Measuring Simplification Error Jonathan Cohen Computer Science Department Johns Hopkins University cohen

Measuring Simplification Error

Measuring Simplification Error

Jonathan Cohen

Computer Science DepartmentJohns Hopkins University

<[email protected]>http://www.cs.jhu.edu/~cohen

Why Measure Error?

• * Guide simplification process– Making better choices produces better

simplifications

• Know quality of results– Object-space error bounds describes quality

• Know when to show a particular LOD– Which LOD for a given screen-space error

• * Balance quality for large environments– What error bound for a given polygon count

Geometric Error Measures

• Promote accurate 3D shape preservation

• Also preserves screen-space shape– Silhouettes– Pixel coverage

Ford Bronco Model

TriangleTriangles: s:

41,855 41,855 27,970 27,970 20,922 20,922 12,939 12,939 8,385 8,385 4,7664,766

courtesy of Division and Viewpointcourtesy of Division and Viewpoint

Performing Simplification

• Measure cost of possible operations according to error measure– Crucial to simplification quality

• Place operations in queue according to error

• Perform operations in queue– After each operation, re-evaluate error

of operations in neighborhood

Classifying Geometric Error Metrics

• Vertex-vertex distance• Vertex-plane distance• Point-surface distance• Surface-surface distance

Vertex-Vertex Distance

• E = max( dist(v1,v3) , dist(v2,v3) ) • Measures the maximum distance

travelled by merging vertices• Appropriate during topology changes

– Rossignac and Borrel 93,– Luebke and Erikson 97

• Loose for topology-preserving collapses

vv11vv22

vv33

Minimizing/Propagating Error

• Error spheres record error at each vertex– Guaranteed bound, but very conservative

• Minimum location lies on edge– Not always at the center

Vertex-Plane Distance

• Store set of planes with each vertex– Error based on distance from vertex to planes– When vertices are merged, merge sets

• Ronfard and Rossignac 96– Store plane sets– Compute max distance

• Error Quadrics - Garland and Heckbert 96– Store quadratic form (symmetric 4x4 matrix)– Compute sum of square distances

aa bb cc

aa

bb

cc

Minimizing/Propagating Error• Minimal position found by solving

4x4 linear system• Error propagated by summing

4x4 matrices of merged vertices

Error quadrics visualized as ellipsoids (Garland/Heckbert 1997)

Memoryless Simplification• Don’t measure error from original

mesh– Measure from the current mesh– Incremental rather than total error– Lindstrom/Turk 98

• New minimization process– Preserve volume and area as

simplification progresses

• Low error demonstrated after-the-fact– Metro - Cignoni et al. 96

Point-Surface Distance

• Used in Hoppe 93 and 96• Map point set to closest points

on simplified surface• Compute sum of square

distances

Minimizing/Propagating Error• Error is minimized by solving a

linear least squares system• Propagation is unnecessary

– Error always measured from original, high-resolution points

– Slows down as more points associated with simplified mesh region

Surface-Surface Distance

• Bound maximum distance between input and simplified surfaces– Tolerance Volumes - Guéziec 96– Simplification Envelopes -

Cohen/Varshney 96– Hausdorf Distance - Klein 96– Mapping Distance - Bajaj/Schikore 96,

Cohen et al. 97

Minimizing/Propagating Error

• Minimum position need not lie on edge• Bounding volume covers surface, not

just vertices• Simplified mesh volume always

contains original mesh

Vertex-Vertex != Surface-Surface

• Error is zero at vertices and exterior edges

• Error is non-zero everywhere else– Be careful! – Not captured by vertex-

vertex or vertex-plane metrics

Edge swapEdge swap

Screen-space Geometric Error

w

d

r

LOD

eye

viewing plane

p

)2tan(2

d

r

w

rp

Attribute Error Metrics

• Attributes include colors, normals, and texture coordinates

• Promote accuracy of final pixel colors

Classifying Attribute Error Metrics

• Vertex-vertex distance

• Vertex-plane distance

• Point-surface distance

• Surface-surface distance

• Image-driven metric

• Perceptually-based metric

Vertex-Vertex Distance

• GAPS point clouds - Erikson/Manocha 98

– Measure sum of square distances from vertex to its constituent vertices (area-weighted)

