analysis in the algorithm of Fig. 4 is described below.
- • Energy distribution analysis : This process is performed each time a 1-level forward DWT is performed. The average energy values of four subbands generated after the DWT are obtained. Next, the subband with the highest energy among the four subbands is selected, and the next level DWT is performed. When the limit is reached, wavelet transform and energy distribution analysis are performed again using the next filter.
- • Energy compaction analysis: After completing the transform for all filters, the energy compaction of the subbands generated for each filter is analyzed. With the least number of subbands, the filter representing the most of the total energy is selected. If the results are the same, the filter with the highest energy compaction of the subband at the highest level is selected.
3.3 Modified zerotree algorithm
After applying the aDWT, the subbands of the wavelet domain are encoded using the modified zerotree algorithm (mZTA). The subbands generated through the aDWT are depicted in Fig. 5(a). Since the subband structure can be made very diverse, we construct a zerotree with high correlation between the upper and lower subbands. Exceptionally, the lowest frequency (LL) subband maintains the Mallet-tree. It is experimentally verified that the structure of the Mallet-tree is better than the proposed energy-based one. Figure 6(b) shows an example in which the LL subband is composed of a Mallet-tree.
Fig. 5. Modified zerotree configuration (a) energy-based scheme, (b) combination of energy-based scheme with Mallat-tree in LL subband.
For subbands excluding the LL subband, the correlation between the upper and lower subbands was calculated using the proposed mSSIM (modified structureal similarity index). The mSSIM is a method of measuring the correlation between coefficients and morphological similarity between upper and lower subbands. The mSSIM is defined as Eq. (1). First, the correlation of wavelet coefficients between subbands is calculated for all subbands upper $N^{th}$ and lower $(N-1)^{th}$ subbands. After analyzing whether one coefficient at the $(x, y)$ position of the $(N)^{th}$ subband and the four coefficients at the $(2x, 2y)$ position of the $(N-1)^{th}$ subband have (almost) zero values, the Correlation of Subband is defined as the number for all cases with zero values. As the features of the upper and lower subbands are similar (the more zerotrees are formed), the compression efficiency of the zerotree algorithm increases. In Eq. (1), $\alpha$ is a proportional constant for adjusting the scale with the SSIM.
Figure 6 shows a modified zerotree algorithm that determines the tree between subbands with high correlation using the mSSIM. ① A subband of $H/2 \times W/2$ size is generated by downsampling the subband of the $(N-1)^{th}$ level with $H \times W$ size. ② the SSIM is calculated for all combinations between the $(N)^{th}$ level subband of $H/2 \times W/2$ size and the $(N-1)^{th}$ level subband of $H/2 \times W/2$ size. Through this process, the morphological similarity between the $(N-1)^{th}$ and $(N)^{th}$ subbands can be obtained. ③ The correlation between the subband of the $(N-1)^{th}$ level of $H \times W$ size without downsampling and the subband of the $(N)^{th}$ level of $H/2 \times W/2$ size is calculated for subband combinations in all cases. Through this process, the correlation of the distribution of zero values between subbands is obtained. ④ Finally, the mSSIM is calculated using Eq. (1) to obtain the modified zerotree.3.4 Encoding and decoding process
To help understand the proposed compression technique, the overall process is shown in Fig. 7 using the resulting figures. There are many different types of wavelet filters. We experimentally analyzed the characteristics of about 70 filters among them, and the five most effective filters for compressing holograms (Daubechies 6, Bi-Orthogonal (4.4), Reverse Bi-Orthogonal (5.5), Reverse Bi-Orthogonal (2.8), Reverse Bi-Orthogonal (4.4)) was selected. After decomposing the hologram using various wavelet filters, filters having the highest energy compaction in the wavelet region were selected. Among these filters, five filters were selected by excluding those having similar characteristics. When a full complex hologram composed of real and imaginary parts is input, energy compaction analysis for various wavelet filters is performed. As a result, various wavelet subbands are output, and a subband structure having the highest compression efficiency is selected among various subband structures (Filter and Subband Decision). The decomposed subband is compressed using the modified zerotree algorithm and SPIHT. Various compression ratios can be obtained by using the zerotree algorithm and SPIHT.
4. Experimental results and discussion
The proposed algorithm was implemented using C/C++, OpenCV, OpenMP and CUDA in Intel I7-7700K CPU @3.6 GHz, 64GB RAM and 64-bit Window10 environment. The full complex holograms used in the experiment were Dices1080p, Ballet1080p, Piano1080p, and Breakdancer1080p, published by JPEG Pleno [5,25–27].
4.1 Effect of filters
We show the results using various combinations of Wavelet filters in Fig. 8. Compression (encoding) and decompression (decoding) experiments were performed using five wavelet filters with different characteristics. In the results of the hologram compression at a low compression ratio of 20:1, the reverse bi-orthogonal (6, 8), a high-frequency compaction filter, shows the PSNR of about 5dB higher than the bi-orthogonal (4, 4) used in JPEG2000. The PSNR and SSIM of the reconstruction result were improved by about 4dB and 0.15, respectively. At a high compression ratio of 200:1, the PSNR of the hologram was improved by about 4dB, and the PSNR and SSIM of the reconstruction result were improved by about 2.5dB and 0.2, respectively. It can be seen that the use of a reverse bi-orthogonal (6, 8) filter that concentrates the high frequency for the hologram shows excellent efficiency in both low and high compression ratio.
