## Abstract

In the transformation based compression algorithms of digital hologram for three-dimensional object, the balance between compression ratio and normalized root mean square (NRMS) error is always the core of algorithm development. The Wavelet transform method is efficient to achieve high compression ratio but NRMS error is also high. In order to solve this issue, we propose a hologram compression method using Wavelet-Bandelets transform. Our simulation and experimental results show that the Wavelet-Bandelets method has a higher compression ratio than Wavelet methods and all the other methods investigated in this paper, while it still maintains low NRMS error.

© 2011 OSA

## 1. Introduction

Holography is a successful technique for recording and reconstructing three-dimensional (3-D) objects [1,2]. With the advent of digital holography and continuing advances in megapixel Charge Coupled Device (CCD) sensors, we can reconstruct 3-D object from hologram with better quality. However, digital holography also has its disadvantage that the corresponding data volume is huge. For example, suppose that we have a digital hologram of 1024 × 1024 pixel with each pixel storing 8 bytes of real information and 8 bytes of imaginary information. Then for a single hologram, we need 16 Mbytes memory in its native format which makes the real-time holography impractical.

Our goal is to compress these holograms for more efficient storage and transmission. Previously, many research groups have investigated this problem and have achieved much progress using transformation such as discrete cosine transform (DCT), and Wavelet transform [3–5]. In addition, A. Shortt et al. also have tried to use quantization compression technique for digital hologram [6] such as nouniform quantization, histogram quantization and mMa quantization. A notable research specialized for the hologram compression was reported by H. Yoshikawa et al. [7,8]. Based on analysis of fringe, they divided a fringe pattern into several segments and transformed them with one-dimensional (1-D) DCT. The resulting coefficients were encoded with moving picture compression technique such as MPEG-4 [9]. This method has been upgraded by Y. H. Seo et al. using 3-D scanning-based compression technique [10].

In this paper we present a new compression method based on combination of the fringe analysis and Wavelet transform. Papers published by Yoshikawa [7,8] and N. Tuck [16] et al (2005) investigate the effects of data compression on fringe hologram using JPEG and Fourier transform. The paper uses only the hologram with very fringe that cover the entire hologram as the object of study. However, this type of fringe patterns is rare in reality. The JPEG compression technique use hologram from a spatial sense and only one frame of hologram can be compressed at one time. Thus the approach does not suitable to compress large number of fringe patterns. However, when we use Bandelets algorithm to determine the fringe on hologram then we can overcome these disadvantages, therefore the analysis of fringe ideas on hologram get approximately the pixel value is applied.

First, we use Wavelet transform to analyze hologram data into low sub-band and high sub-band. Then we use Bandelets transform to analyze Wavelet transformed hologram fringes. Bandelets transform is a special method of transformation [11–14], which decomposes the pixels along multiscale vectors that are elongated in the direction of a geometric flow. This geometric flow indicates the direction in which the pixel values have regular variations; hence approximating the pixel values on each fringe with minimal information loss. Therefore, the proposed method increase compression ratio with minimal increase of the NRMS error.

This paper is organized as follows. Section 2 introduces conventional Wavelet compression and Wavelet-Bandelets transform. In Section 3 we present the proposed compression algorithm using Wavelet-Bandelets transform. Section 4 introduces the experimental setup and discusses the experimental results, and finally section 5 concludes the paper.

## 2. Wavelet transform and Wavelet-Bandelets transform

#### 2.1. Wavelet transform

A continuous Wavelet transform of a function $f(t)$ begins with a function called mother Wavelet $\psi (t)$. After mother Wavelet is selected, the continuous Wavelet transform is defined by

*a, b*denote translation and dilation factors. If we define a function${\psi}_{a,b}(t)$ bythen we get

For discrete data, a discrete Wavelet transform is performed with a selected set of scales and positions [15]. The discrete Wavelet transform has been applied to fields such as signal processing and image compression. In this paper, we use the simplest mother Wavelet function, Haar, to calculate real part and imaginary part of digital holography.

#### 2.2. Bandelets transform

Bandelets transform, introduced by E. L. Pennec and S. Mallat [11–13], in a publication paper, this transform is applied to compress image, these process of compression image is a multiscale geometric analysis arithmetic which to overcome the weakness of wavelet in high-dimensional data by removing the redundancy of a warped wavelet transform. The basic discrete Bandelets transform includes three steps:

- • Implementation of Wavelet transform with data from a digital image.
- • A warped Wavelet transform with a sub-band filtering along the flow lines [14].
- • Computation of Bandelets coefficients from the warped Wavelet coefficients along the flow lines.

