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Speckle evaluation in laser display: From speckle contrast to speckle influence degree
Optics Communications 454 (2020) 124405
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Speckle evaluation in laser display: From speckle contrast to speckle influence degree Yuan Yuan a,b , Yong Bi a ,∗, Min Yuan Sun a,b , Dong Zhou Wang c , Dong Dong Wang a , Wei Nan Gao a , Shuo Zhang a,b a
Center of Applied Laser, Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100190, China c CASIRIS Technology Co., Ltd, Hangzhou 310018, China b
ARTICLE
INFO
Keywords: Laser display Speckle Speckle evaluation Speckle influence degree Human visual perception
ABSTRACT Speckle contrast (SC) is universally used to evaluate speckle in laser display, which has virtues of briefness in calculating and definitude in physical meaning. But the results characterized by SC is usually inconsistent with human visual perception. In this study, an alternative framework for speckle evaluation, speckle influence degree (SID), is developed. It considers the effects of SC, sampling rate, and image resolution on perceived speckle by human eyes. Theoretical analysis is firstly conducted. Then, speckle images of different sampling rate, SC, and image resolution are generated and calculated. Finally, SID is shown to be a more comprehensive metric that could characterize the influence of sampling rate and image resolution and that is better matched with subjective perception of human eyes than SC.
1. Introduction Laser has become a promising solid-state light source for projection displays. Red, green, and blue (RGB) lasers provide a wider color gamut, increased brightness, and longer lifetime than traditional Xenon lamps. However, the main drawback associated with the use of lasers is speckle [1]. Speckle degrades image quality considerably, and it is imperative to reduce the same [2–12]. To reduce laser speckle, it is necessary to develop a measuring and an evaluating system that models the human eyes. From the biological perspective, the human eye forms images of speckle on the retina. Light is then absorbed in the photoreceptors, and the signal is transmitted to the visual cortex for further processing [13]. Accordingly, a device that models the human eyes for speckle measurement and evaluation requires to construct imaging and data processing systems. For a speckle measuring system, the existing research results are as follows. In 2012 and 2014, Roelandt et al. applied some parameters of human eyes to a charge-coupled device (CCD) camera to capture speckle images [14,15]. In 2014, Hsu et al. proposed the recording of a correct speckle image when the speckle width was at least four times the pixel pitch and when the largest speckle intensity was of the order of the saturation level of the camera [16]. In 2014, Kubota simulated the optical transfer function of the human-eye model based on the eye model proposed by Westheimer [17,18]. In 2016, International Electrotechnical Commission (IEC) proposed the observation distance to be ∗
three times the image width for a full high-definition (HD) projector and An observation distance of 5 m for a cinema projector [19]. In 2018, based on those results above, we developed a more accurate speckle measuring system that simultaneously considered the characteristics of human eyes and optical transfer function [20]. Therefore, a device that emulates the imaging features of the human eyes is currently available, which considers the influence of optical power, observation distance, pixel ditch, aperture size, focal length, the optical transfer function and so on. Besides, for a speckle evaluating system, it is essential to be matched to the human eyes. Over the years, speckle has been characterized by speckle contrast (SC), which is defined as the ratio of the standard deviation of the intensity fluctuation to the mean intensity [1]. SC is appealing because it has a clear physical meaning and is mathematically convenient. However, it only considers the intensity fluctuation, ignoring other affecting factors such as the relationship between adjacent pixels, spatial sampling frequencies, and image content. Therefore, results are often shown to be not matched well with human visual perception [11]. In 2004, Wang et al. proposed the use of the structural similarity index (SSI) for the quantification of image quality [21]. We applied SSI in evaluating speckle image. Although SSI considers more factors than SC, it still has some problems. Firstly, because SSI was a full reference algorithm, use of this marker often led to confusing results when a different reference image was chosen. Secondly, SSI could not characterize the effect of sampling rate on subjective perception
Corresponding author. E-mail address: [email protected] (Y. Bi).
https://doi.org/10.1016/j.optcom.2019.124405 Received 3 June 2019; Received in revised form 13 August 2019; Accepted 15 August 2019 Available online 19 August 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
Y. Yuan, Y. Bi, M.Y. Sun et al.
Optics Communications 454 (2020) 124405
of human eyes. Thirdly, although SSI could present the correct trend in terms of mean intensity and image resolution, the sensitivity is low [22]. Therefore, a more accurate speckle metric that is consistent with subjective perception of human eyes needs constructing. In this study, we introduce a new mathematical biomarker referred to as the speckle influence degree (SID) for the evaluation of speckle, which accounts for the sampling rate, speckle contrast, and image resolution. A theoretical analysis of SID is also performed. Three groups of speckle images are used to calculate the SID and SC. The first group includes speckle images with different sampling rate, the second group is obtained by varying the SC based on mean intensity variations, and the third group includes images with different image resolution. 2. Methods
Fig. 1. Theoretical analysis of Eq. (2): SID as a function of (a) sampling rate 𝑤 and image resolution 𝛾 with a speckle contrast SC = 0.5 a.u., (b) SC and 𝑤 with 𝛾 = 1 a.u., (c) SC and 𝛾 with 𝑤 = 10 a.u., (d) 𝑤 with SC = 0.5 a.u. and 𝛾 = 1 a.u., (e) SC with 𝛾 = 1 a.u. and 𝑤 = 10 a.u., and (f) 𝛾 with SC = 0.5 a.u. and 𝑤 = 10 a.u.
