Comparative appraisal of global and local thresholding methods for binarisation of off-axis digital holograms

Optics and Lasers in Engineering 115 (2019) 119–130

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Comparative appraisal of global and local thresholding methods for binarisation of off-axis digital holograms Pavel A. Cheremkhin∗, Ekaterina A. Kurbatova National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe Shosse 31, Moscow, Russia

a r t i c l e

i n f o

Keywords: Digital holography Hologram compression Image binarisation Thresholding Digital micromirror device Digital image processing

a b s t r a c t Binary digital and computer-generated holograms are used in a number of digital micromirror device (DMD) applications, including holographic displays, media characterisation, optical encryption, and others. Binarisation is one of the most simple and efficient methods of hologram compression. In this paper, 12 local and 18 global thresholding techniques of different groups were analysed and compared based on attribute similarity, clustering, entropy, histogram shape, and local adaptive factors. Optically recorded off-axis digital holograms of different objects were binarised using these methods. The amplitude quality of the obtained reconstructed images was compared using PSNR and SSIM values. The clustering-based methods achieved the highest quality. The results of binarisation by global and local methods were comparable on average.

1. Introduction Digital holography is a 2D- and 3D-imaging technique with high spatial and temporal resolution [1–4]. A digital hologram is an image of the interference pattern formed by an object and reference beams. Photo and video cameras with digital photosensors (CMOS, CCD, Foveon X3, and other types) are used to record digital holographic images. Then the obtained digital holograms are processed. An object’s image can be reconstructed numerically (by model propagation of the illumination from the camera’s photosensor plane to the required plane) [5–7] and optically (by displaying the hologram and its illumination on spatial light modulators) [8–11]. The two most popular types of spatial light modulators are liquid crystals and digital micromirrors. Each class has a number of advantages and drawbacks. Liquid crystal spatial light modulators usually provide 256 grey levels or 8 bits and a 60 Hz frame rate [8]. Digital micromirror devices (DMDs) display binary images or holograms but have significantly higher frame rates (up to approximately 40 kHz) [9]. Greyscale images can be displayed using DMD via pulse width modulation [12–13]. But the frame rate is sufficiently decreased [13], and the quality of the reconstructed images from greyscale holograms usually does not exceed binary holograms [14–15]. As a result, binary holograms are more useful for the majority of DMD applications. Binarisation of holograms is required for different optical DMD and digital applications: holographic displays [16–17], scattering media characterisation [18], optical correlators [19], and encryption and watermarking [20], among others. Binary hologram files are smaller than greyscale files and can be rapidly printed [21].

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The first attempts at the binarisation of digital or computergenerated holograms were undertaken by A.W. Lohmann et al. [22–23]. There are presently many methods of hologram binarisation. The fastest and most popular types of methods are based on thresholding [24] the digital images: •

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Local thresholding methods (Niblack [25], Sauvola [26–27], Bradley [28], and others [29–33]) Global thresholding methods (Glasbey [34], Otsu [35], Kapur [36], and others [37–50])

Some of these methods were used for hologram’s binarisation: uniform quantisation [51–54], including Otsu’s method [55–56], Niblack’s method [55], and others. There are several other methods of hologram binarisation based on error diffusion techniques [57–59], using iterative methods of kinoform synthesising [51,60–63], direct binary search algorithms [64–66], sampling [67–70], vector quantisation [54,71], wavelet transform [72], and others. In the literature, a number of papers comparing the binarisation methods are provided for documents [73–74], nondestructive testing images [74], historical archive documents [75], map images [76], Gaussian distributions [28], and others. Many papers compared several methods of producing computer-generated holograms [72,77–79]. However, local and global thresholding methods were not compared for hologram binarisation. Moreover, computer-generated holograms are usually used for binarisation whereas optically recorded digital holograms are not. In this paper, off-axis digital holograms were binarised by various thresholding methods. Holograms of different types of objects were used: almost flat and 3D objects, and transmissive and reflective objects. The amplitude quality of the reconstructed images is compared

Corresponding author. E-mail address: [email protected] (P.A. Cheremkhin).

https://doi.org/10.1016/j.optlaseng.2018.11.019 Received 30 September 2018; Received in revised form 21 November 2018; Accepted 22 November 2018 0143-8166/© 2018 Elsevier Ltd. All rights reserved.

