1. Simplifying Surfaces with Color and Texture using Quadric Error Metrics Michael Garland Paul S. Heckbert Carnegie Mellon University October 1998 Michael Garland Paul S. Heckbert Carnegie Mellon University October 1998

2. Overview Often want to simplify overly complex models Too much for hardware, or simply over-sampledToo much for hardware, or simply over-sampled Common sources: Scanning & ReconstructionCommon sources: Scanning & Reconstruction Quadric-based simplification [SIGGRAPH 97] Fast method producing quality resultsFast method producing quality results Convenient characterization of error/shapeConvenient characterization of error/shape Generalization required to handle properties Color, texture, surface normals, etc.Color, texture, surface normals, etc. Often want to simplify overly complex models Too much for hardware, or simply over-sampledToo much for hardware, or simply over-sampled Common sources: Scanning & ReconstructionCommon sources: Scanning & Reconstruction Quadric-based simplification [SIGGRAPH 97] Fast method producing quality resultsFast method producing quality results Convenient characterization of error/shapeConvenient characterization of error/shape Generalization required to handle properties Color, texture, surface normals, etc.Color, texture, surface normals, etc.

3. Geometric Surface Simplification 1,087,716 faces 20,000 faces 1,000 faces 4 sec. 175 sec.

4. Fundamental Quadric-Based Simplification Algorithm Iterative edge contraction Quadric error metric Each vertex has (conceptual) set of planesEach vertex has (conceptual) set of planes Vertex error = sum of squared distances to planesVertex error = sum of squared distances to planes Combine sets when contracting vertex pairCombine sets when contracting vertex pair Iterative edge contraction Quadric error metric Each vertex has (conceptual) set of planesEach vertex has (conceptual) set of planes Vertex error = sum of squared distances to planesVertex error = sum of squared distances to planes Combine sets when contracting vertex pairCombine sets when contracting vertex pair

5. Measuring Error with Quadrics Given a plane, we can define a quadric Q measuring squared distance to given plane as

6. Measuring Error with Quadrics Each vertex has an associated quadric Measures error at vertexMeasures error at vertex Sum quadrics when contracting pairSum quadrics when contracting pair Sum of endpoint quadrics determines v’ Fixed placement: select v 1 or v 2Fixed placement: select v 1 or v 2 Optimal placement: choose v’ minimizing Q(v’)Optimal placement: choose v’ minimizing Q(v’) Each vertex has an associated quadric Measures error at vertexMeasures error at vertex Sum quadrics when contracting pairSum quadrics when contracting pair Sum of endpoint quadrics determines v’ Fixed placement: select v 1 or v 2Fixed placement: select v 1 or v 2 Optimal placement: choose v’ minimizing Q(v’)Optimal placement: choose v’ minimizing Q(v’)

7. A Simple Example: Contraction & “Planes” in 2D Lines defined by neighboring segments Determine position of new vertexDetermine position of new vertex Error isocontours shown on rightError isocontours shown on right Lines defined by neighboring segments Determine position of new vertexDetermine position of new vertex Error isocontours shown on rightError isocontours shown on right OriginalOriginal After 1 Step v1v1v1v1 v1v1v1v1 v2v2v2v2 v2v2v2v2 v’v’

8. Visualizing Quadrics in 3-D Quadric isosurfaces Are ellipsoids (maybe degenerate)Are ellipsoids (maybe degenerate) Characterize shapeCharacterize shape Stretch in least curved directionsStretch in least curved directions Eigenvalues propor- tional to principal curvaturesEigenvalues propor- tional to principal curvatures

9. Quadrics Give Good Results, But Ignores Attributes Fundamental algorithm works well Simple to implement & simplification is fastSimple to implement & simplification is fast High quality approximationsHigh quality approximations Quadrics record useful information about shapeQuadrics record useful information about shape But many models have additional properties Color, texture, normals, etc.Color, texture, normals, etc. Need to simplify these as wellNeed to simplify these as well Fundamental algorithm works well Simple to implement & simplification is fastSimple to implement & simplification is fast High quality approximationsHigh quality approximations Quadrics record useful information about shapeQuadrics record useful information about shape But many models have additional properties Color, texture, normals, etc.Color, texture, normals, etc. Need to simplify these as wellNeed to simplify these as well

10. Gouraud Shaded Surface: Single RGB Value Per Vertex Surface geometry Radiosity solution

11. Surface Properties as Vertex Attributes Each vertex has a set of properties Each property has one unique value per vertexEach property has one unique value per vertex Attributes are linearly interpolated over facesAttributes are linearly interpolated over faces Primary example: one RGB color per vertexPrimary example: one RGB color per vertex Can’t treat geometry & color separately Position and color are correlatedPosition and color are correlated Optimal position may lie off the surfaceOptimal position may lie off the surface Must synthesize new color for this positionMust synthesize new color for this position Each vertex has a set of properties Each property has one unique value per vertexEach property has one unique value per vertex Attributes are linearly interpolated over facesAttributes are linearly interpolated over faces Primary example: one RGB color per vertexPrimary example: one RGB color per vertex Can’t treat geometry & color separately Position and color are correlatedPosition and color are correlated Optimal position may lie off the surfaceOptimal position may lie off the surface Must synthesize new color for this positionMust synthesize new color for this position

