Comparison of adaptive optical scanning holography based on new evaluation methods

Adaptive Optical Scanning Holography (AOSH) represents a powerful technique that employs an adaptive approach to selectively omit certain lines within holograms, guided by the utilization of Normalized-Mean-Error (NME) as a predictive measure. This approach effectively diminishes scanning time and conserves the storage space required for data preservation. However, there exists alternative methods superior to NME in terms of evaluating the model’s efficacy. This paper introduces two novel methods, namely Normalized-Root-Mean-Square-Error (NRMSE) and Normalized-Mean-Square-Error (NMSE), into the AOSH system, leading to the development of NRMSE-AOSH and NMSE-AOSH. These new systems aim to further minimize duration of holographic recording. Through a comparative analysis of hologram lines between the two newly proposed AOSH systems and the original AOSH, we demonstrate that both NRMSE-AOSH and NMSE-AOSH effectively reduce the number of hologram lines while maintaining the hologram’s informational content. Among the three methods, our two new methods exhibit better performance compared with the original method.

(RMSE) are two commonly used evaluation methods in the analysis of experimental results.In this paper, we aim to demonstrate the performance of AOSH using these two new methods, and compare with the original NME approach.Following this introduction, we present the experimental methodology in Section 2. Section 3 showcases the experimental results and evaluations, comparing the three AOSH approaches using different error evaluation methods.We highlight and compare the results obtained from the three AOSH methods.Finally, Section 4 provides discussions summarizing the findings of the study.

Overall view of the OSH system and AOSH system
Prior to introducing the AOSH system, it is necessary to provide a concise overview of the OSH system.As a comprehensive explanation of OSH can be found in numerous existing literature 6,16,17 , only a succinct elucidation of the OSH principle will be presented.The experimental configuration of OSH is depicted in Fig. 1.The emitted laser of wavelength = 532 nm is split into two by beam splitter BS1.The temporal frequencies of two beams are modulated into ω 0 +� and ω 0 by acoustic-optic modulator1 (AOM1) and acoustic-optic modulator2 (AOM2) , respectively.Thus, the heterodyne frequency between these two beams is introduced.The upper beam is first collimated by beam expander2 (BE2) and then provides a spherical wave on object I 0 (x, y; z) through the focusing action by lens1 (L1).The other beam is collimated by beam expander1 (BE1); hence a plane wave is projected onto the object.The spherical wave and plane wave are combined by beam splitter2 (BS2), generating a heterodyne interference pattern on the scanning mirror.The interference pattern is known as a time-dependent Fresnel zone plate (TD-FZP) 7 .The TD-FZP oscillates at .The scanning of the object is done by the scanning mirror, which can scan the 3D object uniformly in a row-by-row manner.The scattered light transmitted through the object is converged to photodetector1 (PD1) by lens2 (L2).The PD1 collects the light and sends the electrical signal containing the holographic information of the scanned object to the bandpass filter (BPF) which tunes to the electrical signal at frequency .Next, the signal from BPF goes into the lock-in amplifier.In the meantime, photodetector2 (PD2) delivers a heterodyne signal into a Lock-in amplifier as a reference signal.Finally, a complex hologram can be obtained by combining the in-phase output and the quadrature output of the lock-in amplifier.The in-phase and the quadrature-phase outputs of the lock-in amplifier give a sine hologram H sin (x, y) , and cosine hologram H cos (x, y) as follows 7 : and (1) where I 0 (x, y; z) denotes the intensity distribution of the 3-D object, is the wavelength of light in free space, and * denotes 2-D convolution involving x and y 6 .According to Eqs. (1) and (2), the resulting complex hologram H(x, y) in the computer can be expressed as In the OSH configuration, the hologram pixels are obtained sequentially in a line-by-line manner from the lock-in amplifier, synchronized with the movement of the TD-FZP.The AOSH method introduces a selection mechanism for the hologram lines, employing an error evaluation approach to estimate the level of "smoothness" between a pair of hologram lines.This evaluation facilitates the identification and omission of redundant information within the hologram lines.The scanning mechanism of AOSH is illustrated in Fig. 2. The key aspect of the technique lies in the fact that AOSH adjusts the gap between hologram lines by calculating the error evaluation.
The concept of AOSH is explained as follows.We denote the position of rows that will be scanned with the sequence S = s(j) 0≤j<r , where j is the index of S, s(j) is the position of the jth scan row, and r is the total scan rows.The expression of the hologram line located in s(j) is H(x, s(j)) , and its previous row and next row is H(x, s(j − 1)) , and H(x, s(j + 1)) , respectively.The separation between the hologram line at s(j) and the holo- gram line at s(j − 1) and s(j + 1) is denoted as j−1 and j , respectively.Initially, H(x, s(0)) and H(x, s(1)) need to be acquired.Suppose the current scanning hologram line is H(x, s(j)), and the previous hologram line is H(x, s(j − 1)) stored in the buffer.In AOSH, We calculate the error evaluation between the H(x, s(j − 1)) and H(x, s(j)), which can reflect the "smoothness" of a pair of hologram lines, to predict the position of the next scan row s(j + 1) .The predictor estimates j , which is the most significant aspect of AOSH, always need the error evaluation to measure the similarity between the previous hologram line and the current hologram line.As the error evaluation between H(x, s(j − 1)) and H(x, s(j)) is smaller, the gap between s(j) and s(j + 1) becomes wider.The separation between the current and the next scan row is decided by the Predictor as where NEE j denotes the normalized error evaluation of jth scan row, which range is 0 to 1, s and MIN are the factors which decide the scanning speed and the hologram quality.The next position of scan row is expressed as It can be seen that a small NEE j results in large j , and the separation between the current scan row and the next scan row becomes wider.The steps are iteratively performed until the final row of the object scene has been scanned.After capturing all the hologram lines, the regions between adjacent hologram lines are filled with bilinear interpolation as shown in Fig. 3. Given 2 adjacent hologram lines H(x, s(j)) and H(x, s(j + 1)) , the miss- ing hologram line H(x, m) at vertical position 'm' (where s(j) < m < s(j + 1) ) between them is determined as where a and b denote the vertical distance between H(x, m) and H(x, s(j)) and H(x, m) and H(x, s(j + 1)) , respectively.The term b a+b and a a+b are the weight factors, which represent the contribution of H(x, s(j)) and H(x, s(j + 1)) to H(x, m), respectively. (3) Figure 2. Concept of the AOSH scanning mechanism 17 .

