Two-Dimensional Discrete Morphing ⋆ Isameddine Boukhriss, Serge Miguet, and Laure Tougne Universit´ e Lyon 2, Laboratoire LIRIS, Bˆatiment C, 5 av. Pierre Mend` es-France, 69 676 Bron Cedex, France {iboukhri, smiguet, ltougne}@liris.univ-lyon2.fr http://liris.cnrs.fr Abstract. In this article we present an algorithm for discrete object de- formation. This algorithm is a first step for computing an average shape between two discrete objects and may be used for building a statistical atlas of shapes. The method we develop is based on discrete operators and works only on digital data. We do not compute continuous approxima- tions of objects so that we have neither approximations nor interpolation errors. The first step of our method performs a rigid transformation that aligns the shapes as best as possible and decreases geometrical differ- ences between them. The next step consists in searching the progressive transformations of one object toward the other one, that iteratively adds or suppresses pixels. These operations are based on geodesic distance transformation and lead to an optimal (linear) algorithm. 1 Introduction Many medical images are produced every day and their interpretation is a very challenging task. 3D atlases can be of great interest since they allow to help this interpretation by very precise models. Most of the time, these atlases are built manually and represent a considerable amount of work for specialists of the domain. Moreover, they only contain static information corresponding to a single patient or potientially an average shape corresponding to a small set of patients. It would be very useful to compute these atlases in an automated way from a set of images: it would allow to compute not only an average shape between all input data but also statistical measures indicating the interindividual variability of these shapes. This is the basic idea of the statistical atlas [FLD02]. In this paper, our goal is to study the progressive deformation from one object to another one which is the first step for the computation of an average object. For sake of simplicity, we focus on 2D binary images but the proposed approach could easily be generalized to 3D. Our method is decomposed into two steps: the first one consists in making a rigid registration of the two objects and the second one in computing the deformation. ⋆ This work was supported by the RAGTIME project of the Rhˆone-Alpes region. R. Klette and J. ˇ Zuni´ c (Eds.): IWCIA 2004, LNCS 3322, pp. 409–420, 2004. c  Springer-Verlag Berlin Heidelberg 2004