Open Access
Add to BibSonomy
Abstract
Recent years, image enhancement for single-pixel imaging has developed rapidly and provides an image-free way for extracting image information. However, the conventional image enhancement approaches for single-pixel imaging are still based on the discontinuously adjustable operations such as integer-order derivatives, which are frequently used in edge detection but sensitive to the image noise. Therefore, how to balance between two conflicting demands, i.e. edge detection and noise suppression, is a new challenge. To address this issue, we introduce arbitrary-order fractional operations into single-pixel imaging. In experiment, the proposed technique has the capacity to detect image edges with high quality. Compared with integer-order derivative method which amplifies noise significantly while extracting edges, it offers a nice tradeoff between image SNR and performance of edge enhancement. In addition, it also shows good performance of image smoothing and improvement of image quality, if fractional order is negative. The proposed technique provides the adjustable fractional order as a new degree of freedom for edge extraction and image de-noising and therefore makes up for the shortcomings of traditional method for image enhancement.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Single-pixel imaging technology, such as single-pixel camera [1–6] and ghost imaging [7–11], uses a number of spatial patterns to modulate the intensity of the illumination light field or the detected light field. Then, according to the correlation between the patterns and the corresponding measurements of single-pixel detection, the object image could be recovered even in the case of sub-Nyquist sampling [2,12]. Single-pixel imaging only requires a single point detector without spatial resolution, instead of array detector. Therefore, it is an ideal solution of imaging for many wavelength bands, in which array detectors are unavailable or too expensive, such as X-ray [13–16] and Terahertz [17]. Furthermore, it is immune to distortion introduced by scattering [18–20], turbid [21], nonlinear [22,23] and dispersive media [24]. In addition, it is a low-exposure-dose imaging approach and therefore much safer than traditional approaches [16,25]. In conventional way, one has to obtain the object image before extracting the edge information by edge detection methods [26,27]. Differently, the edge detection based on single-pixel imaging provides an image-free way to extract the edge information for unknown objects directly [28–33]. As a result, single-pixel imaging may offer new features to image edge detection, which is important in image processing and has been widely used in image analysis [34,35], pattern recognition [36,37] and other fields.
Generally, the common operators for edge detection of single-pixel imaging are based on first-order and second-order derivatives, such as Sobel operator and Laplace operator [38,39]. However, as an extension of integer-order calculus, fractional calculus has not been introduced into single-pixel imaging for image enhancement till now. The theory of fractional calculus has been established for over 300 years and the first time to mention the concept is in the famous letter from Leibniz to L’Hospital in 1695 [40–42]. It is a branch of mathematical analysis studies the characteristics and applications of differential and integral operators of any order [43–48]. Therefore, it supplies us the fractional order as a new degree of freedom, for the tradeoff between image edges and noise in edge detection applications. Recent years, with the development of computer and intensive study on its application, it has become a hot topic and has been widely used in signal processing [49–54], biomedical engineering [55,56], rheology [57–59], electronics [60,61], control theory [62,63] and image processing [64–66]. Compared with integer-order derivative, the fractional derivative has better performance in many applications [67–70].
In this manuscript, we propose and demonstrate a novel image-enhanced single-pixel imaging, using 2D multi-directional fractional calculus. The proposed technique gives us a new degree of freedom, i.e. the fractional order, for more flexible image-free information processing. Compared with the conventional integer-order derivative method which amplifies the unwanted noise significantly while extracting edges, it offers a nice tradeoff between SNR and performance of edge enhancement. Furthermore, it is also developed for image smoothing and improvement of image quality. Numerical simulations and experimental results are both presented to illustrate the effectiveness of the proposed technique in detail.
