Shape Complexity Based on Mutual Information Jaume Rigau Miquel Feixas Mateu Sbert Institut d’Inform` atica i Aplicacions Universitat de Girona, Girona, Spain {jaume.rigau,miquel.feixas,mateu.sbert}@udg.es Abstract Shape complexity has recently received attention from different fields, such as computer vision and psychology. In this paper, integral geometry and information theory tools are applied to quantify the shape complexity from two differ- ent perspectives: from the inside of the object, we evaluate its degree of structure or correlation between its surfaces (inner complexity), and from the outside, we compute its de- gree of interaction with the circumscribing sphere (outer complexity). Our shape complexity measures are based on the following two facts: uniformly distributed global lines crossing an object define a continuous information chan- nel and the continuous mutual information of this channel is independent of the object discretisation and invariant to translations, rotations, and changes of scale. The measures introduced in this paper can be potentially used as shape descriptors for object recognition, image retrieval, object localisation, tumour analysis, and protein docking, among others. 1. Introduction In the last years, the shape complexity has been anal- ysed from different areas, such as computer vision [13] and psychology [7]. The benefits of a shape complexity theory (with its corresponding measures) would range from ob- ject classification for further database retrieval to improve- ments in cognitive science. The approach taken so far has been to consider the information (measured by Shannon en- tropy [4]) contained in the curvature of an object. Either an a priori distribution for the curvature is given and the ex- trema of the function are shown to contain more informa- tion, or the entropy of histogram of rotated angles (or of angle excess) [13] at the meshed contour or surface of the object is taken as the measure of the complexity of an ob- ject. The first approach depends on the discretisation of the curve or surface, and the second is not suitable to compute the whole complexity. In this paper, we propose two shape complexity mea- sures of an object which are independent of the discretisa- tion and appropriate to compute the partial or global com- plexity of any object. Between the information theoretic measures, one that fulfils this requirement is the continu- ous mutual information (MI), which measures the informa- tion shared between two probability distributions. Shape complexity will be analysed from two different perspectives. First, from the inside of the object, its degree of structure (interdependence between its parts) is evalu- ated. We consider the information shared by the interior contour (or surface) from the object with itself. A differ- ential of contour (or surface) will be related to another dif- ferential of inner contour (or surface) by the uniformly dis- tributed global lines [15] that join them, this is, make them directly visible. Second, from the outside of the object, the degree of interaction between the object and its circum- scribing sphere (”environment“) is calculated. These com- plexity measures could be used as shape descriptors in fields such as object recognition and classification. This paper is organised as follows. In Section 2 we re- view the generation of uniformly distributed global lines and also the MI definition. In Section 3 we present a com- plexity measure which quantifies the degree of (internal) structure of an object. In Section 4, a different approach is introduced in order to evaluate the external complexity of an object. And finally, in Section 5, we present our conclu- sions. 2. Fundamentals In this section, the two basic tools used in this paper are reviewed: generation of uniformly distributed global lines and mutual information definition. 2.1. Global lines From integral geometry [14], a uniform density of lines that is homogeneous and isotropic (invariant under transla- tions and rotations) is defined. An easy way to sample this Proceedings of the International Conference on Shape Modeling and Applications (SMI’05) 0-7695-2379-X/05 $20.00 © 2005 IEEE Authorized licensed use limited to: UNIVERSITAT DE GIRONA. Downloaded on April 27,2010 at 07:15:50 UTC from IEEE Xplore. Restrictions apply.