JAMC

Article http://dx.doi.org/10.26855/jamc.2025.03.010

Laplacian Matrices in Computer Graphics

TOTAL VIEWS: 333

Evanthios Papadopoulos, Ioannis Kougias*

Laboratory of Interdisciplinary Semantic Interconnected Symbiotic Education Environments, Electrical and Computer Engineering Department, Faculty of Engineering, University of Peloponnese, Patras 263 34, Greece.

*Corresponding author:Ioannis Kougias

Published: April 22,2025

Abstract

Linear algebra serves as a foundational pillar in computer graphics programming, providing the essential mathematical framework for representing and manipulating geometric data in both two- and three-dimensional spaces. Vectors and matrices are widely used to perform transformations such as translation, rotation, and scaling, enabling the creation and control of complex visual scenes. In recent years, Laplacian matrices have emerged as powerful tools within this domain, playing a critical role in the development of sophisticated algorithms for mesh processing, surface smoothing, deformation, and simulation. Their applications span various graphical disciplines, including video game development, animation, and virtual reality, where they contribute to the creation of realistic and dynamic environments. This paper is primarily expository in nature, aiming to provide a comprehensive overview of the fundamental concepts and significant results associated with Laplacian matrices. It highlights their practical importance and showcases how they are leveraged to enhance visual fidelity and computational efficiency in modern computer graphics.

References

[1] Papadopoulos E, Kougias I. Algebraic transformations in computer graphics [Internet]. 2017. Available from:
http://repository.library.teiwest.gr/Algebraictransfincomputergra

[2] Vince J. Mathematics for computer graphics. Springer; 2006. p. 31-105. Available from:
https://link.springer.com/book/10.1007/1-84628-283-7

[3] Patel SA, Yildirim A. Overcoming dimensionality constraints: a Gershgorin circle theorem-based feature extraction for weighted Laplacian matrices in computer vision applications. J Imaging. 2024;10:121. doi:10.3390/jimaging10050121

[4] Pang J, Cheung G. Graph Laplacian regularization for image denoising: analysis in the continuous domain [Internet]. 2017. Available from:
https://arxiv.org/pdf/1604.07948.pdf

[5] Rice University. Laplacian edge detection [Internet]. 2020. Available from:
https://www.owlnet.rice.edu/~elec539/Projects97/laplacian.html

[6] University of Edinburgh. Spatial filters-Laplacian/Laplacian of Gaussian [Internet]. 2003. Available from: 

https://homepages.inf.ed.ac.uk/rbf/HIPR2/log.htm

[7] Wu Q, Shi L, Bond S, Yu Y. Laplacian texture synthesis and mixing on surfaces [Internet]. Urbana-Champaign: University of Illinois; 2003. Available from:

https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=266e71a134f16c49b5a658f8690ae96c45c8ef5d

[8] Wobrock D. Image processing using graph Laplacian operator [Internet]. Stockholm: KTH Royal Institute of Technology; 2019. p. 9-20. Available from:
https://kth.diva-portal.org/smash/get/diva2:1412649/FULLTEXT01.pdf

[9] Princeton University. Laplacian mesh editing [Internet]. 2010. Available from: 

https://www.cs.princeton.edu/courses/archive/fall10/cos526/lectures/08-laplacian.pdf

[10] University of Washington. Image processing [Internet]. 2011. Available from:

https://courses.cs.washington.edu/courses/cse457/11au/lectures/image-processing.pdf

[11] Brigham Young University. Image segmentation [Internet]. 2021. Available from: 

https://acme.byu.edu/0000017c-ccff-da17-a5fd-cdff00520000/acmefiles-08-imagesegmentation-2021-pdf

[12] Medvedovsky Y, Treister E, Routtenberg T. Efficient graph Laplacian estimation by a proximal Newton approach [Internet]. 2023. Available from:
https://arxiv.org/pdf/2302.06434.pdf

[13] Washington University in St. Louis. Laplacian deformation [Internet]. 2012. Available from:

https://www.cse.wustl.edu/~taoju/cse554/lectures/lect08_Deformation.pdf

[14] Chung FRK. Spectral graph theory. Providence (RI): American Mathematical Society; 1997. Available from: 

http://mathworld.wolfram.com/LaplacianMatrix.html

[15] Bernstein MN. The graph Laplacian [Internet]. 2020. Available from: https://mbernste.github.io/posts/laplacian_matrix/

[16] Bhairannawar SS. Efficient medical image enhancement technique using transform HSV space and adaptive histogram equalization. In: Soft computing based medical image analysis. 2018. Available from: 

https://www.sciencedirect.com/topics/engineering/laplacian-filter

[17] GeeksforGeeks. Laplacian of Gaussian filter in MATLAB [Internet]. 2022. Available from: 

https://www.geeksforgeeks.org/laplacian-of-gaussian-filter-in-matlab/

[18] MathWorks. What is denoising? [Internet]. 2023. Available from: https://www.mathworks.com/discovery/denoising.html

[19] MathWorks. Smooth surface mesh [Internet]. 2023. Available from: https://www.mathworks.com/help/lidar/ref/smoothsurfacemesh.html

[20] MathWorks. Noise removal [Internet]. 2014. Available from: https://www.mathworks.com/help/images/noise-removal.html

[21] MathWorks. What is image segmentation? [Internet]. 2011. Available from: https://www.mathworks.com/discovery/image-segmentation.html

How to cite this paper

Laplacian Matrices in Computer Graphics

How to cite this paper: Evanthios Papadopoulos, Ioannis Kougias. (2025) Laplacian Matrices in Computer Graphics. Journal of Applied Mathematics and Computation9(1), 75-83.

DOI: http://dx.doi.org/10.26855/jamc.2025.03.010