2. Principle
2.1 Fractional calculus
The Fourier transform (FT) of one-dimensional function u(x) is defined by
where x and k denote the spatial variable and wavenumber, and i represents the imaginary unit. Conversely, u(x) can be reconstructed from û(k) by the inverse Fourier transform (IFT):Supposing the positive integer n-order derivative of u(x) exists, which is written as Dnu(x)=dnu(x)/dxn. Its Fourier transform is
According to the theory of fractional calculus [64], n could be extended to real number. Therefore, the Eq. (3) is further rewritten as
Actually, one-dimensional fractional derivative/integral operation could be performed in only two directions, i.e. the x and -x directions. The corresponding operators in Fourier domain are Ωn(k)=(ik)n and (-ik)n. Further, there are infinite directions available for two-dimensional fractional derivative/integral operation, determined by the gradient direction of the two-dimensional Fourier-domain operator Ωn,θ (kx, ky). Then Eq. (4) is improved as
2.2 Numerical simulations
For simplicity, we set u(x) as sigmoid function and calculate the fractional/first-order derivatives and fractional integrals of u(x), according to Eq. (4). Sigmoid function is in step-like shape and its edge is smooth, as the blue line shown in Fig. 1(a). We do not use step function but sigmoid function as an example, because the former is non-differentiable at x=0. Furthermore, compared with step function, sigmoid function is closer to the image edge in practice, because the point spread function of imaging system is fundamentally smooth. When n=−0.14, −0.12, −0.08, −0.04, 0.2, 0.4, 0.6, 0.8 and 1 respectively, the simulation results are shown in Fig. 1(a). When n>0, the edge of sigmoid function is enhanced with increasing n. Conversely, when n<0, sigmoid function is smoothed with decreasing n. Although the first-order derivative could extract the edge information effectively, the original signal information is therefore lost. Differently, fractional derivative can retain a part of the signal while enhancing the signal edge. In addition, fractional integral could smooth a varying signal with the degree of freedom n.
Fig. 1. The simulation results of fractional/first-order derivatives and fractional integrals of sigmoid function (a) without noise (b) with noise.
In practical applications, the image noise, which is caused by shot noise, dark current noise and readout noise in imaging system, cannot be neglected. Therefore, we simulate the fractional/first-order derivatives and fractional integrals of sigmoid function with random noise under different order n, as shown in Fig. 1(b). The relationship of n and the signal-to-noise ratio (SNR) of simulation results is shown in the inset of Fig. 1(b). Obviously, although first-order derivative provides better edge-enhancing effect than fractional derivatives, the former leads to the noise amplification phenomenon and the corresponding SNR is lower than those of the latter. Interestingly, fractional integral could smooth the noise and improve the SNR, with n=−0.04, −0.08 and −0.12. However, fractional integral can hardly improve the SNR limitlessly, because it reduces the noise as well as the signal significantly if n is small enough. As a result, the SNR of Dnu(x) is lower than that of u(x) when n=−0.14. For one-dimensional images in Fig. 1(b), the SNR is defined as the ratio between the power of image signal and the power of background noise, expressed as
where Asignal is the amplitude of image information and stdnoise the noise standard deviation.3. Experimental results
The experimental setup of image-enhanced single-pixel imaging is shown in Fig. 2. It consists of a commercial digital projector as the illumination device, a photodiode power sensor as the single-pixel detection device and a bear toy as the target object to be imaged. The projector is connected to a computer via USB cable, so that the computer-generated patterns could be projected onto the target object. By adjusting the lens focus of the projector, clear projection pattern could be obtained on the object plane, with illuminated area of 0.15×0.15m2. The computer records the data from the power sensor and generates reconstructions.
Four-step phase-shifting approach is applied for single-pixel imaging to achieve the Fourier spectrum of object image [4]. Fourier base patterns with resolution of 128×128 pixels are projected onto the target object successively. Then the power sensor measures the total light powers from the object, i.e. the Fourier spectrum coefficients of the object û(kx, ky), as shown in Fig. 3(a). Even if the single-pixel detection has no spatial resolution, the object image could still be recovered by two-dimensional IFT of the measurements, as shown in Fig. 3(b). The amplitudes of typical two-dimensional Fourier-domain operators of fractional calculus at θ=0, π/4, π/2 and 3π/4, defined by Eq. (6), are shown in Fig. 3(c)-(f).
Fig. 3. (a) The measured Fourier spectrum of the object image. (b) The object image recovered from the measured Fourier spectrum. (c)-(f) Typical two-dimensional Fourier-domain operators of fractional calculus at θ=0, π/4, π/2 and 3π/4, when n=0.9.
