Optical scanning holography (OSH) is a technique that employs a single-pixel sensor to capture the hologram of a three-dimensional object through a sequential row-by-row scanning process. Being different from standard digital hologram acquisition methods that are based on a two-dimensional camera with restricted capturing area and highly limited spatial resolution, OSH is capable of acquiring holograms of wide-field scenes with high resolution. However, this favorable feature also implies a large data size that inevitably leads to various problems in the transmission and processing of the holographic data. In this paper, we propose a new framework, which we call compressive optical scanning holography (COSH), to handle this problem. Briefly, we incorporate a near computational-free and noniterative method to select the hologram pixels to be included in the optical scanning process, and subsequently to convert the value of each acquired pixel into a 1-bit binary representation at the moment when it is detected by the single-pixel sensor. As such, the data size of the hologram can be reduced by one to two order(s) of magnitude. In addition, in the selection of the pixels with our proposed method, the hologram row that is likely to contain similar content to the previous row is not scanned, hence leading to a considerable reduction in the hologram acquisition time. At the receiving end, the hologram can be recovered through simple interpolation of the compressed data. The compressive OSH capturing system can be realized to operate at video rate with very simple hardware or software implementation. We have demonstrated experimentally that the proposed COSH method is capable of acquiring a hologram with less than 1% of its original data size, and still preserving good fidelity on its contents.
© 2015 Optical Society of America
Electronic acquisition of holographic signals from a physical object scene can be dated back to the pioneering work of Enloe et al. in the mid 1960s . A Mach–Zehnder interferometer is employed to mix the reference laser beam and the object wave such that the interference pattern is recorded by a camera tube, i.e., a vidicon. Subsequently, the object scene can be reconstructed optically or digitally [1,2]. Numerous enhancements such as phase-shifting holography [3,4] have been made based on this fundamental framework. However, these methods are restricted by the size and positioning of the camera that is employed to record the hologram. An alternative solution was proposed by Poon and Korpel in the late 1970s, whereby a “single-pixel holographic camera” instead of a camera tube was suggested . The method was later known as optical scanning holography (OSH) [6,7]. The location and area of the environment to be recorded are flexibly controlled by scanning mirrors that cast a time-varying Fresnel zone plate (FZP) on each point in the object scene, and capture the scattered waves with a single-pixel sensor such as the active area of a photodetector. With such a scanning mechanism, it is possible to capture a digital complex hologram of a scene that is considerably larger than the restricted field of view of a camera or two-dimensional (2D) sensors typically employed in digital holography. The OSH technology has been successfully applied in fluorescence microscopy [8,9], remote sensing , and three-dimensional (3D) image recognition [11,12]. However, the flexibility of OSH in imaging fine details of larger areas also implies that the data size and the acquisition rate of the holographic signal could be overwhelming. For example, consider the acquisition of the hologram at a video rate of 25 frames per second on a small area of around . When the size of the hologram is pixels and each pixel is represented by 32 bits (16 bits for each orthogonal component of a complex hologram), the data-rate will be more than 200 Mbits per second. In this paper, we propose a method, which we refer to as compressive optical scanning holography (COSH), to overcome this problem. COSH is vastly different from, and could easily been mistaken as, the existing hologram compression and compressive sensing technologies, although they share a similar objective of reducing the data size of the holographic signal. For hologram compression methods, such as those reported in [13 –15], compression is applied after the entire hologram has been captured. In general, the compression involves a substantial amount of computation to exploit and remove the redundant information in the holographic signal. For compressive sensing of holograms [16 –20], a single photodetector is employed to record the magnitude of the diffracted object waves that are sampled with a sequence of random patterns or a predefined path. The hologram or its reconstructed image is recovered from the compressed data with an optimization process that involves multiple rounds of iterations. Our proposed COSH method applies a nonuniform downsampling scheme that conducts the compression at the moment when each pixel is acquired by the single-pixel sensor. The decompression process is conducted with simple interpolation instead of optimization. As such, we would like to clarify that our proposed COSH method is different from compressive sensing. With our proposed framework, a hologram can be acquired at a data-rate that is one to two orders of magnitude lower as compared with the straightforward application of OSH, yet preserving favorable quality on the contents of the hologram. In addition, the amount of overhead introduced in the optical scanning process, and the computation involved in the recovery of the hologram at the receiving end, are negligible as the compression technique is noniterative. The organization of the paper is given as follows. Following the introduction, our proposed method is reported in Section 2. Experimental evaluation will be presented in Section 3, where we shall describe the principles of operation of the OSH technology, and how our proposed method can be incorporated into the framework. Finally, a conclusion of the paper will be given in Section 4.
