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Chapter 4

Holographic Data Representations

Abstract

In order to design an effective compression method for a specific type of data, a good understanding of its statistical characteristics is necessary. For digital holographic data, different representations are possible, including intensity-based and complex amplitude-based representations, as we shall see in this chapter. These representations are equivalent in the sense that they all reconstruct the same object. However, different representations may have different statistical characteristics. Therefore, we shall also investigate their suitability for compression.

Keywords

Digital holographic data representation

Intensity

Shifted distance

Real-imaginary

Amplitude-phase

Probability distribution

In order to design an effective compression method for a specific type of data, a good understanding of its statistical characteristics is necessary. For digital holographic data, different representations are possible, as we shall see in this chapter. These representations are equivalent in the sense that they all reconstruct the same object. However, different representations may have different statistical characteristics. Therefore, we also investigate their suitability for compression.

4.1 Representations of Digital Holographic Data

Hereafter, we consider more specifically the case of phase-shifting digital holography (PSDH), as the phase-shifting algorithm can effectively be used for the recording steps in order to obtain reconstructed object images with good quality. However, having three sets of intensity data is not a good target for compression. Therefore, reduced datasets are desirable to mitigate the compression burden. On the other hand, the data should keep all the useful information for reconstruction.

According to the phase-shifting algorithm, the most important information for reconstruction is the complex object field at the hologram plane. If this field can be obtained or expressed by a reduced amount of information, this information can be considered as a representation of interference patterns. So far, a few representations have been mainly used in accordance with the phase-shifting interferometry algorithm, which are introduced in the following.

4.1.1 Intensity-Based Representation

From Eqs. (2.34) and (2.35), the intensity-based representations can be equivalently expressed in two ways.

Intensity information The intensity information, $I_{H} (x, y; φ_{R}), φ_{R} \in \{0, \frac{π}{2}, π\}$ $I_{H} (x, y; φ_{R}), φ_{R} \in \{0, \frac{π}{2}, π\}$ , is the direct representation. However, compressing three sets of data are not optimal in terms of compression.

Shifted distance information To this end, and observing that only two differences of intensity terms are needed in Eq. (2.35), we introduce the difference data D⁽¹⁾ and D⁽²⁾ given by:

$\begin{array}{l} \{\begin{array}{l} D^{(1)} (x, y) = I_{H} (x, y; 0) - I_{H} (x, y; \frac{π}{2}) \\ D^{(2)} (x, y) = I_{H} (x, y; \frac{π}{2}) - I_{H} (x, y; π) . \end{array} \end{array}$ $\begin{array}{l} \{\begin{array}{l} D^{(1)} (x, y) = I_{H} (x, y; 0) - I_{H} (x, y; \frac{π}{2}) \\ D^{(2)} (x, y) = I_{H} (x, y; \frac{π}{2}) - I_{H} (x, y; π) . \end{array} \end{array}$

si2_e (4.1)

In this case, only two sets of data are necessary to reconstruct the complex field in Eq. (2.35). Since the raw data rate is reduced and it can directly be obtained from the difference of intensity information, the definition of D⁽¹⁾ and D⁽²⁾ is highly optimal as one representation of interference patterns. As this representation is defined by the difference of phase-shifted holograms, hence, we referred to it as “shifted distance information.”

4.1.2 Complex Amplitude-Based Representation

Complex amplitude-based representations are derived from different expressions of the complex field. Noted that a complex number has two kinds of expressions, one is defined in Cartesian coordinate system and the other one in polar coordinate system.

Real-imaginary information In Cartesian coordinate system, a complex number can be expressed in the form a + bi, where a and b are the real part and imaginary part, respectively. Thus, the complex field obtained by Eq. (2.35) can be expressed as:

$\begin{array}{l} Û_{O} (x, y) = ℜ (Û_{O} (x, y)) + i \cdot ℑ (Û_{O} (x, y)), \end{array}$ $\begin{array}{l} Û_{O} (x, y) = ℜ (Û_{O} (x, y)) + i \cdot ℑ (Û_{O} (x, y)), \end{array}$

(4.2)

where ℜ(Û_O) and ℑ(Û_O) are, respectively, the real and imaginary parts of the complex object field at the hologram plane.