Mathematically, the arbitrary-order fractional calculus of the object image in arbitrary direction could be obtained by Eq. (5), given by
With the measured spectrum û(kx, ky), we reconstruct fractional derivatives for the object in four directions of θ=0, π/4, π/2 and 3π/4 respectively, with n=0.2, 0.4, 0.6 and 0.8. Obviously, when n=0.2, the reconstruction results basically provide original information of the object. As n increases, the original information of object is suppressed and its edges are enhanced at the direction of θ, as shown in Fig. 4.
Fig. 4. The experimental results of fractional derivatives for object in four directions of θ=0, π/4, π/2 and 3π/4, when n=0.2, 0.4, 0.6 and 0.8 respectively.
The comprehensive object edge information En(x, y) could be extracted by calculating the root-mean-square image of fractional derivatives for object in four directions:
In experiment, the reconstructed images En(x, y) with different n are shown in Fig. 5. As the fractional order increases from 0.1 to 0.5, the object information is reduced and its rough edge can be clearly observed. As the fractional order increases from 0.6 to 1, the object information is almost lost and the edge details and noise are enhanced. The reason is that high-frequency object edges and image noise, caused by shot noise and electrical noise in imaging system, are both amplified significantly by higher-order fractional derivative operation. In other words, higher-order fractional derivative, especially first-order derivative, could effectively extract the edge information of object at the expense of image SNR. As a result, there is actually a trade-off between the edge-enhanced effect and image quality. According to the experimental results, compared with traditional first-order derivative method for single-pixel imaging [38], the proposed technique provides a feasible approach to avoid noise introduction while enhancing image edge details and therefore may show better performance.
Mathematically, fractional integral is the inverse operation of fractional derivative. Therefore, the former smooths the image details, while the latter enhances. With n<0, we could also reconstruct arbitrary-order fractional integral of the object in four directions of θ=0, π/4, π/2 and 3π/4 respectively, according to Eq. (8). The object image is smoothed in four direction respectively, as shown in Fig. 6. Because of the accumulation effect of fractional integral, there are some stripes oriented with the direction of the fractional integral in reconstructed results. Additionally, the smoothing effect increased, as n decreases.
Fig. 6. The experimental results of fractional integrals for object in four directions of θ=0, π/4, π/2 and 3π/4 under different n.
With n<0, the smoothed images En(x, y) are reconstructed by Eq. (10), as shown in Fig. 7. As the fractional order decreases, the image details information and noise are smoothed. However, the images become so blurred that many image detail features are indistinct, if the fractional order is too low. Therefore, there is also a trade-off between image SNR and smoothing effect, which depends directly on the adjustable parameter n.
The experimental relationship of fractional order n and image SNR of the proposed technique is illustrated in Fig. 8. As n decreases from 0 to −0.03, the image SNR is improved over 3 dB, due to the noise reduction caused by smoothing effect of fractional integral operation. But it is worth noting that the image SNR drops as n decrease from −0.03 to −0.06, because the original image signal is also significantly attenuated by smoothing effect if n is too small. On the other hand, the image SNR is degraded, as n increases from 0 to 1. The reason is that the image noise gets strong due to edge enhancement effect provided by fractional derivative operation, with the increase of n. Actually, there is a similar phenomenon in the one-dimensional simulation shown in Fig. 1(b). Compared with the traditional first-order derivative method which amplifies the unwanted noise significantly while extracting edges, the proposed technique could offer a nice balance between image quality and performance of edge enhancement.
Fig. 8. The experimental relationship of fractional order n and image SNR of image-enhanced single-pixel imaging. Note that the image SNRs are calculated by Eq. (7).
4. Discussion and conclusion
The proposed technique consists of 3 steps. 1) Four-step phase-shifting approach is applied for single-pixel imaging to achieve the Fourier spectrum of object image. 2) Then the measured Fourier spectrum is modulated by Fourier-domain fractional calculus operator. 3) Finally, the image-enhanced results are reconstructed by inverse Fourier transform. As a result, the fractional calculus operation has been performed for the unknown object image in Fourier domain before the reconstruction rather than after it. Alternatively, to shorten the computation time, the computer-generated illumination patterns for projector may be modulated by Fourier-domain fractional calculus operator before imaging. Thus the Step 2 of the proposed technique could be skipped. Therefore, fractional calculus has been integrated into single-pixel imaging by the proposed technique. Moreover, for fractional-calculus-based image enhancement, the fractional order could be automatically determined by previously reported methods [71–73]. Therefore, the proposed technique can be applied to many practical areas under different imaging conditions.