12. Vertex Attributes Become Added Dimensions Treat each vertex as a 6-vector [x y z r g b] Assume this 6-D space is EuclideanAssume this 6-D space is Euclidean –Of course, color space is only roughly Euclidean Scale xyz space to unit cube for consistencyScale xyz space to unit cube for consistency Triangle determines a 2-plane in 6-D space Can measure squared distance to this planeCan measure squared distance to this plane Distance along all perpendicular directionsDistance along all perpendicular directions –Generalized Pythagorean Theorem Treat each vertex as a 6-vector [x y z r g b] Assume this 6-D space is EuclideanAssume this 6-D space is Euclidean –Of course, color space is only roughly Euclidean Scale xyz space to unit cube for consistencyScale xyz space to unit cube for consistency Triangle determines a 2-plane in 6-D space Can measure squared distance to this planeCan measure squared distance to this plane Distance along all perpendicular directionsDistance along all perpendicular directions –Generalized Pythagorean Theorem

13. Generalized Quadric Metric Squared distance to 2-plane has same form: A: 6x6 matrix v,b: 6-vectors c: scalar (for RGB)A: 6x6 matrix v,b: 6-vectors c: scalar (for RGB) Underlying algorithm remains the sameUnderlying algorithm remains the same May want to selectively weight channels Relative importance of space & colorRelative importance of space & color Relative importance of red & greenRelative importance of red & green Squared distance to 2-plane has same form: A: 6x6 matrix v,b: 6-vectors c: scalar (for RGB)A: 6x6 matrix v,b: 6-vectors c: scalar (for RGB) Underlying algorithm remains the sameUnderlying algorithm remains the same May want to selectively weight channels Relative importance of space & colorRelative importance of space & color Relative importance of red & greenRelative importance of red & green

14. Generalized Quadric Metric Common property types VertexDimension VertexDimension Color[x y z r g b]6x6 quadricsColor[x y z r g b]6x6 quadrics Texture[x y z s t]5x5 quadricsTexture[x y z s t]5x5 quadrics Normal[x y z u v w]6x6 quadricsNormal[x y z u v w]6x6 quadrics Color+Normal[x y z r g b u v w]9x9 quadricsColor+Normal[x y z r g b u v w]9x9 quadrics Common property types VertexDimension VertexDimension Color[x y z r g b]6x6 quadricsColor[x y z r g b]6x6 quadrics Texture[x y z s t]5x5 quadricsTexture[x y z s t]5x5 quadrics Normal[x y z u v w]6x6 quadricsNormal[x y z u v w]6x6 quadrics Color+Normal[x y z r g b u v w]9x9 quadricsColor+Normal[x y z r g b u v w]9x9 quadrics

15. Simplifying the Dragon Model 50,761 faces 10,000 faces 20 sec.

16. Simplifying the Dragon Model 50,761 faces 1,500 faces 23 sec.

17. Simplified Dragon Mesh 50,761 faces 1,500 faces 23 sec.

18. A Sample Textured Surface

19. Simplifying Geometry Only: Same Texture Coordinates

20. Simplifying with xyzst Quadrics; New Texture Coordinates

21. Related Work Fairly few algorithms address attributes Re-sampling height field attributesRe-sampling height field attributes Mesh optimization methods [Hoppe 96]Mesh optimization methods [Hoppe 96] Attribute map preservation [Cohen et al 98]Attribute map preservation [Cohen et al 98] In comparison, our algorithm is Not limited to simple height field re-samplingNot limited to simple height field re-sampling Faster than other general algorithmsFaster than other general algorithms Still capable of producing quality resultsStill capable of producing quality results Fairly few algorithms address attributes Re-sampling height field attributesRe-sampling height field attributes Mesh optimization methods [Hoppe 96]Mesh optimization methods [Hoppe 96] Attribute map preservation [Cohen et al 98]Attribute map preservation [Cohen et al 98] In comparison, our algorithm is Not limited to simple height field re-samplingNot limited to simple height field re-sampling Faster than other general algorithmsFaster than other general algorithms Still capable of producing quality resultsStill capable of producing quality results

22. Summary Generalized quadric metric Handles surfaces with per vertex attributesHandles surfaces with per vertex attributes Rapidly produces quality approximationsRapidly produces quality approximations Future work Handling attribute discontinuitiesHandling attribute discontinuities Other attribute typesOther attribute types Attribute characteristics necessary for successAttribute characteristics necessary for success Generalized quadric metric Handles surfaces with per vertex attributesHandles surfaces with per vertex attributes Rapidly produces quality approximationsRapidly produces quality approximations Future work Handling attribute discontinuitiesHandling attribute discontinuities Other attribute typesOther attribute types Attribute characteristics necessary for successAttribute characteristics necessary for success

23. Further Details Online Online papers, Sample models, Experimental code http://www.cs.cmu.edu/~garland/quadrics/ Online papers, Sample models, Experimental code http://www.cs.cmu.edu/~garland/quadrics/