Towards a unitary formulation for invariant image description: application to image coding

Here, we introduce a joint topology and harmonic analysis formulation for the extraction of global shape descriptors which are invariant under a given group of geometrical transformations. The topology approach allows the rigorous definition of the notions of shape, shape space, the invariant features space, and a metric between shapes. Therefore a new definition of completeness is given. A stability criterion is defined mathematically. Using harmonic analysis, a unitary operator that is able to separate the shape information and the geometric transformation, allows us to extract a relevant invariant shape descriptors under a given group of transformations. It also gives a robust method for the evaluation ofthe global object motion. In the closed curves, some three-dimensional surfaces and planar gray level image cases, such an operator becomes the Fourier transform on a given group. Therefore, under some assumptions, a complete convergent set of invariant features exists and can be constructed. We derived from this a shape metric. Recent developments in image coding domain for moving pictures offer new perspectives to the application of the image invariant representations of regions and contours. Therefore, we intend to illustrate the importance of our approach in image coding and indexing applications.