SC is defined as the ratio of the standard deviation of the intensity fluctuation to the mean intensity [1], which is defined as, 𝜎 𝑆𝐶 = 𝐼 (1) 𝐼 where 𝜎𝐼 is the standard deviation for the intensity fluctuation, 𝐼̄ is the
be observed that the image resolution has an important influence on the evaluation of the speckle. When the speckle contrast and sampling rate are fixed, with the image resolution value varying, the subjective perception of the image also changes. The higher the image resolution is, the easier it is for the human eye to recognize the image content, and vice versa. Therefore, compared to SC, SID not only takes into account the characterization of SC, but also accounts for the influences of image resolution and sampling rate on the subjective perception of image speckle.
mean intensity of the speckle image. The new SID framework was proposed based on the assumption that the human visual system is considerably adapted to the extraction of structural correlation information from the viewing field. SID accounts for (a) Sampling rate (𝑤), (b) 𝑆𝐶, and for (c) image resolution (𝛾). Its formula is as follows, ( ) 1 SID = 1 − ⋅ SC ⋅ (2 − 𝛾) (2) 𝑤 Sampling rate is defined as the ratio of the recorded speckle size to CCD pixel width. Speckle size is determined as the full-width-at-halfmaximum of the autocovariance function of a speckle image. That is to say, speckle size represents the degree of association between adjacent pixels. When speckle images are recorded by charge coupled device (CCD), sampling theorem requires sampling rate to be larger than 2 (𝑤≥2), i.e., the average speckle size is larger than two times CCD pixel width. In order to match to human eyes, a CCD with a pixel width of 4.4 μm is usually used [20]. When sampling rate is 2, the average speckle size shall be 8.8 μm. Image resolution 𝛾 is characterized based on the graylevels of the ( single color) image. It is defined as, I − Imin 𝛾 = m𝑎𝑥 Im𝑎𝑥 + Imin
3. Results and discussion Single color speckle patterns are used in this study. Three letters ‘‘CAS’’ are chosen as image content, which is the abbreviation of Chinese Academy of Sciences. Each speckle pattern included 255 × 255 (N × N) pixels. Image resolution is the ratio of the mean pixel graylevels for pixels occupied by the letters ‘‘CAS’’ to the mean graylevels for the rest of the pixels in Fig. 1(a). In the calculation of the SID and SC, we assumed an 8 bit graylevels with the highest value of 255. 3.1. Sampling rate
(3)
As explained in the second section, sampling rate represents the ratio of speckle size to CCD pixel width. Fig. 2(a) shows a speckle-free ‘‘CAS’’ image. We generated seven speckle patterns with the sampling rates of 1.77, 3.93, 5.73, 8.09, 9.70, 14.09, and 19.93, respectively, as shown in Fig. 2(b)–(h). The mean intensity is maintained at 80 ± 6% graylevels and image resolution 𝛾 is fixed at one. In Fig. 2(b)–(h), when the sampling rate is small, the image can be clearly displayed. As the sampling rate increases, the image becomes progressively more blurred. Thus, larger sampling rate deteriorate the quality of the displayed images to a larger extent. Fig. 2(i) illustrates the three normalized autocovariance functions of Fig. 2(b), (d), and (e), whose full-width-atmaximum values are 1.77, 5.73, and 8.09 times pixel width, i.e., sampling rates are 1.77, 5.73, and 8.09. Fig. 2(j) depicts the variations of SID and SC as a function of sampling rate. Table 1 lists the exact SID and SC values for comparison. As it can be observed, SID increases from 0.4165 to 0.9403 at increasing sampling rate. As observed, SC fluctuates in the range from 0.87 to 0.88 owing to the random nature of speckle noise, with the exception that SC is 0.8476 when the sampling rate is 1.77. This is caused by the nonconformity of the sampling theorem which requires sampling rate to be at least 2. In the characterization process based on the use of SC, and with the exception of the speckle image with a sampling rate of 1.77, the SC values associated with Fig. 2(b)–(h) are nearly equivalent, while the SID value exhibits significant changes. As shown in Eq. (2), SID could characterize the influence of sampling rate, while SC could not. For the human eyes,
