Skip to main content
SpringerLink
Log in

An Improved Object Detection Algorithm Based on the Hessian Matrix and Conformable Derivative

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

In this paper, a newfangled technique for edge detection that combines the Khalil conformable derivative and the Hessian matrix is developed and experimentally validated. The following main aspects are considered: (i) to attenuate image noise a Gaussian kernel inspired by the Khalil conformable derivative was developed. (ii) The spatial derivatives of the image gray level are calculated via the Khalil derivative, which is suitable for maintaining object contours and texture information even in low-contrast and resolution images. (iii) The conformable Hessian matrix is obtained from these image derivatives, generating continuous, thicker, and brighter edges. Our operator is compared with some existing techniques. Simulation results on different test images confirm a greater robustness to noise by our operator as well as a better visual quality of the edge maps obtained. This statement is validated through a comparative analysis based on peak signal-to-noise ratio and edge-strength-similarity-based image quality. In addition, it is applied in medical image processing and analysis tasks such as mammogram images, computed tomography scans, and magnetic resonance imaging for identification tasks of middle cerebral artery aneurysms, calcifications and breast cancer, proliferative Diabetic Retinopathy, and cerebral arteriovenous malformations. According to our findings, the conformable operator allows better visual identification of the structure of these conditions, improving the accuracy of clinical diagnosis and its subsequent monitoring.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Fig. 35

Similar content being viewed by others

Data Availability

No data were generated or analyzed during the current study.

References

  1. A. Abirami, P. Prakash et al., Fractional diffusion equation for medical image denoising using adi scheme. J. Popul. Ther. Clin. Pharmacol. 30(12), 52–63 (2023). https://doi.org/10.47750/jptcp.2023.30.12.008

    Article  Google Scholar 

  2. R.S. Aldoury, N.M. Al-Saidi, R.W. Ibrahim, H. Kahtan, A new x-ray images enhancement method using a class of fractional differential equation. MethodsX 11, 102264 (2023). https://doi.org/10.1016/j.mex.2023.102264

    Article  Google Scholar 

  3. S. AlShamekh, Arteriovenous malformations. Dermatol. Clin. 40(4), 445–448 (2022). https://doi.org/10.1016/j.det.2022.06.012

    Article  Google Scholar 

  4. Y. An, K. Chen, F. Nie, Echocardiography diagnosis of mitral valve aneurysm complicated with infective endocarditis. Int. Heart J. 64(5), 959–962 (2023). https://doi.org/10.1536/ihj.23-121

    Article  Google Scholar 

  5. J.K. Appati, E. Owusu, M. Agbo Tettey Soli, K.S. Adu-Manu, A novel convolutional Atangana–Baleanu fractional derivative mask for medical image edge analysis. J. Exp. Theor. Artif. Intell. (2022). https://doi.org/10.1080/0952813X.2022.2108147

  6. A. Asokan, J. Anitha, Edge preserved satellite image denoising using median and bilateral filtering, in Recent Trends in Image Processing and Pattern Recognition. ed. by K.C. Santosh, R.S. Hegadi (Springer, Singapore, 2019), pp.688–699. https://doi.org/10.1007/978-981-13-9181-1_59

    Chapter  Google Scholar 

  7. V. Babu, Basilar top and left middle cerebral artery aneurysm. Case study, Radiopaedia.org (2018). Accessed 22 Feb 2024. https://doi.org/10.53347/rID-61965

  8. S. Balochian, H. Baloochian, Edge detection on noisy images using Prewitt operator and fractional order differentiation. Multimedia Tools Appl. 81(7), 9759–9770 (2022). https://doi.org/10.1007/s11042-022-12011-1

    Article  Google Scholar 

  9. A. Ben-loghfyry, A. Hakim, A novel robust fractional-time anisotropic diffusion for multi-frame image super-resolution. Adv. Comput. Math. 49(6), 79 (2023). https://doi.org/10.1007/s10444-023-10079-3

    Article  MathSciNet  Google Scholar 

  10. K. Benyettou, D. Bouagada, On the positivity of fractional two-dimensional systems using the non-conformable derivative. Int. J. Dyn. Control 11(6), 2751–2762 (2023). https://doi.org/10.1007/s40435-023-01137-1

