OVERVIEW OF ADAPTIVE MORPHOLOGY: TRENDS AND PERSPECTIVES Petros Maragos 1 and Corinne Vachier 2 1 School of E.C.E., National Technical University of Athens, Athens 15773, Greece. 2 CMLA, ENS Cachan, CNRS, PRES UniverSud, 61 Av. President Wilson, F-94230 Cachan, France. maragos@cs.ntua.gr corinne.vachier@cmla.ens-cachan.fr ABSTRACT In this paper we briefly overview emerging trends in ‘Adaptive Morphology’, i.e. work related to the theory and/or applications of image analysis filters, systems, or algorithms based on mathematical morphology, that are adaptive w.r.t. to space or intensity or use any other adaptive scheme. We present a new classification of work in this area structured along several major theoretical perspectives. We then sample specific approaches that develop spatially-variant struc- turing elements or intensity level-adaptive operators, modeled and implemented either via conventional nonlinear digital filtering or via geometric PDEs. Finally, we discuss some applications. Index Terms— Adaptive filters, Morphological image analysis 1. INTRODUCTION In image and signal processing, adaptivity of a system or algorithm means its capability to automatically adjust its parameters to the in- put data aiming at optimizing some criterion. In the processing de- sign, this adaptation should take into account the spatial, dynamical and/or temporal and range information which is available or can be computed from the data. The usefulness and necessity of adaptive algorithms are evident if one considers the variability of the signals or images that should commonly be processed by a unique algo- rithm, the internal variability of the data in a single image or image sequence, the a priori knowledge (e.g. context or noise) that needs to be incorporated in the processing, and the requirements of the processing in terms of resulting signal properties (for instance the preservation of certain image structures). We mention a few prac- tical motivations: adapting to the luminance, or the contrast, or the gradient norm or the gradient direction of the image function at every point in space is fundamental if one wants to encourage intra-region smoothing while preserving edges; adapting to a possibly spatially- varying camera perspective is of crucial importance in many surveil- lance situations; the evolution of the image data throughout the time is also a precious information for video compression. There are two fundamental questions when dealing with adap- tive algorithms or adaptive transforms. First, how to mathematically define such operations? Second, how to practically design adaptive transforms, i.e., how to define the link between the transformation parameters and the image data? These questions have received an increasing interest in the image processing community, judging by the great numbers of publications that refer to adaptive algorithms. In this paper, we briefly survey the state of the art on these ques- tions in the field of mathematical morphology (MM) [12, 13, 26], which is a powerful nonlinear methodology for representing and an- alyzing geometrical structures in images and signals based on tools from set and lattice theory, topology and stochastic geometry, with numerous applications in image enhancement, feature extraction, multiscale filtering, detection and segmentation. We discuss three major perspectives and corresponding research directions for adap- tive MM: (i) adaptivity w.r.t. the spatial neighborhood of morpho- logical operators, (ii) algebraic principles such as group and repre- sentation theory to unify important aspects of the adaptive operators, and (iii) adaptivity w.r.t. how the operators process the image level sets at different levels. Our survey includes issues from the theoret- ical, design, computational and applications aspects of these direc- tions. Our discussion of the modeling and implementation aspects of the adaptive operators in categories (i) and (ii) mainly focuses on the conventional filtering view, from which they appear as min-max combinations of nonlinear (sup/inf) spatially-variant convolutions, whereas in category (iii) we also add the viewpoint of partial differ- ential equations (PDEs). Due to the limited paper size, our references are limited and only indicative. More can be found in the the papers we cite. 2. THEORETICAL FRAMEWORKS FOR ADAPTIVE MM Morphological operators, which include well-known rank, median and stack-type nonlinear filters, were originally defined so that they satisfy important properties. Translation invariance is a fundamental one. If f (x) is a real image (or a function) defined on a space do- main E such as R d of Z d ,a translation-invariant (TI) operator is an operator ψ such that for each input f and each (h, v) in E × R ψ(f h,v )=[ψ(f )] h,v , f h,v (x) := f (x − h)+ v The operator is called horizontal-translation-invariant (HTI) or spatially-invariant if it commutes only w.r.t. a horizontal (spatial) shift, and vertical-translation-invariant (VTI) if it commutes only w.r.t. a vertical (value) shift. If we consider only TI operators, then every signal dilation (every increasing operator that distributes with supremum  ) δ and every erosion (every increasing operator that distributes with infimum  ) ε are Minkowski function additions ⊕ and subtractions ⊖; i.e., we can find a fixed function, called the structuring element (SE), g(x) such that δ (f )= f ⊕ g and ε(f )= f ⊖ g where f ⊕g(x)=  y∈B f (x−y)+g(y), f ⊖g(x)=  y∈B f (x+y) −g(y) (1) and B ⊆ E is the support of g(x). If g is flat, i.e. zero over its sup- port, we obtain the flat dilation f ⊕ B and erosion f ⊖ B of f by B. Otherwise, (1) are the weighted dilation and erosion. More complex morphological operators/filters are formed by sup/inf superpositions and/or compositions of the dilations and erosions. The basic oper- ations (1) are nonlinear convolutions, which represent the action of the combined filter as moving-window operations over the spatial 2241 978-1-4244-5654-3/09/$26.00 ©2009 IEEE ICIP 2009