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The Problem of Detail
Graphics systems are awash in model dataGraphics systems are awash in model data• very detailed CAD databasesvery detailed CAD databases
• high-precision surface scanshigh-precision surface scans
Available resources are always constrainedAvailable resources are always constrained• CPU, space, graphics speed, network bandwidthCPU, space, graphics speed, network bandwidth
We need economical modelsWe need economical models• want the minimum level of detail (LOD) requiredwant the minimum level of detail (LOD) required
Graphics systems are awash in model dataGraphics systems are awash in model data• very detailed CAD databasesvery detailed CAD databases
• high-precision surface scanshigh-precision surface scans
Available resources are always constrainedAvailable resources are always constrained• CPU, space, graphics speed, network bandwidthCPU, space, graphics speed, network bandwidth
We need economical modelsWe need economical models• want the minimum level of detail (LOD) requiredwant the minimum level of detail (LOD) required
A Non-Economical Model
424,376 faces424,376 faces 60,000 faces60,000 faces
Automatic Surface Simplification
Automatic Surface Simplification
Produce approximations with fewer trianglesProduce approximations with fewer triangles• should be as similar as possible to originalshould be as similar as possible to original
• want computationally efficient processwant computationally efficient process
Need criteria for assessing model similarityNeed criteria for assessing model similarity• for display, visual similarity is the ultimate goalfor display, visual similarity is the ultimate goal
• similarity of shape is often used insteadsimilarity of shape is often used instead– generally easier to computegenerally easier to compute
– lends itself more to applications other than displaylends itself more to applications other than display
Produce approximations with fewer trianglesProduce approximations with fewer triangles• should be as similar as possible to originalshould be as similar as possible to original
• want computationally efficient processwant computationally efficient process
Need criteria for assessing model similarityNeed criteria for assessing model similarity• for display, visual similarity is the ultimate goalfor display, visual similarity is the ultimate goal
• similarity of shape is often used insteadsimilarity of shape is often used instead– generally easier to computegenerally easier to compute
– lends itself more to applications other than displaylends itself more to applications other than display
Focus on Polygonal Models
Polygonal surfaces are ubiquitousPolygonal surfaces are ubiquitous• only primitive widely supported in hardwareonly primitive widely supported in hardware
• near-universal support in software packagesnear-universal support in software packages
• output of most scanning systemsoutput of most scanning systems
Switching representations is no solutionSwitching representations is no solution• indeed, some suffer from the same problemindeed, some suffer from the same problem
• many applications many applications wantwant polygons polygons
Will always assume models are triangulatedWill always assume models are triangulated
Polygonal surfaces are ubiquitousPolygonal surfaces are ubiquitous• only primitive widely supported in hardwareonly primitive widely supported in hardware
• near-universal support in software packagesnear-universal support in software packages
• output of most scanning systemsoutput of most scanning systems
Switching representations is no solutionSwitching representations is no solution• indeed, some suffer from the same problemindeed, some suffer from the same problem
• many applications many applications wantwant polygons polygons
Will always assume models are triangulatedWill always assume models are triangulated
Historical Background
Function approximation Function approximation [y=f(x)][y=f(x)]• long history in mathematical literaturelong history in mathematical literature
Piecewise linear curve approximationPiecewise linear curve approximation• various fields: graphics, cartography, vision, various fields: graphics, cartography, vision,
……
Height field (i.e., terrain) triangulationHeight field (i.e., terrain) triangulation• research back to at least early 70’sresearch back to at least early 70’s
• important for flight simulatorsimportant for flight simulators
Function approximation Function approximation [y=f(x)][y=f(x)]• long history in mathematical literaturelong history in mathematical literature
Piecewise linear curve approximationPiecewise linear curve approximation• various fields: graphics, cartography, vision, various fields: graphics, cartography, vision,
……
Height field (i.e., terrain) triangulationHeight field (i.e., terrain) triangulation• research back to at least early 70’sresearch back to at least early 70’s
• important for flight simulatorsimportant for flight simulators
Other Related Fields
Geometry compressionGeometry compression• simplification is a kind of lossy compressionsimplification is a kind of lossy compression
Surface smoothingSurface smoothing• reduces geometric complexity of shapereduces geometric complexity of shape
Mesh generationMesh generation• finite element analysis (e.g., solving PDE’s)finite element analysis (e.g., solving PDE’s)
• need appropriate mesh for good solutionneed appropriate mesh for good solution
• overly complex mesh makes solution slowoverly complex mesh makes solution slow
Geometry compressionGeometry compression• simplification is a kind of lossy compressionsimplification is a kind of lossy compression
Surface smoothingSurface smoothing• reduces geometric complexity of shapereduces geometric complexity of shape
Mesh generationMesh generation• finite element analysis (e.g., solving PDE’s)finite element analysis (e.g., solving PDE’s)
• need appropriate mesh for good solutionneed appropriate mesh for good solution
• overly complex mesh makes solution slowoverly complex mesh makes solution slow
Overview ofSimplification Methods
Manual preparation has been widely usedManual preparation has been widely used• skilled humans produce excellent resultsskilled humans produce excellent results
• very labor intensive, and thus costlyvery labor intensive, and thus costly
Most common kinds of automatic Most common kinds of automatic methodsmethods• vertex clusteringvertex clustering
• vertex decimationvertex decimation
• iterative contractioniterative contraction
Manual preparation has been widely usedManual preparation has been widely used• skilled humans produce excellent resultsskilled humans produce excellent results
• very labor intensive, and thus costlyvery labor intensive, and thus costly
Most common kinds of automatic Most common kinds of automatic methodsmethods• vertex clusteringvertex clustering
• vertex decimationvertex decimation
• iterative contractioniterative contraction
Optimal Approximations
Achieve given error with fewest trianglesAchieve given error with fewest triangles• no mesh with fewer triangles meets error limitno mesh with fewer triangles meets error limit
Computationally feasible for curvesComputationally feasible for curves• O(n)O(n) for functions of one variable for functions of one variable
• but but O(nO(n22 log n) log n) for plane curves for plane curves
Intractable for surfacesIntractable for surfaces• NP-hard to find optimal height field NP-hard to find optimal height field [Agarwal–Suri [Agarwal–Suri
94]94]
• must also be the case for surfacesmust also be the case for surfaces
Achieve given error with fewest trianglesAchieve given error with fewest triangles• no mesh with fewer triangles meets error limitno mesh with fewer triangles meets error limit
Computationally feasible for curvesComputationally feasible for curves• O(n)O(n) for functions of one variable for functions of one variable
• but but O(nO(n22 log n) log n) for plane curves for plane curves
Intractable for surfacesIntractable for surfaces• NP-hard to find optimal height field NP-hard to find optimal height field [Agarwal–Suri [Agarwal–Suri
94]94]
• must also be the case for surfacesmust also be the case for surfaces
Vertex Clustering
Partition space into cellsPartition space into cells• grids grids [Rossignac-Borrel][Rossignac-Borrel], spheres , spheres [Low-Tan][Low-Tan], octrees, ..., octrees, ...
