PDE-Driven Adaptive Morphology for Matrix Fields Bernhard Burgeth, Michael Breuß, Luis Pizarro, and Joachim Weickert Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Saarland University, 66041 Saarbrücken, Germany {burgeth,breuss,pizarro,weickert}@mia.uni-saarland.de http://www.mia.uni-saarland.de Abstract. Matrix fields are important in many applications since they are the adequate means to describe anisotropic behaviour in image pro- cessing models and physical measurements. A prominent example is dif- fusion tensor magnetic resonance imaging (DT-MRI) which is a medical imaging technique useful for analysing the fibre structure in the brain. Recently, morphological partial differential equations (PDEs) for dila- tion and erosion known for grey scale images have been extended to three dimensional fields of symmetric positive definite matrices. In this article we propose a novel method to incorporate adaptivity into the matrix-valued, PDE-driven dilation process. The approach uses a structure tensor concept for matrix data to steer anisotropic morpholog- ical evolution in a way that enhances and completes line-like structures in matrix fields. Numerical experiments performed on synthetic and real- world data confirm the gap-closing and line-completing qualities of the proposed method. 1 Introduction Initiated in the sixties by the pioneering research of Serra and Matheron on bi- nary morphology [23, 31], this branch of image processing has developed into a rich field of research. Numerous monographs e.g. [17, 24, 32, 33, 34] and proceed- ings, e.g. [16,18,22] bear witness to the variety in mathematical morphology. The building blocks of morphological operations are dilation and erosion. These are usually realised by algebraic set operations involving a probing set, a so-called structuring element, e.g. [34] for details. An alternative approach to dilation is given [1] by the nonlinear partial differential equation (PDE) ∂ t u = ‖∇u‖ =  |∂ x u| 2 + |∂ y u| 2 (1) with initial condition u(x, y, 0) = f (x, y). The equation mimics the dilation of a grey scale image f with respect to a ball-shaped structuring element of growing radius t. PDEs of this type using a continuous size parameter t for the structuring element give rise to continuous-scale morphology [1,2,6,29,35]. Equation (1) has been extended in two ways: X.-C. Tai et al. (Eds.): SSVM 2009, LNCS 5567, pp. 247–258, 2009. c  Springer-Verlag Berlin Heidelberg 2009