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Computation on Arbitrary Surfaces

Brandon Lloyd

COMP 258October 2002

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Computation on Arbitrary Surfaces

• Mathematical framework for Euclidean geometry enables us to perform important operations

• Usually peformed on regular grid• Triangle meshes have irregular connectivity,

hence irregular sampling• Triangle meshes are general 2-manifolds

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Computation on Arbitrary Surfaces

• Displaced Subdivision Surfaces• Multi-resolution signal processing on meshes• Texture synthesis on meshes

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Displaced Subdivision Surfaces

• Aaron Lee, Henry Moreton and Hughes Hoppe in SIGGRAPH 2000.

• The idea is to use a scalar-valued displacement over a smooth domain mesh

Control mesh Smooth domainsurface

Displacedsubdivision surface

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Representation

• Control mesh for domain surface• Regular scalar displacement mesh

– Subdivided along with the control mesh to produce a continuous displacement function

– Uses parameterization of the domain surface– Displacement map may be used as a bump map

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Conversion Process

• Obtain the control mesh

• Optimize the control mesh to more closely resemble original

• Sample displacement map

Original mesh Displaced subdivisionsurface

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Simplifying the Control Mesh

• Done with edge collapses• Map vertex normals from neighborhood of

candidate edge collapse to Gauss sphere• Disallow collapse if original normals are not

enclosed in spherical triangle

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Optimizing the Domain Surface

• Move vertices of control mesh to more accurately fit the original mesh

• Densely sample original mesh and minimize least squared distances

• Affects normals, but does not produce noticeable problems

Initial control mesh

Optimized controlmesh

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Sampling the Displacement Map

• Perform subdivision on optimized control mesh

• Intersect ray formed by point and normal with original mesh

• Store distance in displacement map

Subdivided Domain surface

Displacement field

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Applications

• Compression – The displacement map contains small, similar values

• Editing – simply modify the scalar field

• Animation – easier to kinematics with control mesh

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Compression Results

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Burt-Adelson Pyramids

• Prefilter and downsample image Ln to produce Ln-1

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Burt-Adelson Pyramids

• Prefilter and downsample image Ln to produce Ln-1

• Upsample Ln-1 and subtract from Ln

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2

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Burt-Adelson Pyramids

• Prefilter and downsample image Ln to produce Ln-1

• Upsample Ln-1 and subtract from Ln

• Repeat till you reach L0

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2

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Burt-Adelson Pyramids

• Prefilter and downsample image Ln to produce Ln-1

• Upsample Ln-1 and subtract from Ln

• Repeat till you reach L0

• The result is an image pyramid with a frequency subband at each level

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2

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Multiresolution Signal Processing for Meshes

• Igor Guskov, Wim Sweldens and Peter Schröder in SIGGRAPH 1999.

• Generalizes basic signal processing tools such as downsampling, upsampling, and filters to irregular triangle meshes

• Uses a non-uniform relaxation procedure for geometric smoothing

• Smoothing combined with hierarchical algorithms are used to build subdivision and pyramid methods

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Importance of Smoothness

• Geometric smoothness measures variance in triangle normals.

• Geometric smoothness implies that there exists a smooth (differentiable) parameterization for the mesh

• Smooth parameterization is important for many algorithms

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Importance of Smoothness

• Refinement can use semi-uniform (weights depend only on local connectivity) subdivision to obtain geometric smoothness

• It is much harder if we want to coarsify a mesh and later refine it again.

• Forced to use non-uniform (weights depend on connectivity and geometry) subdivision.

• The challenge is to ensure smooth local parameterization.

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Divided Differences

• First order divided difference of g(u,v) with discrete gi at the vertices is the gradient:

• The normal to the triangles formed by vertices is given by:

• Second order differences are the the difference between two normals on neighboring triangles. Only the magnitude matters.

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Divided Differences

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Divided Differences

• The magnitude of the second divided difference over edge e is:

• With coefficients

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Relaxation Procedure

• The relaxation operator R minimizes second order differences over a small neighborhood

• Define R as the minimizer of a quadratic energy function E:

• Expand in terms of gi: 2

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Relaxation Procedure

• Set the partial derivative of E with respect to gi to zero and solve:

• Relaxation for geometry use vertices pi in place of gi

• For each divided difference use the hinge map is use for local parameterization. Rotate triangles about their common edge till they lie in the same plane

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Relaxation Procedure

• This non-uniform smoothing operator does not affect triangle shapes much because it takes geometry into account.

• A semi-uniform scheme makes edge lengths as uniform as possible and distorts the mesh

Original mesh Semi-uniform smoothing

Non-uniformsmoothing

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Upsampling and Downsampling

• Uses Hoppe’s Progressive Mesh (PM) approach

• Vertex split provides upsampling• Edge collapse provides

downsampling• Different than most multi-resolution

schemes where downsampling removes a fraction of the samples

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Subdivision

• Starts from a coarse mesh and builds finer, smoother mesh

• Can be viewed as upsampling followed by relaxation

• Coarse mesh comes from PM. • The goal is to start from coarse mesh and

produce a smooth mesh with same connectivity as original with as little distortion as possible

• Performed one vertex at a time

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Subdivision

• The non-uniform scheme has access to parameterization info of the original mesh which guides the subdivision to give the desired result

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Building a Pyramid

• Downsampling: vertex n is removed by an edge collapse

• Subdivision: Points affected by the edge collapse are subdivided

• Detail Computation: Differences between points before edge collapse and after subdivision are stored

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Smoothing and Filtering

• The details from the pyramid are an approximate frequency spectrum

• Scale details for various filtering effects

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Smoothing and Filtering

• The details from the pyramid are an approximate frequency spectrum

• Scale details for various filtering effects

• Smoothing does not mess up texture mapping!

