The mathematical characteristic of the fifth order Laplace contour filters used in digital image processing

• Autorzy: Winnicki Ireneusz , Pietrek Sławomir , Jasinski Janusz , Kroszczynski Krzysztof

Identyfikatory

DOI

10.24425/agg.2022.141300

• Języki publikacji: EN

Abstrakty

EN

The Laplace operator is a differential operator which is used to detect edges of objects in digital images. This paper presents the properties of the most commonly used fifth-order pixels Laplace filters including the difference schemes used to derive them (finite difference method – FDM and finite element method – FEM). The results of the research concerning third-order pixels matrices of the convolution Laplace filters used for digital processing of images were presented in our previous paper: The mathematical characteristic of the Laplace contour filters used in digital image processing. The third order filters is presented byWinnicki et al. (2022). As previously, the authors focused on the mathematical properties of the Laplace filters: their transfer functions and modified differential equations (MDE). The relations between the transfer function for the differential Laplace operator and its difference operators are described and presented here in graphical form. The impact of the corner elements of the masks on the results is also discussed. A transfer function, is a function characterizing properties of the difference schemes applied to approximate differential operators. Since they are relations derived in both types of spaces (continuous and discrete), comparing them facilitates the assessment of the applied approximation method.

Słowa kluczowe

EN

finite element method; finite difference method; modified differential equations; matrices of the fifth-order Laplace filter; Taylor expansion of the transfer function

PL

metoda elementów skończonych; równanie różniczkowe; filtr Laplace'a

• Wydawca: Komitet Geodezji PAN
• Czasopismo: Advances in Geodesy and Geoinformation
• Rocznik: 2022
• Tom: Vol. 71, no. 2
• Strony: art. no. e26, 2022
• Opis fizyczny: Bibliogr. 57 poz., rys., wykr.

