Guillaume Charpiat's Projects in Images

Priors on deformations


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Aim : to set which deformations are probable and which are not; for instance, to favor rigid motions with respect to random noisy ones
Application : morphing a shape to another one, expressing minimisation flows
Method : define a metric on deformations by choosing a inner product that expresses the cost of deformations


Example of use/interest of a prior that favors rigid motion, i.e. translations and rotations.
We want to morph the red shape to the blue one, and do that by moving the red one in order that the distance between the two shapes decreases. We choose here the Hausdorff distance, basically it tends to move red points towards the closest one on the blue shape or towards areas of the blue shape that are far from the red one. The evolution is different depending on the prior chosen on deformations:
Standard (L2) prior hand movement L2 hand movement L2 hand movement L2 hand movement L2 hand movement L2 | hand morphing standard
Rigidifying prior hand movement rigidified hand movement rigidified hand movement rigidified hand movement rigidified hand movement rigidified | hand morphing rigidified
Initia-
-lisation 		Evo- -lu- -tion Conver-
-gence 		| Click for
the movie




Example of a prior that favors rotations of parts of shapes:
Two real human body shapes: human body shape human body shape
human body semilocal rigidification human body semilocal rigidification human body semilocal rigidification human body semilocal rigidification human body semilocal rigidification human body semilocal rigidification | human body semilocal rigidification
Initia-
-lisation 		E- -vo- -lu- -tion Conver-
-gence 		| Click for
the movie




Explanation : priors on deformations that can be applied to a shape.
We aim to associate lower costs to deformations that we consider as more probable.
To associate costs to deformations = to choose a metric in the space of deformations that can be applied to a shape:
a deformation on a shape
When a shape evolution is computed in order to minimize an energy (e.g. the distance between the moving and the target shape), the deformation applied at any step is the negative gradient of the energy. This the steepest direction, i.e. the best (cheapest and most efficient) deformation to apply to the current shape in order to decrease the energy. The gradient depends on the costs chosen, so that different evolutions will be observed for different choices.
different paths for gradient descents
(in blue : the energy; in green : possible pathes to decrease the energy, starting from the top red point)



Comparison of 3 different deformation priors on a simple example.
We minimise the Hausdorff distance to the blue shape with respect to the red one, by gradient descent. The colored figues indicate correspondences with the initial shape (if points are tracked when moving).
shape color initialisation
Standard (L2) prior shape evolution L2 shape evolution L2 shape evolution L2 shape evolution L2 shape evolution L2 shape evolution L2
Smoothing (Gaussian) prior shape evolution L2 shape evolution gaussian shape evolution gaussian shape evolution gaussian shape evolution gaussian shape evolution gaussian shape evolution gaussian
Locally rigidifying prior shape evolution L2 shape evolution rigid shape evolution rigid shape evolution rigid shape evolution rigid shape evolution rigid shape evolution rigid
Initia-
-lisation 		E- -vo- -lu- -tion Conver-
-gence 		Corresp-
-ondences





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