Amplitude-phase information An alternative way to express a complex number is by polar coordinate system, where the complex field can be written by Euler’s formula as

$\begin{array}{l} Û_{O} (x, y) = Â_{O} (x, y) \cdot exp [i \hat{ψ_{O}} (x, y)], \end{array}$ $\begin{array}{l} Û_{O} (x, y) = Â_{O} (x, y) \cdot exp [i \hat{ψ_{O}} (x, y)], \end{array}$

(4.3)

where

$\begin{array}{l} Â_{O} (x, y) = | Û_{O} (x, y) | = \sqrt{ℜ^{2} (Û_{O} (x, y)) + ℑ^{2} (Û_{O} (x, y))}, \end{array}$ $\begin{array}{l} Â_{O} (x, y) = | Û_{O} (x, y) | = \sqrt{ℜ^{2} (Û_{O} (x, y)) + ℑ^{2} (Û_{O} (x, y))}, \end{array}$

(4.4)

$\begin{array}{l} \hat{ψ_{O}} (x, y) = \{\begin{array}{l} arctan \frac{ℑ (Û_{O} (x, y))}{ℜ (Û_{O} (x, y))} & if ℜ (Û_{O} (x, y)) > 0 \\ arctan \frac{ℑ (Û_{O} (x, y))}{ℜ (Û_{O} (x, y))} + π & if ℜ (Û_{O} (x, y)) < 0 and ℑ (Û_{O} (x, y)) \geq 0 \\ arctan \frac{ℑ (Û_{O} (x, y))}{ℜ (Û_{O} (x, y))} - π & if ℜ (Û_{O} (x, y)) < 0 and ℑ (Û_{O} (x, y)) < 0 \\ \frac{π}{2} & if ℜ (Û_{O} (x, y)) = 0 and ℑ (Û_{O} (x, y)) > 0 \\ - \frac{π}{2} & if ℜ (Û_{O} (x, y)) = 0 and ℑ (Û_{O} (x, y)) < 0 . \end{array} \end{array}$ $\begin{array}{l} \hat{ψ_{O}} (x, y) = \{\begin{array}{l} arctan \frac{ℑ (Û_{O} (x, y))}{ℜ (Û_{O} (x, y))} & if ℜ (Û_{O} (x, y)) > 0 \\ arctan \frac{ℑ (Û_{O} (x, y))}{ℜ (Û_{O} (x, y))} + π & if ℜ (Û_{O} (x, y)) < 0 and ℑ (Û_{O} (x, y)) \geq 0 \\ arctan \frac{ℑ (Û_{O} (x, y))}{ℜ (Û_{O} (x, y))} - π & if ℜ (Û_{O} (x, y)) < 0 and ℑ (Û_{O} (x, y)) < 0 \\ \frac{π}{2} & if ℜ (Û_{O} (x, y)) = 0 and ℑ (Û_{O} (x, y)) > 0 \\ - \frac{π}{2} & if ℜ (Û_{O} (x, y)) = 0 and ℑ (Û_{O} (x, y)) < 0 . \end{array} \end{array}$

si6_e (4.5)

The 3D information of the recorded object is fully contained in this representation, especially the phase information, which has a significant influence on reconstruction.

