November  2019, 2(4): 347-359. doi: 10.3934/mfc.2019022

Image enhancement algorithm using adaptive fractional differential mask technique

Northeastern University, 110819, China

* Corresponding author: Xuefeng Zhang

Published  December 2019

Fund Project: The first author is supported by National Natural Science Foundation of P.R.China(61603055).

This paper addresses a novel adaptive fractional order image enhancement method. Firstly, an image segmentation algorithm is proposed, it combines Otsu algorithm and rough entropy to segment image accurately into the objet and the background. On the basis of image segmentation and the knowledge of fractional order differential, an image enhancement model is established. The rough characteristics of each average gray value are obtained by image segmentation method, through these features, we can determine the optimal fractional order of image enhancement. Then image will be enhanced using fractional order differential mask, from which fractional order is obtained adaptively. Several images are used for experiments, the proposed model is compared with other models, and the results of comparison exhibit the superiority of our algorithm in terms of image quality measures.

Citation: Xuefeng Zhang, Hui Yan. Image enhancement algorithm using adaptive fractional differential mask technique. Mathematical Foundations of Computing, 2019, 2 (4) : 347-359. doi: 10.3934/mfc.2019022
References:
[1]

M. R. S. Ammi and I. Jamial, Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 103-117.  doi: 10.3934/dcdss.2018007.

[2]

J. Bai and X. C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process, 16 (2007), 2492-2502.  doi: 10.1109/TIP.2007.904971.

[3]

K. Bashir, X. Tao and S. Gong, Gait Recognition Using Gait Entropy Image, International Conference on Crime Detection and Prevention, London, UK, 2010.

[4]

Y. Q. Chen and B. M. Vinagre, A new IIR-type digital fractional order differentiator, Signal Processing, 83 (2003), 2359-2365.  doi: 10.1016/S0165-1684(03)00188-9.

[5]

D. L. ChenY. Q. Chen and D. Y. Xue, 1-D and 2-D digital fractional-order Savitzky-Golay differentiator, Springer, 6 (2012), 503-511. 

[6]

S. Q. Chen and F. Q. Zhao, The adaptive fractional order differential model for image enhancement based on segmentation, Int. J. Pattern Recognit. Artif. Intell., 32 (2018), 15 pp. doi: 10.1142/S0218001418540058.

[7]

F. F. Dong and Y. M. Chen, A fractional-order derivative based variational framework for image denoising, Inverse Probl. Imaging, 10 (2016), 27-50.  doi: 10.3934/ipi.2016.10.27.

[8]

C. B. Gao and J. L. Zhou, Image enhancement based on quaternion fractional directional differentiation, (Chinese) Acta Automat. Sinica, 37 (2011), 150-159.  doi: 10.3724/SP.J.1004.2011.00150.

[9]

S. F. Gull and J. Skilling, The entropy of an image, SIAM-AMS Proc., Amer. Math. Soc., Providence, RI, 14 (1984), 167–-189.

[10]

J. Guo, C. E. Siong and D. Rajan, Foreground motion detection by difference-based spatial temporal entropy image, Tencon IEEE Region 10 Conference, Chiang Mai, Thailand, 2004.

[11]

F. Y. Hu, An adaptive approach for texture enhancement based on a fractional differential operator with non-integer step and order, Neurocomputing, 158 (2015), 295-306.  doi: 10.1016/j.neucom.2014.10.013.

[12]

G. HuangL. Xu and Y. F. Pu, Summary of research on image processing using fractional calculus, Application Research of Computers, 27 (2012), 1214-1229. 

[13]

Hungenahally and Suresh, Neural Basis for The Design of Fractional-Order Perceptual Filters: Applications in Image Enhancement and Coding, IEEE International Conference on Systems, Vancouver, BC, Canada, 1995.

[14]

K. KimS. Kim and K. S. Kim, Effective image enhancement techniques for fog-affected indoor and outdoor images, IET Image Processing, 12 (2018), 465-471.  doi: 10.1049/iet-ipr.2016.0819.

[15]

KimYunseopR. G. Evans and W. M. Iversen, Remote sensing and control of an irrigation system using a distributed wireless sensor network, IEEE Transactions on Instrumentation and Measurement, 57 (2008), 1379-1387. 

[16]

T. Konno, Selective targeting of anti-cancer drug and simultaneous image enhancement in solid tumors by arterially administered lipid contrast medium, Cancer, 54 (1934), 2367-2374.  doi: 10.1002/1097-0142(19841201)54:11<2367::AID-CNCR2820541111>3.0.CO;2-F.

