## Abstract

A method for fast reconstruction of off-axis digital holograms based on digital multiplexing algorithm is proposed. Instead of the existed angular multiplexing (AM), the new method utilizes a spatial multiplexing (SM) algorithm, in which four off-axis holograms recorded in sequence are synthesized into one SM function through multiplying each hologram with a tilted plane wave and then adding them up. In comparison with the conventional methods, the SM algorithm simplifies two-dimensional (2-D) Fourier transforms (FTs) of four N*N arrays into a 1.25-D FTs of one N*N arrays. Experimental results demonstrate that, using the SM algorithm, the computational efficiency can be improved and the reconstructed wavefronts keep the same quality as those retrieved based on the existed AM method. This algorithm may be useful in design of a fast preview system of dynamic wavefront imaging in digital holography.

© 2014 Optical Society of America

## 1. Introduction

Digital holography and its applications have drawn much attention in resent years with the advanced development of high-performance computing and high-resolution image sensing technology. The applications have been extended to many areas, such as contour measurement [1,2], digital microscopy [3,4], image recognition [5,6], particle test [7,8], terahertz imaging [9,10] and so on. According to the relative orientations between the object beam and the reference adopted in recording geometry, digital holography is often divided into two categories: in-line and off-line (or off-axis). The former is often adopted for reducing the space-bandwidth of the holograms, but it often suffers from the disturbances of the zero-order autocorrelation and the conjugate image, which need to be eliminated or removed by using phase shifting techniques or other time-consuming elimination algorithms [11–14]. For off-axis digital holography, the autocorrelation and the conjugate image could be eliminated simply by a digital spatial filtering [15], although the off-axis geometry will results in a high-resolution demand for the recording materials or imaging sensors. Because the spatial resolution and the pixel number of a digital image sensor have been significantly increased in recent years (for example, the pixel size of a current commercial image sensor can be close to 1 um and the pixel number can reach more than 10 megapixels), off-axis digital holography has become a powerful tool for phase imaging and found many applications in biology, medical diagnosis and metrology [16–19].

To allow faster reconstruction of off-axis digital holograms, some techniques based on computer’s graphic processing unit (GPU) and parallel computing algorithm have been reported [20,21], which allow to reconstruct holograms of 1 megapixel in video rate. Using GPU programming, however, requires special graphic cards and high programming capabilities. Most recently, Girshovitz et al. [22] proposed a simple method for real-time reconstruction of off-axis digital holograms based on a kind of digital angular multiplexing (AM) algorithm. This method realize the quick synchronous reconstruction of two holograms recorded at different time, thus greatly increases the reconstructing speed, which makes it possible to realize real-time holographic reconstruction with a simple personal computer system.

In this paper, we present another multiplexing algorithm for fast reconstruction of off-axis digital holograms. This method utilizes a kind of spatial multiplexing (SM) algorithm, which can synchronously reconstruct 4 holograms without loss of the resolution and the quality of the retrieved images, and thus can simplify two-dimensional (2-D) Fourier transforms (FTs) of four N*N arrays into a 1.25-D FTs of one N*N array, so that the calculation efficiency could be improved.

## 2. Spatial multiplexing in digital reconstruction

Supposing the intensity distribution of an off-axis digital hologram recorded by a digital camera at time t is

*R*and

_{0}*ξ*are the amplitude and spatial frequency of the reference beam,

_{0}*O*and

_{t}*φ*are the amplitude and phase distribution of the sample recorded at time t, respectively. For simplicity, here we assume the reference beam is a plane wave no changing with time and its spatial frequency is unequal to be zero only in

_{t}*x*direction. For an ideal off-axis hologram, the size of

*ξ*is usually set to one and half times of the bandwidth of the recorded samples in order to avoid any interference between the sample frequency and its auto-correlation term in frequency domain. In general, for extracting the complex amplitude, especially the phase, from the hologram expressed by Eq. (1), it has to firstly transform Eq. (1) into its spatial frequency by a 2-D FT, in which the spatial frequency of the sample wave could be separated from the first two terms (corresponding to the auto-correlation terms) and the fourth term (that is the conjugate term) in Eq. (1). After a further spatial filtering and inverse FT, the complex amplitude of the sample wave could be reconstructed finally.

