Stochastic metamorphosis with template uncertainties
This work addresses uncertainty modeling in image analysis for applications like functional data analysis, but it is incremental as it builds on existing metamorphosis and Euler-Poincaré theories.
The paper tackles the problem of modeling uncertainties in image metamorphosis by introducing two stochastic perturbations to the metamorphosis equations, one for random errors in deformation map reconstruction and another combining deformation and template uncertainties, and applies this theory to landmarks, images, and closed curves for functional data analysis.
In this paper, we investigate two stochastic perturbations of the metamorphosis equations of image analysis, in the geometrical context of the Euler-Poincaré theory. In the metamorphosis of images, the Lie group of diffeomorphisms deforms a template image that is undergoing its own internal dynamics as it deforms. This type of deformation allows more freedom for image matching and has analogies with complex fluids when the template properties are regarded as order parameters (coset spaces of broken symmetries). The first stochastic perturbation we consider corresponds to uncertainty due to random errors in the reconstruction of the deformation map from its vector field. We also consider a second stochastic perturbation, which compounds the uncertainty in of the deformation map with the uncertainty in the reconstruction of the template position from its velocity field. We apply this general geometric theory to several classical examples, including landmarks, images, and closed curves, and we discuss its use for functional data analysis.