[17]

H. F. LiZ. G. Yu and C. L. Mao, Fractional differential and variational method for image fusion and super-resolution, Neurocomputing, 171 (2016), 138-148.  doi: 10.1016/j.neucom.2015.06.035.

[18]

B. Li and X. Wei, Image denoising and enhancement based on adaptive fractional calculus of small probability strategy, Neurocomputing, 175 (2016), 704-714.  doi: 10.1016/j.neucom.2015.10.115.

[19]

B. Li and X. Wei, Adaptive fractional differential approach and its application to medical image enhancement, Computers and Electrical Engineering, 45 (2015), 324-335.  doi: 10.1016/j.compeleceng.2015.02.013.

[20]

Y. W. Liu, Remote sensing image enhancement based on fractional differential, 2010 International Conference on Computational and Information Sciences, 2010, 17–19. doi: 10.1109/ICCIS.2010.218.

[21]

A. NamburuK. S. Samayamantula and S. R. Edara, Generalised rough intuitionistic fuzzy c-means for magnetic resonance brain image segmentation, IET Image Processing, 11 (2017), 777-785.  doi: 10.1049/iet-ipr.2016.0891.

[22]

Nobuyuki. Otsu, A threshold selection method from gray-level histograms, IEEE Transactions on Systems, Man, and Cybernetics, 9 (1979), 62-66.  doi: 10.1109/TSMC.1979.4310076.

[23]

S. K. Pal and P. Mitra, Multispectral image segmentation using the rough-set-initialized EM algorithm, IEEE Transactions on Geoscience and Remote Sensing, 40 (2002), 2495-2501.  doi: 10.1109/TGRS.2002.803716.

[24]

S. K. PalB. U. Shankar and P. Mitra, Granular computing, rough entropy and object extraction, Pattern Recognition Letters, 26 (2005), 2509-2517.  doi: 10.1016/j.patrec.2005.05.007.

[25]

W. PanK. Qin and Y. Chen, An adaptable-multilayer fractional Fourier transform approach for image registration, IEEE Trans Pattern Anal Mach Intell, 31 (2008), 400-414. 

[26]

Z. Pawlak, Rough sets, Internat. J. Comput. Inform. Sci., 11 (1982), 341-356.  doi: 10.1007/BF01001956.

[27]

A. Petrosino and G. Salvi, Rough fuzzy set based scale space transforms and their use in image analysis, Internat. J. Approx. Reason, 41 (2006), 212-228.  doi: 10.1016/j.ijar.2005.06.015.

[28]

I. Podlubny, Fractional-order systems and PI$^\lambda$D$^\mu$-Controllers, IEEE Trans. Automat. Control, 44 (1999), 208-214.  doi: 10.1109/9.739144.

[29]

Y. F. Pu, J. L. Zhou and X. Yuan, Fractional differential mask: A fractional differential-based approach for multiscale texture enhancement, IEEE Trans. Image Process, 19 (2010), 491–-511. doi: 10.1109/TIP.2009.2035980.

[30]

Y. F. Pu, P. Siarry, A. Chatterjee, Z. N. Wang, Z. Yi, Y. G. Liu, J. L. Zhou and Y. Wang, A fractional-order variational framework for Retinex: Fractional-order partial differential equation-based formulation for multi-scale nonlocal contrast enhancement with texture preserving, IEEE Trans. Image Process., 27 (2018), 1214–-1229. doi: 10.1109/TIP.2017.2779601.

[31]

Y. F. Pu and W. X. Wang, Fractional differential masks of digital image and their numerical implementation algorithms, Acta Automatica Sinica, 33 (2007), 1128-1135. 

[32]

S. RoyP. ShivakumaraH. A. JalabR. W. IbrahimU. Pal and T. Lu, Fractional poisson enhancement model for text detection and recognition in video frames, Pattern Recognitio, 52 (2016), 433-447.  doi: 10.1016/j.patcog.2015.10.011.

[33]

Shugo, Hamahashi, O. Shuichi and H. Kitano, Detection of nuclei in 4D Nomarski DIC microscope images of early Caenorhabditis elegans embryos using local image entropy and object tracking, Bmc Bioinformatics, 6 (2005), 125 pp.