    Article  MathSciNet  Google Scholar 

  11. W. Burger, M.J. Burge, Digital Image Processing: An Algorithmic Introduction (Springer, London, 2022). https://doi.org/10.1007/978-1-4471-6684-9

    Book  Google Scholar 

  12. J.F. Cavalcante-Neto, G.d.A. Monteiro, A.B. Brandt, G.E.P. Soares, D.J.F. Solla, P.R.L. Leal, G. Cristino-Filho, K.F. da Ponte, Rete middle cerebral artery aneurysm: a case report and systematic review. Asian J. Neurosurg. 18(04), 790–795 (2023). https://doi.org/10.1055/s-0043-1775732

  13. F. Çetinkaya, A review on the evolution of the conformable derivative. Funct. Differ. Equ. 29, 1–2 (2022). https://doi.org/10.26351/FDE/29/1-2/2

    Article  MathSciNet  Google Scholar 

  14. L. Chen, J. Gao, A.M. Lopes, Z. Zhang, Z. Chu, R. Wu, Adaptive fractional-order genetic-particle swarm optimization Otsu algorithm for image segmentation. Appl. Intell. 53(22), 26949–26966 (2023). https://doi.org/10.1007/s10489-023-04969-8

    Article  Google Scholar 

  15. Y. Chen, Y. Yao, H. Yang, Y. Wu, K. Zhang, X. Pan, Curvature-based machine vision method for measuring the dimension of ball screws. IEEE Access 11, 126803–126813 (2023). https://doi.org/10.1109/ACCESS.2023.3328555

    Article  Google Scholar 

  16. D. Cuete, Cerebral arteriovenous malformation. Case study, Radiopaedia.org (2016). Accessed 22 Feb 2024. https://doi.org/10.53347/rID-43729

  17. D. Cuete, Middle cerebral artery aneurysm. Case study, Radiopaedia.org (2014). Accessed 22 Feb 2024. https://doi.org/10.53347/rID-29423

  18. F. Gaillard, Arteriovenous malformation - cerebral. Case study, Radiopaedia.org (2010). Accessed 22 Feb 2024. https://doi.org/10.53347/rID-8172

  19. F. Gaillard, Hemorrhagic hereditary telangiectasia. Case study, Radiopaedia.org (2016). Accessed 22 Feb 2024. https://doi.org/10.53347/rID-29967

  20. F. Gaillard, Spiculated breast cancer. Case study, Radiopaedia.org (2010). Accessed 22 Feb 2024. https://doi.org/10.53347/rID-12608

  21. S. Gamini, V.V. Gudla, C.H. Bindu, Fractional-order diffusion based image denoising model. Int. J. Electr. Electron. Res. (IJEER) 10(4), 837–842 (2022). https://doi.org/10.37391/ijeer.100413

    Article  Google Scholar 

  22. N.M. Hassan, S. Hamad, K. Mahar, Mammogram breast cancer cad systems for mass detection and classification: a review. Multimed. Tools Appl. 81(14), 20043–20075 (2022). https://doi.org/10.1007/s11042-022-12332-1

    Article  Google Scholar 

  23. F.I. Himasa, M. Singhal, A. Ojha, B. Kumar, Prospective for diagnosis and treatment of diabetic retinopathy. Curr. Pharm. Des. 28(7), 560–569 (2022). https://doi.org/10.2174/1381612827666211115154907

    Article  Google Scholar 

  24. A. Hoover, M. Goldbaum, Locating the optic nerve in a retinal image using the fuzzy convergence of the blood vessels. IEEE Trans. Med. Imaging 22(8), 951–958 (2003). https://doi.org/10.1109/TMI.2003.815900

    Article  Google Scholar 

  25. T. Huang, X. Wang, C. Wang, X. Liu, Y. Yu, W. Qiu, Super-resolution reconstruction algorithm for depth image based on fractional calculus. In: 35th Chinese Control and Decision Conference (CCDC), pp. 389–396. IEEE (2023). https://doi.org/10.1109/CCDC58219.2023.10326972

  26. T. Huang, X. Wang, D. Xie, C. Wang, X. Liu, Depth image enhancement algorithm based on fractional differentiation. Fractal Fract. 7(5), 394 (2023). https://doi.org/10.3390/fractalfract7050394

    Article  Google Scholar 

  27. D. Ibrahim, Cerebral arteriovenous malformation. Case study, Radiopaedia.org (2014). Accessed 22 Feb 2024. https://doi.org/10.53347/rID-29531