Fig. 8. Compression result for the combination of wavelet filters; PSNRs of (a) real and (b) imaginary parts, and (c) amplitude of the reconstructed hologram, (d) SSIM of amplitude of the reconstructed hologram.
In order to examine the effects of the high-frequency compaction filters in more detail, we conducted experiments on various filters belonging to the reverse bi-orthogonal series and depicted the comparison results in Fig. 9. At a low compression ratio of 20:1, the PSNRs of holograms showed a difference of up to 3dB. The PSNR and SSIM of the reconstruction results showed a difference of about 4dB and 0.05, respectively. At a high compression ratio of 200:1, the PSNR of holograms showed a difference of up to up to 1.5dB, and the PSNR and SSIM of the reconstruction result showed a difference of up to 0.03dB and 0.3, respectively. Through this result, it can be seen that even with the same wavelet filter, if the length of the tap is long, the energy compaction ratio is high and the efficiency of hologram encoding and decoding is high.
Fig. 9. Compression result according to wavelet filter selection for analysis of high frequency; PSNR of (a) real and (b) imaginary part, and (c) amplitude of the reconstructed hologram, (d) SSIM of amplitude of the reconstructed hologram.
4.2 Effect of DWT level
The number of wavelet transform level affects the subband structure, which affects compression efficiency. Therefore, we tested the performance of compression ratio at various wavelet transform levels, and showed the results in Fig. 10. In the hologram and the reconstruction result, the smaller the number of transform levels, the less the compression efficiency. This tendency is more evident in SSIM.
Fig. 10. Compression result according to wavelet transform level; PSNR of (a) real and (b) imaginary part, and (c) amplitude of the reconstructed hologram, SSIM of (d) SSIM of amplitude of the reconstructed hologram.
4.3 Effect of lowest frequency
Even if the hologram contains a lot of high frequency components and such high frequency components contain a lot of information, the lowest frequency component including the overall average information may be very important. We compared the case of including the lowest frequency component using Mallet-tree and the case of using only the mZTA, and the results are shown in Fig. 11. Overall, the inclusion of the lowest frequency yields better results, and this tendency increases as the compression ratio increases. Also, this trend is very evident in the SSIM of reconstruction.
Fig. 11. Compression result according to processing method of LL subband; PSNR of (a) real and (b) imaginary part, and (c) amplitude of the reconstructed hologram, (d) SSIM of amplitude of the reconstructed hologram.
4.4 Compression efficiency
An experiment was conducted to check whether the proposed algorithm can be applied to various holograms. Figure 12 shows the results of encoding and decoding various holograms provided by JPEG Pleno. From these results, it can be seen that the compression efficiency shows a very large difference according to the characteristics of the hologram. The compression efficiency is lower in the case where the object recorded in the hologram is more complex and contained more object points than the case where it is relatively not. At a compression ratio of 20:1, the PSNR between holograms represents a difference of up to 35dB, and at a compression ratio of 200:1, the PSNR represents a difference of up to 15dB. This difference is even greater in the SSIM of the reconstruction results. In our experiments, SSIM also had a difference of up to 0.7.
Fig. 12. Compression results of several holograms: PSNR of (a) real and (b) imaginary part; (c) amplitude of the reconstructed hologram; and (d) SSIM of amplitude of the reconstructed hologram.
After encoding and decoding the hologram using the proposed compression method, the reconstruction results are shown in Fig. 13. It can be seen that the results were reconstructed at the compression ratios of 50:1, 100:1, and 150:1, and Ballet1080p and Breakdancer1080p without any background were also reconstructed visually with little deterioration.
Fig. 13. Reconstruction image result of the decoded holograms: (a) original; (b) 50:1; (c) 100:1; and (d) 150:1.
4.5 Comparison
The performance improvement was evaluated by comparing the proposed codec with the codec using the Wavelet transform, Mallet-tree and SPIHT. We compared the performance using two representative holograms, Dices1080p (2K) and Piano1080p (2K) in Fig. 14 and Fig. 15.
Fig. 14. Comparison of compression result with Dices 2K hologram; PSNR of (a) real and (b) imaginary part, and (c) amplitude of the reconstructed hologram, (d) SSIM of amplitude of the reconstructed hologram.
Fig. 15. Comparison of compression result with Piano 2K hologram; PSNR of (a) real part, (b) imaginary part, and (c) amplitude of the reconstructed hologram, SSIM of (d) amplitude of the reconstructed hologram.
First, when observing the comparison result using Dices1080p (2K), it can be seen that PSNR results are similar except for SPIHT at most compression rates. However, in the reconstruction results, the proposed method for SSIM shows the best results. This can be estimated as the effect of using the adaptive wavelet transform taking into account the overall shape of the hologram.
