This paper discusses the interest of the Tree of Shapes of an image as a region oriented image representation. The Tree of Shapes offers a compact and structured representation of the family of level lines of an image. This representation has been used for many processing tasks such as filtering, registration, or shape analysis. In this paper we show how this representation can be used for segmentation, rate distortion optimization, and encoding. We address the problem of segmentation and rate distortion optimization using Guigues algorithm on a hierarchy of partitions constructed using the simplified Mumford-Shah multiscale energy. To segment an image, we minimize the simplified Mumford-Shah energy functional on the set of partitions represented in this hierarchy. The rate distortion problem is also solved in this hierarchy of partitions. In the case of encoding, we propose a variational model to select a family of level lines of a gray level image in order to obtain a minimal description of it. Our energy functional represents the cost in bits of encoding the selected level lines while controlling the maximum error of the reconstructed image. In this case, a greedy algorithm is used to minimize the corresponding functional. Some experiments are displayed.
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Coloma Ballester received the Licenciatura degree in mathematics from Barcelona University (UAB) and the Ph.D. degree in computer science from the University of Illes Balears, Spain, in 1995. Currently, she is an associate professor at the Pompeu Fabra University in Barcelona (Spain). Her research interests include image processing and computer vision.
Vicent Caselles received the Licenciatura and Ph.D. degrees in mathematics from Valencia University, Spain, in 1982 and 1985, respectively. Currently, he is professor at the Pompeu Fabra University (Barcelona). He is an associate member of IEEE. His research interests include image processing, computer vision, and the applications of geometry and partial differential equations to both previous fields.
Laura Igual received the degree of Mathematics from the University of Valencia in 2000 and the Ph.D. degree from the Pompeu Fabra University in January 2006. She has been working as research assistant at the Universitat Pompeu Fabra (Barcelona) from 2000 to 2005. Her research interests are several subjects of image processing: image segmentation and compression, motion estimation, and data interpolation on surfaces.
Luis Garrido received the degree of Telecommunication Engineering from the Telecommunication School of the Polytechnic University of Catalonia (UPC), Barcelona, Spain, in 1996. He joined afterwards the Image Processing Group at the UPC to work on his Ph.D. The topic of the work was the study of tree structures for region-based analysis of images and video sequences. Luis Garrido obtained the Ph.D. degree in June 2002.
In January 2003 he joined the Image Processing Group at the Universitat Pompeu Fabra (UPF), Barcelona, Spain. He currently has a Ramon y Cajal contract. His current research interests are contrast invariant motion estimation, tree based image representations and image segmentation.
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Ballester, C., Caselles, V., Igual, L. et al. Level Lines Selection with Variational Models for Segmentation and Encoding. J Math Imaging Vis 27, 5–27 (2007). https://doi.org/10.1007/s10851-006-7252-0
- mathematical morphology
- tree structure
- rate distortion
- morphological encoding
- minimal description length