  28. R.W. Ibrahim, H.A. Jalab, F.K. Karim, E. Alabdulkreem, M.N. Ayub, A medical image enhancement based on generalized class of fractional partial differential equations. Quant. Imaging Med. Surg. 12(1), 172 (2022). https://doi.org/10.21037/qims-21-15

    Article  Google Scholar 

  29. B. Jähne, H. Haussecker, P. Geissler, Handbook of Computer Vision and Applications, vol. 2 (Citeseer, USA, 1999)

    Google Scholar 

  30. M. Joshi, S. Bhosale, V.A. Vyawahare, A survey of fractional calculus applications in artificial neural networks. Artif. Intell. Rev. 56(11), 13897–13950 (2023). https://doi.org/10.1007/s10462-023-10474-8

    Article  Google Scholar 

  31. F.K. Karim, H.A. Jalab, R.W. Ibrahim, R. Ala’a, Mathematical model based on fractional trace operator for COVID-19 image enhancement. J. King Saud Univer.-Sci. 34(7), 102254 (2022). https://doi.org/10.1016/j.jksus.2022.102254

    Article  Google Scholar 

  32. R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014). https://doi.org/10.1016/j.cam.2014.01.002

    Article  MathSciNet  Google Scholar 

  33. M.A. Khan, A. Ullah, Z.-J. Fu, S. Khan, S. Khan, Image restoration via combining a fractional order variational filter and a TGV penalty. Multimedia Tools Appl. (2024). https://doi.org/10.1007/s11042-023-17774-9

  34. M. Khan, P. Kumar, A nonlinear modeling of fractional order based variational model in optical flow estimation. Optik 261, 169136 (2022). https://doi.org/10.1016/j.ijleo.2022.169136

    Article  Google Scholar 

  35. M. Khan, N.K. Mahala, P. Kumar, Caputo derivative based nonlinear fractional order variational model for motion estimation in various application oriented spectrum. Sādhanā 49(1), 1–28 (2024). https://doi.org/10.1007/s12046-023-02318-6

    Article  MathSciNet  Google Scholar 

  36. G. Kruger, Ductal carcinoma in situ in an invasive ductal carcinoma. Case study, Radiopaedia.org (2012). Accessed 22 Feb 2024. https://doi.org/10.53347/rID-19035

  37. G. Kruger, Fat necrosis - breast. Case study, Radiopaedia.org (2013). Accessed 22 Feb 2024. https://doi.org/10.53347/rID-21553

  38. G. Kruger, Invasive ductal carcinoma. Case study, Radiopaedia.org (2013). Accessed 22 Feb 2024. https://doi.org/10.53347/rID-23260

  39. G. Kruger, Liponecrotic breast calcifications. Case study, Radiopaedia.org (2013). Accessed 22 Feb 2024. https://doi.org/10.53347/rID-21895

  40. P. Kumar, M. Khan, N.K. Mahala, A segmentation based robust fractional variational model for motion estimation. In: International Conference on Computer Vision and Image Processing, vol. 1776, pp. 115–128. Springer (2022). https://doi.org/10.1007/978-3-031-31407-0_9

  41. S. Kumar, M.K. Gupta, A.S. Shekhawat, Fractional calculus and special functions with applications in applied mathematics and other sciences—a review. Int. J. Converg. Technol. Manag. 9(1), 74–82 (2023)

    Google Scholar 

  42. J. Lavín-Delgado, J. Gómez-Aguilar, D. Urueta-Hinojosa, Z. Zamudio-Beltrán, J. Alanís-Navarro, An efficient technique for object recognition using fractional Harris–Stephens corner detection algorithm. Multimed. Tools Appl. 83(8), 23173–23199 (2023). https://doi.org/10.1007/s11042-023-16428-0

    Article  Google Scholar 

  43. M. Madaminov, F. Shernazarov, Breast cancer detection methods, symptoms, causes, treatment. Sci. innov. 1(D8), 530–535 (2022). https://doi.org/10.5281/zenodo.7401437

    Article  Google Scholar 

  44. A. Makandar, S. Kaman, R. Biradar, S.B. Javeriya, Impact of edge detection algorithms on different types of images using PSNR and MSE. LC Int. J. STEM 3(4), 1–11 (2022). https://doi.org/10.5281/zenodo.7607059

