Compensated convexity methods for approximations and interpolations of sampled functions in euclidean spaces: Applications to contour lines, sparse data, and inpainting
| Autor: | Antonio Orlando, Kewei Zhang, Elaine Crooks |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Matemáticas
General Mathematics Inpainting INTERPOLATION 02 engineering and technology 01 natural sciences Convexity HIGH DENSITY SALT \& PEPPER NOISE REDUCTION Maximum principle Mathematics - Metric Geometry Euclidean geometry FOS: Mathematics 0202 electrical engineering electronic engineering information engineering Applied mathematics 90C25 90C26 49J52 52A41 65K10 IMAGE INPAINTING 0101 mathematics Mathematics Sparse matrix CONTOUR LINES Quantitative Biology::Biomolecules HAUSDORFF STABILITY Applied Mathematics 010102 general mathematics SCATTERED DATA Regular polygon CONVEX DENSITY RADIUS Metric Geometry (math.MG) Matemática Aplicada INPAINTING Ciencias de la Computación COMPENSATED CONVEX TRANSFORMS Contour line Computer Science::Computer Vision and Pattern Recognition Ciencias de la Computación e Información 020201 artificial intelligence & image processing MAXIMUM PRINCIPLE CIENCIAS NATURALES Y EXACTAS Interpolation APPROXIMATION |
| ISSN: | 1936-4954 |
| DOI: | 10.1137/17M116152X |
| Popis: | This paper is concerned with applications of the theory of approximation and interpolation based on compensated convex transforms developed in [K. Zhang, E. Crooks, A. Orlando, Compensated convexity methods for approximations and interpolations of sampled functions in Euclidean spaces: Theoretical Foundations. SIAM Journal on Mathematical Analysis 48 (2016) 4126-4154]. We apply our methods to $(i)$ surface reconstruction starting from the knowledge of finitely many level sets (or `contour lines'); $(ii)$ scattered data approximation; $(iii)$ image inpainting. For $(i)$ and $(ii)$ our methods give interpolations. For the case of finite sets (scattered data), in particular, our approximations provide a natural triangulation and piecewise affine interpolation. Prototype examples of explicitly calculated approximations and inpainting results are presented for both finite and compact sets. We also show numerical experiments for applications of our methods to high density salt and pepper noise reduction in image processing, for image inpainting and for approximation and interpolations of continuous functions sampled on finitely many level sets and on scattered points. Accepted to appear in SIAM Journal on Imaging Sciences |
| Databáze: | OpenAIRE |
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