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Abstract
Recent advances in phase only liquid crystal on silicon spatial light modulators are making the prospect of commercial fully holographic display a distinct possibility in the near future. There are, however, many remaining challenges when dealing with holographic data. In particular, widespread lossless image compression algorithms like the Lempel-Ziv-Welch and DEFLATE show poor performance when applied to the phase of holographic data from diffuse objects. This effect is caused by the discontinuous nature of phase. We propose an alternative phase representation that reduces these discontinuities. This representation is then processed with common lossless compression algorithms, achieving a significant volume reduction when compared with the direct compression of the original phase. We demonstrate the effectiveness of our proposal using experimentally registered holograms of diffuse objects. Up to a fivefold increase in compression ratio is achieved with our technique. Experimental reconstruction of the compressed holograms is also tested.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Thanks to holography we can register both the phase and amplitude information of an optical field, allowing for complete reconstruction of a 3D object or scene. Holographic techniques have been successfully used in a broad range of applications, from metrology [1], microscopy [2,3], and security [4,5] to more recent fields like holographic displays [6–8], neuronal photostimulation [9,10] and optical trapping [11,12].
The requirements for adequate holographic recording imply that high resolution registering media is needed. In digital holography the registering medium is a digital camera, and as a result a large volume of information must be digitally stored. The same requirements are present when holograms are generated computationally, meaning that, applications involving significant number of holograms must face the challenge of adequately storing and transmitting large amounts of data.
This issue is the focus of significant research efforts, in which the possibility of applying different digital compression techniques to holographic data was explored in depth [13–15]. Among this research, we find works reporting both the compression of holograms with common lossless algorithms, and by using lossy compression, like quantization, digital scaling, and spectral compression.
As result of these works, it was found that lossless compression methods present very low performance when dealing with holographic data. This low performance means that in practice, lossy compression is necessary to obtain the necessary reduction on the volume of holographic data to allow transmission with current technology [16]. However, it was found that the direct application of lossy compression to the phase of the holographic data of diffuse objects can result in a significant degradation in the reconstruction of the hologram in exchange for a relatively small reduction in data volume [17].
Motivated by the lack of effectiveness of traditional compression methods, techniques based in optical schemes were proposed. One of these schemes is an optical scaling technique [17], where the holographic data is scaled using a lens in a virtual setup. Another method consists in sampling with binary masks, which was shown to be an effective compression approach for color 3D holographic data [18].
Additionally, comparative analysis concluded that traditional compression methods, like those found in the JPEG standard, are inefficient when applied to noisy or high entropy data [19]. This helps explain the low effectiveness of JPEG compression when directly applied to the phase information of holographic data of diffuse objects, which have a near random distribution. This is a critical issue, especially when considering that the phase is the part of the optical field that carries the spatial characteristics of the object.
Thus, the problem of holographic compression can be reduced to the problem of phase data compression. Phase compression is hindered by the high entropy of this information when compared with the amplitude data of the optical field. While the amplitude is a continuous function and slowly varies, the phase presents fast variations and is discontinuous, with a random-like distribution found when diffuse objects are involved.
In this paper we present a method to achieve improved phase compression of holographic data. Our proposal consists in suggesting an alternative phase representation (APR); and then process this new representation with common lossless compression algorithms. The APR is chosen to lower the amount of phase discontinuities while at the same time allowing optical reconstruction. The APR can be easily transformed back into the original phase, with only quantization effects [20] as a source of degradation. We demonstrate the effectiveness of our proposal using optically registered holograms of diffuse objects. We also demonstrate the experimental reconstruction of the compressed and uncompressed holograms, and measure the degradation caused by our procedure. Compression ratios of nearly 5 are achieved with minor data loss.
2. Recording of diffuse holograms
As a first step, we will register the hologram of a diffuse object using the off-axis Fourier holographic setup of Fig. 1 [21]. This scheme is chosen to avoid the need of phase-shifting techniques, simplifying the extraction of the phase information from the holographic recording.
Fig. 1. Scheme of the off-axis Fourier holographic setup. (CS: collimation system, BS: Beam splitter, M: mirror).
A CMOS camera registers the hologram of the object, given by
where $O(v,w)$ is the Fourier transform (FT) of the object field $o(x,y)$, $P(v,w)$ is a tilted reference plane wave, and * represents the complex conjugate. This hologram contains all information needed to reconstruct the object, however, it also contains data that results in a DC term and a twin object. This unnecessary data can be discarded by a filtering procedure, resulting in a significant reduction of the data volume [22]. In order to perform this filtering procedure, we take the FT of the registered hologram, given byIn Fig. 2a and b, we show the phase and amplitude of the object optical field in the hologram plane. In Fig. 2c we can see the reconstructed object using the full complex optical field obtained from the hologram after filtering, while in Fig. 2d we can see that reconstruction using only the amplitude fails, resulting in a pattern without any resemblance to the object. On the other hand, reconstruction using only the phase (Fig. 2e) yields a very similar result to that of Fig. 2c. This highlights the importance of the phase information in holographic recording of diffuse objects, and further justifies the need to develop adequate compression techniques for this kind of data.
