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Abstract
In holographic three-dimensional (3D) display, holograms for reconstructing 3D scenes require huge storage space and high transmission bandwidth. Holographic data must be compressed for practical applications. The holograms for 3D display are commonly represented in pure-phase format. Existing hologram compression techniques are generally not designed to handle phase-only holograms. In this paper, we propose a phase-difference-based compression method to compress phase-only holograms for holographic 3D display. Phase-only holograms are decomposed into grayscale images representing the phase distance and binary images containing the sign information of the phase difference. The grayscale images can be better handled by common image compression algorithms since their pixel values are proportional to the phase distance reflecting the distance between complex amplitudes. The binary images are compressed by lossless bi-level image coding. The compressed grayscale images and binary images can be synthesized to recover the phase-only holograms and reconstruct the 3D scenes. The advantages of the proposed method over existing image coding standards are verified by numerical simulations.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Holographic three-dimensional (3D) display is expected to be the most promising 3D display technique with glass-free true-3D features [1]. However, holograms for reconstructing 3D scenes require huge storage space and high transmission bandwidth, especially in the applications of high-definition real-time 3D display.
One way to reduce the amount of holographic data is to carry out compression of the holograms. Lossless data compression techniques, such as Lempel-Ziv, Lempel-Ziv-Welch, Huffman, and Burrows-Wheeler, were applied to digital holograms [2]. However, it is difficult to increase the compression ratio. For a higher compression ratio, a lossy form of compression has to be applied. Lossy quantization was combined with lossless coding techniques to improve the compression ratio [3]. Researchers tried different quantization methods to compress the holographic data, including reduction of quantization levels [4], non-uniform quantization [5–7], and vector quantization [8–10]. Transform coding is another efficient compression method. Commonly used Discrete Cosine Transform (DCT) [11–13], Discrete Wavelet Transform (DWT) [14–16], and their variants [17,18] were introduced into the compression of holographic data. Depending on the representation of holograms, specially designed coding methods, for example, vector lifting scheme [19,20], sparse matrix representation [21], and delta modulation [22,23], were proposed. Existing image and video coding standards can also be used to compress the holographic data. Researchers investigated and showed effectiveness of JPEG [24], JPEG 2000 [24,25], MPEG-4 part 2 [26], Advanced Video Coding (AVC) [27,28], and High Efficiency Video Coding (HEVC) [28] applied to still holograms or hologram sequences.
Due to the lack of amplitude-phase Spatial Light Modulators (SLMs), most holographic 3D displays nowadays employ phase-only SLMs [29–31]. The holograms are represented in pure-phase format while ignoring the changes in amplitude. The vast majority of compression techniques mentioned above are designed to handle digital holograms with real-valued intensity-based representations or real-imaginary complex-amplitude representations. Their compression performances are suboptimal if applied to pure-phase representations of the holograms.
As we all know, encoded pixel values for natural images are proportional to intensities and similar intensities are encoded into similar pixel values. Analogously, encoded pixel values for holograms should reflect complex amplitudes which contribute directly to the reconstruction of 3D scenes and similar complex amplitudes should be encoded into similar pixel values. Specific to phase-only holograms, phases representing similar complex amplitudes should be encoded into similar pixel values.
When phases of a phase-only hologram are wrapped between 0 and 2π, conventional compression methods usually treat phases of 0 and (2π – ε), where ε is a small positive number, as highly different values, even though they actually represent similar complex amplitudes (exp(i0) ≈exp[i(2π – ε)]). This leads to inefficient handling of the wrapped phase data for conventional compression methods. A recent work [32] shows that reconstructed images from JPEG-compressed phase-only holograms suffer from severe quality degradation.
In this paper, we propose a phase-difference-based compression method to compress phase-only holograms for holographic three-dimensional display. Pixel values of the phase-only holograms are regulated between –π and π to represent phase difference. Absolute values of the phase difference, i.e. the phase distance, can be regarded as a rough measure of the distance between complex amplitudes. The phase distance is encoded into a grayscale image with pixel values proportional to the phase distance. Thus, phases representing similar complex amplitudes are encoded into similar pixel values. The grayscale image can be better handled by common image compression algorithms. The sign information of the phase difference is encoded into another binary image which is compressed by lossless bi-level image coding. The compressed grayscale images and binary images can be synthesized to recover the phase-only holograms and reconstruct the 3D scenes. Numerical reconstruction results show that the proposed phase-difference-based compression methods have superior performance over the existing image coding standards.
