Mesh Simplification Katja Bühler Mathematische Methoden der Computergraphik Slides stolen from:...
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Mesh Simplification
Katja Bühler
Mathematische Methoden der Computergraphik
Slides stolen from:
John C. Hart
Who has the slides stolen from:
Michael Garland
![Page 2: Mesh Simplification Katja Bühler Mathematische Methoden der Computergraphik Slides stolen from: John C. Hart Who has the slides stolen from: Michael Garland.](https://reader035.fdocuments.in/reader035/viewer/2022081519/56649d6b5503460f94a4a6bf/html5/thumbnails/2.jpg)
The Problem of Detail
• Graphics systems are awash in model data– very detailed CAD databases
– high-precision surface scans
• Available resources are always constrained– CPU, space, graphics speed, network bandwidth
• We need economical models– want the minimum level of detail (LOD) required
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A Non-Economical Model
424,376 faces424,376 faces 60,000 faces60,000 faces
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Automatic Surface Simplification
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Automatic Surface Simplification
• Produce approximations with fewer triangles– should be as similar as possible to original
– want computationally efficient process
• Need criteria for assessing model similarity– visual similarity
the ultimate goal for display
– similarity of shape is often used instead
• generally easier to compute
• lends itself more to applications other than display
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Focus on Polygonal Models
• Polygonal surfaces are ubiquitous– only primitive widely supported in hardware
– near-universal support in software packages
– output of most scanning systems
• Switching representations is no solution– indeed, some suffer from the same problem
– many applications want polygons
• Will always assume models are triangulated
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Historical Background
• Function approximation [y=f(x)]– long history in mathematical literature
• Piecewise linear curve approximation– various fields: graphics, cartography, vision, …
• Height field (i.e., terrain) triangulation– research back to at least early 70’s
– important for flight simulators
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Other Related Fields
• Geometry compression– simplification is a kind of lossy compression
• Surface smoothing– reduces geometric complexity of shape
• Mesh generation– finite element analysis (e.g., solving PDE’s)
– need appropriate mesh for good solution
– overly complex mesh makes solution slow
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Overview of Simplification Methods
• Manual preparation has been widely used– skilled humans produce excellent results
– very labor intensive, and thus costly
• Most common kinds of automatic methods– vertex clustering
– vertex decimation
– iterative contraction
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Optimal Approximations
• Achieve given error with fewest triangles– no mesh with fewer triangles meets error limit
• Computationally feasible for curves– O(n) for functions of one variable
– but O(n2 log n) for plane curves
• Intractable for surfaces– NP-hard to find optimal height field [Agarwal–Suri 94]
– must also be the case for surfaces
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Vertex Clustering
• Partition space into cells– grids [Rossignac-Borrel], spheres [Low-Tan], octrees, ...
• Merge all vertices within the same cell– triangles with multiple corners in one cell will degenerate
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Vertex Decimation
• Starting with original model, iteratively– rank vertices according to their importance
– select unimportant vertex, remove it, retriangulate hole
• A fairly common technique– Schroeder et al, Soucy et al, Klein et al, Ciampalini et al
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Iterative Contraction
• Contraction can operate on any set of vertices– edges (or vertex pairs) are most common, faces also used
• Starting with the original model, iteratively– rank all edges with some cost metric
– contract minimum cost edge
– update edge costs
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Edge Contraction
• Single edge contraction (v1,v2) v’ is performed by
– moving v1 and v2 to position v’
– replacing all occurrences of v2 with v1
– removing v2 and all degenerate triangles
v1v1
v2v2v’v’
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Vertex Pair Contraction
• Can also easily contract any pair of vertices– fundamental operation is exactly the same
– joins previously unconnected areas
– can be used to achieve topological simplification
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Iterative Edge Contraction
• Currently the most popular technique– Hoppe, Garland–Heckbert, Lindstrom-Turk, Ronfard-Rossignac,
Guéziec, and several others
– simpler operation than vertex removal
– well-defined on any simplicial complex
• Also induces hierarchy on the surface– a very important by-product
– enables several multiresolution applications
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Cost Metrics for Contraction
• Used to rank edges during simplification– reflects amount of geometric error introduced
– main differentiating feature among algorithms
• Must address two interrelated problems– what is the best contraction to perform?
– what is the best position v’ for remaining vertex?