Merge all vertices within the same cellMerge all vertices within the same cell• triangles with multiple corners in one cell will degeneratetriangles with multiple corners in one cell will degenerate
Partition space into cellsPartition space into cells• grids grids [Rossignac-Borrel][Rossignac-Borrel], spheres , spheres [Low-Tan][Low-Tan], octrees, ..., octrees, ...
Merge all vertices within the same cellMerge all vertices within the same cell• triangles with multiple corners in one cell will degeneratetriangles with multiple corners in one cell will degenerate
Vertex Decimation
Starting with original model, iterativelyStarting with original model, iteratively• rank vertices according to their importancerank vertices according to their importance• select unimportant vertex, remove it, retriangulate select unimportant vertex, remove it, retriangulate
holehole
A fairly common techniqueA fairly common technique• Schroeder Schroeder et alet al, Soucy , Soucy et alet al, Klein , Klein et alet al, Ciampalini , Ciampalini et alet al
Starting with original model, iterativelyStarting with original model, iteratively• rank vertices according to their importancerank vertices according to their importance• select unimportant vertex, remove it, retriangulate select unimportant vertex, remove it, retriangulate
holehole
A fairly common techniqueA fairly common technique• Schroeder Schroeder et alet al, Soucy , Soucy et alet al, Klein , Klein et alet al, Ciampalini , Ciampalini et alet al
Iterative Contraction
Contraction can operate on any set of verticesContraction can operate on any set of vertices• edges (or vertex pairs) are most common, faces also edges (or vertex pairs) are most common, faces also
usedused
Starting with the original model, iterativelyStarting with the original model, iteratively• rank all edges with some cost metricrank all edges with some cost metric• contract minimum cost edgecontract minimum cost edge• update edge costsupdate edge costs
Contraction can operate on any set of verticesContraction can operate on any set of vertices• edges (or vertex pairs) are most common, faces also edges (or vertex pairs) are most common, faces also
usedused
Starting with the original model, iterativelyStarting with the original model, iteratively• rank all edges with some cost metricrank all edges with some cost metric• contract minimum cost edgecontract minimum cost edge• update edge costsupdate edge costs
Edge Contraction
Single edge contraction Single edge contraction (v(v11,v,v22) ) v’ v’ is is
performed byperformed by• moving vmoving v11 and v and v22 to position v’ to position v’• replacing all occurrences of vreplacing all occurrences of v22 with v with v11
• removing vremoving v22 and all degenerate triangles and all degenerate triangles
Single edge contraction Single edge contraction (v(v11,v,v22) ) v’ v’ is is
performed byperformed by• moving vmoving v11 and v and v22 to position v’ to position v’• replacing all occurrences of vreplacing all occurrences of v22 with v with v11
• removing vremoving v22 and all degenerate triangles and all degenerate triangles
v1v1
v2v2v’v’
Vertex Pair Contraction
Can also easily contract any pair of verticesCan also easily contract any pair of vertices• fundamental operation is exactly the samefundamental operation is exactly the same
• joins previously unconnected areasjoins previously unconnected areas
• can be used to achieve topological simplificationcan be used to achieve topological simplification
Can also easily contract any pair of verticesCan also easily contract any pair of vertices• fundamental operation is exactly the samefundamental operation is exactly the same
• joins previously unconnected areasjoins previously unconnected areas
• can be used to achieve topological simplificationcan be used to achieve topological simplification
Iterative Edge Contraction
Currently the most popular techniqueCurrently the most popular technique• Hoppe, Garland–Heckbert, Lindstrom-Turk, Ronfard-Hoppe, Garland–Heckbert, Lindstrom-Turk, Ronfard-
Rossignac, Guéziec, and several othersRossignac, Guéziec, and several others
• simpler operation than vertex removalsimpler operation than vertex removal
• well-defined on any simplicial complexwell-defined on any simplicial complex
Also induces hierarchy on the surfaceAlso induces hierarchy on the surface• a very important by-producta very important by-product
• enables several multiresolution applicationsenables several multiresolution applications
Currently the most popular techniqueCurrently the most popular technique• Hoppe, Garland–Heckbert, Lindstrom-Turk, Ronfard-Hoppe, Garland–Heckbert, Lindstrom-Turk, Ronfard-
Rossignac, Guéziec, and several othersRossignac, Guéziec, and several others
• simpler operation than vertex removalsimpler operation than vertex removal
• well-defined on any simplicial complexwell-defined on any simplicial complex
Also induces hierarchy on the surfaceAlso induces hierarchy on the surface• a very important by-producta very important by-product
• enables several multiresolution applicationsenables several multiresolution applications
Cost Metrics for Contraction
Used to rank edges during simplificationUsed to rank edges during simplification• reflects amount of geometric error introducedreflects amount of geometric error introduced
• main differentiating feature among algorithmsmain differentiating feature among algorithms
Must address two interrelated problemsMust address two interrelated problems• what is the best contraction to perform?what is the best contraction to perform?