2. Holographic 3D display
The holograms for holographic 3D display can be calculated using the point source algorithm [33,34] based on the configuration shown in Fig. 1. A 3D object is located behind the hologram plane (x, y). The surface of the 3D object can be segmented into multiple point sources, each of which emits a spherical wave. The complex amplitude distribution at the hologram plane can be obtained by superimposing the wavefronts from all the point sources:
where N is the number of the point sources, λ is the wavelength of the illumination, Aj is the wave amplitude of the j-th point source, φj is the initial phase randomly assigned between 0 and 2π, and rj is the distance between the j-th point source (xj, yj, zj) and the sampling point (x, y, 0) on the hologram plane:There are various ways [35,36] to generate the amplitude and coordinate information from 3D models considering different kinds of depth cues. For simplicity, the distance between the 3D object and the hologram is large enough so that the amplitude information for each point source can be provided by a single shading image of the 3D object as shown in Fig. 2(a), and the coordinate information of the corresponding point sources can be extracted from the depth image as shown in Fig. 2(b).
Fig. 2 (a) Shading image and (b) depth image of a 3D object rendered in Autodesk 3ds Max. Gamma/LUT correction is disabled during rendering to get a linear map between the depth information and the grayscale values of pixels in the depth image. Brighter pixels in the depth image are closer to the hologram plane, and darker ones are farther.
The complex amplitude distribution H(x, y) can be encoded into a phase-only hologram by ignoring the amplitude information [37]. The phase profile ϕ(x, y) of the phase-only hologram satisfies
An example phase profile is shown in Fig. 3. The phase is commonly wrapped between 0 and 2π, and is quantized to 8-bit.The resulting phase-only hologram can be used to numerically or optically reconstruct the 3D object. For numerical reconstruction, inverse Fresnel diffraction can be employed if the wave propagation meets paraxial approximation. The wave field on the reconstruction plane (ξ, η) at a distance zd behind the hologram plane can be written as
where k = 2π/λ is the wavenumber, and R is the amplitude of a plane wave that illuminates the hologram for reconstruction. An example image reconstructed on the middle plane of the 3D object is show in Fig. 4. It can be seen that the quality of the reconstructed image is not as good as the shading image shown in Fig. 2(a) due to the speckle noise caused by the random-phase distribution at the reconstruction plane [38].3. Phase-difference-based compression
Without great influences on the quality of reconstruction for 3D display, the phase-only holograms can be compressed by common image compression algorithms to reduce the amount of holographic data for storage and transmission. However, it is difficult to achieve a high compression ratio because there is a lack of any noticeable structure in the holograms as shown in Fig. 3. Furthermore, most algorithms are not designed to handle wrapped phase data encoded between 0 and 2π. The pixel values of 0 and (2π – ε), where ε is a small positive number, will be considered highly different, even though they are actually very close in phase distance. If phase difference between two pixel values, ϕ1 and ϕ2, is defined as
where m is an integer, then phase distance is the absolute value of Δϕ12 as shown in Fig. 5. For the pixel values of 0 and (2π – ε), the phase distance is |–ε| = ε.Fig. 5 Illustration of the phase difference between two pixel values on the complex plane. Re denotes the real axis and Im denotes the imaginary axis.
