Curve and Surface Regularisation

Scale-Sensitive Tikhonov Stabilizers

Regularisation with Tikhonov stabilizers are often used in computer Vision to solve ill-posed problems. However the stabilizers that enforce the smoothness of the solution are not scale-dependent, while the geometric regularity is clearly scale dependant.

Our approach consists in adding a convolution kernel K(r,v) in the Tikhonov stabilizers that depends on a scale parameter r, this parameter being spatially variable. Thus, the Tikhonov stabilizers written as :

are now generalized as :

The scale parameter r controls the spatial frequency of the solution while the regularization parameter controls the trade-off between smoothness and the data fidelity.

For instance the classical regularising energy used in the snake algorithm :

can now be extended with the following expression :

Intrinsic Polynomial Stabilizers of planar curves

Differential stabilizers are smoothing filters applied to curves or surfaces. A differential stabilizer may not derive from the variation of an energy and therefore it provides a broader concept for studying interesting curve evolutions.

I have introduced a remarkable set of differential stabilizers on planar curves, called the intrinsic polynomial stabilizers

They have the following properties :

• Invariance to translation, rotation, global scale and dependence on an inner-scale r.
• No shrinking effect since all circles are invariant by the application of this set of filters,
• Polynomial smoothing of the curvature profile . Curves that are invariant by those filters are Intrinsic Splines that have a piecewise polynomial curvature profiles as fonction of arc length.

Interpolation with Intrinsic Splines of order 2 (Clothoids or Cornu's Spiral)

Curvature Profile is piecewise linear (curvature continuity)

Regularization of Simplex Meshes

Discrete filters generalizing Intrinsic Polynomial Stabilizers have been defined on simplex meshes with the following properties:

• Tangential component controls the spacing between vertices . Vertices can be evenly spread on the surface or can concentrate on parts of high curvature.
• Ability to smooth mean curvature of a surface mesh with no shrinking effect
• Shape regularization can be performed. . Filter can deformed surfaces towards their original shape up to a similarity transform.
• Continuity control between 3D curves and surface meshes.

Minimal Surface having the symmetry of an icosahedron

Continuity between simplex curves and 3D simplex mesh is enforced

Selected Publications

	Energy functions for regularization algorithm H. Delingette, M. Hébert, and K. Ikeuchi Geometric Methods in Computer Vision, SPIE Vol. 1570, pages 104-115, 1991. SPIE Defines the Intrinsic Polynomial Stabilizers

	Trajectory Generation with Curvature Constraint based on Energy Minimization H. Delingette, M. Hébert, and K. Ikeuchi. In Int. Robotics Systems (IROS`91), 206 - 211 vol.1, Osaka, November 1991 Use Intrinsic Polynomial Stabilizers to define trajectory of robots where curvature is a smooth function of arc length

	On Smoothness Measures of Active Contours and Surfaces H. Delingette In IEEE Workshop on Variational and Level Set Methods in Computer Vision (VLSM 2001), Vancouver, Canada, pages 43-50, July 2001 Provide taxonomy of geometric functionals (such as isoperimetric ratio) that are invariant to translation, rotation and scale.

	Shape and Topology Constraints on Parametric Active Contours H. Delingette and J. Montagnat. Computer Vision and Image Understanding, 83(2):140-171, 2001 Control of vertex spacing on explicit active contours Automatic Topology Change of explicit active contours

Selected Animations