Related Concepts
Definition
The variational method is a way to solve problems given in the form of a variational model, i.e., as an energy minimization problem on an infinite-dimensional space which is typically a function space. It employs tools from the mathematical framework of variational analysis.
Background
Ikeuchi and Horn's shape-from-shading paper and Horn and Schunck's optical flow paper, which appeared in AIJ simultaneously, are the earliest representative works to introduce a variational method to computer vision [7]. Inspired by the extraordinary success of the idea, variational methods have been extensively studied in computer vision and become a very popular tool for a wide variety of problems. They are particularly successful in mathematical image processing, where they are used to describe fundamental low-level problems, like image segmentation [9, 11], denoising [15], and deblurring [3], but have also been employed for high-level...
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ambrosio L, Fusco N, Pallara D (2000) Functions of bounded variation and free discontinuity problems. Oxford University Press, Oxford/New York
Attouch H, Buttazzo G, Michaille G (2006) Variational analysis in Sobolev and BV spaces. MPS-SIAM series on optimization. Society for Industrial and Applied Mathematics, Philadelphia
Chan T, Wong C (1998) Total variation blind deconvolution. IEEE Trans Image Process 7(3):370–375
Faugeras O, Keriven R (1998) Variational principles, surface evolution, PDE's, level set methods, and the stereo problem. IEEE Trans Image Process 7(3):336–344
Gelfand IM, Fomin SV (2003) Calculus of variations. Dover publications reprint of the 1963 edn. Dover Publications Inc., Mineola, NY
Goldluecke B, Ihrke I, Linz C, Magnor M (2007) Weighted minimal hypersurface reconstruction. IEEE Trans Pattern Anal Mach Intell 29(7):1194–1208
Ikeuchi K, Horn BKP (1981) Numerical shape from shading and occluding boundaries. Artif Intell 17: 141–184
Horn B, Schunck B (1981) Determining optical flow. Artif Intell 17:185–203
Kass M, Witkin A, Terzopoulos D (1987) Snakes: active contour models. Int J Comput Vis 1:321–331
Kolev K, Klodt M, Brox T, Cremers D (2009) Continuous global optimization in multiview 3D reconstruction. Int J Comput Vis 84(1):80–96
Mumford D, Shah J (1989) Optimal approximation by piecewise smooth functions and associated variational problems. Commun Pure Appl Math 42:577–685
Nikolova M, Esedoglu S, Chan T (2006) Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J Appl Math 66(5): 1632–1648
Osher S, Fedkiw R (2003) Level set methods and dynamic implicit surfaces. Volume 153 of applied mathematical sciences. Springer, New York
. Overgaard N, Solem J (2007) The variational origin of motion by gaussian curvature. In: Proceedings of the 1st international conference on Scale space and variational methods in computer vision (SSVM)
Rudin LI, Osher S, Fatemi E (1992) Nonlinear total variation based noise removal algorithms. Physica D 60(1–4):259–268
Sharpe RW (1997) Differential geometry. Volume 166 of graduate texts in mathematics. Springer, New York
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this entry
Cite this entry
Goldluecke, B. (2014). Variational Method. In: Ikeuchi, K. (eds) Computer Vision. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-31439-6_684
Download citation
DOI: https://doi.org/10.1007/978-0-387-31439-6_684
Published:
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-30771-8
Online ISBN: 978-0-387-31439-6
eBook Packages: Computer ScienceReference Module Computer Science and Engineering