The phase distance |Δϕ12| can be regarded as a rough measure of the distance between complex amplitudes of two pixels, exp(iϕ1) and exp(iϕ2), which contribute to the integral for reconstruction in Eq. (4), since
The larger the phase distance, the larger the distance between complex amplitudes. Therefore, the encoded pixel values of the hologram should be proportional to the phase distance to a reference pixel to be better handled by common image compression algorithms.For simplicity, the phase of the reference pixel is chosen to be zero. Thus, the phase profile ϕ(x, y) of the hologram determined by Eq. (3) can be encoded between –π and π using Eq. (5) to obtain the phase difference:
Then the absolute value of the phase difference is taken as the phase distance |Δϕ(x, y)|. The phase distance is quantized to 7-bit pixel values and saved as an 8-bit grayscale image. The grayscale image can be compressed by lossy image compression algorithms, such as JPEG, JPEG 2000, AVC intra-frame encoding (AVC-Intra), and HEVC intra-frame encoding (HEVC-Intra).The encoded grayscale image is not sufficient to reconstruct the 3D object because the sign information of the phase difference in Eq. (7) is lost due to the absolute operation when getting the phase distance. So the sign information of the phase difference is encoded into another binary image whose pixel values are 0 if Δϕ(x, y) ≥ 0 and 1 if Δϕ(x, y) < 0. The encoded binary image can be compressed by lossless image compression algorithms, e.g. Joint Bi-level Image Experts Group (JBIG).
To reconstruct the 3D object, the compressed grayscale image and the compressed binary image are decompressed and used to recover the phase difference
where G'(x, y) is the pixel value of the decompressed grayscale image and B'(x, y) is the pixel value of the decompressed binary image. The recovered phase difference can thereby be substituted into Eq. (4) to calculate the optical field distribution on a certain plane. The resulting image can be compared to the reconstructed image from the uncompressed phase profile ϕ(x, y) to evaluate the performance of the compression algorithm.The above phase-difference-based compression procedure is summarized in Fig. 6.
4. Results and discussions
To validate the proposed phase-difference-based compression (PDBC) method, phase-only holograms of three 3D models are compressed with JPEG, JPEG 2000, AVC-Intra, HEVC-Intra, delta modulation [22], wavelet compression [16], and the PDBC methods. The three 3D models are the Stanford dragon, the Stanford bunny, and the Stanford Buddha downloaded from http://www.mrbluesummers.com/. The same optical configuration as shown in Fig. 1 is used to calculate the phase-only holograms. The sizes of the dragon, the bunny, and the Buddha are 45.9mm × 38.2mm × 34.2mm, 42.9mm × 42.8mm × 34.2mm, and 21.4mm × 51.9mm × 21.8mm, respectively. The 3D models are placed 1200mm behind the hologram plane. The holograms have a resolution of 1920 × 1920 with a pixel pitch of 8μm in both x and y directions. The wavelength of the illumination is 532nm. The shading image, the depth image, and the calculated phase-only hologram of the dragon are shown in Figs. 2(a), 2(b), and Fig. 3, respectively. Corresponding images of the bunny and the Buddha are shown in Fig. 7.
Fig. 7 (a) Shading image, (b) depth image, and (c) phase-only hologram of the bunny; (d) shading image, (e) depth image, and (f) phase-only hologram of the Buddha.
In this work, the phase-only holograms are saved as 8-bit grayscale Windows Bitmap (BMP) files whose size is 3,687,478 bytes. JPEG and JPEG 2000 compressions are implemented by the “imwrite” function of MATLAB. The compression ratio is adjusted with the “Quality” parameter for JPEG and the “CompressionRatio” parameter for JPEG 2000. AVC-Intra and HEVC-Intra compressions are implemented by the H.264 and HEVC encoders of FFmpeg. The compression ratio is adjusted with the “CRF” parameter. Delta modulation is implemented using MATLAB code according to the compression scheme in [22]. The step-size is obtained through a trial and error process. The compression ratio for delta modulation is fixed. Wavelet compression is implemented by the “wcompress” function of MATLAB. Parameters in [16] are adopted: 3-level Haar wavelet transform, zeroing 60% of wavelet coefficients, and 5-bit nonuniform quantization. The compression ratio for wavelet compression with these parameters will be determined for a certain hologram.
Part of an example HEVC-Intra-compressed hologram of the dragon is shown in Fig. 8(a) which resides in the same location as the inset in Fig. 3, and the same for the holograms below. The whole HEVC-Intra-compressed hologram has a file size of 522,171 bytes, a compression ratio of 7.06, a Peak Signal-to-Noise Ratio (PSNR) of 18.1 dB, and a Structural Similarity Index (SSIM) of 0.869, with reference to Fig. 3. When the HEVC-Intra-compressed hologram is decompressed and used to reconstruct the dragon, the reconstructed image on the same plane as Fig. 4 is shown in Fig. 8(b) with PSNR = 24.2 dB and SSIM = 0.408 taking Fig. 4 as a reference.