2. PROPOSED COMPRESSIVE OPTICAL SCANNING HOLOGRAPHY
A. Overall View of the OSH and COSH Systems
As mentioned in Section 1, OSH is a technique whereby a 3D object scene is scanned by a time-varying FZP, and the diffracted wave at each instance of the scanning is recorded with a single-pixel sensor such as a photodetector. The output from the detector is a sequence of hologram pixels, each corresponding to a unique point on the hologram plane. A detailed outline of the optical setup of the proposed COSH capturing framework is shown in Fig. 1, and described as follows. To begin with, the laser beam of wavelength goes through a sawtooth-wave driven electro-optic modulator (EOM) at frequency . As the polarization of the input beam is 45° related to the crystal axis of the EOM, two orthogonal modes of lights are generated by the EOM. These two modes of lights, which are subjected to equal modulation but with inverse phase, are split into two beams by polarizing beamsplitter PBS1. Thus the frequency shift between the reflected beam and the transmitted beam is introduced. The transmitted beam is first collimated by beam expander BE2 and then provides a spherical wave on object through the focusing action by lens L2. The reflected beam is collimated by beam expander BE1 and focused to the front focal plane of lens L2 by lens L1. Hence a plane wave is projected onto the object. The transmitted and the reflected beams are combined by a beamsplitter (BS), eventually projecting the interference of the plane and spherical waves, which has become a time-varying FZP, on the object. The scanning of the object is done by mounting the object on an X–Y motorized scanner, as shown in the figure. The light retro-reflected from the object is reflected by polarizing beamsplitter PBS2 and converged to photodetector1 (PD1) by lens L3. Here PD1 is our single-pixel detector. PD1 sends an electrical signal containing holographic information of the scanned object to the lock-in amplifier. The bandpass filter (BPF) tunes in the electrical signal at frequency from PD1. In the meantime, photodetector2 (PD2) delivers a heterodyne signal at frequency , which is used as a reference signal for the lock-in amplifier. The lock-in amplifier delivers the in-phase (I-phase) output as well as the quadrature-phase (-phase) output, which become the sine hologram (imaginary component of the hologram), , and the cosine hologram (real component of the hologram), , respectively. The two holograms will form a complex hologram for twin-image-free reconstruction [6,7]. There are two major factors to note in the OSH system. First, the field of view is dependent on the related movement between the object and the scanning beam. In other words, a wider range of movement of the X–Y scanner will record a larger object scene. Second, the optical resolution is limited by , where NA is the numerical aperture of the spherical wave provided by lens L2.
For the original OSH setup, the hologram pixels are directly recorded at the output of the lock-in amplifier. With our proposed COSH method, the hologram pixels are selected and binarized with a method referred to as “nonuniform sampling delta modulation,” which is represented by the adaptive nonuniform sampling delta modulation (ANSDM) module in Fig. 1. The ANSDM module evaluates the stream of output signals that are acquired by the single-pixel sensor. Through this process, only a small amount of hologram pixels are selected, and binarized to form the reduced output bit-streams and that correspond to the in-phase and quadrature-phase outputs of the lock-in amplifier, respectively. As the optical scanning mechanism is the same for both OSH and COSH, the original and the compressed holograms are recorded concurrently with a single scan of the object. The ANSDM process involves repetitively using a pair of steps, one conducted along the horizontal scan path, and the other along the vertical scan path. We shall describe this pair of steps, and the recovery of the hologram from the compressive data, in the following subsections.