NRMSE and NMSE in AOSH
In the preceding section, we have discussed the utilization of normalized error evaluation (NEE) in AOSH, as depicted in Fig. 2. It is worth noting that the reference 17 introduces AOSH for the first time 17  In the following equations, we denote H(x, s(j)) as the current hologram line; H(x, s(j − 1)) is the previous hologram line which has been stored in the buffer.x represents the coordinate position of the pixel on the hologram line, and X is the total number of pixels on the hologram line.NMSE j and NRMSE j denote the NMSE and NRMSE between the (j − 1) th and (j) th scan row.
The NMSE and NRMSE in AOSH can be respectively expressed as and the NME which is the original method in the AOSH can be expressed as 9 The NMSE and NRMSE ,which both are bounded within the range [0,1], compute the average difference between correspondence pixels between 2 consecutive rows of hologram pixels.However, the two methods do not evaluate the error in the same way.Because NMSE squares the difference between pairs of hologram lines, it will give more penalties to the errors between hologram lines.In other words, NMSE-AOSH is easier to observe the smoothness between holograms, which will help AOSH to skip some similar information more adaptively.www.nature.com/scientificreports/For instance, for the same group of holographic line pairs, if a pair of hologram lines corresponding pixels change smoothly, where the average absolute error is less than one ( 1 X X−1 x=0 |H(x, s(j)) − H(x, s(j − 1))| < 1 ), the value of NMSE j is smaller than that of NME j because the average absolute error is squared, which means that j calculated by NMSE is larger than that calculated by NME, as shown in Eq. ( 2).In other words, the smoothness between the hologram line pairs can be amplified by NMSE, and the AOSH scanned lines will be reduced.The NRMSE, derived from the square root of NMSE, shares a similar characteristic with NMSE and is adept at detecting subtle variations within the data.Therefore, NRMSE-AOSH can also provide an enhancement to the AOSH system.

