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.

对于灰度级函数，传统数学形态学在图像分析中使用的结构性函数与图像相加时无法验证灯光强度变化的特性，因此引入了新的可根据图像幅度调整结构性函数幅度的加法规律，称为对数数学形态学，并提出新的与灯光变化鲁棒性较好的操作符。实验证明，对数数学形态学在一致光照变化的图像中的表现优于传统形态学操作符，并且在非一致光照变化的眼底图像血管分割中也具有比其他三种方法更好的鲁棒性。