Controlled Topology Simplification

(Voxel Based Object Simplification)

T. He, L. Hong, A. Varshney, S. Wang

Udeepta Bordoloi

Why Simplification?

Levels of Detail– High frame rate– Removal of high frequencies (anti-aliasing)

• Moire patterns

Conceptually…

Simplification can be in two areas:– Geometry

• Reduce the number of geometry primitives

– Topology• Reduce the number of holes, tunnels etc.

The factor of Topology

Simplifying– Will change the topology!!!

Preserving– Will limit the amount of simplification possible– Will keep certain high frequencies

Can we take the ‘middle path’?

Pipeline

Sampling

Input models– Polygonal– Range scanned– Volume– Mathematical function

Sample the 3-d space– Get a 3-d signal (volume as we know in sci-viz)

Controlled(?) Filtering

Use low-pass filtering to create frequency-limited versions of the sampled volume– To create a series of LOD volumes, use a series

of low-pass filters.– End result: volume pyramid

Can easily get interpolated volumes for in-between levels

Compare this volume hierarchy with - Locality based clustering of geometry Mip-maps (which use averaging, also a kind

of low-pass filtering)

Pipeline

Reconstruction

Volume reconstruction Surface reconstruction

– Iso-surface• Too many faces

• Can output high frequency patterns

– Need to preserve topology of the iso-surface while simplifying

– Method of choice: improved Splitting Box Algo.

Splitting Box

Vertices above/equal iso-value : black Vertices below iso-value : white Edge is MC if

– Color changes at most once along the sequence of its vertices

Face is MC if all 4 edges are MC Box is MC if all 6 faces are MC

Splitting Box

Start with the whole grid as a single box Recursive division of the box along its

longest edge until– 2x2x2 box– MC box B with satisfactory approximation

• What is a satisfactory approximation?

Satisfactory approximation

Sub-boxes of B, B1 and B2, are MC Approximation triangles intersect the

common face of B1 and B2 in between grid points where there is a color change

Triangle chain of B geometrically coincident with that of B1 and B2 (?)

Thus, approximate surface cannot be more than one sampling distance in error

Adaptive Marching Cubes

Edge is AMC if– Color changes at most once along the sequence

of its vertices

Face is AMC if all 4 edges are AMC Box is AMC if all possible grid lines in the

encompassed 3d space are AMC

How is AMC different from Splitting Box? Splitting Box can have unnecessary splits

– E.g., non-MC box can get split along an MC edge, thus sub-boxes would still be non-MC

– Non-AMC splits are done by finding the axes which have non-AMC edges

Different approximation checking– Within a user-specified distance from the true

(MC) surface

Multi-layered surface rendering

Render multiple surfaces at different opacities

Visibility order determination a problem

Results

Slides