P.A. Cheremkhin and E.A. Kurbatova

Optics and Lasers in Engineering 115 (2019) 119–130

Fig. 1. Fragments of an original experimentally recorded hologram (128 × 128 pixels) before (a) and after binarisation using Huang’s (b), Kittler-Illingworth’s (c), Shanbhag’s (d), Bradley’s (e), and Doyle’s (f) methods.

for all of the objects and thresholding methods. Binarisation methods are described in Section 2. Reconstructed images from holograms binarised by these methods are compared in Section 3. Examples of the obtained images, peak signal-to-noise ratios (PSNR), and structural similarity indexes (SSIM) are presented. The Conclusion provides the main results. 2. Digital hologram binarisation by thresholding The most popular and efficient methods of digital image binarisation by thresholding were chosen for hologram processing. They can be conditionally divided into two groups: global and local. The threshold value remains invariable during binarisation via global methods and is the same for the whole image. In local binarisation, the image is separated into a number of areas with their own threshold values. According to [74], various thresholding methods can be unified into several groups. This paper will follow this classification.

Fig. 2. Fragments of experimentally recorded holograms (a, d, g, and k), sections of the corresponding reconstructed fields (b, e, h, and l), and object images at an average depth of the 3D scene (c, j, and m) and at 3 distances unified in one picture (f).

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2.1. Global thresholding methods Groups of global methods are defined by the following: use of all image histograms (histogram shape-based methods), dividing images into two parts (clustering- and entropy-based methods), and similarity measures (attribute similarity-based methods). 2.1.1. Histogram shape-based thresholding methods This group includes binarisation methods that analyse histogram intensity peaks or other image features. The following methods are considered: •

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histograms contain extremely unequal peaks with strongly different intensities or widths. Zack’s triangle method [38]: This is based on a geometrical approach and does not define skewed histogram data without referencing histogram use. The maximum peak is allocated near one end of the histogram, and the second peak is searched near the other end of the histogram. Thus, problems with the appearance of a maximum not near the histogram peak are solved. In this implementation, the side of the maximum peak with more data is defined. The threshold value is determined in the corresponding range.

2.1.2. Clustering-based thresholding methods In these methods, all of the image pixels are clustered to an object group and to a background group. These two groups are represented by a mixture of two Gaussians. The following methods are used:

Doyle’s method [39]: The threshold is calculated as the image histogram median value. It assumes the fraction of object pixels to be 0.5. Glasbey’s method of mean over histogram [34]: The threshold is the mean value of the image intensity histogram. This method is often used as the initial approximated threshold value for more sophisticated thresholding algorithms. Prewitt’s two methods [37]: These methods assume the use of a bimodal histogram. In both cases, the image histogram is iteratively smoothed to define two local maximums of image histogram values. In Prewitt’s first method, the threshold value is calculated as the average value of the defined local histogram maximum values. In Prewitt’s second method, the threshold value must satisfy a certain inequality. These methods provide poor image quality if the

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Kittler and Illingworth’s method [41]: This is based on minimum error thresholding. The object and background components are divided by comparison with an image brightness threshold value. Both are modelled by Gaussian distribution. The obtained value is a mixture of these two Gaussian distributions. Lloyd–Max method [42–43]: This is based on criterion of a minimum of a mean squared mistake. For a given density function of image values, the mean squared error of thresholding for the obtained borders of an object image and background areas is iteratively calculated while mistakes will not achieve the given accuracy of an approximation. The parameters are the density function of count-

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Optics and Lasers in Engineering 115 (2019) 119–130

Fig. 3. The PSNRs of the reconstructed object images from the hologram (shown in Fig. 2a) binarised using various thresholding methods.

Fig. 4. The SSIMs of the reconstructed object images from the hologram (shown in Fig. 2a) binarised using various thresholding methods.

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age intensity is used as the initial threshold value. The mean values of the object and the background class values are calculated. Then the average values of the pixels at/below and above the threshold are calculated. In the next algorithm step, the threshold value increases and the average values are calculated. This procedure repeats until the threshold value does not exceed the composite average. Two implementations of Ridler–Calvard’s method are considered in this paper. 2.1.3. Entropy-based thresholding methods The entropy of the distribution of the levels of image brightness and the difference in entropy between an object and the background is used for these methods. The following entropy-based methods are considered in this paper: Fig. 5. The reconstructed object images from the hologram (shown in Fig. 2a) binarised using two implementations of Ridler-Calvard’s (a–b), Otsu’s (c), Feng’s (d), Prewitt’s minimum (e), and Bradley’s (f) methods.

