Two frontiers in morphological image analysis: differential evolution models and hybrid morphological/linear neural networks

Abstract

Abstract. In this paper we briefly describe advancements in two broad areas of morphological image analysis. Part I deals with differential morphology and curve evolution. The partial differential equations (PDEs) that model basic morphological operations are first presented. The resulting dilation PDE, numerically implemented by curve evolution algorithms, improves the accuracy of morphological multiscale analysis by Euclidean disks and (its anisotropic/heterogeneous version) is the basic ingredient of PDE models that solve image analysis problems such as gridless halftoning and watershed segmentation based on the eikonal PDE. Part II deals with morphology-related systems for pattern recognition. It presents a general class of multilayer feed-forward neural networks where the combination of inputs in every node is formed by hybrid linear and nonlinear (of the morphological/rank type) operations. For its design a methodology is formulated using ideas from the back-propagation algorithm and robust techniques are developed to circumvent the non-differentiability of rank functions. Experimental results in handwritten character recognition are described and illustrate some of the properties of this new type of neural nets.Keywords: differential morphology, PDEs, curve evolution, neural nets, character recognition. PART I: DIFFERENTIAL MORPHOLOGY AND CURVE EVOLUTIONMorphological image processing has been based traditionally on set and lattice theory. Thus, so far, the two classic approaches to analyze or design deterministic morphological operators have been: (i) geometry by viewing them as image set transformations in Euclidean spaces and (ii) algebra to analyze their properties using set or lattice theory. In parallel to these directions, there is a recently growing part of morphological image processing that uses tools from differential calculus and dynamical systems to model nonlinear multiscale analysis and distance propagation in images. Recently, the multiscale morphological operators of dilation, erosion [1,5,24] and opening, closing [5] were modeled via nonlinear partial differential equations (PDEs) acting in scale-space. These advancements were inspired by previous work in computer vision where multiscale linear convolutions of an image were modeled via the heat PDE. For multiscale flat dilations and erosions of an image f (x, y) by compact convex symmetric structuring sets B⊆ IR 2 at a continuum of scales s ≥ 0, their generating * This work was done mostly at Georgia Tech and was supported by the US NSF under grant MIP-94-21677 and by the ARO under grant DAAH04-96-1-0161. L. Pessoa was also supported by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), Brasília, Brazil, through a Doctoral Fellowship under grant 200.846/92-2. The paper was written while P. Maragos was at I.L.S.P. PDEs have the formwhere Ψ(x, y, s) is the dilation ⊕ or erosion of f by sB, +/− corresponds to dilation/erosion respectively, ||(x, y)||B ≡ sup (a,b)∈B (ax + by), ∇Ψ ≡ (Ψx, Ψy) is the spatial gradient, and Ψx ≡...