• can just choose one of the endpoints
• but can often do better by optimizing position of v’
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Cost Metrics for Contraction
• Simple heuristics– edge length, dihedral angle, surrounding area, …
• Sample distances to original surface– projection to closest point [Hoppe]
– restricted projection [Soucy–Laurendeau, Klein et al, Ciampalini et al]
• Alternative characterization of error– quadric error metrics [Garland–Heckbert]
– local volume preservation [Lindstrom–Turk]
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Measuring Error with Planes
• Each vertex has a (conceptual) set of planes– Error sum of squared distances to planes in set
• Initialize with planes of incident faces– Consequently, all initial errors are 0
• When contracting pair, use plane set union
– planes(v’) = planes(v1) planes(v2)
i
idvError T
i
2)()( vn
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A Simple Example:Contraction & “Planes” in 2D• Lines defined by neighboring segments
– Determine position of new vertex
– Accumulate lines for ever larger areas
OriginalOriginal After 1 StepAfter 1 Step
vv11 vv22 v’v’
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Measuring Error with Planes
• Why base error on planes?– Faster, but less accurate, than distance-to-face
– Simple linear system for minimum-error position
– Efficient implicit form; no sets required
– Drawback: unlike surface, planes are infinite
• Related error metrics– Ronfard & Rossignac — max vs. sum
– Lindstrom & Turk — similar form; volume-based
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The Quadric Error Metric
• Given a plane, we can define a quadric Q
measuring squared distance to the plane as
)d ,d,(c),,(Q 2T nnnbA
c2)Q( TT vbAvvv
2
2
2
2
d
z
y
x
cdbdad2
z
y
x
cbcac
bcbab
acaba
zyx)Q(
v
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The Quadric Error Metric
• Sum of quadrics represents set of planes
• Each vertex has an associated quadric
– Error(vi) = Qi (vi)
– Sum quadrics when contracting (vi,vj) v’
– Cost of contraction is Q(v’)
)(Q)(Q)d(i i
iii
2i
Ti vvvn
)cc,,(QQQ jijijiji bbAA
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The Quadric Error Metric
• Sum of endpoint quadrics determines v’
– Fixed placement: select v1 or v2
– Optimal placement: choose v’ minimizing Q(v’)
– Fixed placement is faster but lower quality
– But it also gives smaller progressive meshes
– Fallback to fixed placement if A is non-invertible
bAvv 1'0)'Q(
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Visualizing Quadrics in 3-D
• Quadric isosurfaces– Are ellipsoids
(maybe degenerate)
– Centered around
vertices
– Characterize shape
– Stretch in least-
curved directions
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Sample Model: Dental Mold
424,376 faces424,376 faces 60,000 faces60,000 faces
50 sec50 sec
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Sample Model: Dental Mold
424,376 faces424,376 faces 8000 faces8000 faces
55 sec55 sec
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Sample Model: Dental Mold
424,376 faces424,376 faces 1000 faces1000 faces
56 sec56 sec
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Must Also Consider Attributes
Mesh for solutionMesh for solution Radiosity solutionRadiosity solution
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Must Also Consider Attributes
50,761 faces50,761 faces 10,000 faces10,000 faces
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Simplification Summary
• Spectrum of effective methods developed– high quality; very slow [Hoppe et al, Hoppe]
– good quality; varying speed[Schroeder et al; Klein et al; Ciampalini et al; Guéziec
Garland-Heckbert; Ronfard-Rossignac; Lindstrom-Turk]
– lower quality; very fast [Rossignac–Borrel; Low–Tan]
– result usually produced by transforming original
• Various other differentiating factors– is topology simplified? restricted to manifolds?– attributes simplified or re-sampled into maps?
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Static Resolution Not Enough• Model used in variety of contexts
– many machines; variable capacity
– projected screen size will vary
• Context dictates required detail– LOD should vary with context
– context varies over time
– with what level of coherence?
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Need Multiresolution Models
• Encode wide range of levels of detail– extract appropriate approximations at run time
– must have low overhead
• space consumed by representation
• cost of changing level of detail while rendering
– can be generated via simplification process
• Image pyramids (mip-maps) a good example– very successful technique for raster images
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Discrete Multiresolution Models
• Given a model, build a set of approximations– can be produced by any simplification system
– at run time, simply select which to render
• Inter-frame switching causes “popping”– can smooth transition with image blending
– or use geometry blending: geomorphing [Hoppe]
• Supported by several software packages– RenderMan, Open Inventor, IRIS Performer, ...