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ing, the accuracy of an approximation, and the maximum number of iterations of an algorithm. Otsu’s method [35]: This is one of the most efficient methods of global binarisation both in terms of quality (mistakes of less than 30%) and processing speed. The threshold value is defined as a weighted sum of variances of the two image classes (object and background). The threshold value is the whole value from 0 to the maximum value of brightness. The best results are achieved when processing images with approximately equal ratios of object and background components. Otsu’s method is one of the most widespread image thresholding algorithms. However, its disadvantages include the degradation of lines, the "coalescing" of objects, especially where they cross, and the loss of thin lines. Ridler–Calvard’s method [40]: This method is based on the iterative separation of object and background parts of an image. The mean im-

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Kapur, Sahoo, and Wong’s maximum entropy thresholding method [36]: The image background and object parts are considered two different signal sources. As a result, the image is thresholded when the sum of the two class entropies reaches a maximum. Two implementations of this method are used in this paper. Li’s minimum cross-entropy thresholding method [47–48]: The thresholding is defined as the minimisation of the distance between an object’s average level of brightness and the background image areas in the initial and thresholded images. Renyi’s entropy thresholding method: This is based on Kapur, Sahoo, and Wong’s maximum entropy method [44,74] but uses Renyi’s entropy calculations. Shanbhag’s fuzzy entropy thresholding method [45]: This method considers the fuzzy indication of the value image brightness that belongs to the background or the object image parts. The farther the brightness level is from an estimated threshold, the higher the potential it belongs to a specific class. Yen’s entropy thresholding method [46]: This is based on a maximum correlation criterion that is a more computationally efficient alternative to entropy measures.

Fig. 6. The PSNRs of the reconstructed object images from the hologram (shown in Fig. 2d) binarised using various thresholding methods. 122

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Fig. 7. The SSIMs of the reconstructed object images from the hologram (shown in Fig. 2d) binarised using various thresholding methods.

2.1.4. Attribute similarity-based thresholding methods These methods define the threshold value based on some attribute quality or similarity measure between the initial and the thresholded images. The following methods are used: •

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Huang’s fuzzy thresholding method [50]: This is a proposed computing index of fuzziness by measuring the distance between the greyscale image and its thresholded binarised version. The optimum threshold is found by minimising the index of fuzziness defined as calculated for an object and the background class medians or means values and membership functions. Tsai’s moment preserving thresholding method [49]: This is based on the idea that the greyscale image is a blurred version of the binary image. Threshold processing is performed in a way that the first three moments of the greyscale image correspond to the first three moments of the binarised image.

2.2. Local adaptive thresholding methods For these methods, the threshold is calculated at each block of a few pixels (or in each pixel) of the image. The local threshold value depends on local statistics such as the range, variance, or surface-fitting parameters of the pixel neighbourhood. The methods can be divided into a number of groups based on the Niblack or Sauvola methods, simple (mean, median, or midgrey) values, and others as follows: •

Niblack’s local adaptive binarisation method [25]: This method allows the high-speed processing of images. It is used for the rapid filtration of contrast images with strongly noisy areas when smoothly varying transitions of brightness are absent. The idea of this method is the variation of the threshold from one point to another on the basis of the local value of a standard deviation. In places with a smoothly varying transition of brightness, this method provides false objects with small noise.

Fig. 8. The reconstructed object images from the hologram (shown in Fig. 2d) binarised using Li’s (a), Phansalkar’s (b), Kittler’s (c), Prewitt’s minimum (d), Yen’s (e), and Soille’s (f) methods.

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Fig. 9. The PSNRs of the reconstructed object images from the hologram (shown in Fig. 2g) binarised using various thresholding methods.