## 3. Proposed method

A general hologram compression algorithm is shown in Fig. 3 . First, hologram data passes through a transformation to create a series of coefficients. These coefficients continue to be quantized to reduce the data storage. This quantized data stream is finally encoded into binary symbols using various coding methods such as Huffman coding and run length coding (RLC).

In this paper, we propose a method shown in Fig. 1 that uses a Wavelet-Bandelets transform. Wavelet-Bandelets method is very efficient in approximating the pixel values, i.e. reducing the data amount, neighboring to the geometric flow in hologram fringes. This geometric flow based approximation ensures higher compression ratio as well as low NRMS error. The proposed method is performed by the following steps:

*First step:*Digital hologram of a 3-D object is recoded using an optical system shown in Fig. 4 . Four holograms with phase shifts of 0, $\pi /2$, π, and $3\pi /2$of reference beam are recorded to yield amplitude and phase of the object optical field based on the principle of phase-shifting digital holography [1,2].*Second step:*Wavelet transform is applied to each of amplitude and phase of the captured hologram with the choice of mother Wavelet function and level of transformation. In our experiment, the simplest Wavelet function, Haar, was applied.*Third step:*Hologram data after Wavelet transform is divided into sub-bands. For each sub-band, quad-tree decomposition algorithm is applied to divide the sub-band into small blocks. The size of the small block is given by${2}^{j}$ × ${2}^{j}$pixels. Block size is important because it affects the performance of the geometric flow detection remarkably. When the block size is small, geometric flow can be determined easily in each block due to small number of pixels, but the large number of blocks makes overall compression ratio low. When the block size is large, on the contrary, determination of the geometric flow direction in each block is not easy since the pixel to pixel variation of the hologram data is usually large. In our experiment, various block sizes from 4 × 4 pixels to 32 × 32 pixels are tested to find optimum size.*Fourth step:*The direction of the geometric flow in each block is determined. In order to find the best direction in a small block, all directions in the block are checked. The number of directions that has to be checked is 2(N-1) for a small block of N × N pixel size. The geometric flow direction is defined as a direction along which the data has minimum variation. The examination of the direction is implemented as follows. In order to examine the data variation along a test direction, the data in the block is first rearranged to 1-D series following the test direction. Haar Wavelet transform is then applied to this 1-D data series to yield 1-D Wavelet coefficient series. The resultant coefficients are compared with a threshold value T. The best direction is determined by a direction which gives minimum number of the coefficients larger than T. This procedure is illustrated in Fig. 5 and Figs. 6(a) and 6(b).- Figure 6(a) describes the fringe of the sub-band after wavelet transform is performed with real information of a digital hologram. First step of this figure is to implement 2-D Wavelet transform with five levels transform. The examination of the fringe is implemented in each sub-band; this figure shows one small block which 8x8 pixels block size in fist sub-band with pixel value is represented in grey level. In the small block, it shows a part of the fringes in the sub-band. By using Bandelets transform, the fringe in each small block was determind. From this, we can approximate value pixel in each small block.
- Figure 6(b) describes the process which find the best direction in a small block, each small block will be checked all directions. In this figure, small block have 8x8 pixels size so it has 2*(8-1) = 14 directions are checked. The first step, the data in the small block is arranged to 1-D series following the test direction after that we applied to 1-D wavelet transform with this data (third picture of Fig. 6(b)). The resultant coefficients of wavelet transform implement absolute values and compare with a threshold value T (the red line that is described as the value T in third picture). The best direction is determined by a direction which gives minimum number of the coefficients larger than T. Cases Test (a), Test (b), Test (c) in this figure describe different cases which have different direction.
*Fifth step:*The data in each block is approximated using the geometric flow direction to reduce the data amount. The original 2-D data is divided into a set of 1-D flow according to the geometric flow direction found in previous step as shown in Fig. 6(c). For the central flow, the minimum pixel value is selected as a reference and other pixels whose deviation from the reference is less than a parameter C are clipped to the reference value. For other flows, the minimum pixel value of the previous flow is selected as a reference and the same process is conducted. The result of this process is Wavelet-Bandelets coefficients and these coefficients are encoded as usual compression algorithms.- Figure 6(c) show the example of pixel value approximation when C = 5. An small block has 4x4 pixel size with the best direction which was diagonal line from pixel position [1,1] to pixel position [4,4]. The directions is remarked (1), (2), (3), (4), (5), (6), (7), the main direction is remarked (1), respectively located above (2), (3), (4) and respectively located lower (5), (6), (7). First step, we approximate with (1) directions (it show in third picture), the minimum of value pixel in this direction is 12 so the value pixel in this direction is approximated equate 12 if the remaining pixel values in the range 12 ± 5, so we see, in this direction have three pixels which change value pixel (the red letter in third picture). Second step, we approximate with the remaining directions include (2), (3), (4), (5), (6) and (7); this process is described as follows: the minimum of value pixel in the (1) direction is approximated for (2) and (5) direction, the minimum of value pixel in the (2) direction is approximated for (3) direction; similarly, the minimum of value pixel in (5) direction is approximated for (6) direction and perform this process until the end direction in small block. In this figure, the minimum of value pixel in the (2) direction is 8 so the value pixel in (3) direction is approximated in the range 8 ± 5, similarly, the minimum of value pixel in the (5) direction is 10 so the value pixel in (6) direction is approximated in the range 10 ± 5. In this figure, the value pixel is approximated is red letter. Finally, we implement run length code with flow direction which we choose.

