PhD proposal : Multi-scale Analysis of Digital Curves and Surfaces


Isabelle Debled-Rennesson ( and Bertrand Kerautret (


From the digital structures studied in the ADAGIo team, the digital sets of the plane or of the space are of main interest. The discrete geometry study these objects and its aim is to define a theoretical framework to transpose in Z^n the base of the euclidean geometry; the digital notions are closed as much as possible to the well known continue version (such as distance, length, convexity, …). We work with the analytic definitions that allows to obtain a compact representation of digital objects, to study objects intrinsically discrete and not only the approximations of continue objects but also to define infinite digital objects. One of the aims of our team is the analyze of the curves and digital surfaces and in particular to determine their geometrical characteristics. The obtained algorithms are used on the digital data extracted from for instance digital scanners, ultrasound or IRM. These data are often noisy and we develop multi-scales analysis tools to take into account the given amount of noise.


The objective of the thesis is to extend to the 3 dimensions the tools developed for the analysis of the 2D curves. The detection of the noise [5, 4] present in a 2D contour combined with the use of if the primitive of the blurred segment [2] allow to define a efficient multi-scale analysis of the contour of a 2D shape represented in an image.

The main idea of the noise level estimation on each contour point is based on the study of the length of the maximal blurred segments which belong to a point at different widths (or scales). The first point given in the thesis will be the detection of the noise level on a 3D curve. The general idea is to adapt the concept developed for the 2D contour to the 3D curves. For this purpose, a primitive of 3D blurred segment will be defined or chosen [9, 10]. For instance it will be possible to exploit the primitive of the 3D segments already implemented in the digital geometry library DGtal [3] or to use the one defined in previous works [9, 10].
According to the quality of the obtained results other definition could be proposed. Then, the variations of the lengths will be studied in order to determine their asymptotic behaviour that could be comparable to the one of the 2D segments. From this analyse, a multi-scale profile could be computed and will determine the noise level for each 3D contour point.

From the obtained results, the extension of the notion of the adaptive tangential cover [6] defined for the 3D curves will be studied and the estimators of geometric characteristics could deduced.

A similar work can be realized on digital surfaces by using a primitive based on digital plane recognition [11, 1] and by study the characteristics of the meaningful scale on each point of a digital surface. A noise level estimator could be deduced from which a multi-scale analysis could proposed with for instance the definition of an adaptive tangential cover for a digital surface. The problem of polyhedrisation of digital surfaces [13, 12] could be revisited and different strategies will be proposed.


[1] E. Charrier, J.-O. Lachaud. Maximal Planes and Multiscale Tangential Cover of 3D Digital Objects, IWCIA 2011: 132-143

[2] I. Debled-Rennesson, F. Feschet, J. Rouyer-Degli. Optimal blurred segments decomposition of noisy shapes in linear time, Computers and Graphics 30(1): 30-36 (2006).

[3] DGtal: Digital Geometry tools and algorithms library,

[4] B. Kerautret, J.O Lachaud : Meaningful Scales Detection along Digital Contours for Unsupervised Local Noise Estimation. IEEE Transactions on Pattern Analysis and Machine Intelligence 34(12), 2379–2392 (Dec 2012)

[5] B. Kerautret, J.O Lachaud : Meaningful Scales Detection: an Unsupervised Noise Detection Algorithm for Digital Contours. Image Processing On Line 4, 98–115 (2014)

[6] P. Ngo and H. Nasser, I. Debled-Rennesson and B. Kerautret. Adaptive Tangential Cover for Noisy Digital Contours, DGCI 2016: 439-451

[7] T.P. Nguyen and I. Debled-Rennesson. A Multi-scale Approach to Decompose a Digital Curve into Meaningful Parts, ICPR 2010: 1072-1075

[8] T.P. Nguyen and I. Debled-Rennesson. A discrete geometry approach for dominant point detection, Pattern Recognition 44(1): 32-44 (2011)

[9] T.P. Nguyen and I. Debled-Rennesson. Curvature and Torsion Estimators for 3D Curves, ISVC (1) 2008: 688-699

[10] T.P. Nguyen and I. Debled-Rennesson. On the Local Properties of Digital Curves, International Journal of Shape Modeling 14(2): 105-125 (2008)

[11] L. Provot, L. Buzer, I. Debled-Rennesson. Recognition of Blurred Pieces of Discrete Planes. DGCI 2006: 65-76

[12] L. Provot, I. Debled-Rennesson. 3D noisy discrete objects: Segmentation and application to smoothing. Pattern Recognition 42(8): 1626-1636 (2009)

[13] I. Sivignon, F. Dupont, Jean-Marc Chassery. Discrete Surface Segmentation into Discrete Planes. 10th International Workshop on Combinatorial Image Analysis, Auckland, New Zealand, Décembre 2004

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