• what is the best position v’ for remaining what is the best position v’ for remaining vertex?vertex?– can just choose one of the endpointscan just choose one of the endpoints
– but can often do better by optimizing position of v’but can often do better by optimizing position of v’
Used to rank edges during simplificationUsed to rank edges during simplification• reflects amount of geometric error introducedreflects amount of geometric error introduced
• main differentiating feature among algorithmsmain differentiating feature among algorithms
Must address two interrelated problemsMust address two interrelated problems• what is the best contraction to perform?what is the best contraction to perform?
• what is the best position v’ for remaining what is the best position v’ for remaining vertex?vertex?– can just choose one of the endpointscan just choose one of the endpoints
– but can often do better by optimizing position of v’but can often do better by optimizing position of v’
Cost Metrics for Contraction
Simple heuristicsSimple heuristics• edge length, dihedral angle, surrounding edge length, dihedral angle, surrounding
area, …area, …
Sample distances to original surfaceSample distances to original surface• projection to closest point projection to closest point [Hoppe][Hoppe]
• restricted projection restricted projection [Soucy–Laurendeau, Klein [Soucy–Laurendeau, Klein et alet al, , Ciampalini Ciampalini et alet al]]
Alternative characterization of errorAlternative characterization of error• quadric error metrics quadric error metrics [Garland–Heckbert][Garland–Heckbert]
• local volume preservation local volume preservation [Lindstrom–Turk][Lindstrom–Turk]
Simple heuristicsSimple heuristics• edge length, dihedral angle, surrounding edge length, dihedral angle, surrounding
area, …area, …
Sample distances to original surfaceSample distances to original surface• projection to closest point projection to closest point [Hoppe][Hoppe]
• restricted projection restricted projection [Soucy–Laurendeau, Klein [Soucy–Laurendeau, Klein et alet al, , Ciampalini Ciampalini et alet al]]
Alternative characterization of errorAlternative characterization of error• quadric error metrics quadric error metrics [Garland–Heckbert][Garland–Heckbert]
• local volume preservation local volume preservation [Lindstrom–Turk][Lindstrom–Turk]
Measuring Error with Planes
Each vertex has a (conceptual) set of planesEach vertex has a (conceptual) set of planes• Error Error sum of squared distances to planes in set sum of squared distances to planes in set
Initialize with planes of incident facesInitialize with planes of incident faces• Consequently, all initial errors are 0Consequently, all initial errors are 0
When contracting pair, use plane set unionWhen contracting pair, use plane set union• planes(v’) = planes(vplanes(v’) = planes(v11) ) planes(v planes(v22))
Each vertex has a (conceptual) set of planesEach vertex has a (conceptual) set of planes• Error Error sum of squared distances to planes in set sum of squared distances to planes in set
Initialize with planes of incident facesInitialize with planes of incident faces• Consequently, all initial errors are 0Consequently, all initial errors are 0
When contracting pair, use plane set unionWhen contracting pair, use plane set union• planes(v’) = planes(vplanes(v’) = planes(v11) ) planes(v planes(v22))
TError( ) ( )i ii
d v n vTError( ) ( )i ii
d v n v
A Simple Example:Contraction & “Planes” in 2D
Lines defined by neighboring segmentsLines defined by neighboring segments• Determine position of new vertexDetermine position of new vertex
• Accumulate lines for ever larger areasAccumulate lines for ever larger areas
Lines defined by neighboring segmentsLines defined by neighboring segments• Determine position of new vertexDetermine position of new vertex
• Accumulate lines for ever larger areasAccumulate lines for ever larger areas
OriginalOriginal After 1 StepAfter 1 Step
vv11 vv22 v’v’
Measuring Error with Planes
Why base error on planes?Why base error on planes?• Faster, but less accurate, than distance-to-faceFaster, but less accurate, than distance-to-face
• Simple linear system for minimum-error positionSimple linear system for minimum-error position
• Efficient implicit form; no sets requiredEfficient implicit form; no sets required
• Drawback: unlike surface, planes are infiniteDrawback: unlike surface, planes are infinite
Related error metricsRelated error metrics• Ronfard & Rossignac — max vs. sumRonfard & Rossignac — max vs. sum
• Lindstrom & Turk — similar form; volume-basedLindstrom & Turk — similar form; volume-based
Why base error on planes?Why base error on planes?• Faster, but less accurate, than distance-to-faceFaster, but less accurate, than distance-to-face