Fig. 2. a) amplitude, and b) phase of the optical field of the object in the hologram plane. c) reconstructed object using both a) and b). d) reconstructed object from a), e) reconstructed object from b).
The result of Fig. 2 was obtained with the recording setup of Fig. 1, where we employed an EO-10012C CMOS camera with a resolution of 3840×2748 pixels. The coherent source was a DPSS laser with 532 nm wavelength with a power output of 150 mW. The dimensions of the 3D object were 28 mm×15 mm×14 mm. The registering lens focal length was 200 mm.
As we can see in the zoomed region of Fig. 2b, the phase has a random-like appearance. We show, to demonstrate this assertion, the histogram of both the phase and the amplitude data. Then, we calculate their corresponding entropy. Since we stored the phase and amplitude as 8-bit grayscale images, the entropy for a histogram with 256 bins is given by
where ${p_i}$ is the normalized count of the histogram in bin i. For reference, the maximum value of the entropy for a 256-bin histogram is 8, which will correspond to a fully uniform distribution of values in the image.In Fig. 3 we show the resulting histograms. As expected from the field of a diffuse object, its amplitude will correspond to a speckle pattern, with nearly Gaussian histogram distribution. The phase, on the other hand, shows a nearly uniform distribution. The entropy of the amplitude was 6.733, while it is for the phase 7.998, close to the maximum possible entropy. As we will show, datasets with high entropy cannot be efficiently compressed by means of a lossless algorithm, since these techniques rely on statistical redundancies.
3. Alternative phase representation
The optical field can be represented as
where $A(v,w)$ is the field amplitude and $\phi (v,w)$ is the phase. Since the phase is the argument of an imaginary exponential function, any measurement of $\phi (v,w)$ will result in a function with discontinuities every $2\pi $. Our hypothesis to allow improved lossless compression of phase data is that reducing the discontinuities present in the phase information can result in a lower entropy representation, which can therefore be subjected to higher degrees of compression [23].In many applications, obtaining a continuous function from the phase is of interest, since it gives information about the 3D geometry of the object. Techniques to achieve this are called phase unwrapping algorithms [24,25]. A phase unwrapping algorithm allows the transformation of a phase function with discontinuities every $2\pi $ into a continuous function. For compression applications, full phase unwrapping does not produce the best results, this is because we are storing the phase information as an image with 8-bit depth, and a fully unwrapped function can have a much larger dynamic range, which cannot be stored with this bit depth while maintaining acceptable fidelity.
Instead of phase unwrapping, we only desire a new phase representation with lower entropy while ensuring object reconstruction. The condition for this is given by
where $\phi (v,w)$ is the original phase and ${\phi _a}(v,w)$ is the APR.The basic algorithm for obtaining an APR from the original phase performs a row-wise one-dimensional operation, which limits discontinuities to values that exceed a maximum selected range, defined as $2R\pi $, with R any positive integer. The steps are as follow
- 1. Take the first row of the matrix containing the phase.
- 2. Select the second pixel of the row and calculate the difference with the first pixel.
- 3. If the difference is greater than $\pi $, subtract $2\pi $ to the rest of the row, and if its less than -$\pi $ add $2\pi $.
- 4. If the resulting value of the pixel is greater than $R\pi $, subtract $2R\pi $ to the remaining pixels of the row, and if the value of the pixel is less than -$R\pi $ add $2R\pi $.
- 5. Repeat this for every pixel of the row, calculating the difference with the previous one.
- 6. Repeat for every row until processing the entire phase array.
In Fig. 4a and b we show the resulting APR with range $50\pi $ and its histogram, which no longer shows a uniform distribution like in Fig. 3b. The calculated entropy for the APR is 7.42. This lower value for entropy is a good indicator that higher lossless compression is possible for the APR. In our proposal, the range is a parameter which will allow to indirectly control the entropy of the resulting representation.
In order to show this effect, we calculate the entropy of the APRs obtained with different ranges. We then reconstruct the object from these APRs and calculate the correlation coefficient between the result and the reconstruction from the original phase shown in Fig. 2e.
In Fig. 5a we can see that the entropy decreases as the range of the APR increases. At the same time, a larger range causes increased quantization error when storing this function as an 8-bit image. It is worth noting that quantization of the phase results in a higher degradation than separately storing the real and imaginary part of the holographic data, however, it allows to perform operations like the ARP presented here, while also ensuring a lower volume of data to be processed.