where 𝐼𝑚𝑎𝑥 and 𝐼𝑚𝑖𝑛 are the highest and the lowest graylevel intensities in speckle-free single color image. The range of 𝛾 is [0, 1]. Besides, SC, SID, 𝑤 and 𝛾 are arbitrary units (a.u.). Similar to SC, larger SID values indicate increased speckle. Fig. 1(a)–(c) illustrates the variations of (a) SID as a function of sampling rate and image resolution with the speckle contrast = 0.5 a.u., (b) SID as a function of speckle contrast and sampling rate with the image resolution = 1 a.u., and (c) SID as a function of speckle contrast and image resolution with the sampling rate fixed at 10. Fig. 1(d)–(f) shows a sectional view of Fig. 1(a)–(c). When the sampling rate increases from 1 to 20, the SID value starts to grow faster, then slowly, and eventually stabilized. The main reason is that Nyquist sampling theorem requires the sampling frequency of CCD to be at least 2. When the sampling rate is less than 2, there is a large quantization error, which makes the SID value smaller and less accurate. As the sampling rate increases, that is, as the speckle size increases, more accurate speckle patterns can be obtained. It can be observed from Fig. 1(a) and (d) that when the sampling rate is greater than 10, a stable speckle pattern is obtained, and SID tends to be stabilized gradually. Fig. 1(b) and (e) show that SID increases linearly as a function of SC. This demonstrates that SID can still effectively characterize the factors that affect speckle contrast. Correspondingly, this new speckle characterizing method based on SID can completely and nondestructively account for the SC. In Fig. 1(c) and (f), SID decreases linearly as a function of image resolution. It can 2
Y. Yuan, Y. Bi, M.Y. Sun et al.
Optics Communications 454 (2020) 124405 Table 2 Comparison between SID and SC in terms of mean intensity. Mean intensity (graylevels)
SC (a.u.)
SID (a.u.)
29 44 58 71 83 104 120 135 155 170
0.9951 0.9795 0.9545 0.9107 0.8719 0.7900 0.7274 0.6731 0.5911 0.5321
0.7419 0.7303 0.7116 0.6790 0.6500 0.5890 0.5423 0.5005 0.4407 0.3967
3.2. Speckle contrast Speckle contrast is varied by changing mean intensity. Computational domain does not include the intensity fluctuation in image content. Fig. 3(a)–(f) depict the speckle patterns with respective mean intensities equal to 29, 58, 83, 120, 170, and 237 graylevels, with sampling rate equal to 3.93, and image resolution 𝛾 equal to 1. Fig. 3(g) shows the variations of SID and SC as a function of mean intensity. In the process of SC characterization, the SC value decreases as a function of the mean intensity. Similarly, the SID value also decreases. Table 1 lists the SID and SC values. When the average intensity of the speckle image is small, the SID value is 0.7409, and the image cannot be easily distinguished. As the average intensity increases, the SID value becomes smaller and the picture becomes gradually clearer. Thus, the increase of the average intensity is conducive to the display image content. Analogously, when the mean intensity is 29 graylevels, speckle noise is considerable, and the SC value is 0.9951. As the mean intensity increases, SC continues to decline to 0.5321. Section 2 has shown that SID is linearly related to SC. When the speckle size and image resolution are fixed, SC is the dominant factor of SID in the characterization of speckle. SID and SC exhibit similar trends. Accordingly, as the mean intensity increases, SID and SC decrease. From the viewpoint of the visual perception of the human eyes, speckle noise becomes progressively smaller in Fig. 3(a)–(f). Therefore, in terms of the influence of the mean intensity on speckle, SC and SID have almost the same ability of characterizing the subjective perception of speckle.
Fig. 2. (a) Free-speckle image. Speckle images at different sampling rates equal to (b) 1.77, (c) 3.93, (d) 65.73, (e) 8.09, (f) 9.70, (g) 14.09, and (h) 19.93, (i) Normalized autocovariance functions (sampling rates) for different speckle patterns, and (j) variations of the speckle influence degree (SID) and SC as a function of sampling rate.
3.3. Image resolution Image resolution is calculated based on Eq. (3). Fig. 4(a)–(d) presents the speckle patterns at various image resolution of 1.00, 0.60, 0.3, and 0.14. In all cases, the mean intensities is maintained at 170 graylevels and the sampling rate is 3.93. Table 3 lists the SC values of Fig. 4(a)–(d). When the image resolution is 1 a.u. (maximum), the image content can be clearly displayed. As the image resolution decreases, SID increases from 0.3967 to 0.7367, and the image content becomes progressively more blurred. That is to say, the higher the image resolution is, the better is the quality and display of the image content. By contrast, the SC remains at 0.5321 as the image resolution decreases. This is because SC only represents the contrast of the background in the image, and does not account for the influence of the image content. The new SID accounts for both the SC and image resolution, which can more accurately emulate the subjective perception of the human eyes to speckle images. Therefore, the image resolution is a critical factor in the evaluation of speckle. The characterization of SID has more advantages compared to SC.
