Morphological signal processing and the slope transform

Abstract

This paper presents the operation of tangential dilation, which describes the touching of differentiable surfaces. It generalizes the classical dilation, but is invertible. It is shown that line segments are eigenfunctions of this dilation, and are parallel transported, and that curvature is additive. We then present the slope transform which provides tangential morphology with the analytical power which the Fourier tansform lends to linear signal processing, in particular: dilation becomes addition (just as under a Fourier transform, convolution becomes multiplication). We give a discrete slope transform suited for implementation, and discuss the relationships to the Legendre transform, the Young-Fenchel conjugate, and the ~¢-transform. We exhibit a logarithmic correspondence of this tangential morphology to linear systems theory, and touch on the consequences for morphological data analysis of a scanning tunnelling microscope. Zusammenfassung Dieser Artikel stellt die Operationen der tangentialen Dilatation vor, die die Beriihrung differnzierbarer Oberfl~ichen beschreiben. Sic verallgemeinert die klassische Dilatation, ist aber umkehrbar. Es wird gezeigt, dab Liniensegmente Eigenfunktionen dieser Dilatation sind und parallel verschoben werden, und dab die Kriimmung additiv ist. Wir pr/isentieren dann die Steigungstransformation, die eine tangentiale Morphologie mit der analytischen Leistung liefert, wie sie die Fouriertransformation ffir die lineare Signalverarbeitung darstellt, speziell wird Dilatation zu Addition (genauso wie unter der Fouriertransformation Faltung zu Multiplikation wird). Wir pr~isentieren eine diskrete Steigungstransformation, die fiir eine Implementierung geeignet ist und besprechen die Beziehungen zur Legendre-Tranformation, zur Young-Fenchel Konjugierten und der ~¢-Transformation. Wir stellen eine logarithmische Obereinstimmung dieser tangentialen Morphologie mit der iinearen Systemtheorie dar und beriihren die Konsequenzen f/Jr die morphologische Datenanalyse eines abtastenden Tunnelmikroskops. R~um~ Cet article pr6sente l'op6ration de dilatation tangentielle, qui d6crit le contact de surfaces d6rivables. Elle g6ngralise la dilatation classique mais est inversible. On d6montre que les droites sont des fonctions propres de cette dilatation, qu'elles sont simplement d6plac6es parall61ement fi elle m6me par cette dilatation et que la courbature est additive. On pr6sente ensuite la transformation de pente qui fournit fi la morphologie tangentielle un outil analytique similaire fi la transformation de Fourier en traitement lin6aire du signal, en particulier: la dilatation devient une addition (de faqon analogue fi la convolution qui devient une multiplication apr6s transformation de Fourier). On donne une version

Morphological Counterparts of Linear Shift-Invariant Scale-Spaces

“…It is well-known that any linear shift-invariant system can be described as a convolution that can be computed elegantly as multiplication in the Fourier domain [40,46]. On the other hand, morphological systems are based on erosions (or dilations) with a concave structuring function, which comes down to additions in the slope domain [42,16]. An explanation for the quasi-logarithmic connection between both worlds has been obtained by Burgeth and Weickert [12]: While linear system theory uses the classical algebra (R, ·, +), they showed that mathematical morphology is a system theory in the min-plus algebra (R ∪ {+∞}, +, min).…”

Transforms in Optics: The Legacy of P.-M. Duffieux

“…Interest in mathematical morphology has grown enormously during the past decade and our understanding of the underlying mathematical structure has improved beyond measure. Nevertheless, it is only since 1993 that the existence of a transform that maps the basic morphological operation into a simpler "product", just as the Fourier transform maps the convolution product into a direct product, has been known [14][15][16][17][18][19]. Furthermore, this is no mere abstract development awaiting applications.…”

“…Furthermore, this is no mere abstract development awaiting applications. Dorst and van den Boomgaard [17] show that image formation in near-field microscopes is intimately related to this new transform; an article by Bonnet et al. [20] [21].…”

A theoretical insight into morphological operations in surface measurement by introducing the slope transform

“…Although morphological operations are straightforward in their spatial processing, more investigations are required to further find out how the structuring element is changing the original surface, just like the way that the weighting function is changing the frequency components of the surface data in a convolution operation. In this paper, we post our initial investigation on the slope transform, based on the work of Dorst and van den Boomgaard (1994) and Maragos (1995). It will be shown that the slope transform can provide an analytical ability for morphological operations, as is the Fourier transform to convolution operations in linear theory.…”