• Simple linear system for minimum-error positionSimple linear system for minimum-error position
• Efficient implicit form; no sets requiredEfficient implicit form; no sets required
• Drawback: unlike surface, planes are infiniteDrawback: unlike surface, planes are infinite
Related error metricsRelated error metrics• Ronfard & Rossignac — max vs. sumRonfard & Rossignac — max vs. sum
• Lindstrom & Turk — similar form; volume-basedLindstrom & Turk — similar form; volume-based
The Quadric Error Metric
Given a plane, we can define a Given a plane, we can define a quadricquadric QQGiven a plane, we can define a Given a plane, we can define a quadricquadric QQ
measuring squared distance to the plane asmeasuring squared distance to the plane as
( )
a ab ac x x
Q x y z ab b bc y ad bd cd y d
ac bc c z z
v ( )
a ab ac x x
Q x y z ab b bc y ad bd cd y d
ac bc c z z
v
T T( )Q c v v Av bvT T( )Q c v v Av bv
T( , , ) ( , , )Q c d d A b nn nT( , , ) ( , , )Q c d d A b nn n
The Quadric Error Metric
Sum of quadrics represents set of planesSum of quadrics represents set of planes
Each vertex has an associated quadricEach vertex has an associated quadric• Error(Error(vvii) = ) = QQii (v (vii))
• Sum quadrics when contracting Sum quadrics when contracting (v(vii, v, vjj) ) v’ v’
• Cost of contraction is Cost of contraction is QQ(v’)(v’)
Sum of quadrics represents set of planesSum of quadrics represents set of planes
Each vertex has an associated quadricEach vertex has an associated quadric• Error(Error(vvii) = ) = QQii (v (vii))
• Sum quadrics when contracting Sum quadrics when contracting (v(vii, v, vjj) ) v’ v’
• Cost of contraction is Cost of contraction is QQ(v’)(v’)
T( ) ( ) ( )i i i ii i i
d Q Q
n v v vT( ) ( ) ( )i i i ii i i
d Q Q
n v v v
( , , )i j i j i j i jQ Q Q c c A A b b( , , )i j i j i j i jQ Q Q c c A A b b
The Quadric Error Metric
Sum of endpoint quadrics determines v’Sum of endpoint quadrics determines v’• Fixed placement: select Fixed placement: select vv1 1 or or vv22
• Optimal placement: choose Optimal placement: choose v’v’ minimizing minimizing QQ(v’)(v’)
• Fixed placement is faster but lower qualityFixed placement is faster but lower quality
• But it also gives smaller progressive meshesBut it also gives smaller progressive meshes
• Fallback to fixed placement if Fallback to fixed placement if AA is non- is non-invertibleinvertible
Sum of endpoint quadrics determines v’Sum of endpoint quadrics determines v’• Fixed placement: select Fixed placement: select vv1 1 or or vv22
• Optimal placement: choose Optimal placement: choose v’v’ minimizing minimizing QQ(v’)(v’)
• Fixed placement is faster but lower qualityFixed placement is faster but lower quality
• But it also gives smaller progressive meshesBut it also gives smaller progressive meshes
• Fallback to fixed placement if Fallback to fixed placement if AA is non- is non-invertibleinvertible
( ')Q v v A b( ')Q v v A b
Visualizing Quadrics in 3-D
Quadric Quadric isosurfacesisosurfaces• Are ellipsoidsAre ellipsoids
(maybe (maybe degenerate)degenerate)
• Centered around Centered around vertices vertices
• Characterize Characterize shapeshape
• Stretch in least-Stretch in least-curved directionscurved directions
Quadric Quadric isosurfacesisosurfaces• Are ellipsoidsAre ellipsoids
(maybe (maybe degenerate)degenerate)
• Centered around Centered around vertices vertices
• Characterize Characterize shapeshape
• Stretch in least-Stretch in least-curved directionscurved directions
Sample Model: Dental Mold
424,376 faces424,376 faces 60,000 faces60,000 faces
50 sec50 sec
Sample Model: Dental Mold
424,376 faces424,376 faces 8000 faces8000 faces
55 sec55 sec
Sample Model: Dental Mold
424,376 faces424,376 faces 1000 faces1000 faces
56 sec56 sec
Must Also Consider Attributes
Mesh for solutionMesh for solution Radiosity solutionRadiosity solution
Must Also Consider Attributes
50,761 faces50,761 faces 10,000 faces10,000 faces
Simplification Summary
Spectrum of effective methods developedSpectrum of effective methods developed• high quality; very slow high quality; very slow [Hoppe et al, Hoppe][Hoppe et al, Hoppe]• good quality; varying speedgood quality; varying speed
[Schroeder et al; Klein et al; Ciampalini et al; Guéziec[Schroeder et al; Klein et al; Ciampalini et al; GuéziecGarland-Heckbert; Ronfard-Rossignac; Lindstrom-Turk]Garland-Heckbert; Ronfard-Rossignac; Lindstrom-Turk]
• lower quality; very fast lower quality; very fast [Rossignac–Borrel; Low–Tan][Rossignac–Borrel; Low–Tan]
• result usually produced by transforming result usually produced by transforming originaloriginal
Various other differentiating factorsVarious other differentiating factors• is topology simplified? restricted to manifolds?is topology simplified? restricted to manifolds?• attributes simplified or re-sampled into maps?attributes simplified or re-sampled into maps?
