  Functional Asplund's metrics were recently introduced to perform pattern
matching robust to lighting changes thanks to double-sided probing in the
Logarithmic Image Processing (LIP) framework. Two metrics were defined, namely
the LIP-multiplicative Asplund's metric which is robust to variations of object
thickness (or opacity) and the LIP-additive Asplund's metric which is robust to
variations of camera exposure-time (or light intensity). Maps of distances-i.e.
maps of these metric values-were also computed between a reference template and
an image. Recently, it was proven that the map of LIP-multiplicative As-plund's
distances corresponds to mathematical morphology operations. In this paper, the
link between both metrics and between their corresponding distance maps will be
demonstrated. It will be shown that the map of LIP-additive Asplund's distances
of an image can be computed from the map of the LIP-multiplicative Asplund's
distance of a transform of this image and vice-versa. Both maps will be related
by the LIP isomorphism which will allow to pass from the image space of the
LIP-additive distance map to the positive real function space of the
LIP-multiplicative distance map. Experiments will illustrate this relation and
the robustness of the LIP-additive Asplund's metric to lighting changes.