B. First Step: Pixel Selection along the Horizontal Scanning Direction
The pixel selection along each row of the scanned hologram signal is based on a kind of predictive coding technique known as ANSDM [21,22]. Past research has demonstrated that the method is effective in providing a moderate amount of compression to a one-dimensional (1D) binary output bit-stream. However, when dealing with holographic data, there exists a certain smoothness between several horizontal lines. The rationale is that in holograms the “image” is made up of an interference pattern, which kind of “looks” random, or the information presented seems to more or less spread out evenly, and for this reason, the term “smoothness” mentioned earlier has been used. For this reason we believe ANSDM should work well with holographic data.
The concept of ANSDM can be illustrated with the flowchart shown in Fig. 2(a). For simplicity of explanation, we shall only describe the formation of the output bit-stream for a single row of pixels of the real component of the complex hologram, with the understanding that the imaginary output bit-stream, as well as other rows of hologram pixels, are obtained with the same procedure. Furthermore, we have employed a simple example based on a short input sequence of data denoted by , as shown in Fig. 2(b), representing the real part of six hologram pixels that have been acquired in a row of scanning. Each sample in is quantized to 16 bits. A fixed quantity, , known as the “step-size,” is defined, and displayed together with the sequence in Fig. 2(b). In this example, we have assigned an arbitrary value of 5 to the step-size . The sequence of hologram pixels is to be compressed into a short binary output bit-stream , where . From the output bit-stream, an approximate version of the source sequence, which is referred to as the “decoded sequence,” will be reconstructed. All elements in the sequences and are indices with the symbol . The process of applying the method to the input sequence is further explained as follows. In the coding process of ANSDM, samples in the input sequence are selected by a decision maker (which we shall explain later) to form the output bit-stream. If the amplitude of a selected sample is not less than the previous sample in , a bit “1” will be added to ; otherwise a bit “0” will be included instead. By default, the first element in is always selected and contributed to the output bit-stream. In the current example, we have assumed that all the samples before and (i.e., and , which are hypothetical data in a casual system) are zero.
Referring to the figure we observe that the first sample (which is always included) is larger than , and a bit “1” is added to the output bit-stream as the element . Next, the first element of the decoded sequence is obtained by adding to the previous sample of , as shown in Fig. 3(a). As is set to zero at the start of the ANSDM process, we have . Subsequently the next sample in to be evaluated is determined by the decision maker. Suppose the decision is to skip the next sample , so that is to be evaluated. We note that is larger than the previous decoded sample , so a bit “1” is appended to the output bit-stream as the element . The corresponding decoded sample is determined by adding to , resulting in as depicted in Fig. 3(b). The decision maker is applied again, and suppose for this round the next two samples are skipped and the evaluation is passed to the last sample , which is smaller than . A bit “0” is added to as the element , and the decoded value is obtained by subtracting from , resulting in . The final results showing the compressed bit-stream, i.e., , and the decoded sequence, i.e., are shown in Fig. 3(c). It can be seen that the original sequence has been compressed into a 3-bit binary number in the output bit-stream . An approximation of the source sequence can be reconstructed with simple interpolation on the samples in , resulting in the dotted curve overlaying on the samples of the decoded sequence as shown in Fig. 3(c).
From the above descriptions, it can be inferred that the encapsulation of the input sequence into the output sequence will result in compression. In the above example, we have six samples in , each quantized into 16 bits, and 3 bits in the output bit-string . The compression ratio will be times. In general, the compression ratio increases with larger step-size as the output bit-stream can track the input signal with fewer sampling points. It is also noted that the compression ratio is not a function of the sampling interval and is based on the number of samples that are recorded, as compared with the total number of samples. As such the distance between sample points is not needed.
C. Decision Maker
The decision maker is formed by the two decision blocks that are enclosed in the dotted rectangle at the bottom part of the flowchart in Fig. 2(a). Suppose that and denote the position of the latest pixel that has been selected in the hologram sequence and the sampling interval between the current and the next selected samples, respectively. The next pixel to be selected will be the one at the scanning instance . All pixels in between will not contribute to the output bit-stream. The decision maker determines the value of , with set to unity. After evaluating the current hologram pixel, the sampling interval is updated in the decision maker based on the current and the past two samples in the output bit-stream as follows:1) determines the sampling interval and Eq. (2) recovers the hologram samples from the output bit-steam . To further illustrate the operation of the ANSDM, we have listed, in Table 1, the step-by-step the formation of the output bit-stream of the previous example based on the short sequence in Fig. 2(b), and step-size . The rows highlighted in gray in each column of the table depict the input data of the decision maker, and the corresponding sampling intervals that are determined as a consequence.