According to Eq. (4), the Fourier-domain operator of one-dimensional fractional calculus Ωn(k) could be regarded as the transfer function for filtering. Its amplitude is written as
In principle, this amplitude is nearly proportional to |k|, if n is close to 1, as shown in Fig. 9. Thus, after performing the higher-order fractional derivative/first-order derivative, high-frequency noise is amplified too much, while the low-frequency component including object pattern is attenuated. However, if n is relatively small, the gain for low-frequency component becomes close to that for high-frequency component. Therefore, the best balance between image SNR and performance of edge enhancement is reached when n is around 0.6, as shown in Fig. 5. Moreover, with n<0, |Ωn(k)| is close to ∞ as |k| is small, as shown in Fig. 9. Thus the gain for low-frequency component is far greater than that for high-frequency component. In other words, the fractional integral is actually a kind of low-pass filtering. As a result, the high-frequency information such as texture and noise is lost after the fractional integral operation, as shown in Fig. 7.
Fig. 9. The amplitude of Fourier-domain operator of one-dimensional fractional calculus under different n.
We demonstrate a novel image-enhanced single-pixel imaging for edge extraction and image smoothing. Compared with conventional single-pixel imaging using first-order derivative, the proposed technique offers an additional degree of freedom, i.e. the fractional order, to meet various requirements of edge enhancement in practice. With the adjustable fractional order, it has provided a nice balance between the image SNR and the performance of edge enhancement in experiment. In addition, fractional integral is also applied to single-pixel imaging for flexible image de-noising. We believe this research will lead the development of next-generation multipurpose single-pixel imaging technology and may have many important applications in feature extraction, object recognition and motion tracking for biomedical, military and industrial uses.
Funding
National Natural Science Foundation of China (61975017, 61905015); Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics (KF201908); Open Project of National Engineering Laboratory for Forensic Science (2020NELKFKT01).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. M. P. Edgar, G. M. Gibson, and M. J. Padgett, “Principles and prospects for single-pixel imaging,” Nat. Photonics 13(1), 13–20 (2019). [CrossRef]
2. M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Proc. Mag. 25(2), 83–91 (2008). [CrossRef]
3. M. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7(1), 12010 (2016). [CrossRef]
4. Z. Zhang, X. Ma, and J. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6(1), 6225 (2015). [CrossRef]
5. Z. Zhang, X. Wang, G. Zheng, and J. Zhong, “Hadamard single-pixel imaging versus Fourier single-pixel imaging,” Opt. Express 25(16), 19619–19639 (2017). [CrossRef]
6. R. Li, J. Hong, X. Zhou, C. Wang, Z. Chen, B. He, Z. Hu, N. Zhang, Q. Li, P. Xue, and X. Zhang, “SNR study on Fourier single-pixel imaging,” New J. Phys. 23(7), 073025 (2021). [CrossRef]
7. B. I. Erkmen and J. H. Shapiro, “Ghost imaging: from quantum to classical to computational,” Adv. Opt. Photon. 2(4), 405–450 (2010). [CrossRef]
8. P. Moreau, E. Toninelli, T. Gregory, and M. J. Padgett, “Ghost imaging using optical correlations,” Laser Photonics Rev. 12(1), 1700143 (2018). [CrossRef]