Fig. 8 Compression results using HEVC-Intra for the phase-only hologram of the dragon: (a) part of the compressed hologram; (b) reconstructed image.
For the PDBC method, each phase-only hologram is first decomposed into a grayscale image representing the phase distance and a binary image containing the sign information of the phase difference. An example decomposition of the phase-only hologram of the dragon is shown in Figs. 9(a) and 9(b). The grayscale image in Fig. 9(a) is darker than the inset in Fig. 3 because the pixel values in Fig. 9(a) are actually 7-bit values instead of 8-bit ones. The uncompressed grayscale image and binary image can be recomposed into a hologram as shown in Fig. 9(c) which is identical to the original hologram. Then, the grayscale image is compressed by JPEG, JPEG 2000, AVC-Intra, and HEVC-Intra, and the compression ratio is adjusted with the “Quality”, “CompressionRatio”, or “CRF” parameters. The binary image is compressed by the JBIG encoder of NConvert. Part of an example HEVC-Intra-compressed grayscale image of the dragon is shown in Fig. 9(d). The whole HEVC-Intra-compressed grayscale image has a file size of 181,111 bytes. Plus the file size of the JBIG-compressed binary image of 312,286 bytes, the total file size of the PDBC-compressed hologram of the dragon is 493,397 bytes, and the corresponding compression ratio is 7.47. Finally, the compressed grayscale image and binary image can be synthesized to recover the phase difference using Eq. (8) and reconstruct the 3D object. Part of the recovered phase difference distribution regulated between 0 and 2π, i.e. the recovered hologram, of the dragon, is shown in Fig. 9(e). The whole recovered hologram has a PSNR of 21.4 dB and a SSIM of 0.943. The reconstructed image on the same plane as Fig. 4 is shown in Fig. 9(f) with PSNR = 27.6 dB and SSIM = 0.590.
Fig. 9 Compression results using PDBC-HEVC-Intra for the phase-only hologram of the dragon: (a) grayscale image representing the phase distance; (b) binary image containing the sign information of the phase difference; (c) recomposed hologram from uncompressed grayscale image and binary image; (d) part of the HEVC-Intra-compressed grayscale image; (e) recovered hologram; (f) reconstructed image.
The results of the above two compression methods, HEVC-Intra and PDBC utilizing HEVC-Intra (PDBC-HEVC-Intra), are summarized in Table 1. We can see that the compressed file size of PDBC-HEVC-Intra is smaller than that of HEVC-Intra, leading to a higher compression ratio, while the PSNRs and the SSIMs of the recovered hologram and the reconstructed image for PDBC-HEVC-Intra are all larger than those for HEVC-Intra. So PDBC-HEVC-Intra has better compression performance than HEVC-Intra.
Table 1. Compression Results of HEVC-Intra and PDBC-HEVC-Intra for Similar Compression Ratio
We also compare the compression performances of JPEG and PDBC utilizing JPEG (PDBC-JPEG), JPEG 2000 and PDBC utilizing JPEG 2000 (PDBC-JPEG 2000), AVC-Intra and PDBC utilizing AVC-Intra (PDBC-AVC-Intra), HEVC-Intra and PDBC-HEVC-Intra, delta modulation, and wavelet compression indicated by PSNRs and SSIMs of reconstructed images as a function of compression ratio for the phase-only holograms of the dragon, the bunny, and the Buddha as shown in Fig. 10, Fig. 11, and Fig. 12, respectively. From these figures, we can see that: (1) Each PDBC method has superior performance over the corresponding existing image coding standard with a PSNR gain of approximately 3 dB; (2) All the PDBC methods have higher performances than existing image coding standards except for PDBC-JPEG when the compression ratio is lower than about 3; (3) Among the existing image coding standards, HEVC-Intra has the highest performance, JPEG has the lowest performance, and JPEG 2000 has similar performance to AVC-Intra; (4) Among the PDBC methods, PDBC-HEVC-Intra has the highest performance, PDBC-JPEG has the lowest performance, and PDBC-JPEG 2000 has similar performance to PDBC-AVC-Intra; (5) Delta modulation has a single compression ratio of about 8 for each hologram and lower performance than the PDBC methods; (6) Wavelet compression has a single compression ratio of about 3.2 for each hologram and similar performance to JPEG. It is not surprising that PDBC-HEVC-Intra achieves the best performance among the 10 compression methods since HEVC-Intra is currently the most advanced image compression algorithm and is able to handle the wrapped phase data with the help of the proposed phase-difference-based decomposition.