Spectrum of effective methods developedSpectrum of effective methods developed• high quality; very slow high quality; very slow [Hoppe et al, Hoppe][Hoppe et al, Hoppe]• good quality; varying speedgood quality; varying speed
[Schroeder et al; Klein et al; Ciampalini et al; Guéziec[Schroeder et al; Klein et al; Ciampalini et al; GuéziecGarland-Heckbert; Ronfard-Rossignac; Lindstrom-Turk]Garland-Heckbert; Ronfard-Rossignac; Lindstrom-Turk]
• lower quality; very fast lower quality; very fast [Rossignac–Borrel; Low–Tan][Rossignac–Borrel; Low–Tan]
• result usually produced by transforming result usually produced by transforming originaloriginal
Various other differentiating factorsVarious other differentiating factors• is topology simplified? restricted to manifolds?is topology simplified? restricted to manifolds?• attributes simplified or re-sampled into maps?attributes simplified or re-sampled into maps?
Static Resolution Not Enough
Model used in variety of contextsModel used in variety of contexts• many machines; variable capacitymany machines; variable capacity
• projected screen size will varyprojected screen size will vary
Context dictates required detailContext dictates required detail• LOD should vary with contextLOD should vary with context
• context varies over timecontext varies over time
• with what level of coherence?with what level of coherence?– generally high coherence in viewgenerally high coherence in view
– possibly poor coherence in loadpossibly poor coherence in load
Model used in variety of contextsModel used in variety of contexts• many machines; variable capacitymany machines; variable capacity
• projected screen size will varyprojected screen size will vary
Context dictates required detailContext dictates required detail• LOD should vary with contextLOD should vary with context
• context varies over timecontext varies over time
• with what level of coherence?with what level of coherence?– generally high coherence in viewgenerally high coherence in view
– possibly poor coherence in loadpossibly poor coherence in load
Need Multiresolution Models
Encode wide range of levels of detailEncode wide range of levels of detail• extract appropriate approximations at run timeextract appropriate approximations at run time
• must have low overheadmust have low overhead– space consumed by representationspace consumed by representation
– cost of changing level of detail while renderingcost of changing level of detail while rendering
• can be generated via simplification processcan be generated via simplification process
Image pyramids (mip-maps) a good Image pyramids (mip-maps) a good exampleexample• very successful technique for raster imagesvery successful technique for raster images
Encode wide range of levels of detailEncode wide range of levels of detail• extract appropriate approximations at run timeextract appropriate approximations at run time
• must have low overheadmust have low overhead– space consumed by representationspace consumed by representation
– cost of changing level of detail while renderingcost of changing level of detail while rendering
• can be generated via simplification processcan be generated via simplification process
Image pyramids (mip-maps) a good Image pyramids (mip-maps) a good exampleexample• very successful technique for raster imagesvery successful technique for raster images
Discrete Multiresolution Models
Given a model, build a set of approximationsGiven a model, build a set of approximations• can be produced by any simplification systemcan be produced by any simplification system
• at run time, simply select which to renderat run time, simply select which to render
Inter-frame switching causes “popping”Inter-frame switching causes “popping”• can smooth transition with image blendingcan smooth transition with image blending
• or use geometry blending: geomorphing or use geometry blending: geomorphing [Hoppe][Hoppe]
Supported by several software packagesSupported by several software packages• RenderMan, Open Inventor, IRIS Performer, ...RenderMan, Open Inventor, IRIS Performer, ...
Given a model, build a set of approximationsGiven a model, build a set of approximations• can be produced by any simplification systemcan be produced by any simplification system
• at run time, simply select which to renderat run time, simply select which to render
Inter-frame switching causes “popping”Inter-frame switching causes “popping”• can smooth transition with image blendingcan smooth transition with image blending
• or use geometry blending: geomorphing or use geometry blending: geomorphing [Hoppe][Hoppe]
Supported by several software packagesSupported by several software packages• RenderMan, Open Inventor, IRIS Performer, ...RenderMan, Open Inventor, IRIS Performer, ...
Limits of Discrete Models
We may need varying LOD over surfaceWe may need varying LOD over surface• large surface, oblique view (eg. on terrain)large surface, oblique view (eg. on terrain)
– need high detail near the viewerneed high detail near the viewer– need less detail far awayneed less detail far away
• single LOD will be inappropriatesingle LOD will be inappropriate– either excessively detailed in the distance (wasteful)either excessively detailed in the distance (wasteful)– or insufficiently detailed near viewer (visual artifacts)or insufficiently detailed near viewer (visual artifacts)
Doesn’t really exploit available coherenceDoesn’t really exploit available coherence• small view change may cause large model small view change may cause large model
changechange
We may need varying LOD over surfaceWe may need varying LOD over surface• large surface, oblique view (eg. on terrain)large surface, oblique view (eg. on terrain)
– need high detail near the viewerneed high detail near the viewer– need less detail far awayneed less detail far away
• single LOD will be inappropriatesingle LOD will be inappropriate– either excessively detailed in the distance (wasteful)either excessively detailed in the distance (wasteful)– or insufficiently detailed near viewer (visual artifacts)or insufficiently detailed near viewer (visual artifacts)
Doesn’t really exploit available coherenceDoesn’t really exploit available coherence• small view change may cause large model small view change may cause large model
changechange
Progressive Meshes
We get more than just final approximationWe get more than just final approximation• sequence of contractionssequence of contractions
• corresponding intermediate approximationscorresponding intermediate approximations
Re-encode as progressive mesh (PM) Re-encode as progressive mesh (PM) [Hoppe][Hoppe]