D. Second Step: Downsampling along the Vertical Direction
Referring to the previous description, it can be inferred that after a row of hologram pixels is processed, an -bit output bit-stream will result. The quantity “” is the number of sample points along the row of hologram pixels, which also reflects the smoothness of the holographic signal. Fewer sample points indicates that the row of hologram pixels is generally smooth, so that it can be represented with sparsely spaced sample points, and vice versa. On this basis, we posit that if the holographic signal is smooth along the horizontal direction, it is also likely smooth along the vertical direction. As such, if is smaller than a given threshold , the next (where ) row(s) of hologram pixels will be considered to be similar to the current row of pixels, and will be skipped and excluded from the output bit-stream. This mechanism is equivalent to adaptive downsampling along the vertical direction. Apparently, whenever a row of the hologram pixels is excluded in the scanning process, the time taken to acquire the hologram will also be decreased proportionally. A status bit (where denotes the current scan row) is appended as the last element of the output bit-stream of a row of pixels that has been scanned, with and indicating that the next row of pixels will be excluded, or included, from the output bit-stream, respectively. A general expression of the compression ratio CR can be established as given by3, the downsampling processes along the horizontal and vertical directions, together with the binarization of the sampled hologram samples, will lead to a reduction of one to two orders of magnitude of the data size of the hologram using our proposed noniterative compressive technique.
E. Hologram Recovery Process
At the receiving end, the holographic data acquired with our proposed COSH method can be recovered in a row-by-row manner. Before decoding the “th” row of pixels, the status bit of the last row is examined. If the status bit , the next row(s) of decoded hologram pixels will simply duplicate the last row. Otherwise, the output bit-stream of the current row is evaluated sequentially, starting from the first bit with the process shown in Fig. 4. Referring to the flowchart, recovery of the amplitude of the sample points and its position “” from the encoded data in the previous example we employed in Section 2.B is shown in Figs. 5(a)–5(c). Briefly, the value of the first hologram pixel is obtained by adding the step-size “” to the value of the previous hologram pixel (which is set to “0”) as . Next, the current and the last two bits of the output bit-stream are passed to the decision maker [Eq. (1)]. The result determined by the decision is , indicating that the next sample to be recovered should be located at . In Fig. 2(b), we observe that , and hence is obtained by adding the step-size to . The process repeats until an entire row of hologram pixels has been recovered. For the sake of clarity, we also have listed the detailed steps of recovering the sequence from the bit-stream in Table 2. Similar to Table 1, the part associated with the decision maker is highlighted in gray. After the sequence of recovered sample points is determined, the missing gaps between the samples are reconstructed with simple bilinear interpolation.
Finally, we would like to provide some insight into the effect of the step-size on the compression ratio and the quality of the compressed hologram. From Eq. (2), each sample in the decoded sequence can be expressed as
3. EXPERIMENTAL RESULTS
The experimental setup of the proposed COSH system in Fig. 1 is outlined as follows. The wavelength of the laser beam is 633 nm, and the EOM is driven by a 10 KHz sawtooth wave, giving a heterodyne signal of 10 KHz. The size of the time-varying FZP light beam at the object plane is 3 mm in diameter, and the effective NA of the system is 0.01.
To evaluate the effectiveness of our proposed method, a optical hologram representing the image of a stack of two coins is captured with the optical settings listed in Table 3. For this sample, the pixel size of the hologram is . The stacked coins are parallel to and located at a distance of and from the back focal plane of lens L2, thereby giving . Note that the NA of the optical system in this case is small in that the two coins are resolved basically at the same plane along the direction. The cosine and the sine holograms, and the reconstructed image at the focused plane, are shown in Figs. 6(a)–6(c), respectively.