The above representations are equivalent, up to the precision of computation, in the sense that they all reconstruct the same object. The intensity of the interference patterns, Eq. (2.34), can be rewritten as

$\begin{array}{l} I_{H} (x, y; φ_{R}) = A_{R}^{2} (x, y) + A_{O}^{2} (x, y) + 2 A_{R} (x, y) A_{O} (x, y) cos (φ_{R} - φ_{O}) . \end{array}$ $\begin{array}{l} I_{H} (x, y; φ_{R}) = A_{R}^{2} (x, y) + A_{O}^{2} (x, y) + 2 A_{R} (x, y) A_{O} (x, y) cos (φ_{R} - φ_{O}) . \end{array}$

(4.6)

It can be noticed that the third term contains all effective information in the sense that the first and second terms will be eliminated during the subtraction operations in Eq. (2.35). Thus, by only retaining the third term, three signals can be defined as follows:

$\begin{array}{l} I^{(1)} (x, y) & = 2 A_{R} (x, y) A_{O} (x, y) cos (0 - φ_{O} (x, y)) \\ = 2 A_{R} (x, y) A_{O} (x, y) cos (φ_{O} (x, y)) \\ = 2 A_{R} (x, y) ℜ (U_{O} (x, y)) \\ I^{(2)} (x, y) & = 2 A_{R} (x, y) A_{O} (x, y) cos (\frac{π}{2} - φ_{O} (x, y)) \\ = 2 A_{R} (x, y) A_{O} (x, y) sin (φ_{O} (x, y)) \\ = 2 A_{R} (x, y) ℑ (U_{O} (x, y)) \\ I^{(3)} (x, y) & = 2 A_{R} (x, y) A_{O} (x, y) cos (π - φ_{O} (x, y)) \\ = - I^{(1)} (x, y) . \end{array}$ $\begin{array}{l} I^{(1)} (x, y) & = 2 A_{R} (x, y) A_{O} (x, y) cos (0 - φ_{O} (x, y)) \\ = 2 A_{R} (x, y) A_{O} (x, y) cos (φ_{O} (x, y)) \\ = 2 A_{R} (x, y) ℜ (U_{O} (x, y)) \\ I^{(2)} (x, y) & = 2 A_{R} (x, y) A_{O} (x, y) cos (\frac{π}{2} - φ_{O} (x, y)) \\ = 2 A_{R} (x, y) A_{O} (x, y) sin (φ_{O} (x, y)) \\ = 2 A_{R} (x, y) ℑ (U_{O} (x, y)) \\ I^{(3)} (x, y) & = 2 A_{R} (x, y) A_{O} (x, y) cos (π - φ_{O} (x, y)) \\ = - I^{(1)} (x, y) . \end{array}$

si8_e (4.7)

Note that D⁽¹⁾ and D⁽²⁾ given by Eq. (4.1) can be re-expressed as follows:

$\begin{array}{l} \{\begin{array}{l} D^{(1)} (x, y) = I^{(1)} (x, y) - I^{(2)} (x, y) \\ D^{(2)} (x, y) = I^{(1)} (x, y) + I^{(2)} (x, y) . \end{array} \end{array}$ $\begin{array}{l} \{\begin{array}{l} D^{(1)} (x, y) = I^{(1)} (x, y) - I^{(2)} (x, y) \\ D^{(2)} (x, y) = I^{(1)} (x, y) + I^{(2)} (x, y) . \end{array} \end{array}$

si9_e (4.8)

Therefore, we can be assured that real-imaginary information has similar features as shifted distance information.

Selecting a suitable representation for digital holographic data is very important for further processing, for example, compression, transmission, and storage. Indeed, in order to design an effective compression method for a specific type of data, a good understanding of its statistical characteristics is necessary. In this context, it is important to address the following questions:

1. Which representation is more suitable for compression? What are its main statistical characteristics?

2. For a given representation, does one of the information components affect more the reconstruction quality?

3. How to design a compression scheme? Should it be specifically designed for a given representation?

A thorough study and analysis is needed in order to characterize different holographic data representations and to start addressing the above issues. This is the subject of the two following sections.

4.2 Study of Probability Distributions

In this section, we investigate the probability distributions for each holographic data representation.