[34]

C. StudholmeD. L. G. Hill and D. J. Hawkes, An overlap invariant entropy measure of 3D medical image alignment, Pattern Recognition, 32 (1999), 71-86.  doi: 10.1016/S0031-3203(98)00091-0.

[35] R. TaoL. Qi and Y. Wang, Theory and Applications of The Fractional Fourier Transform, Tsinghua University Press, 2004. 
[36]

R. TaoB. Deng and Y. Wang, Research progress of the fractional Fourier in signal processing, Sci. China Ser. F, 49 (2006), 1-25.  doi: 10.1007/s11432-005-0240-y.

[37]

C. C. WangB. C. JiangY. S. Chou and C. C. Chu, Multivariate analysis-based image enhancement model for machine vision inspection, International Journal of Production Research, 49 (2011), 2999-3021.  doi: 10.1080/00207541003801242.

[38]

Q. YangY. Z. ZhangT. B. Zhao and Y. Q. Chen, Single image super-resolution using self-optimizing mask via fractional-order gradient interpolation and reconstruction, ISA Transactions, 82 (2018), 163-171.  doi: 10.1016/j.isatra.2017.03.001.

[39]

X. F. Zhang and L. L. Shang, Application of Image Segmentation Algorithm Based on VPRS-PSO Method, Control Engineering of China, 18 2011.

[40]

X. F. Zhang and J. K. Shang, Image segmentation algorithm based on Monte Carlo methods and rough entropy standard, Journal of Petrochemical Universities, 22 (2009), 94-98. 

[41]

W. Z. ZhuH. L. JiangE. WangY. HouL. D. Xian and J. Debnath, X-ray image global enhancement algorithm in medical image classification, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 1297-1309. 

show all references

References:
[1]

M. R. S. Ammi and I. Jamial, Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 103-117.  doi: 10.3934/dcdss.2018007.

[2]

J. Bai and X. C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process, 16 (2007), 2492-2502.  doi: 10.1109/TIP.2007.904971.

[3]

K. Bashir, X. Tao and S. Gong, Gait Recognition Using Gait Entropy Image, International Conference on Crime Detection and Prevention, London, UK, 2010.

[4]

Y. Q. Chen and B. M. Vinagre, A new IIR-type digital fractional order differentiator, Signal Processing, 83 (2003), 2359-2365.  doi: 10.1016/S0165-1684(03)00188-9.

[5]

D. L. ChenY. Q. Chen and D. Y. Xue, 1-D and 2-D digital fractional-order Savitzky-Golay differentiator, Springer, 6 (2012), 503-511. 

[6]

S. Q. Chen and F. Q. Zhao, The adaptive fractional order differential model for image enhancement based on segmentation, Int. J. Pattern Recognit. Artif. Intell., 32 (2018), 15 pp. doi: 10.1142/S0218001418540058.

[7]

F. F. Dong and Y. M. Chen, A fractional-order derivative based variational framework for image denoising, Inverse Probl. Imaging, 10 (2016), 27-50.  doi: 10.3934/ipi.2016.10.27.

[8]

C. B. Gao and J. L. Zhou, Image enhancement based on quaternion fractional directional differentiation, (Chinese) Acta Automat. Sinica, 37 (2011), 150-159.  doi: 10.3724/SP.J.1004.2011.00150.

[9]

S. F. Gull and J. Skilling, The entropy of an image, SIAM-AMS Proc., Amer. Math. Soc., Providence, RI, 14 (1984), 167–-189.

[10]

J. Guo, C. E. Siong and D. Rajan, Foreground motion detection by difference-based spatial temporal entropy image, Tencon IEEE Region 10 Conference, Chiang Mai, Thailand, 2004.

[11]

F. Y. Hu, An adaptive approach for texture enhancement based on a fractional differential operator with non-integer step and order, Neurocomputing, 158 (2015), 295-306.  doi: 10.1016/j.neucom.2014.10.013.

[12]

G. HuangL. Xu and Y. F. Pu, Summary of research on image processing using fractional calculus, Application Research of Computers, 27 (2012), 1214-1229. 

[13]

Hungenahally and Suresh, Neural Basis for The Design of Fractional-Order Perceptual Filters: Applications in Image Enhancement and Coding, IEEE International Conference on Systems, Vancouver, BC, Canada, 1995.

[14]

K. KimS. Kim and K. S. Kim, Effective image enhancement techniques for fog-affected indoor and outdoor images, IET Image Processing, 12 (2018), 465-471.  doi: 10.1049/iet-ipr.2016.0819.