• take final approximation to be base meshtake final approximation to be base mesh
• reverse of contraction sequence is split sequencereverse of contraction sequence is split sequence
• can reconstruct any intermediate modelcan reconstruct any intermediate model
• allow for progressive transmission & compressionallow for progressive transmission & compression
We get more than just final approximationWe get more than just final approximation• sequence of contractionssequence of contractions
• corresponding intermediate approximationscorresponding intermediate approximations
Re-encode as progressive mesh (PM) Re-encode as progressive mesh (PM) [Hoppe][Hoppe]
• take final approximation to be base meshtake final approximation to be base mesh
• reverse of contraction sequence is split sequencereverse of contraction sequence is split sequence
• can reconstruct any intermediate modelcan reconstruct any intermediate model
• allow for progressive transmission & compressionallow for progressive transmission & compression
PM’s a Limited Multiresolution
More flexibility is requiredMore flexibility is required• local addition/subtraction of triangleslocal addition/subtraction of triangles
– as conditions change, make small updates in LODas conditions change, make small updates in LOD
– this is the multi-triangulation framework this is the multi-triangulation framework
– may require novel approximationsmay require novel approximations
Must encode dependency of contractionsMust encode dependency of contractions• PM’s imply dependency on earlier contractionsPM’s imply dependency on earlier contractions
• but we can reorder non-overlapping but we can reorder non-overlapping contractions contractions
More flexibility is requiredMore flexibility is required• local addition/subtraction of triangleslocal addition/subtraction of triangles
– as conditions change, make small updates in LODas conditions change, make small updates in LOD
– this is the multi-triangulation framework this is the multi-triangulation framework
– may require novel approximationsmay require novel approximations
Must encode dependency of contractionsMust encode dependency of contractions• PM’s imply dependency on earlier contractionsPM’s imply dependency on earlier contractions
• but we can reorder non-overlapping but we can reorder non-overlapping contractions contractions
Structure Induced on Surface
Every vertex on approximation corresponds toEvery vertex on approximation corresponds to• a connected set of vertices on the originala connected set of vertices on the original• hence a region on the surface: the union of neighborhoodshence a region on the surface: the union of neighborhoods
Initial conditionsInitial conditions• every vertex set is a singleton, every region a neighborhoodevery vertex set is a singleton, every region a neighborhood
Every vertex on approximation corresponds toEvery vertex on approximation corresponds to• a connected set of vertices on the originala connected set of vertices on the original• hence a region on the surface: the union of neighborhoodshence a region on the surface: the union of neighborhoods
Initial conditionsInitial conditions• every vertex set is a singleton, every region a neighborhoodevery vertex set is a singleton, every region a neighborhood
Structure Induced on Surface
A contraction merges corresponding vertex setsA contraction merges corresponding vertex sets• remaining vertex accumulates larger surface regionremaining vertex accumulates larger surface region
When merging regions, can link them by mesh edgeWhen merging regions, can link them by mesh edge• as shown on left hand sideas shown on left hand side
A contraction merges corresponding vertex setsA contraction merges corresponding vertex sets• remaining vertex accumulates larger surface regionremaining vertex accumulates larger surface region
When merging regions, can link them by mesh edgeWhen merging regions, can link them by mesh edge• as shown on left hand sideas shown on left hand side
Structure Induced on Surface
Links within single region form spanning treeLinks within single region form spanning tree• links within all regions form spanning forestlinks within all regions form spanning forest
• any contraction order within regions is (topologically) validany contraction order within regions is (topologically) valid
Regions always completely partition original surfaceRegions always completely partition original surface
Links within single region form spanning treeLinks within single region form spanning tree• links within all regions form spanning forestlinks within all regions form spanning forest
• any contraction order within regions is (topologically) validany contraction order within regions is (topologically) valid
Regions always completely partition original surfaceRegions always completely partition original surface
Structure Induced on Surface
Pair-wise merging forms hierarchyPair-wise merging forms hierarchy• binary tree of verticesbinary tree of vertices
• also a binary tree of surface regionsalso a binary tree of surface regions
Pair-wise merging forms hierarchyPair-wise merging forms hierarchy• binary tree of verticesbinary tree of vertices
• also a binary tree of surface regionsalso a binary tree of surface regions
Example:Initial Vertex Neighborhoods
Example:99% of vertices removed
Example:99.9% of vertices removed
Vertex Hierarchies
A cut through the treeA cut through the tree• contract all below cutcontract all below cut• leaves are “active”leaves are “active”• determines partitiondetermines partition• and an approximationand an approximation
Encodes dependenciesEncodes dependencies• PM’s assume total orderPM’s assume total order• disjoint subtrees indep.disjoint subtrees indep.• get novel approximationsget novel approximations
• but must avoid fold-overbut must avoid fold-over
A cut through the treeA cut through the tree• contract all below cutcontract all below cut• leaves are “active”leaves are “active”• determines partitiondetermines partition• and an approximationand an approximation
Encodes dependenciesEncodes dependencies• PM’s assume total orderPM’s assume total order• disjoint subtrees indep.disjoint subtrees indep.• get novel approximationsget novel approximations
• but must avoid fold-overbut must avoid fold-over
Vertex Hierarchies forView-Dependent Refinement
Multiresolution representation for displayMultiresolution representation for display• incrementally move cut between framesincrementally move cut between frames
[Xia-Varshney, Hoppe, Luebke-Erickson][Xia-Varshney, Hoppe, Luebke-Erickson]
