The Tree of Shapes of an Image

This chapter presents the tree of shapes of an image, a mix of the component trees of upper and lower level sets. Its existence under fairly weak assumptions and its completeness are proven. Ignoring the small details of the image, we show the essentially finite nature of the tree. Finally, we illustrate these theoretical results with a direct application to gray level quantization.

[1]  M. Wertheimer.  Untersuchungen zur Lehre von der Gestalt. II , 1923 .

[2]  R. Oppermann.  Elements of the topology of plane sets of points: by M. H. A. Newman. 221 pages, illustrations, 15 × 23 cms. New York, The Macmillan Company, 1939. Price $3.50 , 1939 .

[3]  M. Golubitsky,et al.  Stable mappings and their singularities , 1973 .

[4]  G. Matheron.  Random Sets and Integral Geometry , 1976 .

[5]  Jean Serra,et al.  Image Analysis and Mathematical Morphology , 1983 .

[6]  E. Giorgi,et al.  Un nuovo tipo di funzionale del calcolo delle variazioni , 1988 .

[7]  D. Mumford,et al.  Optimal approximations by piecewise smooth functions and associated variational problems , 1989 .

[8]  S. Beucher,et al.  Morphological segmentation , 1990, J. Vis. Commun. Image Represent..

[9]  Luc Vincent,et al.  Watersheds in Digital Spaces: An Efficient Algorithm Based on Immersion Simulations , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[10]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[11]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[12]  P. Lions,et al.  Axioms and fundamental equations of image processing , 1993 .

[13]  Philippe Salembier,et al.  Connected operators and pyramids , 1993, Optics & Photonics.

[14]  L. Vincent.  Grayscale area openings and closings, their efficient implementation and applications , 1993 .

[15]  Philippe Salembier,et al.  Morphological multiscale segmentation for image coding , 1994, Signal Process..

[16]  Luc Vincent,et al.  Morphological Area Openings and Closings for Grey-scale Images , 1994 .

[17]  G. Sapiro,et al.  On affine plane curve evolution , 1994 .

[18]  Philippe Salembier,et al.  Flat zones filtering, connected operators, and filters by reconstruction , 1995, IEEE Trans. Image Process..

[19]  J. W. Modestino,et al.  Flat Zones Filtering, Connected Operators, and Filters by Reconstruction , 1995 .

[20]  Corinne Vachier,et al.  Valuation of image extrema using alternating filters by reconstruction , 1995, Optics + Photonics.

[21]  Z. Grande.  On functions of two variables equicontinuous in one variable , 1996 .

[22]  Montse Pardàs,et al.  Morphological operators for image and video compression , 1996, IEEE Trans. Image Process..

[23]  Curtis R. Vogel,et al.  Iterative Methods for Total Variation Denoising , 1996, SIAM J. Sci. Comput..

[24]  Gene H. Golub,et al.  A Nonlinear primal dual method for TV-based image restoration , 1996 .

[25]  Murray Eden,et al.  Fundamentals of Digital Optics , 1996 .

[26]  Simon Masnou.  Filtrage et désocclusion d'images par méthodes d'ensembles de niveau , 1998 .

[27]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[28]  Petros Maragos,et al.  Morphological Scale-Space Representation with Levelings , 1999, Scale-Space.

[29]  Jesse Freeman,et al.  in Morse theory, , 1999 .

[30]  Guillermo Sapiro,et al.  Shape preserving local histogram modification , 1999, IEEE Trans. Image Process..

[31]  Pascal Monasse,et al.  Contrast invariant registration of images , 1999, 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258).

[32]  Yann Gousseau,et al.  Scales in Natural Images and a Consequence on their Bounded Variation Norm , 1999, Scale-Space.

[33]  Jacques Froment,et al.  A Compact and Multiscale Image Model Based on Level Sets , 1999, Scale-Space.

[34]  Lionel Moisan,et al.  Affine Invariant Mathematical Morphology Applied to A Generic Shape Recognition Algorithm , 2000, ISMM.

[35]  Pascal Monasse,et al.  Scale-Space from a Level Lines Tree , 2000, J. Vis. Commun. Image Represent..

[36]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

[37]  S. Durand,et al.  IMAGE DEBLURRING, SPECTRUM INTERPOLATION AND APPLICATION TO SATELLITE IMAGING , 2000 .

[38]  Philippe Salembier,et al.  Binary partition tree as an efficient representation for image processing, segmentation, and information retrieval , 2000, IEEE Trans. Image Process..

[39]  Pascal Monasse,et al.  Fast computation of a contrast-invariant image representation , 2000, IEEE Trans. Image Process..

[40]  Philippe Salembier,et al.  Region-Based Filtering of Images and Video Sequences , 2000 .

[41]  Françoise Dibos,et al.  Total variation minimization by the fast level sets transform , 2001, Proceedings IEEE Workshop on Variational and Level Set Methods in Computer Vision.

[42]  J. Morel,et al.  Connected components of sets of finite perimeter and applications to image processing , 2001 .

[43]  V. Caselles,et al.  The M-components of level sets of continuous functions in WBV , 2001 .

[44]  Refractor.  Vision , 2000, The Lancet.

[45]  Jean-Michel Morel,et al.  Topographic Maps and Local Contrast Changes in Natural Images , 1999, International Journal of Computer Vision.

[46]  Lionel Moisan,et al.  Edge Detection by Helmholtz Principle , 2001, Journal of Mathematical Imaging and Vision.

[47]  Guillermo Sapiro,et al.  Affine invariant scale-space , 1993, International Journal of Computer Vision.