Twórcy

autor

Winnicki Ireneusz

• Military University of Technology, Warsaw, Poland

• https://orcid.org/0000-0001-9170-422X

autor

Pietrek Sławomir

• Military University of Technology, Warsaw, Poland

• https://orcid.org/0000-0001-9890-8487

autor

Jasinski Janusz

• Military University of Technology, Warsaw, Poland

• https://orcid.org/0000-0001-5634-3430

autor

Kroszczynski Krzysztof

• Military University of Technology, Warsaw, Poland

• https://orcid.org/0000-0003-1197-9915

Bibliografia

• 1. Appadu, A.R., Dauhoo, M.Z., Rughooputh, S.D.D.V. (2008). Control of numerical effects of dispersion and dissipation in numerical schemes for efficient shock-capturing through an optimal Courant number. Comput. Fluids, 37(6), 767–783. DOI: 10.1016/j.compfluid.2007.07.018.
• 2. Appadu, A.R., and Dauhoo, M.Z. (2011). The Concept of Minimized Integrated Exponential Error for Low Dispersion and Low Dissipation Schemes. Int. J. Numerical Methods Fluids, 65(5), 578–601. DOI: 10.1002/fld.2206.
• 3. Appadu, A.R. (2014). Optimized Low Dispersion and Low Dissipation Runge–Kutta Algorithms in Computational Aeroacoustics. Appl. Math. Inf. Sci., 8(1), 57–68. DOI: 10.12785/amis/080106.
• 4. Borawski, M. (2004). Preliminary digital image processing. In: Methods of Comparative Navigation (in Polish), 88–119. Gdynia: Andrzej Stateczny GTN.
• 5. Burger, W., and Burge, M.J. (2008). Digital Image Processing. An Algorithmic Introduction Using Java. Texts in Computer Science. New York: Springer Science ¸ Business Media.
• 6. Burger, W., and Burge, M.J. (2009a). Principles of Digital Image Processing. Core Algorithms. London: Springer–Verlag.
• 7. Burger, W., and Burge, M.J. (2009b). Principles of Digital Image Processing. Fundamental Techniques. London: Springer–Verlag.
• 8. Fryskowska, A., Kedzierski, M., Wierzbicki, D. et al. (2019). Analysis of imagery interpretability of open sources radar satellite imagery. In: Proceedings of 12th Conference on Reconnaissance and Electronic Warfare Systems (CREWS). Oltarzew (2019). DOI: 10.1117/12.2525013.
• 9. Gates, W.L. (1956). On the truncation error, stability, and convergence of difference solutions of the barotropic vorticity equation. J. Meteorol., 16, 556–568.
• 10. Gates, W.L. (1961). The reduction of truncation error by extrapolation techniques. Mon. Wea. Rev., 89, 115–124. DOI: 10.1175/1520-0493(1961)089<0115:TROTEB>2.0.CO;2.
• 11. Gonzalez, R.C., and Woods, R.E. (2018). Digital Image Processing, 4 edn. Pearson International Edition. New Jersey: Pearson Education.
• 12. Harris, C., and Stephens, M. (1988). A combined corner and edge detector. In Taylor C.J. (Eds.) Proceedings of the Alvey Vision Conference, p. 23. 1–23.6. DOI: 10.5244/C.2.23.
• 13. Iserles, A. (2009). A First Course in the Numerical Analysis of Differential Equations. Cambridge: University Press.
• 14. Jähne, B., Scharr, H., Körkel, S. et al. (1999). Principles of filter design. In: Handbook of computer vision and applications, 2, 125–151. Academic Press.
• 15. Jähne, B. (2002). Digital Image Processing. Berlin: Springer–Verlag.
• 16. Jasinski, J., Kroszczynski, K., Rymarz, C. et al. (1999). Satellite Images of Atmospheric Processes Moulding Weather (in Polish). Warsaw: PWN.
• 17. Knighting, E. (1955). Reduction of truncation errors in symmetrical operators. Technical Memorandum of the Joint Numerical Weather Prediction Unit, 3(5).
• 18. Krawczyk, K., Winnicki, I., Pietrek, S. et al. (2012). Spectral properties of fifth order Laplace filters used for digital data processing. Bulletin of the Military University of Technology, 61(1), 145–170.
• 19. Kupidura, A, and Kupidura, P. (2009). Investigating urban sprawl using remote sensing and GIS technology. In: Proceedings of 29th EARSeL Symposium, 224–230. Chania, Greece (2009).
• 20. Kupidura, P., Koza, P., Marciniak, J. (2010). Mathematical Morphology in Remote Sensing (in Polish). Warsaw: PWN.
• 21. Kurczynski, Z., Rozycki, S., Bylina, P. (2017). Mapping of polar areas based on high-resolution satellite images: the example of the Henryk Arctowski Polish Antarctic Station. Reports Geod. Geoinf., 104(1), 65–78. DOI: 10.1515/rgg-2017-0016.
• 22. LeVeque, R.J. (2007). Finite Difference Methods for Ordinary and Partial Differential Equations. Steady-State and Time-Dependent Problems. Philadelphia: SIAM. DOI: 10.1137/1.9780898717839.
• 23. Li, J., and Yang, Z. (2011). Heuristic modified equation analysis on oscillations in numerical solutions of conservation laws. SIAM J. Numer. Anal., 49, 2386–2406. DOI: 10.1137/110822591.
• 24. Li, J., and Yang, Z. (2013). The von Neumann analysis and modified equation approach for finite difference schemes. Appl. Math. Comput., 225, 610–621. DOI: 10.1016/j.amc.2013.09.046.
• 25. Lynch, D.R. (2010). Numerical Partial Differential Equations for Environmental Scientists and Engineers: A First Practical Course. New York: Springer Science ¸ Business Media. DOI: 10.1121/1.1577548.
• 26. Mallat, S. (2009). A Wavelet Tour of Signal Processing. The Sparse Way. Amsterdam: Academic Press.
• 27. Marr, D., and Hildreth, E. (1980). Theory of edge detection. Proc. Royal Soc., 207, 187–217. DOI: 10.1098/rspb.1980.0020.
