[Japanese | Thesis | Researches in Minoh Lab | Minoh Lab]

A FFD Control Points Decision Method from Repeated Simple Deformations of Curved Surface

Recently, in the world of computer graphics, it becomes important to design 3-D geometric models including curved surfaces. In most cases, a simple deformation operation, i.e. selecting one partition of a curved surface and modifying the partition,is defined. A user applies several simple deformations to the surface to obtain a desired curved surface.When the user applies the same deformation to another surface, it is difficult for the user to repeat the simple operations from the beginning.

By registering the simple deformations as a united deformation, we can reduce the user's burden of repeating the operations. This paper presents a method for converting the simple deformations to the united deformation. Here, as a basic discussion, we treat only surfaces on which all points can be seen from one view direction. More specifically, these surfaces have a following nature; (i) corresponding one axis of 3-axes orthogonal coordinate system to the view direction, the surfaces are represented by one-valued function whose parameters are coordinates of the other two axes, and (ii) are smooth. And we restrict the user's deformations so that the surfaces must preserve the above nature.

It is important to keep the smoothness of a curved surface while deforming the surface. Free-Form Deformation (FFD) is a method of deforming a surface smoothly by (i) enclosing the surface by control points and (ii) translating the control points. Using FFD, users can deform surfaces smoothly by translating the control points. We propose approximating user's deformations as a united deformation by FFD, registering translations of control points of FFD as the united deformation and applying the registered deformation to another surface.

We consider the view direction to be the direction of the deformations. The registered deformation is represented by a deformation along W-axis of the local O-UVW orthogonal coordinate of FFD. First, we arrange the control points of two B-Spline surfaces which correspond to those of FFD along U-coordinate and V-coordinate. Second, we approximate two surfaces(a pre-deformed surface and a post-deformed surface) by the above B-Spline surface respectively. We calculate the differences of the W-coordinates of the control points of these B-Spline surfaces and make the differences the translations of the control points of FFD. We can apply this registered deformation by FFD to another surface and deform this surface in a similar way. The registered deformation is expressed as a deformation by the simplest FFD that uses a rectangular parallelepiped bounding box.

In order to keep the smoothness of a model around borders of deformation, we fix the control points around the borders. Since we use B-Spline function of order 4 for a basis function of FFD, around the borders, 3 control points along each axis effect deformation of a surface. Consequently we fix 3 control points along each axis around the borders.

We adjust translations of control points of FFD for the registered deformation in order to apply the registered deformation to a surface whose size and direction are different with those of the original. A user specifies the size and direction of a bounding box and applies the registered deformation. We adjust the calculated coordinates of control points to be in harmony with the size and direction of the bounding box. Then the user can apply a registered deformation to a surface, whose size and direction are different with those of the original.

In order to show the availability of our method, we gave examples of surfaces deformed by our method. We conducted a comparative experiment. We let some testees try two kinds of deformations; (i) the deformations using our method; and (ii) the deformations by translating the control points of FFD manually. The questionnaire results showed that our method could be useful.

Go back to Thesis Page