Logarithmic Mathematical Morphology: theory and applications
Guillaume Noyel
Classically, in Mathematical Morphology, an image (i.e., a grey-level
function) is analysed by another image which is named the structuring element
or the structuring function. This structuring function is moved over the image
domain and summed to the image. However, in an image presenting lighting
variations, the analysis by a structuring function should require that its
amplitude varies according to the image intensity. Such a property is not
verified in Mathematical Morphology for grey level functions, when the
structuring function is summed to the image with the usual additive law. In
order to address this issue, a new framework is defined with an additive law
for which the amplitude of the structuring function varies according to the
image amplitude. This additive law is chosen within the Logarithmic Image
Processing framework and models the lighting variations with a physical cause
such as a change of light intensity or a change of camera exposure-time. The
new framework is named Logarithmic Mathematical Morphology (LMM) and allows the
definition of operators which are robust to such lighting variations. In images
with uniform lighting variations, those new LMM operators perform better than
usual morphological operators. In eye-fundus images with non-uniform lighting
variations, a LMM method for vessel segmentation is compared to three
state-of-the-art approaches. Results show that the LMM approach has a better
robustness to such variations than the three others.