# 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

 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
 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
 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.
 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

Automatic topological changes based on explicit active contours
Automatic refinement for explicit active contours
Simplex-angle continuity regularisation of 2-simplex mesh
Shape regularisation of 2-simplex mesh