Next, we have applied our proposed COSH method to capture the hologram of the coins, based on six different values of step-sizes () that are set from 0.08 to 0.03, in steps of 0.01, of the dynamic range of the hologram. As mentioned previously, the holograms of OSH and COSH are acquired concurrently with the same scanning. In all three cases for COSH, the number of skipped lines is set to 2. The threshold is assigned to be half the number of pixels (i.e., 256 for a hologram) in a row of the hologram. As such, if the number of sample points in the last scan row is less than 256, the next two rows will be skipped. In Table 4 we show, for each step-size, the compression ratio of the reconstructed image (taking into account the inclusion of the status bits in each scanned row) and the number of lines that have been scanned. We have also included, in the last column of Table 4, the correlation scores that quantitatively reflect the similarity between the reconstructed image of the compressed COSH hologram and that obtained with the traditional uncompressed OSH. From the table, we have observed that the compression ratio is increased with step-size, while the fidelity is decreased accordingly. In addition, the number of lines that has been scanned is inversely proportional to the step-size. The reconstructed images at the focused plane for step-sizes of 0.08, 0.05, and 0.03 are shown in Figs. 7(a)–7(c), respectively. It can be seen that despite the degradation, the visual quality of the reconstructed image is still favorable even at a high compression ratio of more than 100 times.
To illustrate 3D imaging, a similar evaluation is conducted on the digital hologram of a pair of Chinese characters that are separated by 3 mm. In this case, the hologram has been acquired in the transmission OSH geometry, with the experimental setup that has been described in more detail in . The diameter of the scanning beam is about 12 mm, and the effective NA is about 0.06. The size of the hologram pixel is . The size of the hologram is , and the cosine and sine holograms, together with the reconstructed images at the focused planes, are shown in Figs. 8(a)–8(d). In Figs. 8(c) and 8(d), we have observed that when one of the Chinese characters is in focus, the other one is defocused and blurred in appearance. Similar to the first example, three COSH holograms are captured with step-sizes from 0.08 to 0.03, in steps of 0.01. The compression ratios, and the number of lines being scanned, are listed in Table 5. Quantitative evaluations based on the correlation scores that reflect the similarity between the reconstructed images of the compressed COSH hologram and the original OSH hologram at the two focused planes, are listed in the last column of the table. It can be seen that favorable visual quality and high correlation scores are preserved at a high compression ratio of more than 100. The reconstructed images at the two focused planes obtained for step-size , 0.05, and 0.03 are shown in Figs. 9(a)–9(f).
In this paper we presented an enhanced framework for OSH technology, which we referred to as COSH. Being different from standard OSH, we have applied a nonuniform sampling scheme so that only a small number of hologram samples are selected to represent the hologram. In addition, each selected hologram sample is converted into a single-bit binary value, which further reduces the data size of the hologram. From the compression ratios obtained with different step-sizes as shown in Tables 4 and 5, we note that with a step-size of of the dynamic range, and , the COSH method is capable of acquiring a hologram with less than 1% of its original data size, and still preserving good fidelity on its contents. It is also important to note that apart from the compression, our proposed method also decreases the number of lines that need to be scanned. As the time taken to acquire an OSH is proportional to the number of lines being scanned, our method is capable of speeding up the hologram acquisition process. On top of the above favorable factors, the sampling process involves a negligible amount of computation, and is conducted at the moment when a hologram pixel is acquired in the scanning process. Likewise, the hologram can be recovered from the compressed bit-stream with a very small number of arithmetic operations. As such, COSH incorporates a low-complexity, symmetric compression (i.e., similar computation efficiency for both the compression and the decompression processes) mechanism into the original OSH framework. We anticipate that the favorable features of our proposed COSH method should bring forth significant advancements in the realm of large-scale dynamic hologram acquisition, where the size of the hologram has to be increased substantially to represent a wide-field object scene. While our proposed method is particularly suitable for the OSH system, we anticipate that it is also useful in providing on-the-fly compression on holograms that are generated at video rate (e.g., [24 –26]).
Ministry of Science and Technology of Taiwan (MOST-103-2221-E-035-037-MY3).
The authors thank Mr. Chieh-Cheng Lee and Mr. Wei-Jen Siao for their help in performing the OSH experiments.
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