Next, comparing the compression results using Piano1080p, the proposed method showed the best compression efficiency when both real and imaginary parts were considered. The real part showed similar results when compared to JPEG2000 and HEVC Intra. However, in the high compression rate, the proposed method tends to be superior. In the imaginary part, the proposed method showed better results at all compression ratios than other methods. It can be seen that the PSNR of the reconstruction result is almost similar due to the effect of the real part. However, the SSIM result is consistently superior to the proposed method at all compression rates.
In order to verify the compression effect on holograms of various resolutions, compression was performed on Dices holograms with 4K$\times$2K and 8K$\times$8K resolutions, and the results are shown in Fig. 16 and Fig. 17, respectively.
Fig. 16. Comparison of compression result with Dices 4K hologram; PSNR of (a) real and (b) imaginary part, and (c) amplitude of the reconstructed hologram, (d) SSIM of amplitude of the reconstructed hologram.
Fig. 17. Comparison of compression result with Dices 8K hologram; PSNR of (a) real and (b) imaginary part, and (c) amplitude of the reconstructed hologram, (d) SSIM of amplitude of the reconstructed hologram.
The compression results for Dices 4K are shown in Fig. 16. The proposed method for a hologram with 4K$\times$2K resolution showed superior results than SPIHT and JPEG2000 in all cases. Compared with HEVC Intra, HEVC Intra generally shows superior results in low compression of 100:1 or less, and the proposed method shows better results in high compression. The compression results for Dices 8K are shown in Fig. 17. Even at the high resolution of 8K$\times$8K, it can be seen that the proposed method has better compression efficiency than JPEG2000 in all results.
We compressed and decompressed the 8K hologram, and then divided the decompressed hologram. Next, we numerically reconstruct the divided holograms, and observe the visual characteristics of each part, as shown in Fig. 18. In our experiment, the hologram was divided into 9 positions (viewpoints) and the reconstructed images at each position were numerically observed. This can indirectly confirm the effect of compression on the viewing angle. From the experimental results, we can confirm that compression does not affect the viewing angle.
In Fig. 19, we compared the visual quality of the reconstruction results of the proposed method and other codecs. The compression was performed at a compression ratio of 100:1, and the results for a total of 4 holograms were summarized. Observing the reconstructed quality, it can be seen that the proposed scheme and HEVC Intra are better than other schemes. In addition, in the case of the Ballet hologram, it can be observed that the proposed method is more similar to the original than HEVC Intra.
Fig. 19. Reconstruction result of Dices, Piano, Ballet and Breakdancer with compression ratio of about 100:1 (a) original, (b) SPIHT, (c) JPEG2000, (d) HEVC Intra, (e) Ours.
4.6 Coding time
We measured how much time each compression algorithm takes to encode and decode holograms. The measured average time is summarized in Table 1. As for the encoding time, the proposed method takes the shortest time, and the HEVC Intra method takes the most time. The encoding time is directly proportional to the complexity of the encoding algorithm. That is, since HEVC Intra is composed of a very complex encoding algorithm compared to other methods, it takes a long time to be difficult to compare with other methods. It may take about 62 hours (estimated) to compress one 8K hologram. The proposed scheme takes only 0.6% of the time compared to the encoding speed of HEVC Intra. JPEG2000 takes the next most encoding time due to the complexity of EBCOT (embedded block coding with optimized truncation). When comparing the proposed method and SPIHT, since the proposed method compresses subbands divided into small units, it takes a little less encoding time than SPIHT. In the case of decoding time, the HEVC Intra takes the shortest time, and the proposed scheme takes the next shortest time. In consideration of actual applications, still image compression should enable encoding and decoding within a very fast time. In this respect, since HEVC Intra takes too long encoding time, it is not actually used for compressing still images.
Table 1. Encoding and decoding time (sec)
5. Conclusion
In this paper, we proposed a new codec that compresses a full-complex hologram using a zerotree of multi-resolution wavelet subbands. There were two main methods proposed. One was to build a subband structure adaptive to the hologram using various wavelet filters. Through this technique, it was possible to compact important information of the hologram into small data. The second was the mZTA. We proposed a method to extract the zerotree from various types of subband structures using the mSSIM. Using the mZTA, it was possible to efficiently compress the frequency information of the hologram compacted in the high-frequency subband. In the experimental results, we showed the compression effect of the proposed codec according to the type of filter, the length of the filter tap, the level of wavelet transform, and the influence of the lowest frequency subband. Next, the generality of the proposed codec was verified by analyzing the compression tendency for various holograms. Lastly, by comparing the results of the SPIHT-based wavelet codec, JPEG2000 and HEVC Intra, it was shown that the proposed algorithms contributed to an increase in the compression efficiency.
Funding
Giga KOREA Project (GK19D0100); National Research Foundation of Korea (2018R1D1A1B07043220).
Acknowledgments
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07043220). This work was supported by Giga KOREA project [GK19D0100, Development of Telecommunications Terminal with Digital Holographic Table-top Display].
Disclosures
The authors declare no conflicts of interest.
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