Haydar KILIC, Salim CEYHAN

Görüntü İşleme Uygulamaları İçin Finsler Eğri Evrim Modeli İncelemesi

Bu çalışmada Finsler geometrisi ile kurulan ve varyasyonlar hesabı ile eğri evrim modeline dönüşen Finsler eğri evrim modeli incelenecektir. Bu model, Riemann eğri evrim modelinde olduğu gibi sadece görüntü uzayındaki konumlar değil yönlerde dikkate alınarak anizotropik bir uzayda kurulmuştur. Model, görüntüye izotropik bir yapı olarak bakan Riemann modelinin aksine anizotropik yapı olarak bakarak daha esnek bir çalışma alanı sunar. Bu nedenle görüntü işleme üzerine çalışan araştırmacıların sıklıkla üzerinde çalıştıkları bir model haline gelmiştir. Bu çalışmada ele alınan Finsler eğri evrim modelinin matematiksel analizi kullanılarak gelecekteki görüntü işleme alanında yapılacak çalışmalara katkı sağlanması amaçlanmaktadır.

Anahtar Kelimeler:

Finsler metrik, görüntü işleme, varyasyonlar hesabı, eğri evrimi

Finsler Curve Evolution Analysis for Image Processing

In this study, the Finsler curve evolution model, which was established with Finsler geometry and turned into a curved evolution model with the calculation of variations, will be examined. This model is not constructed by considering only locations in the image space as in the Riemann curve evolution model. It was established in an anisotropic space by considering both locations and directions. The model offers a flexible operation in the image by eliminating the necessity of isotropic spaces. Therefore, it has become a model that researchers working on image processing mostly work on. It is aimed to contribute to future studies in the field of image processing by using the mathematical analysis of the Finsler curve evolution model discussed in this study.

Keywords:

Finsler metric, image processing, variational calculus, curve evolution. Finsler metrik, görüntü işleme, varyasyonlar hesabı, eğri evrimi,

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Boyd, S., Boyd, S. P., & Vandenberghe, L. (2004). Convex optimization. Cambridge university press.
Caselles V, Kimmel R, Sapiro G, 1997. Geodesic Active Contours. International Journal of Computer Vision, 22 (1): 61–79.
Chen, D., Mirebeau, J., & Cohen, L.D. (2016). A New Finsler Minimal Path Model with Curvature Penalization for Image Segmentation and Closed Contour Detection. 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 355-363.
Dokur, E., Ceyhan, S., & Kurban, M. (2017). Finsler Geometry for Two-Parameter Weibull Distribution Function. Mathematical Problems in Engineering, 2017, 1-6.
Estellers V, Zosso D, Bresson X, Thiran JP, 2013. Harmonic active contours. IEEE Transactions on Image Processing, 23 (1): 69–82.
Jelena S, 2015. Anisotropic frameworks for dynamical systems and image processing. Ph. D. thesis, Univerzitet u Novom Sadu, Prirodnomatematički fakultet u Novom Sadu.
Kilic H, Ceyhan S, 2021. Riemann anlamında eğri evrim modeli incelemesi: Görüntü segmentasyonu uygulaması. Bilecik Şeyh Edebali Üniversitesi Fen Bilimleri Dergisi 8 (2): 693–721.
Kılıç, H. & Ceyhan, S. (2022). Yeni Bir Anizotropik Metrik Kullanılarak Renkli Görüntü İyileştirme. 30. IEEE Sinyal İşleme ve İletişim Uygulamaları Kurultayı, (Yayın no, 276).
Kimmel R, Sochen N, Malladi R, 1997. From high energy physics to low level vision. In International Conference on Scale-Space Theories in Computer Vision, 236–247.
Kolmogorov V, Boykov Y, 2005. What metrics can be approximated by geo-cuts, or global optimization of length/area and flux. In Tenth IEEE International Conference on Computer Vision (ICCV’05), 564–571.
Kühnel W, 2015. Differential geometry, Curves - Surfaces - Manifolds, Volume 77. American Mathematical Soc., 3th edition.
Lee JM, 2006. Riemannian manifolds: An introduction to curvature, Volume 176. Springer Science & Business Media.
Melonakos J, Pichon E, Angenent S, Tannenbaum A, 2008. Finsler active contours. IEEE Transactions on Pattern Analysis and Machine Intelligence, 30 (3): 412–423.
Pichon E, 2005. Novel methods for multidimensional image segmentation. Ph. D. thesis, Georgia Institute of Technology.
Rund H, 2012. The differential geometry of Finsler spaces, Volume 101. Springer Science & Business Media. Shen YB, Shen Z, 2016. Introduction to modern Finsler geometry. World Scientific Publishing Company.
Yajima, T., & Nagahama, H. (2009). Finsler geometry of seismic ray path in anisotropic media. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 465, 1763 - 1777.
Yajima, T., & Nagahama, H. (2015). Finsler geometry for nonlinear path of fluids flow through inhomogeneous media. Nonlinear Analysis-real World Applications, 25, 1-8.
Young JW, 1930. Projective geometry, Volume 4. American Mathematical Soc.
Zach, C., Shan, L., & Niethammer, M. (2009). Globally Optimal Finsler Active Contours. Pattern recognition: DAGM Symposium, proceedings. DAGM (Organization), 5748, 552-561.
Zucchini R, 1991. A polyakov action on riemann surfaces. Physics Letters B, 260 (3-4): 296–302.
Zucchini R, 1993. A polyakov action on riemann surfaces (ii). Communications in mathematical physics, 152 (2): 269–297.