Fig. 3. Speckle patterns of different mean intensities equal to (a) 29, (b) 58, (c) 83, (d) 120, (e) 170, and (f) 237 graylevels. (g) Comparison between SID and SC in terms of mean intensity.
Table 1 Comparison between Speckle Influence Degree (SID) and Speckle Contrast (SC) in terms of sampling rate. Sampling rate (a.u.)
SC (a.u.)
SID (a.u.)
1.77 3.93 5.73 8.09 9.70 14.09 19.93
0.8476 0.8719 0.8810 0.8717 0.8718 0.8810 0.8658
0.3687 0.6500 0.7272 0.7639 0.7819 0.8185 0.8224
the perception of speckle patterns depicted in Fig. 2(b)–(h) become
4. Conclusions
progressively more intense, which is consistent with the variation of the values of SID. Therefore, the SID could express in a better manner
In this study, we have introduced the biomarker SID for the evaluation of speckle. SID accounts for sampling rate, speckle contrast, and image resolution. The results show that SID increases as a function of
the influence of sampling rate on the visual effect of speckle image compared to SC (see Table 2). 3
Y. Yuan, Y. Bi, M.Y. Sun et al.
Optics Communications 454 (2020) 124405
References [1] J.W. Goodman, Speckle Phenomena in Optics: Theory and Application, Roberts and Company Publishers, 2007. [2] M.N. Akram, V. Kartashov, Z.M. Tong, Speckle reduction in line-scan laser projectors using binary phase codes, Opt. Lett. 35 (2010) 444–446. [3] M.N. Akram, Z. Tong, G. Ouyang, X. Chen, V. Kartashov, Laser speckle reduction due to spatial and angular diversity introduced by fast scanning micromirror, Appl. Opt. 49 (2010) 3297–3304. [4] M. Elbaum, M. Greenebaum, M. King, A wavelength diversity technique for reduction of speckle size, Opt. Commun. 5 (1972) 171–174. [5] B. Redding, M.A. Choma, H. Cao, Speckle-free laser imaging using random laser illumination, Nat. Photonics 6 (2012) 355–359. [6] Y. Yuan, D. Wang, D. Wang, M. Sun, W. Gao, S. Zhang, Y. Bi, Influence and evaluation of vibrating screen methods on subjective speckle reduction, Proc. SPIE (2019) 11023. [7] H. Yamada, K. Moriyasu, H. Sato, H. Hatanaka, Effect of incidence/observation angles and angular diversity on speckle reduction by wavelength diversity in laser projection systems, Opt. Express 25 (2017) 32132–32141. [8] Ø. Svensen, X. Chen, M.N. Akram, Speckle reduction in laser projection displays through angle and wavelength diversity, Appl. Opt. 55 (2016) 1267–1274. [9] J.L.J. Duan, B. Hao, T. Zhao, Q. Gao, X. Duan, Formulas of partially spatial coherent light and design algorithm for computer-generated holograms, Opt. Express 26 (17) (2018) 22284–22295. [10] J.L.X. Duan, X. Li, G. Xue, T. Zhao, J. Duan, Novel computer-generated hologram encoding method based on partially temporal coherent light, Opt. Express 27 (5) (2019) 6851–6862. [11] J.L.T. Zhao, J. Duan, X. Li, Y. Wang, Image quality enhancement via gradientlimited random phase addition in holographic display, Opt. Commun. 442 (2019) 84–89. [12] J.L.H. Ma, M. Yang, X. Li, G. Xue, Y. Wang, Influence of limited random-phase of objects on the image quality of 3D holographic display, Opt. Commun. 385 (2017) 153–159. [13] F.W. Cambell, R.W. Gubisch, Optical quality of the eye, J. Physiol. 186 (1966) 558–578. [14] S. Roelandt, Y. Meuret, A. Jacobs, K. Willaert, P. Janssens, H. Thienpont, G. Verschaffelt, Human speckle perception threshold for still images from a laser projection system, Opt. Express 22 (2014) 23965–23979. [15] S. Roelandt, Y. Meuret, G. Craggs, G. Verschaffelt, P. Janssens, H. Thienpont, Standardized speckle measurement method matched to human speckle perception in laser projection systems, Opt. Express 20 (2012) 8770–8783. [16] .W.F. Hsu, M.C. Chou, Optimizing the parameters for measuring laser speckle and speckle contrast, Appl. Opt. 53 (2014) E26–32. [17] G. Westheimer, The eye as an optical instrument, in: L.K.R. Boff, a.J.P.T. Kaufman (Eds.), Handbook of Perception and Human Performance, Wiley, 1986, 4/1–4/20. [18] S. Kubota, Simulating the human eye in measurements of speckle from laser-based projection displays, Appl. Opt. 53 (2014) 3814–3820. [19] I.E. Comission, Laser display devices, in: Optical Measuring Methods of Speckle Contrast, 2016. [20] Y. Yuan, T. Fang, M.Y. Sun, W.N. Gao, S. Zhang, Y. Bi, Speckle measuring instrument based on biological characteristics of the human eyes and speckle reduction with advanced electromagnetic micro-scanning mirror, Laser Phys. 28 (2018) 075002. [21] A.C.B.Z. cWang, H.R. Sheikh, E.P. Simoncelli, Image quality assessment from error visibility to structural similarity, IEEE Trans. Image Process. 13 (2004). [22] Y. Yuan, Yong Bi, Speckle Measurement, Evaluation and Reduction in Laser Display, University of Chinese academy of science, 2019.