[15]

KimYunseopR. G. Evans and W. M. Iversen, Remote sensing and control of an irrigation system using a distributed wireless sensor network, IEEE Transactions on Instrumentation and Measurement, 57 (2008), 1379-1387. 

[16]

T. Konno, Selective targeting of anti-cancer drug and simultaneous image enhancement in solid tumors by arterially administered lipid contrast medium, Cancer, 54 (1934), 2367-2374.  doi: 10.1002/1097-0142(19841201)54:11<2367::AID-CNCR2820541111>3.0.CO;2-F.

[17]

H. F. LiZ. G. Yu and C. L. Mao, Fractional differential and variational method for image fusion and super-resolution, Neurocomputing, 171 (2016), 138-148.  doi: 10.1016/j.neucom.2015.06.035.

[18]

B. Li and X. Wei, Image denoising and enhancement based on adaptive fractional calculus of small probability strategy, Neurocomputing, 175 (2016), 704-714.  doi: 10.1016/j.neucom.2015.10.115.

[19]

B. Li and X. Wei, Adaptive fractional differential approach and its application to medical image enhancement, Computers and Electrical Engineering, 45 (2015), 324-335.  doi: 10.1016/j.compeleceng.2015.02.013.

[20]

Y. W. Liu, Remote sensing image enhancement based on fractional differential, 2010 International Conference on Computational and Information Sciences, 2010, 17–19. doi: 10.1109/ICCIS.2010.218.

[21]

A. NamburuK. S. Samayamantula and S. R. Edara, Generalised rough intuitionistic fuzzy c-means for magnetic resonance brain image segmentation, IET Image Processing, 11 (2017), 777-785.  doi: 10.1049/iet-ipr.2016.0891.

[22]

Nobuyuki. Otsu, A threshold selection method from gray-level histograms, IEEE Transactions on Systems, Man, and Cybernetics, 9 (1979), 62-66.  doi: 10.1109/TSMC.1979.4310076.

[23]

S. K. Pal and P. Mitra, Multispectral image segmentation using the rough-set-initialized EM algorithm, IEEE Transactions on Geoscience and Remote Sensing, 40 (2002), 2495-2501.  doi: 10.1109/TGRS.2002.803716.

[24]

S. K. PalB. U. Shankar and P. Mitra, Granular computing, rough entropy and object extraction, Pattern Recognition Letters, 26 (2005), 2509-2517.  doi: 10.1016/j.patrec.2005.05.007.

[25]

W. PanK. Qin and Y. Chen, An adaptable-multilayer fractional Fourier transform approach for image registration, IEEE Trans Pattern Anal Mach Intell, 31 (2008), 400-414. 

[26]

Z. Pawlak, Rough sets, Internat. J. Comput. Inform. Sci., 11 (1982), 341-356.  doi: 10.1007/BF01001956.

[27]

A. Petrosino and G. Salvi, Rough fuzzy set based scale space transforms and their use in image analysis, Internat. J. Approx. Reason, 41 (2006), 212-228.  doi: 10.1016/j.ijar.2005.06.015.

[28]

I. Podlubny, Fractional-order systems and PI$^\lambda$D$^\mu$-Controllers, IEEE Trans. Automat. Control, 44 (1999), 208-214.  doi: 10.1109/9.739144.

[29]

Y. F. Pu, J. L. Zhou and X. Yuan, Fractional differential mask: A fractional differential-based approach for multiscale texture enhancement, IEEE Trans. Image Process, 19 (2010), 491–-511. doi: 10.1109/TIP.2009.2035980.

[30]

Y. F. Pu, P. Siarry, A. Chatterjee, Z. N. Wang, Z. Yi, Y. G. Liu, J. L. Zhou and Y. Wang, A fractional-order variational framework for Retinex: Fractional-order partial differential equation-based formulation for multi-scale nonlocal contrast enhancement with texture preserving, IEEE Trans. Image Process., 27 (2018), 1214–-1229. doi: 10.1109/TIP.2017.2779601.

[31]

Y. F. Pu and W. X. Wang, Fractional differential masks of digital image and their numerical implementation algorithms, Acta Automatica Sinica, 33 (2007), 1128-1135. 

[32]

S. RoyP. ShivakumaraH. A. JalabR. W. IbrahimU. Pal and T. Lu, Fractional poisson enhancement model for text detection and recognition in video frames, Pattern Recognitio, 52 (2016), 433-447.  doi: 10.1016/j.patcog.2015.10.011.