Three virtual objects (“Luigi,” “Girl,” and “Bunny”) shown in Fig. 4.1 are selected in the experiments, and the corresponding holograms are computer-generated using PSDH based on the methods described in Chapter 2.

f04-01-9780128028544 — Figure 4.1 Test objects: “Luigi,” “Girl,” and “Bunny.”

Examples of the three representations: shifted distance information, real-imaginary information, and amplitude-phase information, obtained from the same virtual object, are shown in Figs. 4.2–4.4, respectively. First, we can observe that the patterns in the two components of the shifted distance information are visually similar. The same observation also holds for the real-imaginary information. However, in the amplitude-phase representation, the two patterns are rather dissimilar. It can be inferred that more inter-component redundancies exist in shifted distance information or real-imaginary information than in the amplitude-phase information. As a consequence, compression methods exploiting inter-component redundancies are expected to be more effective in the shifted distance and read-imaginary cases, but rather ineffective in the amplitude-phase case. However, these observations need to be quantitatively verified by experiments.

f04-02-9780128028544 — Figure 4.2 Example of shifted distance representation D⁽¹⁾ and D⁽²⁾.

f04-03-9780128028544 — Figure 4.3 Example of real-imaginary representation.

f04-04-9780128028544 — Figure 4.4 Example of amplitude-phase representation.

4.3 Comparative Study of Different Representations

In this section, comparative studies of compressing different holographic data representations with various quantization methods are described, following the work in Xing et al. (2014b). More specifically, the impact of components on the reconstruction quality is first investigated. Then, redundancies across components are analyzed.

Next, probability distributions for each representation are studied, as shown in Fig. 4.5. It is straightforward that, except for the phase component, all other components have clear nonuniform distributions. Besides, the real-imaginary components and the shifted distance components share very similar distributions.

Hereafter, some common methods of quantization tools are used to study and analyze the characteristics of different holographic data representations. In particular, SQ processes input components independently, whereas VQ processes them jointly.

Impact analysis of each component on reconstruction quality We first investigate the impact of each component on the quality of the reconstructed object. For this purpose, each component is individually quantized using scalar quantization, and the corresponding object is then reconstructed.
For the sake of simplicity, only the peak signal-to-noise ratio (PSNR) index is reported hereafter, whose definition is given as follows:

$\begin{array}{l} \begin{matrix} PSNR = 10 {log}_{10} (\frac{max {(I_{ref})}^{2} - min {(I_{ref})}^{2}}{MSE}), \\ MSE = \frac{1}{M N} \sum_{m = 1}^{M} \sum_{n = 1}^{N} {[I_{ref} (m, n) - I_{rec} (m, n)]}^{2}, \end{matrix} \end{array}$ $\begin{array}{l} \begin{matrix} PSNR = 10 {log}_{10} (\frac{max {(I_{ref})}^{2} - min {(I_{ref})}^{2}}{MSE}), \\ MSE = \frac{1}{M N} \sum_{m = 1}^{M} \sum_{n = 1}^{N} {[I_{ref} (m, n) - I_{rec} (m, n)]}^{2}, \end{matrix} \end{array}$

si10_e (4.9)

where I_ref is the M × N reference image (or the image reconstructed from uncompressed holographic data), and $max (I_{ref})$ $max (I_{ref})$ and $min (I_{ref})$ $min (I_{ref})$ are, respectively, the maximum and minimum pixel values of the reference image.I_rec is the approximated image reconstructed from compressed holographic data and MSE is the mean square error. The reference and approximated images correspond to the reconstructed intensity rather than amplitude. Comparative studies, experimental results, and discussions are described hereafter.
In the first set of experiments, three representations are separately quantized by uniform scalar quantization (USQ). More precisely, each component is quantized with bit levels from 2 to 7. Tables 4.1–4.3 evaluate the reconstruction quality of the object “Luigi.” Figures 4.6–4.8 show corresponding illustrations.