9. J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inf. Process. 11(4), 949–993 (2012). [CrossRef]
10. Y. Shih, “The physics of ghost imaging: nonlocal interference or local intensity fluctuation correlation?” Quantum Inf. Process. 11(4), 995–1001 (2012). [CrossRef]
11. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 061802 (2008). [CrossRef]
12. O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95(13), 131110 (2009). [CrossRef]
13. D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental X-ray ghost imaging,” Phys. Rev. Lett. 117(11), 113902 (2016). [CrossRef]
14. H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117(11), 113901 (2016). [CrossRef]
15. J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. 92(9), 093903 (2004). [CrossRef]
16. A. Zhang, Y. He, L. Wu, L. Chen, and B. Wang, “Tabletop x-ray ghost imaging with ultra-low radiation,” Optica 5(4), 374–377 (2018). [CrossRef]
17. R. Stantchev, B. Sun, S. Hornett, P. Hobson, G. Gibson, M. Padgett, and E. Hendry, “Non-invasive, near-field terahertz imaging of hidden objects using a single pixel detector,” Sci. Adv. 2(6), e1600190 (2016). [CrossRef]
18. M. Bina, D. Magatti, M. Molteni, A. Gatti, L. A. Lugiato, and F. Ferri, “Backscattering differential ghost imaging in turbid media,” Phys. Rev. Lett. 110(8), 083901 (2013). [CrossRef]
19. Y. Xu, W. Liu, E. Zhang, Q. Li, H. Dai, and P. Chen, “Is ghost imaging intrinsically more powerful against scattering?” Opt. Express 23(26), 32993–33000 (2015). [CrossRef]
20. X. Zhang, H. Yin, R. Li, J. Hong, S. Ai, W. Zhang, C. Wang, J. Hsieh, Q. Li, and P. Xue, “Adaptive ghost imaging,” Opt. Express 28(12), 17232–17240 (2020). [CrossRef]
21. R. E. Meyers, K. S. Deacon, and Y. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett. 100(13), 131114 (2012). [CrossRef]
22. X. Zhang, H. Yin, R. Li, J. Hong, Q. Li, and P. Xue, “Ghost network analyzer,” New J. Phys. 22(1), 013040 (2020). [CrossRef]
23. X. Zhang, H. Yin, R. Li, J. Hong, C. Wang, S. Ai, J. Hsieh, B. He, Z. Chen, Q. Li, and P. Xue, “Distortion-free frequency response measurements,” J. Phys. D: Appl. Phys. 53(39), 39LT02 (2020). [CrossRef]
24. P. Ryczkowski, M. Barbier, A. T. Friberg, J. M. Dudley, and G. Genty, “Ghost imaging in the time domain,” Nat. Photon. 10(3), 167–170 (2016). [CrossRef]
25. P. Morris, R. Aspden, J. Bell, R. Boyd, and M. Padgett, “Imaging with a small number of photons,” Nat. Commun. 6(1), 5913 (2015). [CrossRef]
26. P. Bao, L. Zhang, and X. Wu, “Canny edge detection enhancement by scale multiplication,” IEEE Trans. Pattern. Anal. 27(9), 1485–1490 (2005). [CrossRef]
27. L. Zhang and P. Bao, “Edge detection by scale multiplication in wavelet domain,” Pattern Recogn. Lett. 23(14), 1771–1784 (2002). [CrossRef]
28. C. Zhou, G. Wang, H. Huang, L. Song, and K. Xue, “Edge detection based on joint iteration ghost imaging,” Opt. Express 27(19), 27295–27307 (2019). [CrossRef]
29. X. Liu, X. Yao, R. Lan, C. Wang, and G. Zhai, “Edge detection based on gradient ghost imaging,” Opt. Express 23(26), 33802–33811 (2015). [CrossRef]
30. L. Wang, L. Zou, and S. Zhao, “Edge detection based on subpixel-speckle-shifting ghost imaging,” Opt. Commun. 407, 181–185 (2018). [CrossRef]
31. S. Yuan, D. Xiang, X. Liu, X. Zhou, and P. Bing, “Edge detection based on computational ghost imaging with structured illuminations,” Opt. Commun. 410, 350–355 (2018). [CrossRef]
32. H. Ren, L. Wang, and S. Zhao, “Efficient edge detection based on ghost imaging,” OSA Continuum 2(1), 64–73 (2019). [CrossRef]