_{0}We know operation of a 2-D FT of a picture is a time-consuming process, especially when the picture has large number of pixels, because calculating a 2-D FT of an N*N array usually needs to operate 2N 1-D FTs of 1*N arrays. In Girshovitz’s method [22], they introduced the angular multiplexing often adopted in recording procedure into digital reconstruction process. They firstly construct a multiplexing function by adding the hologram *H _{t}* with the next hologram

*H*after rotating 90 degree. This multiplexing function can be expressed as

_{t + 1}For further speedup of the digital processing, here we build the multiplexing function based on a SM algorithm instead of the above AM model. In this SM algorithm, a spatial multiplexing function is defined as

*H*,

_{t}*H*,

_{t + 1}*H*and

_{t + 2}*H*are four of the sequential holograms recorded by the digital camera;

_{t + 3}*B*is the bandwidth of the holograms. For an ideal off-axis hologram, the size of

*B*is equal to four times the bandwidth of the recorded sample. From Eq. (3) we can see that, in our method, the SM function is constructed by adding four holograms, each of which is multiplied by a plane wave tilted in

*y*direction (orthogonal to the tilt direction of the reference beam in holographic recording).

For comparison of the difference among Eqs. (1), (2) and (3), we show, respectively, their spatial frequencies in Fig. 1(a), 1(b) and 1(c). We can see that the AM algorithm [22] as shown in Fig. 1(b) only stack two holograms without influence on the final imaging quality, while our SM method as shown in Fig. 1(c) can multiplex four holograms at the same time. This improves the reconstruction speed because of decreasing the redundant computing of zero pads. And, because the interested information in the spatial frequency of the SM function is only located in one quarter columns of the spatial frequency array, we can further simplify a 2-D FT of a N*N array into a 1.25-D FT (which is so called because it only includes 1.25N 1-D FTs of 1*N arrays). So the reconstruction speed could be further increased.

Figure 2 presents the flowchart of our SM algorithm for
fast reconstruction of off-axis holograms, which mainly contains the following steps: (A) Sum
four holograms recorded in sequence at time t, t + 1, t + 2 and t + 3, each multiplied by a
plane wave tilted in the direction orthogonal to the reference beam used in recording setup, to
form the SM function *H* as represented in Eq. (3); (B) Transform the SM function *H* into the spatial frequency
domain using a 1.25-D fast FTs (FFTs) algorithm, that is, firstly calculate N 1-D FFTs of all
the rows of the SM function arrays to get a new array and then further calculate N/4 1-D FFTs of
the interested one quarter columns in the new array, which results in a N*N/4 array only
containing the cross-correlation terms of the four holograms; (C) Crop the N*N/4 array into four
subarrays, each with N/4*N/4 pixels and containing one of the spatial frequencies of the four
samples recorded at different time; (D) Respectively transform the four N/4*N/4 cropped spatial
frequencies back to their image domains by using a 2-D IFFT, resulting in the reconstructed
complex amplitudes of the four holograms; (E) If required, unwrap the reconstructed wrapped
phase distribution; (F) Enlarge each N/4*N/4 unwrapped matrix into a suitable size for display,
for example, to its original N*N size.

## 3. Experiments

We carried out experiments to demonstrate the method described above. The experimental arrangement is essentially a Mach-Zehnder interferometer, similar with that shown in Fig. 1 of reference [15]. In our experiments, a He-Ne laser with a wavelength of 632.8 nm is used as the light source. The recorded object is a dynamic refractive-index field produced by a flame, which is imaged on the recording plane by a 4-f imaging system and interferes with a tilted plane wave. The holograms were recorded by a CCD image sensor with pixel size of 6.45*6.45 um and pixel number of 1024*1024. Figures 3(a)-3(d) show examples of four off-axis holograms recorded in sequence in the experiments. The detailed interferometric fringe structure of the holograms can be seen from the zoom-in picture shown in Fig. 3(a). The digital reconstruction was performed in a personal computer (Intel i3-3110, 2.4 GHz CPU, 4 GB RAM), without using GPU or parallel processing (only a single core was utilized).

