Preserving Sharp Edges in Geometry Images
-
Upload
sloane-fitzgerald -
Category
Documents
-
view
22 -
download
0
description
Transcript of Preserving Sharp Edges in Geometry Images
![Page 1: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/1.jpg)
MATHIEU GAUTHIERPIERRE POULIN
LIGUM, DEPT. I.R.O.UNIVERSITÉ DE MONTRÉAL
GRAPHICS INTERFACE 2009GRAPHICS INTERFACE 2009
Preserving Sharp Edges in Geometry Images
![Page 2: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/2.jpg)
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
![Page 3: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/3.jpg)
Geometry ImagesHow to create them?
Original ModelOriginal Model CutCut SamplingSampling Geometry ImageGeometry Image
ReconstructionReconstructionSampling GridSampling GridParameterizationParameterization
![Page 4: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/4.jpg)
Motivation
…And there in lies the problem: The regular grid used to sample the parameterization cannot capture sharp features
The problem
![Page 5: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/5.jpg)
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
![Page 6: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/6.jpg)
MotivationSimple example
Slightly perturbating the grid, such as done in dual contouring [JLSW02], might simply and more easily resolve some alignment problems
![Page 7: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/7.jpg)
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
![Page 8: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/8.jpg)
Grid Alignment to the Sharp FeaturesCorner & Edge Snapping - Part 1
![Page 9: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/9.jpg)
Grid Alignment to the Sharp FeaturesCorner & Edge Snapping - Part 2
![Page 10: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/10.jpg)
Grid Alignment to the Sharp FeaturesCorner & Edge Snapping - Part 3
![Page 11: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/11.jpg)
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?
![Page 12: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/12.jpg)
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)
![Page 13: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/13.jpg)
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
![Page 14: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/14.jpg)
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
![Page 15: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/15.jpg)
SamplingBack to Our Example
12
3
4 5
6
78
![Page 16: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/16.jpg)
SamplingBack to Our Example
1
2
3
4 5
6
7
8
![Page 17: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/17.jpg)
SamplingResult
1 Position Image (9x9) 8 Normal Images (9x9)
![Page 18: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/18.jpg)
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
![Page 19: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/19.jpg)
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
![Page 20: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/20.jpg)
RemeshingExample
![Page 21: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/21.jpg)
ResultsSquare Torus (Original Model)
![Page 22: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/22.jpg)
ResultsSquare Torus (Comparison)
![Page 23: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/23.jpg)
ResultsSquare Torus (Position and Normal images)
![Page 24: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/24.jpg)
ResultsFandisk (Original Model)
![Page 25: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/25.jpg)
ResultsFandisk (Remeshings)
129×129 Geometry Images33×33 Geometry Images
![Page 26: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/26.jpg)
ResultsFandisk (129×129 Position and Normal images)
![Page 27: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/27.jpg)
ResultsCSG (Orignal Model and 257×257 Remeshing)
![Page 28: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/28.jpg)
Results257×257 Positon and Normal Geometry Images
![Page 29: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/29.jpg)
Results
Start!
Video
![Page 30: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/30.jpg)
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
![Page 31: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/31.jpg)
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
![Page 32: Preserving Sharp Edges in Geometry Images](https://reader035.fdocuments.us/reader035/viewer/2022070401/56813673550346895d9dfe49/html5/thumbnails/32.jpg)
Thank You!
Questions? Comments?