– Used for colors, normals, and texture coordinates

• Normal cones

– Luebke/Erikson 97, Xia et al. 97

Vertex-Plane Distance

• Higher-dimensional error quadrics– Garland and Heckbert 98

– Vertices live in higher-dimensional position + attribute space

– Planes defined in this space

• Multiple attribute quadrics– Hoppe 99

– Decouples affects of position and attributes

– Reduces storage and computational complexity

Point-Surface Distance• Extension of geometric point-surface

distance

– Hoppe 96

• Geometric correspondences found between original surface samples and simplified surface

• Sum of square attribute distances minimized

• Used primarily for vertex colors

Surface-Surface Distance

• Bajaj / Schikore 96– Geometric projections provide

local mappings– Maximum distance of scalar

attributes measured over surface

Image-driven Simplification

• Measure error by rendering– Compare resulting images– Lindstrom/Turk 2000

• Captures attribute and shading error, as well as texture content

12 cameras used to capture quality of bunny simplification (Lindstrom/Turk 2000)

Perceptually-guided Simplification• Use contrast sensitivity function to guide

simplification– Gives some handle on perceptibility– Williams et al. 2003

• Measure contrast and spatial frequency of changes induced by operation

Lit, textured model Texture contrast Lit texture contrast

Screen-space Attribute Error?

• Normal error controls dynamic refinement around highlights– Xia et al. 97, Klein 98– Doesn’t allow more simplification as

objects recede• Color control?• Texture coordinates work like

geometric error – Cohen et al. 98

• Perceptual metrics can weigh relative importance of color, normal, textures, lighting and silhouettes – for a price

Appearance Preserving Simplification

• Preserve three appearance attributes:– Surface Position– Surface Curvature– Material Color

• Each may require different sampling

Normals Undersampled

13,433 triangles 1,749 triangles10 pixel surface deviation

Normals Properly Sampled

13,433 triangles 1,749 triangles, 10 pixel deviation

v1, c1, n1

v = vertex coordinate = (x,y,z)c = color = (r,g,b)n = normal = (nx,ny,nz)

v2, c2, n2

v3, c3, n3

Traditional Polygonal Representation

Traditional Simplification

• Filters surface position, colors, and normals

• Must filter all three equally

texture map

normal map

v1, t1

v2, t2

v3, t3

v = vertex coordinate = (x,y,z)t = texture coordinate = (u,v)c = color = (r,g,b)n = normal vector = (nx,ny,nz)

c1

c2

c3

n1

n2n3

Decoupled Representation

Decoupled Approach

• Simplification filters surface position and texture coordinates

• Color and normal attributes filtered per-pixel (mip-mapping, etc.)

Sample Normal Map

polygonal surface patch normal map

Normal Map vs. Bump Map

• Normal map– Absolute normals in object space– Constant as object is simplified

• Same normal map okay for all LODs

• Bump map– Perturbations of triangle normal– Changes as object is simplified

• Need different bump map for each LOD

Texture Deviation Metric

• Distance between corresponding 3D points– Same 2D texture coordinates

– Projects at run time to 2D pixel deviation

• Intuitive error tolerance specification– Pixels of deviation for both surface

position and texture error!

Point Correspondence

mesh Mmesh Mii mesh Mmesh Mi+1i+1

2D texture domain2D texture domain

(i+1)(i+1)stst edge collapse edge collapse

XXii XXi+1i+1

xxei,i+1(x) = dist(Xi, Xi+1)

Ei,i+1 = max ei,i+1(x) xP

P

Hardware Requirements

• Texture and normal (or bump) map capability– Bandwidth for attribute map lookups– Per-pixel lighting computation

• Demonstrated originally on PixelFlow

• Possible on most modern graphics hardware

Model Requirements• Parameterized model• Ideal parameterization

– Each patch fills [0,1] texture space– Allows each patch to simplify to 2 triangles– Allows equal texture resolution along patch

boundaries– Allows standard mip-mapping

• Acceptable parameterization– Each patch occupies simple polygon in

texture space– Simplification limited by polygon sides– Mip-mapping problematic

• Want to minimize required resolution

APS Level-of-detail Hierarchy

model courtesy of Stanford and Caltech

7,809 tris

3,905 tris

1,951 tris

975 tris

488 tris

250,000 TrisOriginal

62,000 TrisPhong Shading

3 pixel error


62,000 TrisNormal Map

3 pixel error



15 pixel error



15 pixel error



78 pixel error



78 pixel error

Conclusions• Geometric error measured in object-space

– Used during simplification and at run-time

• Attribute error measured in attribute space– Used during simplification, and more limited use

at run-time

• Appearance-preserving simplification captures attributes and simplifies geometry– Texture deviation used during simplification and

at run-time (same as geometric error)– Attributes “simplified” using mip-mapping

References– Bajaj, Chandrajit and Daniel Schikore. Error-bounded

Reduction of Triangle Meshes with Multivariate Data. SPIE. vol. 2656. 1996. pp. 34-45.

