Two Frontiers in Morphological Image Analysis: Differential Evolution Models and Hybrid Morphological/linear Neural Networks

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. 1 PART I: DIFFERENTIAL MORPHOLOGY AND CURVE EVOLUTION Morphological image processing has been based traditionally on modeling images as sets or as points in a complete lattice of functions and viewing morphological image transformations as set or lattice operators. Thus, so far, the two classic approaches to analyze or design the deterministic systems of mathematical morphology 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, 3, 23] and opening, closing [3] 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 a signal/image were modeled via the heat PDE. For multiscale dilations and erosions of 2D signals/images f(x, y) by typical compact convex flat structuring elements (sets) B⊆ IR at a continuum of scales This work was done 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 Cient́ıfico e Tecnológico), Braśılia, Brazil, through a Doctoral Fellowship under grant 200.846/92-2. The paper was written while P. Maragos was at I.L.S.P. s ≥ 0 the generating PDEs have the form Ψs = ±||∇Ψ||B , Ψ(x, y, 0) = f(x, y) (1) where 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, Ψs ≡ ∂Ψ/∂s and Ψx ≡ ∂Ψ/∂x denote the scale and spatial partial derivatives. For example, if B is the unit disk, || · ||B is the Euclidean norm || · || and Ψs = ±||∇Ψ|| = ± √