LUSAC EA 2607 1 A graph approach to color mathematical morphology Olivier Lezoray, Cyril Meurie, Abderrahim Elmoataz University of Caen France IEEE Symposium on Signal Processing and Information Technology 

LUSAC EA 2607 2 Outline é Color Image Processing é Color Mathematical Morphology é Graph vector erosion and dilatation ò Determining the Æ and the Ç ? ò Proposed Algorithm ò Examples é Hierarchy of partitions ò Morphological gradient ò Non parametric watershed hierarchy é Conclusion 

LUSAC EA 2607 3 Color Image Processing é Very important area of research which is a particular case of the processing of multichannel images defined by a vector of scalar values é No natural ordering on a set of color vectors and more generally of multivariate data é several possible types of multidimensional vector ordering: ò marginal ordering, ò reduced ordering, ò partial ordering, ò conditional ordering, ò total ordering. 

LUSAC EA 2607 4 How to order colors ? 

LUSAC EA 2607 5 Recall on ordering 

LUSAC EA 2607 6 Different orderings 

LUSAC EA 2607 7 Color Mathematical Morphology (1) 

LUSAC EA 2607 8 Influence of the first component L S 

LUSAC EA 2607 9 Color Mathematical Morphology (2) é Erosion and dilatation operations can be defined by : What is essential is to determine the two extreme values corresponding to the Ç and the Æ which are respectively higher and lower than all the other colors of the lattice 

LUSAC EA 2607 10 Determining the Æ and the Ç ? é Consider the structuring element B as a Region Adjacency Graph G0 where each pixel of B is a node of the graph é Total ordering : path from Æ to Ç é Solution :Prune the RAG to obtain such a path é The Minimum Spanning Tree (MST) of the RAG G0 is a good approximation 

LUSAC EA 2607 11 Proposed Algorithm é The MST is close to the wanted solution é Internal nodes cannot be potential Æ ot Ç é Only external nodes are potential candidates é This assumption can be used to build another graph from the leaves of the MST é The MST of this new graph can be again computed and so on until finding a two-node path é Æ : closest to a reference color 

LUSAC EA 2607 12 Complete Algorithm Æ Ç 

LUSAC EA 2607 13 Examples 

LUSAC EA 2607 14 Influence of the reference color black red green blue yellow 

LUSAC EA 2607 15 Examples Erosion Dilatation 

LUSAC EA 2607 16 Color space influence RGB HSL L*a*b* 

LUSAC EA 2607 17 Hierarchy of partitions 

LUSAC EA 2607 18 Waterfall hierarchies 

LUSAC EA 2607 19 Morphological Graph gradients 

LUSAC EA 2607 20 Gradients examples Lexicographic sup graph sup Graph L1 Graph L2 

LUSAC EA 2607 21 Waterfall examples Lexicographic sup graph sup Graph L1 Graph L2 

LUSAC EA 2607 22 é We proposed a new method based on graph decimation for determining the inf and the sup of a set of colors : better results than lexicographic ordering é The traditional morphological definition of morphological gradient has been extended and new waterfall hierarchies can be obtained é Perspectives : ò Use the cascade of MSTs to find an ordering of colors ò Compare the proposed method with lexicographic and bit-mixing on different color spaces Conclusion 

LUSAC EA 2607 23 é Thank you for your attention. é Any questions ?