Abstract
This paper addresses the problem of enhancement of noisy scalar and vector fields, when they are known to be constrained to a manifold. As an example, we address selective smoothing of orientation using the geometric Beltrami framework. The orientation vector field is represented accordingly as the embedding of a two dimensional surface in the spatial-feature manifold. Orientation diffusion is treated as a canonical example where the feature (orientation in this case) space is the unit circle S 1. Applications to color analysis are discussed and numerical experiments demonstrate again the power of this framework for non-trivial geometries in image processing.
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References
T. Chan and J. Shen, “Variational restoration of non-flat image features: Models and algorithms”, SIAM J. Appl. Math., to appear.
R. Kimmel and N. Sochen, “How to Comb a Porcupine?”, ISL Technical Report, Technion Tel-Aviv University report, Israel, June 2000. Accepted to special issue on PDEs in Image Processing, Computer Vision, and Computer Graphics, Journal of Visual Communication and Image Representation.
R. Kimmel and R. Malladi and N. Sochen, “Images as Embedding Maps and Minimal Surfaces: Movies, Color, Texture, and Volumetric Medical Images”, International Journal of Computer Vision, 39(2) (2000) 111–129.
E. Kreyszing, “Differential Geometry”, Dover Publications, Inc., New York, 1991.
A. M. Polyakov, “Quantum geometry of bosonic strings”, Physics Letters, 103B (1981) 207–210.
P. Perona, “Orientation Diffusion” IEEE Trans. on Image Processing, 7 (1998) 457–467.
“Geometry Driven Diffusion in Computer Vision”, Ed. B. M. ter Haar Romeny, Kluwer Academic Publishers, 1994.
L. Rudin and S. Osher and E. Fatemi, “Nonlinear total variation based noise removal algorithms”, Physica D, 60 (1991) 259–268.
A. R. Smith, “Color gamut transform pairs”, SIGGRAPH’79, 12–19
N. Sochen and R. M. Haralick and Y. Y. Zeevi, “A Geometric functional for Derivatives Approximation” EE-Technion Report, April 1999.
N. Sochen and R. Kimmel and R. Malladi, “From high energy physics to low level vision”, Report, LBNL, UC Berkeley, LBNL 39243, August, Presented in ONR workshop, UCLA, Sept. 5 1996.
N. Sochen and R. Kimmel and R. Malladi, “A general framework for low level vision”, IEEE Trans. on Image Processing, 7 (1998) 310–318.
N. Sochen and Y. Y. Zeevi, “Representation of colored images by manifolds embedded in higher dimensional non-Euclidean space”, IEEE ICIP’98, Chicago, 1998.
B. Tang and G. Sapiro and V. Caselles, “Direction diffusion”, International Conference on Computer Vision, 1999.
B. Tang and G. Sapiro and V. Caselles, “Color image enhancement via chromaticity diffusion” Technical report, ECE-University of Minnesota, 1999.
J. Weickert, “Coherence-enhancing diffusion in colour images”, Proc. of VII National Symposium on Pattern Rec. and Image Analysis, Barcelona, Vol. 1, pp. 239–244, 1997.
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© 2001 Springer-Verlag Berlin Heidelberg
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Kimmel, R., Sochen, N. (2001). Using Beltrami Framework for Orientation Diffusion in Image Processing. In: Arcelli, C., Cordella, L.P., di Baja, G.S. (eds) Visual Form 2001. IWVF 2001. Lecture Notes in Computer Science, vol 2059. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45129-3_31
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DOI: https://doi.org/10.1007/3-540-45129-3_31
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