• move up/down where less/more detail neededmove up/down where less/more detail needed• relies on frame-to-frame coherencerelies on frame-to-frame coherence• can accommodate geomorphing can accommodate geomorphing [Hoppe][Hoppe]
Common application of vertex hierarchyCommon application of vertex hierarchy• hierarchy only guides active front evolutionhierarchy only guides active front evolution• more flexibility & overhead vs. discrete multiresmore flexibility & overhead vs. discrete multires
Multiresolution representation for displayMultiresolution representation for display• incrementally move cut between framesincrementally move cut between frames
[Xia-Varshney, Hoppe, Luebke-Erickson][Xia-Varshney, Hoppe, Luebke-Erickson]
• move up/down where less/more detail neededmove up/down where less/more detail needed• relies on frame-to-frame coherencerelies on frame-to-frame coherence• can accommodate geomorphing can accommodate geomorphing [Hoppe][Hoppe]
Common application of vertex hierarchyCommon application of vertex hierarchy• hierarchy only guides active front evolutionhierarchy only guides active front evolution• more flexibility & overhead vs. discrete multiresmore flexibility & overhead vs. discrete multires
Further Refinement inVertex Hierarchies
Also support synthetic refinementAlso support synthetic refinement• edge contraction is an inverse of edge splitedge contraction is an inverse of edge split
• can synthesize temporary levels in tree by splitting can synthesize temporary levels in tree by splitting edgesedges
• fractal extrapolation of terrain surface, for examplefractal extrapolation of terrain surface, for example
Also support synthetic refinementAlso support synthetic refinement• edge contraction is an inverse of edge splitedge contraction is an inverse of edge split
• can synthesize temporary levels in tree by splitting can synthesize temporary levels in tree by splitting edgesedges
• fractal extrapolation of terrain surface, for examplefractal extrapolation of terrain surface, for example
Applications Beyond Display
Other important applications are Other important applications are appearingappearing• surface editing surface editing [Guskov [Guskov et al et al 99]99]
• surface morphing surface morphing [Lee [Lee et al et al 99]99]
• multiresolution radiosity multiresolution radiosity [Willmott [Willmott et al et al 99]99]
Still others seem promisingStill others seem promising• hierarchical bounding volumeshierarchical bounding volumes
• object matchingobject matching
• shape analysis / feature extractionshape analysis / feature extraction
Other important applications are Other important applications are appearingappearing• surface editing surface editing [Guskov [Guskov et al et al 99]99]
• surface morphing surface morphing [Lee [Lee et al et al 99]99]
• multiresolution radiosity multiresolution radiosity [Willmott [Willmott et al et al 99]99]
Still others seem promisingStill others seem promising• hierarchical bounding volumeshierarchical bounding volumes
• object matchingobject matching
• shape analysis / feature extractionshape analysis / feature extraction
Multiresolution Model Summary
Representations are available to supportRepresentations are available to support• progressive transmissionprogressive transmission
• view-dependent refinementview-dependent refinement
• hierarchical computation (e.g., radiosity)hierarchical computation (e.g., radiosity)
But limitations remainBut limitations remain• vertex hierarchies may over-constrain vertex hierarchies may over-constrain
adaptationadaptation
• adaptation overhead not suitable for all casesadaptation overhead not suitable for all cases
• interacting multiresolution objects ignoredinteracting multiresolution objects ignored
Representations are available to supportRepresentations are available to support• progressive transmissionprogressive transmission
• view-dependent refinementview-dependent refinement
• hierarchical computation (e.g., radiosity)hierarchical computation (e.g., radiosity)
But limitations remainBut limitations remain• vertex hierarchies may over-constrain vertex hierarchies may over-constrain
adaptationadaptation
• adaptation overhead not suitable for all casesadaptation overhead not suitable for all cases
• interacting multiresolution objects ignoredinteracting multiresolution objects ignored
Looking Ahead
We’ve reached a performance plateauWe’ve reached a performance plateau• broad range of methods for certain situationsbroad range of methods for certain situations
• incremental improvement of existing methodsincremental improvement of existing methods
Major progress may require new techniquesMajor progress may require new techniques• broader applicability of simplificationbroader applicability of simplification
• higher quality approximationshigher quality approximations
Needs better understanding of performanceNeeds better understanding of performance• how well, in general, does an algorithm perform?how well, in general, does an algorithm perform?
We’ve reached a performance plateauWe’ve reached a performance plateau• broad range of methods for certain situationsbroad range of methods for certain situations
• incremental improvement of existing methodsincremental improvement of existing methods
Major progress may require new techniquesMajor progress may require new techniques• broader applicability of simplificationbroader applicability of simplification
• higher quality approximationshigher quality approximations
Needs better understanding of performanceNeeds better understanding of performance• how well, in general, does an algorithm perform?how well, in general, does an algorithm perform?
Greater Generality
Many applications require non-rigid surfacesMany applications require non-rigid surfaces• articulated models for animationarticulated models for animation
Other model types have complexity issuesOther model types have complexity issues• tetrahedral volumes, spline patch surfaces, ...tetrahedral volumes, spline patch surfaces, ...
Need to handle extremely large data setsNeed to handle extremely large data sets• precise scans on the order of 10precise scans on the order of 1099 triangles triangles
• this is where simplification is needed the mostthis is where simplification is needed the most
• even at 10even at 1066 triangles, many algorithms fail triangles, many algorithms fail
Many applications require non-rigid surfacesMany applications require non-rigid surfaces• articulated models for animationarticulated models for animation
Other model types have complexity issuesOther model types have complexity issues• tetrahedral volumes, spline patch surfaces, ...tetrahedral volumes, spline patch surfaces, ...