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Nick’s method [30]: This is based on Niblack’s method. Its authors contend that it can provide good results for dark images with low contrast. Concomitantly, false objects do not appear on the binary image. Feng’s method [31]: This is another variation on Niblack’s method. The local mean value and the value of the minimum and mean square deviations are first calculated. Then the local window moves and changes size and similar parameters are calculated for the next window. The optimum size of a local window can be found by repeating these steps of the algorithm. Sauvola’s method [26–27]: This is an improvement to Niblack’s method. It prevents the imposition of noise on an object and provides more precise separation of an object from the background noise. Instead of a square window, a round window is used in Sauvola’s method. It surpasses Niblack’s method for thresholding sharp and contrast images. However, it provides inferior results with lowcontrast images when the values of an object’s pixel brightness are close to one another. Phansalkar’s method [32]: This is a modification of Sauvola’s thresholding method and works with low-contrast images. Mean local thresholding method [24]: The threshold values are defined as the mean values of the brightness of image pixels in an analysed image part. Median local thresholding method [24]: The threshold values are defined as the median values of the brightness of image pixels in an analysed image part. Midgrey local thresholding method [24]: The threshold values are calculated as the mid-grey of the local greyscale image distribution. Bernsen’s local thresholding method [29]: This is often used for diagrammatic and cartographical representations. For each pixel, the threshold of brightness is chosen as the average value between the minimum and maximum values of brightness of pixels in the local vicinity. However, Bernsen’s method has some disadvantages. For example, strong error noise can form after processing the monotonic areas of brightness. In certain cases, it results in the appearance of

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false black spots. However, Bernsen’s method is the fastest of the analogue approaches. Bradley’s method (Bradley–Roth) [28]: This method defines an optimum threshold based on the integral image. One of properties of an integral representation is the very rapid calculation of the sum of the pixels of the arbitrary area in a matrix. Using the obtained matrix, the image can be divided into n∗ n areas. The sum of the sizes is calculated for each area. The respective threshold values are also calculated. The advantages of Bradley-Roth’s method are its simplicity and high speed of realisation. There is no need to choose parameters for most cases. It provides high-quality compression for images with nonuniform backgrounds. One limitation is poor sensitivity to low-contrast image details. Otsu’s local thresholding method [24,80]: This is based on Otsu’s global thresholding method but is implemented in local image areas. The algorithm searches for the threshold that minimises the intraclass variance that is defined as a weighted sum of the variances of the two classes. Soille’s contrast thresholding method [33]: This is based on a simple contrast toggle. The pixel value is nullified or is equal to the maximum value depending on whether its current value is closest to the local maximum or minimum, respectively.

3. Applying the binarisation methods to digital holograms 3.1. Comparing the binarisation methods The described binarisation methods were applied to experimentally recorded digital holograms. The setup for registering these digital holograms and some examples of the objects used are provided in [81– 82]. The digital holograms had 2048 × 2048 pixels. The pixel size was 9 𝜇m × 9 𝜇m. Off-axis digital holograms of different types of objects were used during these experiments. A fragment (128 × 128 pixels) of a greyscale optically recorded hologram is shown in Fig. 1a. Examples of binarisation of the same region are shown in Fig. 1b–f. The follow124

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Fig. 10. The SSIMs of the reconstructed object images from the hologram (shown in Fig. 2g) binarised using various thresholding methods.

ing methods were used: Huang (based on attribute similarity), KittlerIllingworth (based on clustering), Shanbhag (based on entropy), Bradley (local adaptive method), and Doyle (based on histogram shape). These methods were chosen because they demonstrated median values of reconstruction quality across their groups (see Section 3.2). Images of 3D cross-sections were reconstructed using the direct calculation of the Fresnel diffraction method [1,5–6]: ( ) { [ ( )]} exp (𝑖𝑘𝑧) 𝑖𝑘 𝑖𝑘 2 𝑅(𝑥, 𝑦, 𝑧) = exp , (1) 𝐹 𝐹 𝑇 𝐻 (𝑢, 𝑣, 0) exp 𝑢 + 𝑣2 𝑖 2𝑧 2𝑧 where FFT{…} is the fast Fourier transform, k is the wave number, z is the distance between the cross-section of the 3D-scene and the hologram plane, (u,v,0) are the coordinates in the hologram plane, (x,y,z) are the coordinates in the reconstruction plane, and H(u,v,0) is the hologram image. Fragments of experimentally recorded holograms of 2D and 3D objects, the full reconstruction field, and the obtained object images are shown in Fig. 2. Different types of objects were used: •

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where MAXf is the maximum signal value of the reconstructed object image from the original digital hologram and MSE is the mean squared error. 𝑀𝑆𝐸 =