Fig. 4. Speckle images at different image resolutions 𝛾 of (a) 1.00, (b) 0.60, (c) 0.33, and (d) 0.14 a.u., (e) Comparison between SID and SC in terms of image resolution.
Table 3 Comparison between SID and SC in terms of image resolution. Image resolution (a.u.)
SC (a.u.)
SID (a.u.)
1.00 0.60 0.33 0.14
0.5321 0.5321 0.5321 0.5321
0.3967 0.5554 0.6625 0.7379
sampling rate and image resolution, and deceases as a function of mean intensity, and that SID could emulate the perception of the human eyes to speckle images. But SC could not characterize the influence of sampling rate and image resolution. Therefore, SID is considered as a more comprehensive marker than SC in characterizing speckle evaluation. This study proposes an effective way to evaluate speckle in specklereducing device or laser display device. The deficiency of the study is that the validity of SID is only demonstrated by algorithm simulation. In the future work, the effectiveness of SID will be researched in more application conditions.
Funding National Key Research and Development Program of China (2016YFB0401902), Basic Research on Key Technologies of LD Laser Display Machine Oriented to Three Primary Colors, New System of Laser Display and Design of Corresponding Light Source; and National Key Research and Development Program of China (2016YFB0402001).
4
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Speckle evaluation in laser display: From speckle contrast to speckle influence degree Yuan Yuan a,b , Yong Bi a ,∗, Min Yuan Sun a,b , Dong Zhou Wang c , Dong Dong Wang a , Wei Nan Gao a , Shuo Zhang a,b a
Center of Applied Laser, Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100190, China c CASIRIS Technology Co., Ltd, Hangzhou 310018, China b
ARTICLE
INFO
Keywords: Laser display Speckle Speckle evaluation Speckle influence degree Human visual perception
ABSTRACT Speckle contrast (SC) is universally used to evaluate speckle in laser display, which has virtues of briefness in calculating and definitude in physical meaning. But the results characterized by SC is usually inconsistent with human visual perception. In this study, an alternative framework for speckle evaluation, speckle influence degree (SID), is developed. It considers the effects of SC, sampling rate, and image resolution on perceived speckle by human eyes. Theoretical analysis is firstly conducted. Then, speckle images of different sampling rate, SC, and image resolution are generated and calculated. Finally, SID is shown to be a more comprehensive metric that could characterize the influence of sampling rate and image resolution and that is better matched with subjective perception of human eyes than SC.
1. Introduction Laser has become a promising solid-state light source for projection displays. Red, green, and blue (RGB) lasers provide a wider color gamut, increased brightness, and longer lifetime than traditional Xenon lamps. However, the main drawback associated with the use of lasers is speckle [1]. Speckle degrades image quality considerably, and it is imperative to reduce the same [2–12]. To reduce laser speckle, it is necessary to develop a measuring and an evaluating system that models the human eyes. From the biological perspective, the human eye forms images of speckle on the retina. Light is then absorbed in the photoreceptors, and the signal is transmitted to the visual cortex for further processing [13]. Accordingly, a device that models the human eyes for speckle measurement and evaluation requires to construct imaging and data processing systems. For a speckle measuring system, the existing research results are as follows. In 2012 and 2014, Roelandt et al. applied some parameters of human eyes to a charge-coupled device (CCD) camera to capture speckle images [14,15]. In 2014, Hsu et al. proposed the recording of a correct speckle image when the speckle width was at least four times the pixel pitch and when the largest speckle intensity was of the order of the saturation level of the camera [16]. In 2014, Kubota simulated the optical transfer function of the human-eye model based on the eye model proposed by Westheimer [17,18]. In 2016, International Electrotechnical Commission (IEC) proposed the observation distance to be ∗
three times the image width for a full high-definition (HD) projector and An observation distance of 5 m for a cinema projector [19]. In 2018, based on those results above, we developed a more accurate speckle measuring system that simultaneously considered the characteristics of human eyes and optical transfer function [20]. Therefore, a device that emulates the imaging features of the human eyes is currently available, which considers the influence of optical power, observation distance, pixel ditch, aperture size, focal length, the optical transfer function and so on. Besides, for a speckle evaluating system, it is essential to be matched to the human eyes. Over the years, speckle has been characterized by speckle contrast (SC), which is defined as the ratio of the standard deviation of the intensity fluctuation to the mean intensity [1]. SC is appealing because it has a clear physical meaning and is mathematically convenient. However, it only considers the intensity fluctuation, ignoring other affecting factors such as the relationship between adjacent pixels, spatial sampling frequencies, and image content. Therefore, results are often shown to be not matched well with human visual perception [11]. In 2004, Wang et al. proposed the use of the structural similarity index (SSI) for the quantification of image quality [21]. We applied SSI in evaluating speckle image. Although SSI considers more factors than SC, it still has some problems. Firstly, because SSI was a full reference algorithm, use of this marker often led to confusing results when a different reference image was chosen. Secondly, SSI could not characterize the effect of sampling rate on subjective perception
Corresponding author. E-mail address: [email protected] (Y. Bi).
