Asplünd's metric defined in the logarithmic image processing (LIP) framework for colour and multivariate images

Abstract

Asplünd's metric, which is useful for pattern matching, consists in a double-sided probing, i.e. the over-graph and the sub-graph of a function are probed jointly. It has previously been defined for grey-scale images using the Logarithmic Image Processing (LIP) framework. LIP is a non-linear model to perform operations between images while being consistent with the human visual system. Our contribution consists in extending the Asplünd's metric to colour and multivariate images using the LIP framework. Asplünd's metric is insensitive to lighting variations and we propose a colour variant which is robust to noise.

Double-Sided Probing by Map of Asplund’s Distances Using Logarithmic Image Processing in the Framework of Mathematical Morphology

“…The current reasoning can be extended to the double-sided probing by Asplund's distances for colour and multivariate images using the LIP multiplicative or the LIP additive framework [23,11,24]. This will be presented in a coming paper.…”

“…Importantly, the probe has a non convex domain shape and is not flat. We compute the map of Asplund's distances between the probe B and the image f d with a tolerance, As × B,p f d , as introduced in [14,23]. This metric, robust to noise, is computed by discarding p = 30% of the points which are the closest to the least upper bounds and to the greatest lower bounds.…”

“…Notice: by replacing the dilation and erosion by rank-filters [26] one can compute the map of Asplund's distances with a tolerance [14,23].…”

“…Jourlin et al [13,14] have extended this metric to functions and to grey-level images in the framework of the Logarithmic Image Processing (LIP) [15,16] using a multiplicative or an additive LIP law [11]. Then, Asplund's metric has been extended to colour and multivariate images by a marginal approach in [23,11] or by a spatio-colour (i.e. vectorial [21,22]) approach in [24].…”

Spatio-Colour Asplünd’s Metric and Logarithmic Image Processing for Colour Images (LIPC)

“…In [13], an Asplünd's metric between colour images was defined with the LIP model by using a marginal approach (i.e. channel by channel) [11,12] .…”