33. R. Li, J. Hong, X. Zhou, Q. Li, and X. Zhang, “Fractional Fourier single-pixel imaging,” Opt. Express 29(17), 27309–27321 (2021). [CrossRef]
34. P. Xu, Z. He, T. Qiu, and H. Ma, “Quantum image processing algorithm using edge extraction based on Kirsch operator,” Opt. Express 28(9), 12508–12517 (2020). [CrossRef]
35. M. Mittal, A. Verma, I. Kaur, B. Kaur, M. Sharma, L. Goyal, S. Roy, and T. Kim, “An Efficient Edge Detection Approach to Provide Better Edge Connectivity for Image Analysis,” IEEE Access 7, 33240–33255 (2019). [CrossRef]
36. M. Iqbal, M. Abdullah-Al-Wadud, B. Ryu, F. Makhmudkhujaev, and O. Chae, “Facial Expression Recognition with Neighborhood-Aware Edge Directional Pattern (NEDP),” IEEE Trans. Affect. Comput. 11(1), 125–137 (2020). [CrossRef]
37. S. Yi, D. Labate, G. R. Easley, and H. Krim, “A Shearlet Approach to Edge Analysis and Detection,” IEEE Trans. Image Process. 18(5), 929–941 (2009). [CrossRef]
38. H. Ren, S. Zhao, and J. Gruska, “Edge detection based on single-pixel imaging,” Opt. Express 26(5), 5501–5511 (2018). [CrossRef]
39. T. Mao, Q. Chen, W. He, Y. Zou, H. Dai, and G. Gu, “Speckle-Shifting Ghost Imaging,” IEEE Photonics J. 8, 1–10 (2016). [CrossRef]
40. V. Kiryakova, “A brief story about the operators of the generalized fractional calculus,” Fract. Calc. Appl. Anal. 11(2), 203–220 (2008).
41. J. Tenreiro Machado and V. Kiryakova, “The Chronicles of Fractional Calculus,” Fract. Calc. Appl. Anal. 20(2), 307–336 (2017). [CrossRef]
42. B. Ross, “The development of fractional calculus 1695–1900,” Hist. Math. 4(1), 75–89 (1977). [CrossRef]
43. J. A. T. Machado, A. M. S. F. Galhano, and J. J. Trujillo, “On development of fractional calculus during the last fifty years,” Scientometrics 98(1), 577–582 (2014). [CrossRef]
44. D. Valério, J. T. Machado, and V. Kiryakova, “Some pioneers of the applications of fractional calculus,” Fract. Calc. Appl. Anal. 17(2), 552–578 (2014). [CrossRef]
45. J. Tenreiro Machado, A. M. Galhano, and J. J. Trujillo, “Science metrics on fractional calculus development since 1966,” Fract. Calc. Appl. Anal. 16(2), 479–500 (2013). [CrossRef]
46. B. Ross, “Fractional Calculus,” Mathematics Magazine 50(3), 115–122 (1977). [CrossRef]
47. J. Canny, “A Computational Approach To Edge Detection,” IEEE Trans. Pattern. Anal. PAMI PAMI-8(6), 679–698 (1986). [CrossRef]
48. B. Gardiner, S. A. Coleman, and B. W. Scotney, “Multiscale Edge Detection Using a Finite Element Framework for Hexagonal Pixel-Based Images,” IEEE Trans. Image Process. 25(4), 1 (2016). [CrossRef]
49. R. Panda and M. Dash, “Fractional generalized splines and signal processing,” Signal Process. 86(9), 2340–2350 (2006). [CrossRef]
50. M. D. Ortigueira and J. A. T. Machado, “Fractional signal processing and applications,” Signal Process. 83(11), 2285–2286 (2003). [CrossRef]
51. M. D. Ortigueira and J. A. T. Machado, “Fractional calculus applications in signals and systems,” Signal Process. 86(10), 2503–2504 (2006). [CrossRef]
52. K. Razminia, A. Razminia, and J. J. Trujilo, “Analysis of radial composite systems based on fractal theory and fractional calculus,” Signal Process. 107, 378–388 (2015). [CrossRef]
53. X. Zhang, C. Wang, W. Zhang, S. Ai, W. Liao, J. Hsieh, B. He, Z. Chen, Z. Hu, N. Zhang, and P. Xue, “High-speed all-optical processing for spectrum,” Opt. Express 29(1), 305–314 (2021). [CrossRef]
54. J. Hong, X. Zhou, N. Xin, Z. Chen, B. He, Z. Hu, N. Zhang, Q. Li, P. Xue, and X. Zhang, “Theoretical and experimental study of hybrid optical computing engine for arbitrary-order FRFT,” Opt. Express 29(24), 40106–40115 (2021). [CrossRef]