Results
In this section, we will assess and compare the performance of NMSE and NRMSE with the original method (NME) in AOSH.The experimental setup in Fig. 1 is outlined as follows.The wavelength of the laser beam is 532 nm, and the center frequency of AOM1 and AOM2 is 120 MHZ, providing a heterodyne signal of 10 kHz.To facilitate a comprehensive evaluation of these methods, we have employed three AOSH approaches to capture holograms of two objects: the United States Air Force resolution chart and a dice.The object United States Air Force resolution chart is a transmissive object and located at around 150 mm from the scanning mirrors.Its physical size is 20 mm × 20 mm.The dice is a reflection type object located at around 320 mm from the scanning mirrors.Its physical size is 25 mm × 25 mm.The classical OSH technique is utilized to record holograms of the two objects.The physical parameters in the experiment are outlined in Table 1.
The cosine and the sine holograms of the 2 objects, and their reconstructed images at the focused plane, are shown in Figs.4a-d and 5a,b, respectively.Next, we apply AOSH method using NRMSE and NMSE respectively   to capture the hologram of these two objects, based on MIN = 1 , s is changed from 2 to 7. In the case of a fixed MIN , as the value of s increases in AOSH, the number of scanned lines decreases, leading to a decrease in the quality of the reconstructed image.On the other hand, the variability of s serves as an indicator of the robustness of NRMSE-AOSH and NMSE-AOSH in the context of AOSH.To assess and compare the performance of NRMSE-AOSH and NMSE-AOSH with the original method (NME-AOSH), we employ the NME-AOSH technique to capture holograms of the two objects under identical experimental conditions.The reconstructed images obtained from three different methods of AOSH are depicted in Fig. 6.It is evident that the reconstructed images achieved through NRMSE-AOSH and NMSE-AOSH (Fig. 6a-d) closely resemble the image obtained using the original approach (NME-AOSH) (Fig. 6e,f).Apart from the visual inspection, we evaluate the difference in compression performance between the two new methods and AOSH based on the original method (NME-AOSH).The compression performance is evaluated by two aspects: the fidelity of the reconstructed image and the number of scanned lines for the hologram.The fidelity of the reconstructed images is measured in Peak-Signal-to-Noise-Ratio (PSNR) to compare with the reconstructed image of the holograms acquired with classical OSH.We then evaluate improvement of compression rate (scanned lines) between new methods (NRMSE-AOSH and NMSE-AOSH) and the original method (NME-AOSH) by evaluating the ratio of the difference between the number of scanned lines in original method and the two new methods to the number of scanned lines in the original method.The compression rate R is expressed as For the dice and United States Air Force resolution chart, the compression performance of the two new methods (NRMSE-AOSH, NMSE-AOSH) and the original method (NME-AOSH) in different s are listed in Tables 2 and 3.
According to the results in Tables 2 and 3, we conclude that the new method of AOSH (NRMSE-AOSH, NMSE-AOSH) have a better compression performance compared to the original method of AOSH (NME-AOSH) since the number of scanned lines required by the new method for AOSH is less than that of the original method (where R is 15-23% for the dice and 38-56% for the United States Air Force (USAF) resolution chart).This is not surprising because the USAF chart tends to have more similarity than the dice because it is made up of regular groups of horizontal strips and vertical strips.Therefore, R is object-dependent.We also note that R corresponds to the reduction of time by the same rate, assuming the time to acquire successive y-scan is the same, which is true in practice.The degradation of the reconstructed image of the two new methods, however, is extremely close to that of the original method.Also, as we previously mentioned, the factor s in the AOSH system decides the quality of the reconstructed image represented by the scanned lines.
For a given object and method, a larger s corresponds to a reduced number of scanned lines in AOSH.The rate R exhibits slight variations when modifying s within the same AOSH method, indicating the stability and validity of the proposed new AOSH techniques.Moreover, it is noteworthy to note that the proposed two methods display markedly distinct R values when applied to two different types of objects (reflective and transmissive), providing further evidence of the adaptability of the new methods in selectively capturing pertinent information.

Discussion
We have presented two enhanced methods for AOSH technology, which we refer to as NRMSE-AOSH and NMSE-AOSH.The major difference between the new methods (NRMSE-AOSH, NMSE-AOSH) and the original AOSH (NME-AOSH) is that the two of new methods use a better error evaluation method in model evaluation instead of using NME.Theoretically, both NRMSE and NMSE could better show the similarity between hologram lines, thereby improving the compression performance of AOSH.As such, both the overall time required to scan the scene and the storage space needed are reduced in NRMSE-AOSH and NMSE-AOSH.The improvement in the hologram acquisition process is extremely important for wide-field applications, in which case lengthy capturing time in the original AOSH method is needed.We have evaluated our proposed new (
by employing Normalized-Mean-Error (NME) as a specific NEE method.NME is the normalized Mean-Absolute-Error(MAE) and owns the same characteristics as MAE.Nevertheless, it has been suggested that Root-Mean-Square-Error (RMSE) exhibits superior performance to MAE in the evaluation of models.MAE is affected by a large number of average error values, and cannot fully reflect some large errors compared with RMSE19  .Considering this, we hypothesize that normalized Root-Mean-Square Error (NRMSE) may exhibit superior performance within the AOSH framework.Additionally, Mean-Square Error (MSE) represents another commonly used error evaluation method, capturing the quadratic nature of errors and providing detailed insights into error analysis results.Furthermore, Normalized Mean-Square Error (NMSE) serves as a normalized variant of MSE.Both NRMSE and NMSE present alternative approaches to normalized error evaluation (NEE) within the AOSH context.In this section, we will introduce the specific formulations of these two error evaluation methods in AOSH and elucidate their significance.To facilitate clarity, we will refer NRMSE-based AOSH as NRMSE-AOSH, NMSEbased AOSH as NMSE-AOSH, and NME-based AOSH as NME-AOSH.

Figure 3 .
Figure 3. Filling a row of pixels between a pair of hologram lines.

Figure 4 .
Figure 4. (a) Cosine hologram of the dice; (b) Sine hologram of the dice; (c) Cosine hologram of the United States Air Force resolution chart; (d) Sine hologram of the United States Air Force resolution chart.

Figure 5 .
Figure 5. (a) Reconstructed image of the dice; (b) Reconstructed image of the United States Air Force resolution chart.

Figure 6 .
Figure 6.(a, b) Reconstructed image of the dice and the resolution chart with NRMSE-AOSH; (c, d) reconstructed image of the dice and the resolution chart with NMSE-AOSH; (e, f) reconstructed image of the dice and the resolution chart with original method (NME-AOSH).

Table 1 .
Parameters in the OSH/AOSH acquisition process.

Table 2 .
Compression performance of the reconstructed image of the dice.The symbol "-" means that the original method does not compare with itself.

Table 3 .
Compression performance of the reconstructed image of the United States Air Force resolution chart.The symbol "-" means that the original method does not compare with itself.