# Non-manifold Implicit Surfaces

Implicit surfaces in 3D geometric modeling are limited to two manifolds because the corresponding implicit fields are usually defined by real-valued functions which bisect space into interior and exterior. We present a novel method of modeling non-manifold surfaces by implicit representation. Our method allows discontinuity of the field function and assesses the special meaning of the locus where the function is not differentiable. The enhancement can yield a non-manifold surface with such features as holes and boundaries. The discontinuous field function also enables multiple classification of the field, which makes it possible to represent branches and intersections of the implicit surfaces. The implicit field is polygonized by the algorithm based on the marching cubes algorithm, which is extended to treat discontinuous fields correctly. We also describe an efficient implementation of converting a surface model into a set of discrete samples of field function, and present the result of the non-manifold surfaces reproduced by our method. The implicit surfaces are directly visualized at interactive frame rates independent of surface complexity by the hardware-accelerated volume rendering method. We also developed a system for visualizing the implicit surfaces and have confirmed that it can render surfaces at sufficient quality and speed.

## Publications

Shuntaro Yamazaki, Kiwamu Kase and Katsushi Ikeuchi,

“
Non-manifold implicit surfaces based on discontinuous implicitization
and polygonization
”,

In Proc. Geometric Modeling and Processing 2002, pp.138-146,

June 2002

Shuntaro Yamazaki, Kiwamu Kase and Katsushi Ikeuchi,

“Interactive visualization of non-manifold implicit surfaces using pre-integrated volume rendering”,

In Proc. Pacific Graphics, pp.475-476,

October 2002

Shuntaro Yamazaki, Kiwamu Kase and Katsushi Ikeuchi,

“Hardware-accelerated visualization of volume-sampled distance fields”,

In Proc. Shape Modeling International, pp.264-271,

May 2003

Shuntaro Yamazaki, Kiwamu Kase and Katsushi Ikeuchi,

“Interactive visualization of implicit surfaces using PC graphics hardware”,

IEICE Transactions on Information and Systems D-II, vol.J87-D-II, no.9, pp.1823-1833,

Sep. 2004

(Japanese)

Shuntaro Yamazaki, Kiwamu Kase and Katsushi Ikeuchi,

“High-Quality Interactive Volume Rendering on Standard PC Graphics Hardware”,

In Proc. Computer Vision and Image Media, 2004-CVIM-130(10), pp.71-78,

November 2001

(Japanese)

## Supplementary Material

### Figures in Paper

We propose an interactive volume rendering method using flexible texture mapping capability of standard PC graphics hardware. This method can be extended to rendering of implicit surfaces.

Multiple classification for nonmanifold implicit surfaces: The left figure shows the vertex generation diagram corresponding the classification.

Rendering Non-manifold Implicit Surfaces: (Top) Input triangle meshes. (Bottom) Visualized surfaces represented in volume-sampled distance fields.