Fig. 5. a) Entropy of the APR as a function of the allowed range, b) correlation coefficient between the reconstructed object from the original phase and the object obtained from the APRs, as a function of the allowed range.
While this effect decreases the reconstruction quality, as shown in Fig. 5b and the accompanying indented reconstructions, a visually identifiable object is still achieved from an APR with a range of $100\pi $.
4. Digital APR compression
We now apply standard compression methods both to the initial phase extracted from our hologram, and to the APRs obtained with the above described procedure. We will be using the Lempel-Ziv-Wech (LZW) [26] coding, the DEFLATE and the PACKBITS algorithms, all of which are commonly used for image compression, and are part of the TIFF format. The LZW algorithm is a modification of the classic LZ77 [23] adaptative compression technique, which finds repeated strings of characters or values and replaces them with a codeword. The DEFLATE algorithm is a combination of the LZ77 technique with Huffman codding [27], which first replaces each character with a codeword whose length will increase as the character becomes less frequent. Finally, PACKBITS performs a straightforward run-length encoding [28], where repeated characters or values are replaced by a number that represents the times the corresponding character is repeated.
Accordingly, to perform this test, we compress APRs with different ranges using the mentioned algorithms, and then we measure the obtained compression ratio C, defined as
where ${V_i}$ is the data volume of the original phase, and ${V_f}$ is the data volume of the APR after compression.In Fig. 6, we show the result of the proposed test. When the APR range increases, we obtain a higher compression ratio with all three algorithms. These results are consistent with the decreasing entropy of the APRs shown in Fig. 5a. In particular, the DEFLATE algorithm offers the highest compression ratio for all tested ranges. This is expected, since DEFLATE includes two techniques for lossless compression. For comparison, compression of the original phase yields a compression ratio of 0.7839 for LZW, of 1 for the DEFLATE and of 0.9926 for PACKBITS. Compression ratios lower than 1 represent an increase of the data volume after processing, highlighting the inefficiency of these algorithms when applied to very high entropy data.
Fig. 6. Compression ratio of the APR with different ranges when applying the DEFLATE, LZW, and PACKBITS algorithms.
5. Experimental results
We now proceed to experimentally reconstruct the phase of Fig. 2b and the APR obtained with range $50\pi $, after compression with the DEFLATE algorithm. For this purpose, we used the reconstruction setup of Fig. 7. In this scheme, the phase we want to reconstruct is projected in a phase only spatial light modulator (SLM), multiplied with a phase grating to ensure that the reconstructed object does not suffer from crosstalk with the undiffracted light coming from the SLM.
Fig. 7. Scheme for experimental reconstruction of objects from phase only data (M: mirror, BS: beam splitter, L: lens, SLM: spatial light modulator).
The intensity of the reconstructed object is recorded using an EO-10012C CMOS camera with 3848 $\times $ 2748 pixels of resolution and a pixel pitch of 1.67 $\mu m \times $ 1.67 $\mu m$. The SLM used is a PLUTO-2-VIS-016, with 1920 $\times $ 1080 resolution and a pixel pitch of 8 $\mu m$, calibrated for a linear phase response in the range $0 - 2\pi $. As light source, we used a diode-pumped DPSS laser with a wavelength of 532 $nm$ and an output power of 150 $mW$. The transforming lens had a focal length of 150 $mm$.
In Fig. 8 we show experimental results. As we can see, there is no visual difference between the result with the original phase and the result with the APR. The correlation coefficient between these two results is 0.9023. It is worth noting that in these experimental results there are other sources of degradation, like mechanical vibrations, which explain the lower correlation when compared with the numerical test of Fig. 5b.
Fig. 8. Reconstructed objects from the original phase and the DEFLATE compressed APR with range $50\pi $.
For the tested APR, we obtained a compression ratio of 2.64 with respect to the original phase of the filtered optical field extracted from the hologram
6. Conclusions
In this paper we have shown a novel technique to optimize the compression of phase when using conventional algorithms applied to holographic data. We first demonstrate that the entropy of the phase from a diffuse object is close to the maximum possible entropy for a 2D function, when stored as an 8-bit image. We introduce an APR with a lower entropy than the original phase, while ensuring object reconstruction. Depending on the range we set during the APR generation, we can obtain different results. Higher ranges produce lower entropy representations of the phase, but suffer increased quantization error when stored as 8-bit images. We then apply common lossless compression methods to the APRs, resulting in a significant increase in compression ratio over direct application to the original phase. This proposal is an interesting proof of concept, which we hope to expand seeking more optimized alternative representations for the phase, and at the same time exploring combinations with other compression techniques, both digital and optical.
Funding
Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) (0849/16); Universidad Nacional de La Plata (UNLP) (11/I215).
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