https://doi.org/10.1016/j.optcom.2019.124405 Received 3 June 2019; Received in revised form 13 August 2019; Accepted 15 August 2019 Available online 19 August 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
Y. Yuan, Y. Bi, M.Y. Sun et al.
Optics Communications 454 (2020) 124405
of human eyes. Thirdly, although SSI could present the correct trend in terms of mean intensity and image resolution, the sensitivity is low [22]. Therefore, a more accurate speckle metric that is consistent with subjective perception of human eyes needs constructing. In this study, we introduce a new mathematical biomarker referred to as the speckle influence degree (SID) for the evaluation of speckle, which accounts for the sampling rate, speckle contrast, and image resolution. A theoretical analysis of SID is also performed. Three groups of speckle images are used to calculate the SID and SC. The first group includes speckle images with different sampling rate, the second group is obtained by varying the SC based on mean intensity variations, and the third group includes images with different image resolution. 2. Methods
Fig. 1. Theoretical analysis of Eq. (2): SID as a function of (a) sampling rate 𝑤 and image resolution 𝛾 with a speckle contrast SC = 0.5 a.u., (b) SC and 𝑤 with 𝛾 = 1 a.u., (c) SC and 𝛾 with 𝑤 = 10 a.u., (d) 𝑤 with SC = 0.5 a.u. and 𝛾 = 1 a.u., (e) SC with 𝛾 = 1 a.u. and 𝑤 = 10 a.u., and (f) 𝛾 with SC = 0.5 a.u. and 𝑤 = 10 a.u.
SC is defined as the ratio of the standard deviation of the intensity fluctuation to the mean intensity [1], which is defined as, 𝜎 𝑆𝐶 = 𝐼 (1) 𝐼 where 𝜎𝐼 is the standard deviation for the intensity fluctuation, 𝐼̄ is the
be observed that the image resolution has an important influence on the evaluation of the speckle. When the speckle contrast and sampling rate are fixed, with the image resolution value varying, the subjective perception of the image also changes. The higher the image resolution is, the easier it is for the human eye to recognize the image content, and vice versa. Therefore, compared to SC, SID not only takes into account the characterization of SC, but also accounts for the influences of image resolution and sampling rate on the subjective perception of image speckle.
