  One of the most challenging problems in the domain of 2-D image or 3-D shape
is to handle the non-rigid deformation. From the perspective of transformation
groups, the conformal transformation is a key part of the diffeomorphism.
According to the Liouville Theorem, an important part of the conformal
transformation is the Mobius transformation, so we focus on Mobius
transformation and propose two differential expressions that are invariable
under 2-D and 3-D Mobius transformation respectively. Next, we analyze the
absoluteness and relativity of invariance on them and their components. After
that, we propose integral invariants under Mobius transformation based on the
two differential expressions. Finally, we propose a conjecture about the
structure of differential invariants under conformal transformation according
to our observation on the composition of the above two differential invariants.
