Shape Learning with Function-Described Graphs Gerard Sanrom` a a , Francesc Serratosa a , Ren´ e Alqu´ ezar b a Departament d’Enginyeria Inform`atica i Matem`atiques, Universitat Rovira i Virgili, Spain {gsanroma@gmail.com} {francesc.serratosa@urv.cat} b Departament de Llenguatges i Sistemes Inform`atics, Universitat Polit` ecnica de Catalunya, Spain {alquezar@lsi.upc.es} Abstract. A new method for shape learning is presented in this paper. This method incorporates abilities from both statistical and structural pattern recognition approaches to shape analysis. It borrows from sta- tistical pattern recognition the capability of modelling sets of point co- ordinates, and from structural pattern recognition the ability of dealing with highly irregular patterns, such as those generated by points miss- ingness. To that end we use a novel adaptation of Procrustes analysis, designed by us to align sets of points with missing elements. We use this information to generate sets of attributed graphs (AGs). From each set of AGs we synthesize a function-described graph (FDG), which is a type of compact representation that has the capability of probabilistic mod- elling of both structural and attribute information. Multivariate normal probability density estimation is used in FDGs instead of the originally used histograms. Comparative results of classification performance are presented of structural vs. attributes + structural information. Key Words: intermittently present landmarks, missing data, shape analysis, procrustes analysis, attributed graph, function-described graph 1 Introduction Shape analysis has become increasingly important in the last decade. Applica- tions to shape learning, matching and clustering have been presented that face the problem from both the statistical and structural pattern recognition fields. From the statistical pattern recognition field some approaches to shape learn- ing solve shape-related optimization and inference problems by using Riemma- nian geometry [1] Other approaches face the problem as one of shape matching [2] [3]. Through the use of rigid (e.g. similarity) and non-rigid (e.g. splines) transformations a dense correspondence between shapes is stablished. Once cor- respondences are set, the use of landmark-based shape statistics such as Point Distribution Models [4] is straightforward. Moreover, some authors recast the problem of finding sets of correspondent points as an optimization one of build- ing the best model [5] [6].