A Discrete Reaction-Diffusion Operator for Moving Curves and Edge Detection Juan Manuel Rend´ on, Marcos Capistr´ an and Bruno Lara Facultad de Ciencias, UAEM. Cuernavaca, Mor. Mexico. {rendon, marcos, bruno.lara}@uaem.mx Abstract This paper introduces a discrete Reaction-Diffusion op- erator for iterative curve moving. The operator is a lin- ear combination of an average operator and a maximum- minimum operator. It approximates curvature motion and affine motion. Furthermore, it features properties of level set active contours, namely topological changes of the curve are handled automatically and it does not require the initial curve to be situated close to searched objects. The operator here presented is stable and simple to program. An appli- cation to edge detection is shown. 1 Introduction Active contours have become a very active field of re- search in recent years. The consequent mathematical meth- ods typically move an initial curve until it stops at some searched feature in an image (e.g. the edges of an object in the image) while numerical stability throughout curve prop- agation is enforced. Among the major works in this field are the following: Contour Active Models or Snakes [7]. This is a precise and fast method, however, it can not be used directly if changes in the topology of the snake are required (i.e. if the curve ought to divide or merge). Geodesic Active Con- tours (GAC) [2][3] are a geometric alternative to snakes. This method does not have limitations with respect to topo- logical changes or the initial position of the evolving curve. Much work has been done improving this model, from ex- plicit implementations [11][6] to a semi-implicit approach in Additive Operator Splitting (AOS) [5][18]. On the other hand, Merriman et al. [8][12] developed a method to prop- agate interfaces based on convolution with a gaussian ker- nel plus a thresholding. This method automatically handles topological changes of the curve, with the limitation of hav- ing pixelic resolution. This paper is organized as follows. In section 2 are briefly described Snakes, GAC, the implementation of level sets and the main idea of the method of Merriman et al. In section 3 are presented the Discrete Reaction-Diffusion Operator and the analysis of its stability. In Section 4 the Reaction-Diffusion operator is used to approximate curva- ture motion and affine motion and to perform edge detec- tion in images. Finally, in section 5 we offer conclusions and perspectives. 2 Standard methods 2.1 Active Contour Model (Snakes) Active contours [7] is a method that depends on a set of parameters and an initial curve. The method consists of an evolving curve that deforms itself until it matches some im- age features (e.g. edges of an object in the image). Snakes use a variational framework to obtain an evolving equation for the moving curve. The idea is to minimize an energy functional where several parameters must be tuned accord- ing to the problem in question. As represented in Eq. (1) the first two terms are the internal energy (first and second derivatives of the curve) and the third one is an external term (e.g. edge detector). E(C(s)) = α ∂C(s) ∂s 2 ds +β ∂ 2 C(s) ∂s 2 2 ds - λ |∇I (C(s))| 2 ds (1) The first and second derivatives determine curve behav- ior and the external term stops the curve evolution over searched image features. Snakes use a particle model to implement the curve, making it a fast algorithm. Prob- lems such as stereo matching or motion detection can be addressed with some modifications of the method. On the other hand, this approach has difficulties when topological changes of the curve occur (e.g. when the curve divides). Furthermore, the initial guess or initial position of the curve are required to be very close to the searched features of the