[33]

Shugo, Hamahashi, O. Shuichi and H. Kitano, Detection of nuclei in 4D Nomarski DIC microscope images of early Caenorhabditis elegans embryos using local image entropy and object tracking, Bmc Bioinformatics, 6 (2005), 125 pp.

[34]

C. StudholmeD. L. G. Hill and D. J. Hawkes, An overlap invariant entropy measure of 3D medical image alignment, Pattern Recognition, 32 (1999), 71-86.  doi: 10.1016/S0031-3203(98)00091-0.

[35] R. TaoL. Qi and Y. Wang, Theory and Applications of The Fractional Fourier Transform, Tsinghua University Press, 2004. 
[36]

R. TaoB. Deng and Y. Wang, Research progress of the fractional Fourier in signal processing, Sci. China Ser. F, 49 (2006), 1-25.  doi: 10.1007/s11432-005-0240-y.

[37]

C. C. WangB. C. JiangY. S. Chou and C. C. Chu, Multivariate analysis-based image enhancement model for machine vision inspection, International Journal of Production Research, 49 (2011), 2999-3021.  doi: 10.1080/00207541003801242.

[38]

Q. YangY. Z. ZhangT. B. Zhao and Y. Q. Chen, Single image super-resolution using self-optimizing mask via fractional-order gradient interpolation and reconstruction, ISA Transactions, 82 (2018), 163-171.  doi: 10.1016/j.isatra.2017.03.001.

[39]

X. F. Zhang and L. L. Shang, Application of Image Segmentation Algorithm Based on VPRS-PSO Method, Control Engineering of China, 18 2011.

[40]

X. F. Zhang and J. K. Shang, Image segmentation algorithm based on Monte Carlo methods and rough entropy standard, Journal of Petrochemical Universities, 22 (2009), 94-98. 

[41]

W. Z. ZhuH. L. JiangE. WangY. HouL. D. Xian and J. Debnath, X-ray image global enhancement algorithm in medical image classification, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 1297-1309. 

Figure 1.  Amplitude - frequency characteristic curves of fractional differential operators (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Figure 2.  The superposition of partial differential mask by 8 directions
Figure 3.  Segmentation results of Lena
Figure 4.  Segmentation results of Fishing boat
Figure 5.  The block diagram of the proposed model in this paper
Figure 6.  The original images
Figure 7.  Enhancement results of Lena
Figure 8.  Enhancement results of the moving head
Figure 9.  Enhancement results of the medical image
Figure 10.  Enhancement results of the aerial image
Figure 11.  Enhancement results of the airplane image
Table 1.  The information entropy of images
information entropy
original $ 0.2 - $ $ 0.8 - $ AFDA our
Fig. image order order method method
7 5.0572 5.0803 5.2640 5.1225 15.2047
8 3.5754 3.5843 3.6526 3.6005 3.6358
9 4.9163 4.9366 5.0044 4.9196 4.9273
10 5.1387 5.2031 4.9300 5.2184 5.3338
11 5.5089 5.5302 6.7563 5.6435 5.9013
information entropy
original $ 0.2 - $ $ 0.8 - $ AFDA our
Fig. image order order method method
7 5.0572 5.0803 5.2640 5.1225 15.2047
8 3.5754 3.5843 3.6526 3.6005 3.6358
9 4.9163 4.9366 5.0044 4.9196 4.9273
10 5.1387 5.2031 4.9300 5.2184 5.3338
11 5.5089 5.5302 6.7563 5.6435 5.9013
Table 2.  The average gradient of images
average gradient
original $ 0.2 - $ $ 0.8 - $ AFDA our
Fig. image order order method method
7 3.0202 3.6840 33.3047 4.6871 10.5149
8 2.2055 2.3810 6.1382 3.9755 4.0240
9 1.5844 1.7256 5.3264 4.3742 2.5745
10 9.3186 12.9233 50.3865 19.8006 23.5776
11 4.3996 5.7272 38.6371 7.5458 14.5694
average gradient
original $ 0.2 - $ $ 0.8 - $ AFDA our
Fig. image order order method method
7 3.0202 3.6840 33.3047 4.6871 10.5149
8 2.2055 2.3810 6.1382 3.9755 4.0240
9 1.5844 1.7256 5.3264 4.3742 2.5745
10 9.3186 12.9233 50.3865 19.8006 23.5776
11 4.3996 5.7272 38.6371 7.5458 14.5694
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