## 4. Experimental results

In experiment, we captured digital holograms of the 3-D objects based on phase-shifting principle using the optical setup shown in Fig. 2
and applied the proposed compression method. The laser beam with a 532 nm wavelength and output power of 150 mW is expanded, collimated, and divided into object and reference beam. The former illuminates an object placed at a distance of 490 mm and 810 mm from the CCD camera. The CCD camera has 1024 × 1024 pixels, with each pixel having an area of 9 × 9 µm^{2}, and 8 bit resolution, giving an output video signal with 256 gray levels. Hence each pixel of the captured hologram has 8 bytes of real information and 8 bytes of imaginary information. The objects are cubic of 10 × 10 × 10 mm^{3} and screw of 20 × 5 × 5 mm^{3}. The reference beam was phase-shifted by the retardation plates at amounts of 0, $\pi /2$, π and $3\pi /2$.

In our performance analysis, the NRMS error is defined as follows:

*U*and

*U*′ are numerically reconstructed object from the original and compressed holograms respectively and

*N*and

_{x}*N*are dimensions of the digital hologram. The compression ratio is calculated as follows:

_{y}In Fig. 7
, we compare three different methods of transformation for compression of digital hologram with data is recorded by phase-shifting method of cubic object and the distance from CCD to object is 490mm. For the first method, DCT transform has been applied. The DCT block size was 8 × 8 pixels and the coefficients were quantified by uniform method with 3 bits level quantization. For the second method, Wavelet transform has been applied. Haar function was used as a mother Wavelet function with five levels of transform. The coefficients were quantified by uniform method with 3 bits level quantization. For the third method, the proposed Wavelet-Bandelets transform has been applied. Haar function with 5 levels of transform has been used in the first Wavelet transform of the proposed method. In Fig. 7, points P_{A}, P_{B} and P_{C} correspond to the following parameters: 3 bit quantization, 8 × 8 pixels block size, T = 40 for P_{A}, T = 35 for P_{B} and T = 30 for P_{C}. Points P_{D}, P_{E}, P_{F}, P_{G}, and P_{H} are obtained by the parameters of 3 bit quantization, 4 × 4 pixels block size, T = 25 for P_{D}, T = 10 for P_{E}, T = 20 for P_{F}, T = 15 for P_{G} and T = 30 for P_{H}. Finally, points P_{I}, P_{J}, and P_{K} correspond to the following parameters: 2 bit quantization, 8 × 8 pixels block size, T = 25 for P_{I}, T = 35 for P_{J} and T = 15 for P_{K}. In the pixel value approximation processes of all points, the parameter C = 5 was used.

From Fig. 7, it can be seen that the proposed method outperforms other two methods in terms of compression ratio and NRMS error.