For reconstruction of the holograms using the SM algorithm described above, we first got the SM function of four holograms based on Eq. (3), and then calculated the 1-D FFTs of all the horizontal rows of the SM function array. Figure 3(e) shows the picture after the 1-D FFTs. In the next step, instead of calculating the 1-D FFTs of all the vertical columns of the function array shown in Fig. 3(e), we simply calculated the 1-D FFTs of the columns in the region between the two dashed lines shown in Fig. 3(e) to get the spatial frequencies required for further reconstruction. Figure 3(f) shows the spatial frequencies of the samples recorded by the four holograms from the above-mentioned calculations. Then the wavefronts recorded by the four holograms can be retrieved, respectively, by inverse FFT of the four frequency parts shown in Fig. 3(f).

Figures 4(a)-4(d) present the phase profiles reconstructed from the holograms as shown in Figs. 3(a)-3(d) by using the SM algorithm. For quantitative comparison of the phase reconstruction quality, we also reconstructed the holograms by using the AM algorithm [22]. Figure 4(e) shows the unwrapped phase profiles of the sample along the dashed line in Fig. 4(a) reconstructed, respectively, by using the SM algorithm (red dashed line) and the AM algorithm (black line). We can see that the reconstructed phase distribution based on the SM algorithm is consistent with the result using the AM algorithm, despite various noises existed in real experiments.

In reference [22], a detailed comparison of the average run times for reconstruction of a hologram using the AM algorithm with using the conventional algorithms has been made. Here we further compared the average run time with the SM algorithms and the AM algorithm. Because the main differences between the two methods are only in steps A, B and C of the procedure as shown in Fig. 2, we just compared the run time in these calculating processes. After 100 repetitions under the same computing and programming condition, we found that the run time using AM algorithm is about 86 ms per hologram in average, while the corresponding run time using our SM algorithm is only about 39 ms per hologram. The processing rate using the SM algorithm is more than twice that using the AM algorithm.

In addition, it should be indicated that the position of the marked region for further calculation shown in Fig. 3(e) depends on the size of the reference frequency adopted in real experiments. For the ideal case of the reference frequency equal to three-eighth of the hologram bandwidth as described in section 2, such a region should be located in the rightmost part of four equal parts of the picture shown in Fig. 3(e). In experiments, however, the reference frequency is often less than the ideal case. So the real calculation region (corresponding to the first diffraction order) must shift an amount to the left, and the region width ought to be smaller than its idea width of N/4 for avoiding the disturbance of the zero diffraction order. Usually this region width can be configured to two-thirds of the reference frequency adopted in real experiments.

## 4. Conclusion

In summary, we demonstrated that the off-axis holograms can be fast reconstructed by using the SM algorithm, in which four off-axis holograms are firstly synthesized into one 2-D SM function before digital reconstruction of them. In comparison with the conventional method used in off-axis holography, the new SM algorithm simplifies the 2-D FFTs of four N*N arrays into a 1.25-D FFT of one N*N array, thus a great deal of unnecessary calculations can be cut out and the computational efficiency is greatly improved. We think that this improved algorithm may be useful in implementation of real-time holographic visualization in a simple personal computer system. Especially, it could be a good choice in design of a preview system for dynamic wavefront imaging in digital holographic microscopy.

## Acknowledgments

The work is supported in part by National Natural Science Foundation of China (NNSFC) under Grant Nos. 11474186 and 11074152 as well as the Research Foundation for the Doctoral Program of Higher Education of China under Grant No. 20113704110002.

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