    Article  Google Scholar 

  45. M. Mansouri, E. Therasse, E. Montagnon, Y.O. Zhan, S. Lessard, A. Roy, L.-M. Boucher, O. Steinmetz, E. Aslan, A. Tang, et al., Ct analysis of aortic calcifications to predict abdominal aortic aneurysm rupture. Eu. Radiol. (2023). https://doi.org/10.1007/s00330-023-10429-1

  46. D. Martin, C. Fowlkes, D. Tal, J. Malik, et al., A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics. In: Proceedings 8th IEEE International Conference on Computer Vision, vol. 2, pp. 416–423. IEEE (2001). https://doi.org/10.1109/ICCV.2001.937655

  47. C.J. Miosso, A. Bauchspiess, Fuzzy inference system applied to edge detection in digital images. In: Proceedings of the V Brazilian Conference on Neural Networks, pp. 481–486 (2016). https://doi.org/10.21528/CBRN2001-102

  48. M. Mortazavi, M. Gachpazan, M. Amintoosi, Improving canny edge detection algorithm using fractional-order derivatives. J. Math. Model. 10(4), 495–514 (2022). https://doi.org/10.22124/JMM.2022.21875.1921

    Article  MathSciNet  Google Scholar 

  49. M.A.S. Murad, Modified integral equation combined with the decomposition method for time fractional differential equations with variable coefficients. Appl. Math.—A J. Chin. Univer. 37(3), 404–414 (2022). https://doi.org/10.1007/s11766-022-4159-5

    Article  MathSciNet  Google Scholar 

  50. D. Nicoletti, Middle cerebral artery aneurysm. Case study, Radiopaedia.org (2016). Accessed 22 Feb 2024. https://doi.org/10.53347/rID-43168

  51. C. O’Sullivan, S. Coveney, X. Monteys, S. Dev, The effectiveness of edge detection evaluation metrics for automated coastline detection. In: 2023 Photonics and Electromagnetics Research Symposium (PIERS), pp. 31–40. IEEE (2023). https://doi.org/10.1109/PIERS59004.2023.10221292

  52. R. Parvaz, Image restoration with impulse noise based on fractional-order total variation and framelet transform. SIViP 17(5), 2455–2463 (2023). https://doi.org/10.1007/s11760-022-02462-2

    Article  Google Scholar 

  53. R.J. Rahme, R. Singh, N. De La Pena, E.L. Turcotte, B.R. Bendok, Arteriovenous malformations: Treatment and management, in Introduction to Vascular Neurosurgery. ed. by J.R. Mascitelli, M.J. Binning (Springer, Switzerland, 2022), pp.389–410. https://doi.org/10.1007/978-3-030-88196-2_20

    Chapter  Google Scholar 

  54. K.V. Rani, M.E. Prince, P.S. Therese, P.J. Shermila, E.A. Devi, Content-based medical image retrieval using fractional Hartley transform with hybrid features. Multimedia Tools Appl. 83(9), 27217–27242 (2023). https://doi.org/10.1007/s11042-023-16462-y

    Article  Google Scholar 

  55. Y. Saijo, T. Orihara, N. Kanno, H. Yagami, T. Ishii, Estimation of fractional flow reserve in coronary artery based on serial intravascular ultrasound images. In: 2023 IEEE International Ultrasonics Symposium (IUS), pp. 1–3. IEEE (2023). https://doi.org/10.1109/IUS51837.2023.10307832

  56. R. Sharma, B. Goyal, A. Dogra, Advancement in diabetic retinopathy diagnosis techniques: automation and assistive tools. Open Neuroimaging J. 16(1), 1–10 (2023). https://doi.org/10.2174/18744400-v16-230831-2022-1

    Article  Google Scholar 

  57. H. Singh, H. Srivastava, R. Pandey, Special Functions in Fractional Calculus and Engineering, 1st edn. (CRC Press, Boca Raton, 2023)

    Book  Google Scholar 

  58. R. Sun, T. Lei, Q. Chen, Z. Wang, X. Du, W. Zhao, A.K. Nandi, Survey of image edge detection. Front. Signal Process. 2, 826967 (2022). https://doi.org/10.3389/frsip.2022.826967