In Fig. 8
, we test the relationship between compression ratio and NRMS error when changing the mother function of wavelet transform, we still use cubic object with similar parameters in the case Haar function in Fig. 7. Meaning the following condition: With points P_{A}, P_{B} and P_{C} correspond to the following parameters: 3 bit quantization, 8 × 8 pixels block size, T = 40 for P_{A}, T = 35 for P_{B} and T = 30 for P_{C}. Points P_{D}, P_{E}, P_{F}, P_{G}, and P_{H} are obtained by the parameters of 3 bit quantization, 4 × 4 pixels block size, T = 25 for P_{D}, T = 10 for P_{E}, T = 20 for P_{F}, T = 15 for P_{G} and T = 30 for P_{H}. Finally, points P_{I}, P_{J}, and P_{K} correspond to the following parameters: 2 bit quantization, 8 × 8 pixels block size, T = 25 for P_{I}, T = 35 for P_{J} and T = 15 for P_{K.}

In the proposed method, compression ratio and NRMS error depend on three factors, i.e. the threshold parameter T, size of block in sub-band and the level of coefficient quantization. The effects of these 3 factors are shown in Figs. 9 –11 .

Figure 9 shows effect of the threshold parameter T. For Figs. 9(a) and 9(b), cubic object located at 490 mm from the CCD was used with block size 4 × 4 and level of quantization 2 bit and 3 bit. For Figs. 9(c) and 9(d), screw object located at 810 mm from the CCD was used instead of cubic object.

Figure 9 shows that there is a general tendency that small T results in higher compression ratio. When T is less than 10, compression ratio of real and imaginary part of hologram is almost unchanged. In this range, T is always smaller than the minimum value of Wavelet coefficients in the small block; hence the geometrical flow in each small block cannot be determined exactly. This leads to many errors in approximating the coefficients, which increases the NRMS error in final reconstructed hologram. Similarly, when T is larger than 35, compression ratio is unchanged again since T is always larger than the maximum of Wavelet coefficients in each small block. In this case, compression ratio of the proposed method converges to the compression ratio of the conventional Wavelet method.

Figure 10 shows the dependency of the block size to NRMS error and compression ratio. In Fig. 10, level of quantization is 3 bits and distance from object to CCD is 490 mm. It can be observed that the block size of 4 × 4 or 8 × 8 has the best compression ratio ranging from 30 times to 60 times and NRMS error from 0.43 to 0.6. The size of block over 16 × 16 or 32 × 32 is too large to determine the best geometric flow direction. Since the pixel to pixel variation in hologram is usually huge, the geometric flow in such a large block cannot be determined exactly, which results in low compression ratio. In case of block size 2 × 2, each block has only 4 pixels, hence geometrical flow direction can be determined easily. Large number of blocks, however, creates many geometrical flows for the given hologram, consequently compression ratio in this case is generally lower than the cases where the size of block is 4 × 4 or 8 × 8.

Figure 11 shows the variation of NRMS and compression ratio caused by different quantization level. As expected, quantization with higher bits results in low compression ratio and low NRMS error. In Fig. 11, the experiment was performed with T = 20, 4 × 4 block size, cubic object located at 490mm from the CCD and the Haar mother Wavelet.

Finally, the above performance analysis was confirmed visually as shown in Fig. 12 . Figures 12(a), 12(b), and 12(c) show the reconstruction results with different block sizes, i.e. 32 × 32, 4 × 4 and 8 × 8 for Figs. 12(a), 12(b), and 12(c), respectively. The other parameters are T = 20, d = 810 mm, quantization of level 3bit and Haar Wavelet function. Similarly, Figs. 12(d), 12(e), and 12(f) show reconstruction results for different T values of 40, 30 and 10 with block size 4 × 4, d = 490 mm, quantization of level 3 bit and Haar Wavelet function. Finally effects of quantization level are shown in Figs. 12(g), 12(h), and 12(k) with block size 4 × 4, d = 810 mm, T = 20, and Haar Wavelet function. Figure 12 confirms the variations of the reconstruction quality according to various parameters which was analyzed above.

## 5. Conclusion

A novel hologram compression method using the Wavelet-Bandelets transform was proposed. The performance of the proposed method was analyzed with varying three parameters; the threshold parameter T, block size in sub-band of Wavelet-transformed hologram data, and the level of quantization. Change of these parameters was shown to have ability to increase compression ratios with little increase in reconstruction error. In addition, the method using Wavelet-Bandelets transform is also proven to have higher compression ratio than conventional Wavelet methods, and DCT methods with similar NRMS error.

## Acknowledgment

This work was partly supported by the grant of the Korean Ministry of Education, Science and Technology (The Regional Core Research Program/Chungbuk BIT Research-Oriented University Consortium). This research was also partly by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0015491).

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