• 28. Miyakoda, K. (1960). Numerical calculation of Laplacian and Jacobian using 9 and 25 gridpoint systems. J. Meteorol. Soc. Japan, 2, 94–105.
• 29. Ogura, Y. (1958). On the truncation error which arises from the use of finite differences in the Laplacian operator. J. Meteorol., 15(5), 475–480. DOI: 10.1175/1520-0469(1958)015<0475:OTTEWA>2.0.CO;2.
• 30. Parker, J.R. (2011). Algorithms for Image Processing and Computer Vision. Indianapolis: Wiley.
• 31. Pavel, S., Segeth, K., Dolezel, I. (2003). Higher-Order Finite Element Methods. Chapman & Hall/CRC.
• 32. Petrou, M., and Petrou, C. (2011). Image Processing. The Fundamentals. Chichester: John Wiley & Sons.
• 33. Peyret, R., and Taylor, T.D. (1983). Computational Methods for Fluid Flow. New York: Springer-Verlag.
• 34. Pitas, I. (2000). Digital Image Processing Algorithms and Applications. New York: John Wiley & Sons.
• 35. Pokonieczny, K., and Moscicka, A. (2018). The influence of the shape and size of the cell on developing military passability maps. ISPRS Int. J. Geoinf., 7(7). DOI: 10.3390/ijgi7070261.
• 36. Pratt, W.K. (2007). Digital Image Processing. New York: Wiley-Interscience.
• 37. Prewitt, J.M.S. (1970). Object enhancement and extraction. In: B.S. Lipkin and A. Rosenfeld. (Eds.) Picture Processing and Psychopictorics, 75–150. New York: Academic Press.
• 38. Reda, K., and Kedzierski, M. (2020). Detection, classification, and boundary regularization of buildings in satellite imagery using faster edge region convolutional neural networks. Remote Sens., 12(14). DOI: 10.3390/rs12142240.
• 39. Richardson, L.F. (1910). The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Philos. Proc. R. Soc. London Ser. A, 210, 307–357.
• 40. Scharr, H. (2000). Optimal operators in digital image processing. Ph.D. thesis. Interdisciplinary Center for Scientific Computing, University of Heidelberg, Germany.
• 41. Scharr, H., and Weickert, J. (2000). An Anisotropic Diffusion Algorithm with Optimized Rotation Invariance. In: Proceedings of 22th DAGM-Symposium, Kiel, Germany, 460–467. DOI: 10.1007/978-3-642-59802-9.
• 42. Shokin, Y., Winnicki, I., Jasinski, J. et al. (2020a). High order modified differential equation of the Beam-Warming method. Part I: The dispersive features. Russ. J. Numer. Anal. Math. Model., 35(2), 83–94. DOI: 10.1515/rnam-2020-0007.
• 43. Shokin, Y., Winnicki, I., Jasinski, J. et al. (2020b). High order modified differential equation of the Beam-Warming method. Part II: The dissipative features. Russ. J. Numer. Anal. Math. Model., 35(3), 175–185. DOI: 10.1515/rnam-2020-0014.
• 44. Shuman, G.F. (1956). The computational difficulties fundamental to the balance equation. Technical Memorandum of the Joint Numerical Weather Prediction Unit, 3(6).
• 45. Stateczny, A., and Nowakowski, M. (2006). Analytical radar image compression methods for comparative navigation. Scientific Journals of the Maritime University of Szczecin, 8, 93–101.
• 46. Stateczny, A., Kazimierski, W., Kulpa, K. (2020). Radar and sonar imaging and processing. Remote Sens., 12(11). DOI: 10.3390/rs12111811.
• 47. Stateczny, A., Kazimierski, W., Kulpa, K. (2021). Radar and sonar imaging and processing (2nd edition). Remote Sens., 13(22). DOI: 10.3390/rs13224656.
• 48. Strang, G., and Fix, G. (2008). An Analysis of the Finite Element Method. New Edition, 2nd edn. Wellesley–Cambridge Press.
• 49. Strikwerda, J.C. (2004). Finite Difference Schemes and Partial Differential Equations. Philadelphia: SIAM.
• 50. Tadeusiewicz, R., and Korohoda, P. (1997). Computer Analysis and Image Processing (in Polish). Progress of Telecommunication Foundation Publishing House, Krakow (1997). Retrieved from http://winntbg.bg.agh.edu.pl/skrypty2/0098/.
• 51. Thompson, P.D. (1955a). Reduction of truncation errors in the computation of geostrophic advection and other Jacobians. Technical Memorandum of the Joint Numerical Weather Prediction Unit, 1(12).
• 52. Thompson, P.D. (1955b). Reduction of truncation errors in the computation of geostrophic vorticity, the Laplacian operator, and its inverse. Technical Memorandum of the Joint Numerical Weather Prediction Unit, 2(9).
• 53. Torre, V., and Poggio, T. (1984). On edge-detection. IEEE Trans. Pattern Anal. Machine Intelligen., 8, 147–163. DOI: 10.1109/TPAMI.1986.4767769.
• 54. Warming, R.F., and Hyett, B.J. (1974). The modified equation approach to the stability and accuracy of finite difference methods. J. Comp. Phys., 14(2), 159–179. DOI: 10.1016/0021-9991(74)90011-4.
• 55. Winnicki, I., Jasinski, J., Pietrek, S. (2019). New approach to the Lax–Wendroff modified differentia equation for linear and non-linear advection. Numer. Methods Partial Differ. Equ., 35(6), 2275–2304. DOI: 10.1002/num.22412.
• 56. Winnicki, I., Pietrek, S., Jasinski, J. et al. (2022). The mathematical characteristic of the Laplace contour filters used in digital image processing. The third order filters. Adv. Geod. Geoinf., 71(2), article no. e23. DOI: 10.24425/gac.2022.141176.
• 57. Wojtkowska, M., Kedzierski, M., Delis, P. (2021). Validation of terrestrial laser scanning and artificial intelligence for measuring deformations of cultural heritage structures. Measurement, 167. DOI: 10.1016/j.measurement.2020.108291.