    Article  Google Scholar 

  59. K.G. Vishnu, K.E. Bhageerath, A.V. Pallanti, A comparative analysis of edge detection techniques for processing of a video signal. Int. J. Adv. Netw. Appl. 13(4), 5029–5036 (2022)

    Google Scholar 

  60. Y. Wang, F. Xiong, J. Leach, E. Kao, B. Tian, C. Zhu, Y. Zhang, M. Hope, D. Saloner, D. Mitsouras, Contrast-enhanced CT radiomics improves the prediction of abdominal aortic aneurysm progression. Eur. Radiol. 33(5), 3444–3454 (2023). https://doi.org/10.1007/s00330-023-09490-7

    Article  Google Scholar 

  61. X.-J. Wu, Q. Zhao, M.-Y. Gao, S.-K. Xu, S.-X. Liu, Image reconstruction algorithm of electromagnetic tomography based on fractional Kalman filter. Flow Meas. Instrum. 86, 102198 (2022). https://doi.org/10.1016/j.flowmeasinst.2022.102198

    Article  Google Scholar 

  62. Y. Yang, H.H. Zhang, Fractional Calculus with Its Applications in Engineering and Technology, 1st edn. (Springer, Switzerland, 2022)

    Google Scholar 

  63. D. Zhang, Y. Lin, H. Chen, Z. Tian, X. Yang, J. Tang, K.T. Cheng, Deep learning for medical image segmentation: tricks, challenges and future directions. arXiv preprint arXiv:2209.10307 (2022)

  64. X. Zhang, X. Feng, W. Wang, W. Xue, Edge strength similarity for image quality assessment. IEEE Signal Process. Lett. 20(4), 319–322 (2013). https://doi.org/10.1109/LSP.2013.2244081

    Article  Google Scholar 

Download references

Acknowledgements

J.E. Lavín-Delgado, J.E. Solís-Pérez, J.F. Gómez-Aguilar and José R. Razo-Hernández acknowledges the support provided by SNI-CONAHCyT. The fifth and sixth authors would like to thank Azarbaijan Shahid Madani University. Also, the authors would like to thank dear reviewers for their constructive comments to improve the quality of the paper.

Funding

No funds were received for this paper.

Author information

Authors and Affiliations

  1. Universidad Politécnica del Estado de Guerrero (UPEG), Dirección de Ingeniería en Redes y Telecomunicaciones, Puente Campuzano, Carretera Federal Iguala-Taxco K.M. 105, Taxco de Alarcón, 40321, Guerrero, Mexico

    J. E. Lavín-Delgado

  2. Escuela Nacional de Estudios Superiores Unidad Juriquilla, Universidad Nacional Autónoma de México, Blvd. Juriquilla 3001, 76230, Juriquilla, Querétaro, Mexico

    J. E. Solís-Pérez

  3. Centro de Investigación en Ingeniería y Ciencias Aplicadas (CIICAp-IICBA), Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Col. Chamilpa, 62209, Cuernavaca, Morelos, Mexico

    J. F. Gómez-Aguilar

  4. CA-Fuentes Alternas y Calidad de la Energía Eléctrica, Departamento de Ingeniería Electromecánica, Tecnológico Nacional de México, Instituto Tecnológico Superior de Irapuato (ITESI), Carr. Irapuato-Silao km 12.5, Colonia El Copal, Irapuato, 36821, Guanajuato, Mexico

    J. R. Razo-Hernández

  5. Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran

    Sina Etemad & Shahram Rezapour

  6. Insurance Research Center, Tehran, Iran

    Shahram Rezapour

  7. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

    Shahram Rezapour

Authors
  1. J. E. Lavín-Delgado
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. E. Solís-Pérez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. F. Gómez-Aguilar
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. R. Razo-Hernández
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Sina Etemad
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Shahram Rezapour
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.

Corresponding authors

Correspondence to J. F. Gómez-Aguilar or Shahram Rezapour.

Ethics declarations

Conflict of interest

The authors declare that they have no Conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lavín-Delgado, J.E., Solís-Pérez, J.E., Gómez-Aguilar, J.F. et al. An Improved Object Detection Algorithm Based on the Hessian Matrix and Conformable Derivative. Circuits Syst Signal Process (2024). https://doi.org/10.1007/s00034-024-02669-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00034-024-02669-3

Keywords

Search

Navigation