𝑚−1 𝑛−1 1 ∑ ∑‖ ‖2 ‖𝑂 − 𝐵𝜁,𝜂 ‖ , ‖ 𝑚𝑛 0 0 ‖ 𝜁,𝜂

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where O𝜁 , 𝜂 and B𝜁 , 𝜂 are the object images reconstructed from the original and the binarised digital hologram, respectively, m,n are the number of rows and columns of pixels in the images, and 𝜁 ,𝜂 are the indexes of the rows and columns. The structural similarity index (SSIM) [84] was used as another measurement of the quality of the reconstructed object image B𝜁 , 𝜂 from the binarised digital hologram: ( )( ) 2 𝜇𝑂 𝜇𝐵 + 𝐶 1 2 𝜎 𝑂 𝐵 + 𝐶 2 𝑆 𝑆 𝐼𝑀 = ( (4) )( 2 ), 2 + 𝜇2 + 𝐶 2 𝜇𝑂 1 𝜎𝑂 + 𝜎𝐵 + 𝐶 2 𝐵 where 𝜇 O and 𝜇 B are the local mean values for the object images O𝜁 , 𝜂 and B𝜁 , 𝜂 , respectively; 𝜎 O and 𝜎 B are the local standard deviations for O𝜁 , 𝜂 and B𝜁 , 𝜂 , respectively; 𝜎 OB is the local covariance for O𝜁 , 𝜂 and B𝜁 , 𝜂 ; and C1 and C2 are constants: ( )2 ( )2 𝐶1 = 𝐾1 𝐿 , 𝐶2 = 𝐾2 𝐿 , (5)

Transmissive and an almost flat object (Fig. 2a–c; the distance between the object and the hologram was 1315 mm) Transmissive 3D scene that consisted of 3 almost flat elements (Fig. 2d–f; the total scene depth was 480 mm; the distances between the objects and the hologram were 850 mm (Fig. 2e), 1190 mm, and 1330 mm; the focused object images are shown in Fig. 2f) Reflective 3D object with 1 mm scene depth (Fig. 2g–j; the distance between the object and the hologram was 990 mm) Reflective 3D objects with 12 mm scene depth (Fig. 2k–m; the distance between the elements of the 3D scene and the hologram was from 1000 mm to 1012 mm; a reconstructed image is shown in Fig. 2m for the average distance of 1006 mm)

where L is the dynamic range of the pixel values and K1 < < 1 and K2 < < 1 are small constants. The PSNR or SSIM for the almost flat object was defined as a single value. The PSNR or SSIM for the 3D objects or 3D scenes was defined as the average value over the PSNRs or SSIMs for the differently focused 2D elements. 3.2. Results of the binarisation of digital holograms

The peak signal-to-noise ratio (PSNR) value [83] was used as a measurement of the reconstruction quality: ( ) 𝑀𝐴𝑋𝑓 𝑃 𝑆𝑁𝑅 = 20log10 √ , (2) 𝑀𝑆𝐸

All 30 of the aforementioned methods of global and local thresholding were tested via the binarisation of optically recorded holograms. The results of the reconstruction and the obtained PSNR and 125

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SSIM values were also achieved using Otsu’s local method, Niblack’s method, and Li’s entropy-based method. The lowest average PSNR and SSIM values were obtained using Prewitt’s minimum method. However, no single method demonstrated the best or worst results for all of the reconstructed images. The global and local methods demonstrated results comparable to one other on average. The obtained results were similar to those demonstrated in [55]. In [55], holographic filters were binarised using Niblack’s, uniform, and Otsu’s methods, and the results were compared. For each reconstructed image, the maximum PSNR and SSIM values were considered 100%. The relative PSNR and SSIM values were then calculated for each method. The average values obtained are shown in Figs. 15 and 16. The average difference between the maximum and obtained PSNR values was less than 5% for 9 algorithms. The highest relative PSNR values were achieved via the two implementations of the Ridler–Calvard methods (98.8% and 98.6% relative maximum PSNR for all of the holograms), Otsu’s method (98.5%), and Kittler-Illingworth’s method (97.9%). Robust results were also obtained using Li’s entropy-based method (97.2%), Tsai’s method (97.2%), Otsu’s local adaptive method (96.2%), and Glasbey’s (96.1%) and Niblack’s (95.2%) methods. Ten methods demonstrated results between 90% and 95%. Eleven methods produced worse results. The methods demonstrated similar results for the SSIM values. The average difference between the maximum and obtained SSIM values was less than 5% for 18 algorithms. The highest relative PSNR values were achieved via the clustering-based methods of Kittler–Illingworth (99.2% relatively maximum SSIM for all holograms), Otsu (98.3%), Ridler–Calvard (98.5%), Otsu’s local method (99.4%), Niblack (99.3%), the mean value (98.9%), and Phansalkar (98.7%). High results were also obtained using Li’s entropy-based method (99.4%) and Huang’s attribute-based method (98.6%). Five methods demonstrated results between 90% and 95%. Seven methods produced worse results. An increase in the complexity of the thresholding algorithms resulted in a decrease in the rate of hologram binarisation. Numerical experi-