55. Y. Zhang, Y. Wang, W. Zhang, F. Lin, Y. Pu, and J. Zhou, “Statistical iterative reconstruction using adaptive fractional order regularization,” Biomed. Opt. Express 7(3), 1015–1029 (2016). [CrossRef]
56. Y. Zhang, W. Zhang, Y. Lei, and J. Zhou, “Few-view image reconstruction with fractional-order total variation,” J. Opt. Soc. Am. A 31(5), 981–995 (2014). [CrossRef]
57. S. Larsson and F. Saedpanah, “The continuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity,” IMA J. Numer. Anal. 30(4), 964–986 (2010). [CrossRef]
58. R. L. Bagley and P. J. Torvik, “On the Fractional Calculus Model of Viscoelastic Behavior,” J. Rheol. 30(1), 133–155 (1986). [CrossRef]
59. F. C. Meral, T. J. Royston, and R. Magin, “Fractional calculus in viscoelasticity: An experimental study,” Commun. Nonlinear Sci. 15(4), 939–945 (2010). [CrossRef]
60. X. Yang, J. A. T. Machado, C. Cattani, and F. Gao, “On a fractal LC-electric circuit modeled by local fractional calculus,” Commun. Nonlinear Sci. 47, 200–206 (2017). [CrossRef]
61. R. Caponetto, G. Maione, A. Pisano, M. R. Rapaić, and E. Usai, “Analysis and shaping of the self-sustained oscillations in relay controlled fractional-order systems,” Fract. Calc. Appl. Anal. 16(1), 93–108 (2013). [CrossRef]
62. G. Zhong, H. Deng, and J. Li, “Chattering-freevariable structure controller design via fractional calculus approach and itsapplication,” Nonlinear Dynam. 81(1-2), 679–694 (2015). [CrossRef]
63. G. Si, L. Diao, and J. Zhu, “Fractional-order charge-controlled memristor: theoretical analysis and simulation,” Nonlinear Dynam. 87(4), 2625–2634 (2017). [CrossRef]
64. R. Herrmann, “Fractional Calculus in Multidimensional Space — 2D-Image Processing,” in Fractional Calculus An Introduction for Physicists2nd Edition (World Scientific, 2014).
65. J. Hu, Y. Pu, and J. Zhou, “Fractional Integral Denoising Algorithm and Implementation of Fractional Integral Filter,” Journal of Computational Information Systems 7(3), 729–736 (2011). [CrossRef]
66. Y. Pu, J. Zhou, and X. Yuan, “Fractional Differential Mask: A Fractional Differential-Based Approach for Multiscale Texture Enhancement,” IEEE Trans. Image Process. 19(2), 491–511 (2010). [CrossRef]
67. J. Zhang and Z. Wei, “A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising,” Appl. Math. Model. 35(5), 2516–2528 (2011). [CrossRef]
68. B. Mathieu, P. Melchior, A. Oustaloup, and C. Ceyral, “Fractional differentiation for edge detection,” Signal Process. 83(11), 2421–2432 (2003). [CrossRef]
69. J. Liu, S. Chen, and X. Tan, “Fractional order singular value decomposition representation for face recognition,” Pattern Recogn. 41(1), 378–395 (2008). [CrossRef]
70. J. Bai and X. Feng, “Fractional-Order Anisotropic Diffusion for Image Denoising,” IEEE Trans. Image Process. 16(10), 2492–2502 (2007). [CrossRef]
71. Q. Yang, D. Chen, T. Zhao, and Y. Chen, “Fractional calculus in image processing: a review,” Fractional Calculus and Applied Analysis 19(5), 1222–1249 (2016). [CrossRef]
72. M. Ganga, N. Janakiraman, A. K. Sivaraman, A. Balasundaram, R. Vincent, and M. Rajesh, “Survey of Texture Based Image Processing and Analysis with Differential Fractional Calculus Methods,” in 2021 International Conference on System, Computation, Automation and Networking (ICSCAN)2021), pp. 1–6.
73. S. Khanna and V. Chandrasekaran, “Fractional derivative filter for image contrast enhancement with order prediction,” in IET Conference on Image Processing (IPR 2012)2012), pp. 1–6.
