1 Introduction

Deformable surfaces have many uses in computer vision. They have been used for medical image analysis, segmentation, shape representation and modelling. For a recent survey see [13].

A major drawback of most current deformable surfaces is that they have a fixed mesh topology. This is because they are based on a tensor product representation and consequently they cannot represent surfaces of arbitrary topology. It is also not possible to adaptively distribute control points, which is important when dealing with objects with long protrusions or areas of fine detail. For these reasons the topic of arbitrary topology curves and surfaces has received recent attention [12, 18]. It should be noted that at the cost of sacrificing continuity the topology problem is made considerably easier because a polyhedral mesh can be used [6]. ( continuity means smoothly varying tangent plane, for a precise definition see [5].)

In previous work [17] we introduced a deformable surface called `Slime' that can take on arbitrary topology while maintaining first order geometric ( ) continuity throughout. In this paper we present recent progress that makes the surface as easy and fast to use as a conventional B-Spline surface. In particular we have now addressed the following points

• The formalism has been recast into the standard linear form used for B-Spline curves.
• A matrix form has been developed allowing other users to compute the surface using precomputed matrices, thereby bypassing some complex operations that are difficult to implement.
• The data force has been coupled directly to the surface in place of the previous approximate scheme which coupled the data to the control points.
• A lookup based scheme for computing surface points has been developed which makes all computations as fast as the traditional B-Spline scheme.
• A new seeding scheme based on decimated meshes is demonstrated.

It is clear that spline based representations are more efficient than piecewise linear representations for representing smooth curved surfaces. One possible use of this representation is for transmission of 3D models over the web where the number of bytes required is an important consideration. In this paper we apply the deformable surface to produce spline based representations of dense polygonal meshes.

Having solved the problem of fitting with an arbitrary topology deformable surface it has now become possible to address a variety of problems. However it soon becomes apparent that new strategies are necessary to control the mesh topology. This remains for future work.