Fig. 11. The reconstructed object images from the hologram (shown in Fig. 2g) binarised using local Otsu’s (a), Niblack’s (b), Ridler–Calvard’s (c), Prewitt’s (d), Zack’s (e), and Renyi’s (f) methods.

SSIM values are provided in Figs. 3–14. According to the PSNR values (Figs. 3, 6, 9, and 12), the 3 best and 3 worst results for each reconstructed object are shown in Fig. 5a–c, 8a–c, 11a–c, and 14a–c, and in Fig. 5d–f, 8d–f, 11d–f, and 14d–f, respectively. The PSNR and SSIM values demonstrated very similar results. For all of the cases, high PSNR and SSIM values were achieved using the following cluster-based group methods: both implementations of Ridler–Calvard’s method, Kittler-Illingworth’s method, and Otsu’s global method. This may be explained by the similarity of the registered holograms of the diffuse objects as the sum of random phasors and a Gaussian-mixture representation of the clustering-based methods. High

Fig. 12. The PSNRs of the reconstructed object images from the hologram (shown in Fig. 2k) binarised using various thresholding methods. 126

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Fig. 13. The SSIMs of the reconstructed object images from the hologram (shown in Fig. 2k) binarised using various thresholding methods.

example, the processing time using Glasbey’s method was 0.03 s. The clustering-based group produced the slowest global methods. In this group and the local methods, the average hologram processing time ranged from a fraction of a second to approximately 10 s. For faster hologram binarisation, accelerated computation techniques should be used, including programme acceleration, graphic processing units [85– 86], and field-programmable gate arrays [87], among others. [88].

4. Conclusion In this paper, optically recorded off-axis digital holograms (2048 × 2048 pixels) of transmissive and reflective objects and 3D scenes were binarised using various methods. Overall, 12 local and 18 global thresholding techniques of different groups were analysed based on attribute similarity, clustering, entropy, histogram shape, and local adaptive methods. The results of the reconstruction were compared using the PSNR and SSIM values. Examples of the object images obtained were shown. No single method demonstrated the best or worst results for all of the digital holograms. According to the PSNRs and SSIMs, the quality of reconstruction varied up to 2 times between the different methods. Significant differences between the global and local methods were not obtained on average. Eight methods demonstrated good results for both metric values. Ten methods showed average results. Eleven methods demonstrated satisfactory results. The highest quality of image reconstruction was achieved using the clustering-based methods of Ridler–Calvard, Kittler– Illingworth, and Otsu and Li’s entropy-based method. For Ridler– Calvard’s method, the average PSNR and SSIM value was 98.7% of the relative maximum value among all of the methods. For Kittler– Illingworth’s, Otsu’s, and Li’s methods, this value was 98.6%, 98.4%, and 98.3%, respectively. High results were also achieved using the second implementation of Ridler–Calvard’s clustering-based method and Otsu’s local and Niblack’s methods.

Fig. 14. The reconstructed object images from the hologram (shown in Fig. 2k) binarised using local midgrey (a), Ridler–Calvard’s (b), Bernsen’s (c), Nick’s (d), Zack’s (e), and Yen’s (f) methods.

ments were performed using a MATLAB environment on a computer with an Intel Core i5-5200 CPU at 2.2 GHz and 4 Gb RAM. The three global groups provided the fastest methods of image thresholding based on attribute similarity, entropy, and histogram shape. Holograms were binarised using these methods in less than a fraction of a second. For 127

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Fig. 15. The average relative maximum PSNR values among all of the holograms for the methods utilised.

Fig. 16. The average relative maximum SSIM values among all of the holograms for the methods utilised.

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The results obtained can be used to select the appropriate thresholding method and for fast digital hologram binarisation for storage, transfer, or use in DMD applications.

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