Fig. 10 Compression performances of 10 compression methods indicated by (a) PSNRs and (b) SSIMs of reconstructed images for the phase-only hologram of the dragon.
Fig. 11 Compression performances of 10 compression methods indicated by (a) PSNRs and (b) SSIMs of reconstructed images for the phase-only hologram of the bunny.
Fig. 12 Compression performances of 10 compression methods indicated by (a) PSNRs and (b) SSIMs of reconstructed images for the phase-only hologram of the Buddha.
Generally, the computational complexity of JPEG, JPEG 2000, AVC-Intra, and HEVC-Intra increases in turn. So does their encoding time. We record the encoding time of each hologram for existing image coding standards and PDBC methods as shown in Fig. 13. As expected, JPEG and PDBC-JPEG are the fastest methods, JPEG 2000 and PDBC-JPEG 2000 are faster than AVC-Intra and PDBC-AVC-Intra, and HEVC-Intra and PDBC-HEVC-Intra are the slowest ones. Due to the additional lossless compression of the binary image for PDBC methods, the encoding time for PDBC-JPEG is longer than that for JPEG, and the encoding time for PDBC-JPEG 2000 is longer than that for JPEG 2000. However, the encoding time for PDBC-AVC-Intra is shorter than or similar to that for AVC-Intra, and the encoding time for PDBC-HEVC-Intra is shorter than or similar to that for HEVC-Intra. The reason might be that the decomposed 7-bit grayscale images are easier to be compressed by AVC-Intra and HEVC-Intra than the original 8-bit phase-only holograms. Encoding time of one hologram for delta modulation is approximately 0.1s which is the shortest among the 10 compression methods since its complexity is the lowest. Encoding time of one hologram for wavelet compression is approximately 7s which is the longest among the 10 compression methods possibly because of its high complexity and MATLAB implementation.
Fig. 13 Encoding time for existing image coding standards and PDBC methods as a function of compression ratio for phase-only holograms of (a) the dragon, (b) the bunny, and (c) the Buddha.
Due to the lossless compression of the binary image, the PDBC methods have an upper limit on the compression ratio, although more efficient algorithms than JBIG are possible. The compressed binary image alone can be used to reconstruct the 3D object. However, its performance might not be as good as HEVC-Intra. An example reconstructed image from the JBIG-compressed binary image of the dragon is shown in Fig. 14(a). Its PSNR is lower than that of the reconstructed image from the HEVC-Intra-compressed hologram with similar compression ratio as shown in Fig. 14(b).
Fig. 14 (a) Reconstructed image from the JBIG-compressed binary image of the dragon; (b) comparison of PSNRs of reconstructed images from the JBIG-compressed binary image and the HEVC-Intra-compressed holograms of the dragon.
5. Conclusion
A phase-difference-based compression method to compress phase-only holograms for holographic three-dimensional display is proposed. Phase-only holograms are decomposed into grayscale images representing the phase distance and binary images containing the sign information of the phase difference. The grayscale images can be better handled by common image compression algorithms, since their pixel values are proportional to the phase distance reflecting the distance between complex amplitudes. Simulation results show that the proposed method has superior performance over the corresponding existing image coding standard with a PSNR gain of approximately 3 dB, while introducing little or none additional computational complexity. The phase-difference-based compression method provides an efficient way to reduce the amount of holographic data.
Funding
National Natural Science Foundation of China (NSFC) (61205013); National Key Basic Research Program of China (2013CB329202).
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