– Cohen, Jonathan, Dinesh Manocha, and Marc Olano. Simplifying Polygonal Models using Successive Mappings. Proceedings of IEEE Visualization '97. pp. 395-402.

– Cohen, Jonathan, Marc Olano, and Dinesh Manocha. Appearance-Preserving Simplification. Proceedings of ACM SIGGRAPH 98. pp. 115-122.

– Cohen, Jonathan, Amitabh Varshney, Dinesh Manocha, Gregory Turk, Hans Weber, Pankaj Agarwal, Frederick Brooks, and William Wright. Simplification Envelopes. Proceedings of SIGGRAPH 96. pp. 119-128.

– DeFloriani, Leila, Paola Magillo, and Enrico Puppo. Building and Traversing a Surface at Variable Resolution. Proceedings of IEEE Visualization ‘97. pp. 103-110.

– Erikson, Carl and Dinesh Manocha. GAPS: General and Automatic Polygonal Simplification. Proceedings of 1999 ACM Symposium on Interactive 3D Graphics. pp. 79-88.

References– Garland, Michael and Paul Heckbert. Simplifying Surfaces

with Color and Texture using Quadric Error Metrics. Proceedings of IEEE Visualization '98. pp. 263-269, 542.

– Garland, Michael and Paul Heckbert. Surface Simplification using Quadric Error Bounds. Proceedings of SIGGRAPH 97. pp. 209-216.

– Guéziec, André. Surface Simplification with Variable Tolerance. Proceedings of Second Annual International Symposium on Medical Robotics and Computer Assisted Surgery (MRCAS '95). pp. 132-139.

– Hoppe, Hugues. Progressive Meshes. Proceedings of SIGGRAPH 96. pp. 99-108.

– Hoppe, Hugues. View-Dependent Refinement of Progressive Meshes. Proceedings of SIGGRAPH 97. pp. 189-198.

– Hoppe, Hugues, Tony DeRose, Tom Duchamp, John McDonald, and Werner Stuetzle. Mesh Optimization. Proceedings of SIGGRAPH 93. pp. 19-26.

– Hoppe, Hugues H. New Quadric Metric for Simplifying Meshes with Appearance Attributes. Proceedings of IEEE Visualization '99. pp. 59-66.

References– Klein, Reinhard, Gunther Liebich, and Wolfgang

Straßer. Mesh Reduction with Error Control. Proceedings of IEEE Visualization '96.

– Lindstrom, Peter and Greg Turk. Fast and Memory Efficient Polygonal Simplification. Proceedings of IEEE Visualization '98. pp. 279-286, 544.

– Lindstrom, Peter and Greg Turk. Image-Driven Simplification. ACM Transactions on Graphics. vol 19(3). pp. 204-241.

– Luebke, David and Carl Erikson. View-Dependent Simplification of Arbitrary Polygonal Environments. Proceedings of SIGGRAPH 97. pp. 199-208.

– Luebke, David, Martin Reddy, Jonathan Cohen, Benjamin Watson, Amitabh Varshney, and Robert Huebner. Level of Detail for 3D Computer Graphics. Morgan-Kaufman. 2002.

– Ronfard, Remi and Jarek Rossignac. Full-range Approximation of Triangulated Polyhedra. Computer Graphics Forum. vol. 15(3). 1996. pp. 67-76 and 462.

References– Rossignac, Jarek and Paul Borrel. Multi-

Resolution 3D Approximations for Rendering. Modeling in Computer Graphics. Springer-Verlag. 1993. pp. 455-465.

– Williams, Nathaniel, David Luebke, Jonathan Cohen, Mike Kelly, and Brenden Schubert. Perceptually-guided Simplification of Lit, Textured Meshes. Proceedings of 2003 ACM Symposium on Interactive 3D Graphics.

– Xia, Julie C., Jihad El-Sana, and Amitabh Varshney. Adaptive Real-Time Level-of-Detail-Based Rendering for Polygonal Models. IEEE Transactions on Visualization and Computer Graphics. vol. 3(2). 1997. pp. 171-183.

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Measuring Simplification Error Jonathan Cohen Computer Science Department Johns Hopkins University cohen