mean intensity of the speckle image. The new SID framework was proposed based on the assumption that the human visual system is considerably adapted to the extraction of structural correlation information from the viewing field. SID accounts for (a) Sampling rate (𝑤), (b) 𝑆𝐶, and for (c) image resolution (𝛾). Its formula is as follows, ( ) 1 SID = 1 − ⋅ SC ⋅ (2 − 𝛾) (2) 𝑤 Sampling rate is defined as the ratio of the recorded speckle size to CCD pixel width. Speckle size is determined as the full-width-at-halfmaximum of the autocovariance function of a speckle image. That is to say, speckle size represents the degree of association between adjacent pixels. When speckle images are recorded by charge coupled device (CCD), sampling theorem requires sampling rate to be larger than 2 (𝑤≥2), i.e., the average speckle size is larger than two times CCD pixel width. In order to match to human eyes, a CCD with a pixel width of 4.4 μm is usually used [20]. When sampling rate is 2, the average speckle size shall be 8.8 μm. Image resolution 𝛾 is characterized based on the graylevels of the ( single color) image. It is defined as, I − Imin 𝛾 = m𝑎𝑥 Im𝑎𝑥 + Imin
3. Results and discussion Single color speckle patterns are used in this study. Three letters ‘‘CAS’’ are chosen as image content, which is the abbreviation of Chinese Academy of Sciences. Each speckle pattern included 255 × 255 (N × N) pixels. Image resolution is the ratio of the mean pixel graylevels for pixels occupied by the letters ‘‘CAS’’ to the mean graylevels for the rest of the pixels in Fig. 1(a). In the calculation of the SID and SC, we assumed an 8 bit graylevels with the highest value of 255. 3.1. Sampling rate
(3)
As explained in the second section, sampling rate represents the ratio of speckle size to CCD pixel width. Fig. 2(a) shows a speckle-free ‘‘CAS’’ image. We generated seven speckle patterns with the sampling rates of 1.77, 3.93, 5.73, 8.09, 9.70, 14.09, and 19.93, respectively, as shown in Fig. 2(b)–(h). The mean intensity is maintained at 80 ± 6% graylevels and image resolution 𝛾 is fixed at one. In Fig. 2(b)–(h), when the sampling rate is small, the image can be clearly displayed. As the sampling rate increases, the image becomes progressively more blurred. Thus, larger sampling rate deteriorate the quality of the displayed images to a larger extent. Fig. 2(i) illustrates the three normalized autocovariance functions of Fig. 2(b), (d), and (e), whose full-width-atmaximum values are 1.77, 5.73, and 8.09 times pixel width, i.e., sampling rates are 1.77, 5.73, and 8.09. Fig. 2(j) depicts the variations of SID and SC as a function of sampling rate. Table 1 lists the exact SID and SC values for comparison. As it can be observed, SID increases from 0.4165 to 0.9403 at increasing sampling rate. As observed, SC fluctuates in the range from 0.87 to 0.88 owing to the random nature of speckle noise, with the exception that SC is 0.8476 when the sampling rate is 1.77. This is caused by the nonconformity of the sampling theorem which requires sampling rate to be at least 2. In the characterization process based on the use of SC, and with the exception of the speckle image with a sampling rate of 1.77, the SC values associated with Fig. 2(b)–(h) are nearly equivalent, while the SID value exhibits significant changes. As shown in Eq. (2), SID could characterize the influence of sampling rate, while SC could not. For the human eyes,
where 𝐼𝑚𝑎𝑥 and 𝐼𝑚𝑖𝑛 are the highest and the lowest graylevel intensities in speckle-free single color image. The range of 𝛾 is [0, 1]. Besides, SC, SID, 𝑤 and 𝛾 are arbitrary units (a.u.). Similar to SC, larger SID values indicate increased speckle. Fig. 1(a)–(c) illustrates the variations of (a) SID as a function of sampling rate and image resolution with the speckle contrast = 0.5 a.u., (b) SID as a function of speckle contrast and sampling rate with the image resolution = 1 a.u., and (c) SID as a function of speckle contrast and image resolution with the sampling rate fixed at 10. Fig. 1(d)–(f) shows a sectional view of Fig. 1(a)–(c). When the sampling rate increases from 1 to 20, the SID value starts to grow faster, then slowly, and eventually stabilized. The main reason is that Nyquist sampling theorem requires the sampling frequency of CCD to be at least 2. When the sampling rate is less than 2, there is a large quantization error, which makes the SID value smaller and less accurate. As the sampling rate increases, that is, as the speckle size increases, more accurate speckle patterns can be obtained. It can be observed from Fig. 1(a) and (d) that when the sampling rate is greater than 10, a stable speckle pattern is obtained, and SID tends to be stabilized gradually. Fig. 1(b) and (e) show that SID increases linearly as a function of SC. This demonstrates that SID can still effectively characterize the factors that affect speckle contrast. Correspondingly, this new speckle characterizing method based on SID can completely and nondestructively account for the SC. In Fig. 1(c) and (f), SID decreases linearly as a function of image resolution. It can 2
Y. Yuan, Y. Bi, M.Y. Sun et al.
Optics Communications 454 (2020) 124405 Table 2 Comparison between SID and SC in terms of mean intensity. Mean intensity (graylevels)
SC (a.u.)
SID (a.u.)
