By Jessica

Animation -- Deformation

There are two kinds of animation: rigid body animation and elastic deformation. This report focuses on the second one, which can be divided into three steps.

· Volumetric geometric mesh construction

· Deformation Calculation

· Display (movie)

Volumetric geometric mesh construction (fat surface)

According to different given conditions, three kinds of geometric mesh construction method are discussed here.

a) Perturbation Method

Given one surface, another parallel surface is constructed by using perturbation method. That is to give a perturbation along the normal direction for each given vertex. The thickness between the two surfaces can be controlled, either homogeneous or variant in different area. Prism element can be constructed between the two parallel surfaces.

Fig.1 Head model – fat surface by perturbation

b) Extraction directly from volume data (Seedset Algorithm)

Given two iso-values < , which are corresponding to the outer and inner iso-surfaces, to extract tetrahedral mesh directly from volume data (.rawiv). Suppose corresponds to the inner surface and corresponds to the outer one,

Case 1: Tetrahedral mesh between the two iso-surfaces is extracted.

Case 2: if is set to a small number (e.g. at least smaller than the minimum iso-value of the volume data), then the volume inside the outer surface will be extracted, or a solid model is obtained. If corresponds to the outer surface, what we need to do is to set to be a larger number (e.g. at least larger than the maximum iso-value of the volume data).

The detailed description is given in “Volumetric Tetrahedral Mesh Extraction”. Fig.2~4 give several examples.

Inner-outer surfaces	Tetrahedral mesh	Outer surface	Tetrahedral mesh
Fig.2 Tetrahedral mesh between two concentrated spheres		Fig.3 Tetrahedral mesh inside the “eight” surface

Fig.4 Tetrahedral mesh inside the “head” surface

c) Specified boundary

Similar to part (b), except that one surface is specified. The other surface is extracted from volume data according to one iso-value.

Deformation Calculation

There are two methods to calculate the deformation of objects. One simple way is to scale the geometric mesh by defining one deformation function. This method can control the deformed shape easily to obtain what you want. The other way is to calculate deformation using finite element method (some FE software are available, e.g. ABAQUS). This method attributes the object’s deformation some certain physically meaning, which is also the traditional method to solve engineering problem. The deformation can reflect natural mode of each eigen-value, response of some certain impulse forces, or strain/stress distribution properties.

a) Scaling

The deformation function can be defined based on your requirements. Here, I chose the square of distance as my deformation function. First, I choose one original point, which can be the center point of your object. One simple way to get the center point is to average the minimum and maximum values in x, y, z directions.

Deformation Function = (x – x0) * (x – x0) + (y – y0) * (y – y0) + (z – z0) * (z – z0)

Where (x0, y0, z0) is the original point

Before deformation	After deformation
Fig.5 Heart-beating

Before deformation	Deformation 1	Deformation 2	Deformation 3
Fig.6 Head expansion

b) Finite element method

The detailed description of finite element method is given in “Volumetric Tetrahedral Mesh Extraction”.

Before deformation	After deformation
Fig.7 Head deformation (natural mode)

Display (movie)

Several 3D-movies are generated using Anthony’s viewer. Here are 2D version movie.

Heart Head0

Head1 Head2

Head_abaqus

Reference

1. C. Bajaj, G. Xu, “Smooth Shell  Construction  with Mixed Prism Fat Surfaces”, Geometric Modeling, Springer Verlag, Computing Supplementum 14, 2001, pg 19 - 36

2. FütterlingS.,Klein,R.,Strasser,W.,Weber,H., “Automated Finite Element of Human Mandible with Dental Implants”, The Sixth International Conference in Central Europe on Computer Graphics and Visualization’98.

3. Vijay Natarajan, Herbert Edelsbrunner, “Simplificaiton of 3-Dimensional Density Maps”, Category: Technical Paper

4. W. E. Lorensen, H. E. Cline, “Marching Cubes: A High Resolution 3D Surface Construction Algorithm”, Computer Graphics, vol.21, no.4, pp.163-169, July 1987.

5. I. Fujishiro, Y. Maeda H. Sato and Y. Takeshima, “Volumetric Data Exploration Using Interval Volume”, IEEE Transactions on Visualization and Computer Graphics, Vol. 2, No. 2, pp. 144-155, June 1996.

6. Nielson, G.M., Junwon Sung, “Interval volume tetrahedrization”, Proceedings of the 8th IEEE Visualization '97 conference.