Table 4.1

PSNR Obtained for “Luigi” by Quantizing Amplitude-Phase Information Using USQ (Am: Amplitude, Ph: Phase)

Number of Bits		Ph
Number of Bits		2	3	4	5	6	7
	2	24.66	30.79	31.27	31.22	31.26	31.27
	3	25.14	35.06	38.45	39.04	39.47	39.58
Am	4	25.24	36.43	42.71	44.79	46.76	47.34
	5	25.25	36.67	44.00	47.29	51.89	54.15
	6	25.26	36.72	44.25	47.85	53.80	58.10
	7	25.26	36.73	44.30	47.97	54.38	59.80

t0010

Table 4.2

PSNR Obtained for “Luigi” by Quantizing Real-Imaginary Information Using USQ (Re: Real Part, Im: Imaginary Part)

Number of Bits		Im
Number of Bits		2	3	4	5	6	7
	2	20.24	25.30	26.70	26.91	26.93	26.94
	3	25.26	32.05	35.50	36.27	36.42	36.47
Re	4	26.63	35.51	40.42	42.70	43.38	43.56
	5	26.82	36.29	42.67	46.74	48.77	49.50
	6	26.85	36.46	43.37	48.83	52.75	54.85
	7	26.86	36.51	43.57	49.56	54.77	58.81

t0015

Table 4.3

PSNR Obtained for “Luigi” by Quantizing Shifted Distance Information Using USQ

Number of Bits		D⁽²⁾
Number of Bits		2	3	4	5	6	7
	2	20.43	25.45	26.86	27.06	27.09	27.10
	3	25.48	32.22	35.66	36.45	36.63	36.68
D⁽¹⁾	4	26.84	35.62	40.47	42.73	43.45	43.66
	5	27.02	36.35	42.70	46.77	48.85	49.57
	6	27.05	36.52	43.40	48.82	52.85	54.93
	7	27.06	36.57	43.61	49.55	54.89	58.88

t0020

f04-05-9780128028544 — Figure 4.5 Probability distribution of different representations.

f04-06-9780128028544 — Figure 4.6 Figure corresponding to Table 4.1.

f04-07-9780128028544 — Figure 4.7 Figure corresponding to Table 4.2.

f04-08-9780128028544 — Figure 4.8 Figure corresponding to Table 4.3.

• From Fig. 4.6, the different contributions of amplitude and phase components can easily be observed. At a given bit level for the amplitude component, the change of PSNR value is sharper than that at the same bit level for the phase component. Similarly, it is implied in the numbers in Table 4.1 that the phase component contributes more to the reconstruction than the amplitude component. This different behavior of the amplitude and phase components is not unexpected, given their obviously different distributions as shown in Fig. 4.5.

• For the real and imaginary components, a quite balanced contribution can be observed from Fig. 4.7. This conclusion is also confirmed by the numerical evaluation in Table 4.2.

• Similarly, for the shifted distance components, a quite balanced contribution can be observed from Fig. 4.8 and Table 4.3.

In the second set of experiments, USQ is replaced by adaptive scalar quantization (ASQ). Considering the probability distributions in Fig. 4.5, ASQ is expected to improve performance further when the distributions are nonuniform. In the experiments, the Lloyd-Max algorithm (Max, 1960; Lloyd, 1982) is adopted for ASQ.
Table 4.4 compares the PSNR obtained from ASQ and USQ on the amplitude-phase and shifted distance information of three objects, with same bit level on each sample. The two kinds of representations similarly affect the reconstruction quality when each sample has the same bit level. However, ASQ is less effective on the amplitude-phase information due to the nearly uniform distribution of phase information.