29 44 58 71 83 104 120 135 155 170
0.9951 0.9795 0.9545 0.9107 0.8719 0.7900 0.7274 0.6731 0.5911 0.5321
0.7419 0.7303 0.7116 0.6790 0.6500 0.5890 0.5423 0.5005 0.4407 0.3967
3.2. Speckle contrast Speckle contrast is varied by changing mean intensity. Computational domain does not include the intensity fluctuation in image content. Fig. 3(a)–(f) depict the speckle patterns with respective mean intensities equal to 29, 58, 83, 120, 170, and 237 graylevels, with sampling rate equal to 3.93, and image resolution 𝛾 equal to 1. Fig. 3(g) shows the variations of SID and SC as a function of mean intensity. In the process of SC characterization, the SC value decreases as a function of the mean intensity. Similarly, the SID value also decreases. Table 1 lists the SID and SC values. When the average intensity of the speckle image is small, the SID value is 0.7409, and the image cannot be easily distinguished. As the average intensity increases, the SID value becomes smaller and the picture becomes gradually clearer. Thus, the increase of the average intensity is conducive to the display image content. Analogously, when the mean intensity is 29 graylevels, speckle noise is considerable, and the SC value is 0.9951. As the mean intensity increases, SC continues to decline to 0.5321. Section 2 has shown that SID is linearly related to SC. When the speckle size and image resolution are fixed, SC is the dominant factor of SID in the characterization of speckle. SID and SC exhibit similar trends. Accordingly, as the mean intensity increases, SID and SC decrease. From the viewpoint of the visual perception of the human eyes, speckle noise becomes progressively smaller in Fig. 3(a)–(f). Therefore, in terms of the influence of the mean intensity on speckle, SC and SID have almost the same ability of characterizing the subjective perception of speckle.
Fig. 2. (a) Free-speckle image. Speckle images at different sampling rates equal to (b) 1.77, (c) 3.93, (d) 65.73, (e) 8.09, (f) 9.70, (g) 14.09, and (h) 19.93, (i) Normalized autocovariance functions (sampling rates) for different speckle patterns, and (j) variations of the speckle influence degree (SID) and SC as a function of sampling rate.
3.3. Image resolution Image resolution is calculated based on Eq. (3). Fig. 4(a)–(d) presents the speckle patterns at various image resolution of 1.00, 0.60, 0.3, and 0.14. In all cases, the mean intensities is maintained at 170 graylevels and the sampling rate is 3.93. Table 3 lists the SC values of Fig. 4(a)–(d). When the image resolution is 1 a.u. (maximum), the image content can be clearly displayed. As the image resolution decreases, SID increases from 0.3967 to 0.7367, and the image content becomes progressively more blurred. That is to say, the higher the image resolution is, the better is the quality and display of the image content. By contrast, the SC remains at 0.5321 as the image resolution decreases. This is because SC only represents the contrast of the background in the image, and does not account for the influence of the image content. The new SID accounts for both the SC and image resolution, which can more accurately emulate the subjective perception of the human eyes to speckle images. Therefore, the image resolution is a critical factor in the evaluation of speckle. The characterization of SID has more advantages compared to SC.
Fig. 3. Speckle patterns of different mean intensities equal to (a) 29, (b) 58, (c) 83, (d) 120, (e) 170, and (f) 237 graylevels. (g) Comparison between SID and SC in terms of mean intensity.
Table 1 Comparison between Speckle Influence Degree (SID) and Speckle Contrast (SC) in terms of sampling rate. Sampling rate (a.u.)
SC (a.u.)
SID (a.u.)
1.77 3.93 5.73 8.09 9.70 14.09 19.93
0.8476 0.8719 0.8810 0.8717 0.8718 0.8810 0.8658
0.3687 0.6500 0.7272 0.7639 0.7819 0.8185 0.8224
the perception of speckle patterns depicted in Fig. 2(b)–(h) become
4. Conclusions
progressively more intense, which is consistent with the variation of the values of SID. Therefore, the SID could express in a better manner
In this study, we have introduced the biomarker SID for the evaluation of speckle. SID accounts for sampling rate, speckle contrast, and image resolution. The results show that SID increases as a function of
the influence of sampling rate on the visual effect of speckle image compared to SC (see Table 2). 3
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Optics Communications 454 (2020) 124405
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Fig. 4. Speckle images at different image resolutions 𝛾 of (a) 1.00, (b) 0.60, (c) 0.33, and (d) 0.14 a.u., (e) Comparison between SID and SC in terms of image resolution.
Table 3 Comparison between SID and SC in terms of image resolution. Image resolution (a.u.)
SC (a.u.)
SID (a.u.)
1.00 0.60 0.33 0.14
0.5321 0.5321 0.5321 0.5321
0.3967 0.5554 0.6625 0.7379
sampling rate and image resolution, and deceases as a function of mean intensity, and that SID could emulate the perception of the human eyes to speckle images. But SC could not characterize the influence of sampling rate and image resolution. Therefore, SID is considered as a more comprehensive marker than SC in characterizing speckle evaluation. This study proposes an effective way to evaluate speckle in specklereducing device or laser display device. The deficiency of the study is that the validity of SID is only demonstrated by algorithm simulation. In the future work, the effectiveness of SID will be researched in more application conditions.
Funding National Key Research and Development Program of China (2016YFB0401902), Basic Research on Key Technologies of LD Laser Display Machine Oriented to Three Primary Colors, New System of Laser Display and Design of Corresponding Light Source; and National Key Research and Development Program of China (2016YFB0402001).
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