MATHIEU GAUTHIERPIERRE POULIN

LIGUM, DEPT. I.R.O.UNIVERSITÉ DE MONTRÉAL

GRAPHICS INTERFACE 2009GRAPHICS INTERFACE 2009

Preserving Sharp Edges in Geometry Images

Geometry Images

Simple mesh representation data structure Encodes mesh geometry and connectivity in

an image-like array

What are they?

257 × 257 Geometry Image

Reconstruction

Vertices PositionsVertices Positions4 Neighbours = 1 Quad

4 Neighbours = 1 Quad

Geometry ImagesHow to create them?

Original ModelOriginal Model CutCut SamplingSampling Geometry ImageGeometry Image

ReconstructionReconstructionSampling GridSampling GridParameterizationParameterization

Motivation

…And there in lies the problem: The regular grid used to sample the parameterization cannot capture sharp features

The problem

Motivation

Add constraints such that sharp features align with the sampling grid in the parameterization domain

It makes the process very difficult to converge

Non-linear, energy function is not smooth, infinity or no good solution

One solution

MotivationSimple example

Slightly perturbating the grid, such as done in dual contouring [JLSW02], might simply and more easily resolve some alignment problems

Grid Alignment to the Sharp FeaturesIdentifying sharp features

Input 3D ModelInput 3D Model ParameterizationParameterization

Sharp Edge Sharp CornerSharp Corner

Chain of Sharp Edges = Sharp Segment

Grid Alignment to the Sharp FeaturesCorner & Edge Snapping - Part 1

Grid Alignment to the Sharp FeaturesCorner & Edge Snapping - Part 2

Grid Alignment to the Sharp FeaturesCorner & Edge Snapping - Part 3

Sampling

UVs coordinates are no longer implicitWe can no longer use 1 normal per vertex, we

need more, especially for lighting.

What about UVs and normals?

SamplingNormals

Because of the regular structure of the geometry image and the way we remesh, we will never need more than 8 normals around a vertex (one per octant)

SamplingNormals of Corners

To sample the normals around a sharp corner, we simply iterate in CCW order between sharp edges, compute the angle-weighted normal and assign it to all the associated octants

Sampling

For a sample snapped to a sharp edge, the procedure is very similar, only two normals will be computed and stored, in the respective octant

Normals of Sharp Edges

SamplingBack to Our Example

12

3

4 5

6

78

SamplingBack to Our Example

1

2

3

4 5

6

7

8

SamplingResult

1 Position Image (9x9) 8 Normal Images (9x9)

RemeshingAlgorithm

Remeshing from geometry images is very similar to the original method

A vertex is created for each image pixelFor each group of four pixels, two triangles

are created…But since we have up to 8 normals per

vertex, more vertices may need to be createdFaces must also be connected to the

appropriate vertices

Remeshing

1. For each image pixel, we create as many vertices as there are different normals (up to 8) and store them in an array[8]

2. When creating the faces, we use the following rule to select which vertex to connect.

Algorithm

RemeshingExample

ResultsSquare Torus (Original Model)

ResultsSquare Torus (Comparison)

ResultsSquare Torus (Position and Normal images)

ResultsFandisk (Original Model)

ResultsFandisk (Remeshings)

129×129 Geometry Images33×33 Geometry Images

ResultsFandisk (129×129 Position and Normal images)

ResultsCSG (Orignal Model and 257×257 Remeshing)

Results257×257 Positon and Normal Geometry Images

Results

Start!

Video

Conclusion

Simple and efficient techniqueDoes not over-constrain the parameterization

processCan be added to any geometry image

generation pipelineCan only encode a maximum of 8 normalsMust store these 8 normals and 1 uv

coordinates

Wrap up

Future Work

Once the grid is snapped to sharp features, it may be possible to add an extra relaxation step to deform the parameterization and bring back the grid to a regular shape

Try something other than a greedy algorithm, maybe something like a quadric error metric could help find a better overall solution

Thank You!

Questions? Comments?