Table 4.4

PSNR Obtained From ASQ and USQ on Amplitude-Phase (Am, Ph) and Shifted Distance (D⁽¹⁾,D⁽²⁾)

Number of Bits	Luigi				Girl				Bunny
	Am, Ph		D⁽¹⁾,D⁽²⁾		Am, Ph		D⁽¹⁾,D⁽²⁾		Am, Ph		D⁽¹⁾,D⁽²⁾
	ASQ	USQ	ASQ	USQ	ASQ	USQ	ASQ	USQ	ASQ	USQ	ASQ	USQ
2	24.74	24.66	25.63	20.43	25.82	26.01	26.71	19.71	19.36	18.77	22.81	15.65
3	35.12	35.06	34.61	32.22	35.73	35.67	34.53	31.63	28.69	28.28	29.71	25.38
4	42.83	42.71	42.44	40.47	43.61	43.32	42.51	40.36	35.26	34.83	36.23	32.33
5	47.27	47.29	48.77	46.77	48.16	48.09	49.35	47.07	41.11	40.82	41.85	38.51
6	53.82	53.80	53.71	52.85	54.72	54.72	54.02	53.04	46.91	46.82	45.73	44.51
7	59.80	59.80	58.89	58.88	60.54	60.57	59.15	59.08	52.90	52.90	50.53	50.52

t0025

Redundancy analysis In the third set of experiments, we investigate the inter-component redundancies for the different representations. From Figs. 4.2–4.4, we have a first hint that the real-imaginary and shifted distance information exhibit similar visual patterns.
For this purpose, 2D inter vector quantization is applied to each representation, where an input vector consists of one element from one component and the corresponding element from the other component. Linde-Buzo-Gray algorithm (LBG-VQ) (Linde et al., 1980) is used in the experiments. Before applying LBG-VQ, it is necessary to normalize the amplitude and phase information, due to the huge difference in their range of values. In the encoding process, the codebook is produced by training all vectors until the MSE is lower than a threshold τ = 0.001.Table 4.5 evaluates the performance of LBG-VQ on amplitude-phase and shifted distance information of three objects. The average bit level varies from 1 to 5.
Reconstructed images for the object “Luigi” obtained by LBG-VQ applied on amplitude-phase and shifted distance representations are given in the left and right column of Fig. 4.9, respectively, at bit level 2, 3, and 4.

Table 4.5

PSNR Obtained From LBG-VQ on Amplitude-Phase (Am, Ph) and Shifted Distance Information of Three Objects

Number of Bits	Luigi		Girl		Bunny
Number of Bits	Am, Ph	D⁽¹⁾,D⁽²⁾	Am, Ph	D⁽¹⁾,D⁽²⁾	Am, Ph	D⁽¹⁾,D⁽²⁾
1	16.15	19.54	16.40	16.99	13.92	16.05
2	17.97	27.61	27.64	29.56	22.39	23.38
3	34.10	35.75	36.36	37.36	29.17	30.52
4	40.62	43.18	42.90	45.15	35.46	37.54
5	46.72	50.26	49.12	52.47	41.58	44.07

t0030

f04-09-9780128028544 — Figure 4.9 Reconstructed images at bit levels of 2, 3, and 4 from amplitude-phase information (left column) and shifted distance information (right column) of “Luigi” object.

The results are summarized as follows:

• It is verified that with increasing codebook size, shifted distance information outperforms amplitude-phase information.

• Comparing with the evaluations of ASQ (see Table 4.4), the impact of applying LBG-VQ on amplitude-phase information is rather limited with gains below 1 dB. In contrast, LBG-VQ applied on shifted distance information leads to significant improvements with gains 3 dB. In other words, higher inter-component redundancies existing in shifted distance information are experimentally confirmed.

• The better performance of shifted distance information is also verified by the visual difference of reconstructed intensities, as shown in Fig. 4.9. Moreover, different from the distortions that could appear in natural images coded by VQ, the intensity of reconstructed objects from quantized holographic information shows surface roughness at high compression.

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Table of Contents for Chapter 4: Holographic Data Representations

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4.1 Representations of Digital Holographic Data

4.1.1 Intensity-Based Representation

4.1.2 Complex Amplitude-Based Representation

4.2 Study of Probability Distributions

4.3 Comparative Study of Different Representations

Table of Contents for
Chapter 4: Holographic Data Representations