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Limits of Discrete Models
• We may need varying LOD over surface– large surface, oblique view (eg. on terrain)
• need high detail near the viewer
• need less detail far away
– single LOD will be inappropriate
• either excessively detailed in the distance (wasteful)
• or insufficiently detailed near viewer (visual artifacts)
• Doesn’t really exploit available coherence– small view change may cause large model change
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Progressive Meshes
• We get more than just final approximation– sequence of contractions
– corresponding intermediate approximations
• Re-encode as progressive mesh (PM) [Hoppe]
– take final approximation to be base mesh
– reverse of contraction sequence is split sequence
– can reconstruct any intermediate model
– allow for progressive transmission & compression
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PM’s a Limited Multiresolution
• More flexibility is required– local addition/subtraction of triangles
• as conditions change, make small updates in LOD
• this is the multi-triangulation framework
• may require novel approximations
• Must encode dependency of contractions– PM’s imply dependency on earlier contractions
– but we can reorder non-overlapping contractions
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Structure Induced on Surface
• Every vertex on approximation corresponds to– a connected set of vertices on the original
– hence a region on the surface: the union of neighborhoods
• Initial conditions– every vertex set is a singleton, every region a neighborhood
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Structure Induced on Surface
• A contraction merges corresponding vertex sets– remaining vertex accumulates larger surface region
• When merging regions, can link them by mesh edge– as shown on left hand side
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Structure Induced on Surface
• Links within single region form spanning tree– links within all regions form spanning forest
– any contraction order within regions is (topologically) valid
• Regions always completely partition original surface
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Structure Induced on Surface
• Pair-wise merging forms hierarchy– binary tree of vertices
– also a binary tree of surface regions
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Example:Initial Vertex Neighborhoods
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Example:99% of vertices removed
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Example:99.9% of vertices removed
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Vertex Hierarchies
• A cut through the tree– contract all below cut
– leaves are “active”
– determines partition
– and an approximation
• Encodes dependencies– PM’s assume total order
– disjoint subtrees indep.
– get novel approximations
– but must avoid fold-over
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Vertex Hierarchies forView-Dependent Refinement• Multiresolution representation for display
– incrementally move cut between frames[Xia-Varshney, Hoppe, Luebke-Erickson]
– move up/down where less/more detail needed– relies on frame-to-frame coherence– can accommodate geomorphing [Hoppe]
• Common application of vertex hierarchy– hierarchy only guides active front evolution– more flexibility & overhead vs. discrete multires
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Further Refinement inVertex Hierarchies• Also support synthetic refinement
– edge contraction is an inverse of edge split
– can synthesize temporary levels in tree by splitting edges
– fractal extrapolation of terrain surface, for example
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Applications Beyond Display
• Other important applications are appearing– surface editing [Guskov et al 99]
– surface morphing [Lee et al 99]
– multiresolution radiosity [Willmott et al 99]
• Still others seem promising– hierarchical bounding volumes
– object matching
– shape analysis / feature extraction
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Multiresolution Model Summary
• Representations are available to support– progressive transmission
– view-dependent refinement
– hierarchical computation (e.g., radiosity)
• But limitations remain– vertex hierarchies may over-constrain adaptation
– adaptation overhead not suitable for all cases
– interacting multiresolution objects ignored
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Looking Ahead
• We’ve reached a performance plateau– broad range of methods for certain situations
– incremental improvement of existing methods
• Major progress may require new techniques– broader applicability of simplification
– higher quality approximations
• Needs better understanding of performance– how well, in general, does an algorithm perform?
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Greater Generality
• Many applications require non-rigid surfaces– articulated models for animation
• Other model types have complexity issues– tetrahedral volumes, spline patch surfaces, ...
• Need to handle extremely large data sets– precise scans on the order of 109 triangles
– this is where simplification is needed the most
– even at 106 triangles, many algorithms fail
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Too Large for Many Methods
1,765,388 faces1,765,388 faces 80,000 faces80,000 faces
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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
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Better Topological Simplification
• Imperceptible holes & gaps can be removed– most methods do this only implicitly
• Few if any methods provide good control– when exactly are holes removed?
– will holes above a certain size be preserved?
• Requires better understanding of the model– when to simplify geometry vs. topology
– seems to benefit from more volumetric approach
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Better Performance Analysis
• Better criteria for evaluating similarity– image-based metric more appropriate for display
– metrics which accurately account for attributes
• Most analysis has been case-based– measure/compare performance on 1 data set
• More thorough analysis is required– theoretical analysis of quality [Heckbert-Garland 99]
– provably good approximations possible?
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Higher Quality Approximations
• Poor performance at extreme reductions– algorithms do much worse than humans– perhaps because all transform original into result
• Simple iterative method quite short-sighted– only look one step ahead and never reconsider– many consider only the local effect of operation
• Consider separating analysis & synthesis– firstbuild multi-level knowledge of surface shape– then proceed with simplification
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Alternative Frameworks
• Greedy simplification convenient but limited– directly produce contraction sequence
– poor choices can never be reconsidered
• Other, albeit expensive, approaches possible– should produce a single sequence of contractions
– graph partitioning builds sequence in reverse
– more explicit optimization methods
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Conclusions
• Substantial progress since 1992– simplification of 3D surfaces
– multiresolution representations (PM, hierarchies)
– application of multiresolution in different areas
• There remains much room for improvement– more effective, more general simplification
– better analysis and understanding of results
– other multiresolution representations