Need to handle extremely large data setsNeed to handle extremely large data sets• precise scans on the order of 10precise scans on the order of 1099 triangles triangles
• this is where simplification is needed the mostthis is where simplification is needed the most
• even at 10even at 1066 triangles, many algorithms fail triangles, many algorithms fail
Too Large for Many Methods
1,765,388 faces1,765,388 faces 80,000 faces80,000 faces
Really Too Large …
8.2 million triangles(2 mm resolution)
6.8 million triangles(¼ mm resolution)
Complete data sets:0.3–1.0 billion
triangles
Better Topological Simplification
Imperceptible holes & gaps can be removedImperceptible holes & gaps can be removed• most methods do this only implicitlymost methods do this only implicitly
Few if any methods provide good controlFew if any methods provide good control• when exactly are holes removed?when exactly are holes removed?
• will holes above a certain size be preserved?will holes above a certain size be preserved?
Requires better understanding of the modelRequires better understanding of the model• when to simplify geometry vs. topologywhen to simplify geometry vs. topology
• seems to benefit from more volumetric approachseems to benefit from more volumetric approach
Imperceptible holes & gaps can be removedImperceptible holes & gaps can be removed• most methods do this only implicitlymost methods do this only implicitly
Few if any methods provide good controlFew if any methods provide good control• when exactly are holes removed?when exactly are holes removed?
• will holes above a certain size be preserved?will holes above a certain size be preserved?
Requires better understanding of the modelRequires better understanding of the model• when to simplify geometry vs. topologywhen to simplify geometry vs. topology
• seems to benefit from more volumetric approachseems to benefit from more volumetric approach
Better Performance Analysis
Better criteria for evaluating similarityBetter criteria for evaluating similarity• image-based metric more appropriate for displayimage-based metric more appropriate for display
• metrics which accurately account for attributesmetrics which accurately account for attributes
Most analysis has been case-basedMost analysis has been case-based• measure/compare performance on 1 data setmeasure/compare performance on 1 data set
More thorough analysis is requiredMore thorough analysis is required• theoretical analysis of quality theoretical analysis of quality [Heckbert-Garland 99][Heckbert-Garland 99]
• provably good approximations possible?provably good approximations possible?
Better criteria for evaluating similarityBetter criteria for evaluating similarity• image-based metric more appropriate for displayimage-based metric more appropriate for display
• metrics which accurately account for attributesmetrics which accurately account for attributes
Most analysis has been case-basedMost analysis has been case-based• measure/compare performance on 1 data setmeasure/compare performance on 1 data set
More thorough analysis is requiredMore thorough analysis is required• theoretical analysis of quality theoretical analysis of quality [Heckbert-Garland 99][Heckbert-Garland 99]
• provably good approximations possible?provably good approximations possible?
Higher Quality Approximations
Poor performance at extreme reductionsPoor performance at extreme reductions• algorithms do much worse than humansalgorithms do much worse than humans• perhaps because all transform original into resultperhaps because all transform original into result
Simple iterative method quite short-sightedSimple iterative method quite short-sighted• only look one step ahead and never reconsideronly look one step ahead and never reconsider• many consider only the local effect of operationmany consider only the local effect of operation
Consider separating analysis & synthesisConsider separating analysis & synthesis• firstbuild multi-level knowledge of surface shapefirstbuild multi-level knowledge of surface shape• then proceed with simplificationthen proceed with simplification
Poor performance at extreme reductionsPoor performance at extreme reductions• algorithms do much worse than humansalgorithms do much worse than humans• perhaps because all transform original into resultperhaps because all transform original into result
Simple iterative method quite short-sightedSimple iterative method quite short-sighted• only look one step ahead and never reconsideronly look one step ahead and never reconsider• many consider only the local effect of operationmany consider only the local effect of operation
Consider separating analysis & synthesisConsider separating analysis & synthesis• firstbuild multi-level knowledge of surface shapefirstbuild multi-level knowledge of surface shape• then proceed with simplificationthen proceed with simplification
Alternative Frameworks
Greedy simplification convenient but limitedGreedy simplification convenient but limited• directly produce contraction sequencedirectly produce contraction sequence
• poor choices can never be reconsideredpoor choices can never be reconsidered
Other, albeit expensive, approaches possibleOther, albeit expensive, approaches possible• should produce a single sequence of contractionsshould produce a single sequence of contractions
• graph partitioning builds sequence in reversegraph partitioning builds sequence in reverse
• more explicit optimization methodsmore explicit optimization methods
Greedy simplification convenient but limitedGreedy simplification convenient but limited• directly produce contraction sequencedirectly produce contraction sequence
• poor choices can never be reconsideredpoor choices can never be reconsidered
Other, albeit expensive, approaches possibleOther, albeit expensive, approaches possible• should produce a single sequence of contractionsshould produce a single sequence of contractions
• graph partitioning builds sequence in reversegraph partitioning builds sequence in reverse
• more explicit optimization methodsmore explicit optimization methods
Conclusions
Substantial progress since 1992Substantial progress since 1992• simplification of 3D surfacessimplification of 3D surfaces
• multiresolution representations (PM, hierarchies)multiresolution representations (PM, hierarchies)
• application of multiresolution in different areasapplication of multiresolution in different areas
There remains much room for improvementThere remains much room for improvement• more effective, more general simplificationmore effective, more general simplification
• better analysis and understanding of resultsbetter analysis and understanding of results
• other multiresolution representationsother multiresolution representations
Substantial progress since 1992Substantial progress since 1992• simplification of 3D surfacessimplification of 3D surfaces
• multiresolution representations (PM, hierarchies)multiresolution representations (PM, hierarchies)
• application of multiresolution in different areasapplication of multiresolution in different areas
There remains much room for improvementThere remains much room for improvement• more effective, more general simplificationmore effective, more general simplification
• better analysis and understanding of resultsbetter analysis and understanding of results
• other multiresolution representationsother multiresolution representations