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Enhancement

• Single resolution scheme– Smooth mesh a number

of times– Extrapolate differences

between smoothed and original mesh

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Enhancement

• Single resolution scheme– Smooth mesh a number

of times– Extrapolate differences

between smoothed and original mesh

• Multi-resolution scheme– Scale details with values

greater than 1

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Multiresolution Editing

• User manipulates vertices at a level in the pyramid and system adds finer details back in

• Details are defined with respect to local frames

Original Coarsest Coarsestedit

Multi-resreconst.

Single-resreconst.

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Texture Synthesis on Surfaces

• Paper by Greg Turk, SIGGRAPH 2001• Motivation

– Texture greatly improves appearance– The ideal system is texture-from-sample– Existing texture synthesis operates on 2D

grids– It is difficult to map 2D texture– It is not always possible to use 3D texture

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Texture Synthesis on Surfaces

• Solution = Synthesize texture directly on a mesh.

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Texture Synthesis Algorithm

• Based on work by Wei and Levoy [Wei00]

• Initialize destination image to white noise

• Process pixels in raster scan order

• Find best match for neighborhood in original image

• Replace pixel in destination with pixel from original image

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Texture Synthesis Algorithm

• Use multi-resolution to get a full neighborhood.

• Create an image pyramid• Start at the top of the

pyramid• Use data from higher

levels to fill in unprocessed pixels in full neighborhood

• Multi-resolution improves quality

1 Level 2 Levels 3 Levels

Full 5x5 neighborhood

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Creating the Mesh Hierarchy

• Place uniform random points on the surface of the mesh– Choose a random

polygon based on area– Choose a random point

within the polygon

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Creating the Mesh Hierarchy

• Use repulsion forces to get an even distribution– Map nearby points to a

plane– Compute forces within

the plane

• Compute Voronoi diagram to establish sample connectivity– Once again map nearby

points to a plane– Compute diagram within

the plane

• Details in [Turk91]

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Creating the Mesh Hierarchy

• Start by placing n points on the mesh

• Add an additional 3n points and repel them from themselves and original n

• Create a new mesh including all 4n points

• Repeat to create several levels

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Operations on Mesh Hierarchy

• Interpolation• Low-pass filtering• Downsampling• Upsampling

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Operations on Mesh Hierarchy:Interpolation

• Take a weighted average of all mesh vertices vi within a particular radius r of point p

• The weighting function causes vertex contributions to falloff smoothly to 0 as distance from p approaches r.

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Operations on Mesh Hierarchy:Low-pass Filtering

• Borrow techniques from mesh smoothing to modify vertex positions. Weights wi are inverse edge length:

• Similarly for colors of adjacent mesh vertices

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Operations on Mesh Hierarchy:Low-pass Filtering

• Regrouping terms, the expression can be simplified to:

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Operations on Mesh Hierarchy:Upsampling and Downsampling

• Vertices store a color for each level they are in.

• To downsample from mesh Mk the vertices in the lower resolution mesh Mk+1 inherit the blurred mesh colors in Mk.

• To upsample from mesh Mk+1, the colors of the vertices in Mk are interpolated from Mk+1.

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Vector Field Creation

• Vector field is needed to establish orientation• A few vectors are assigned at lowest level. All

others set to zero• Vectors are “pulled” to coarsest mesh by a

number of downsamplings• Remaining zero vectors are “filled” by

diffusing vectors over the mesh• Vectors are “pushed” to lower levels by

repeated upsampling

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Surface Sweeping

• Similar to raster scanning in original algorithm• All vertices are assigned a sweep distance

s(v). • Anchor vertex has a sweep distance of 0

User-assigned vectors

Resulting vector field

Sweep values

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Surface Sweeping

• Calculate an estimate Δw of how much further downstream a vertex is than its assigned neighbors:

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Surface Sweeping

• Assign a sweep distance using estimates:

• Propogate sweep distances over coarsest mesh

• Assign sweep distances to finer meshes with upsampling

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Neighborhood Colors

• Neighbor positions are defined as offsets (i,j) from the current sample.

• Local orientation is provided by the vector field• Step to neighborhood location by marching

along mesh in the direction of the vector field for i and perpendicular to it for j.

• Each step is the length of the average distance between vertices

• Retrieve color by interpolation

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Complete Algorithm

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Texture Synthesis over Arbitrary Manifold Surfaces

• Paper by Li-Yi Wei and Marc Levoy also in SIGGRAPH 2001 [WL01]

• Major differences from [Turk01]– Synthesis ordering – Vertices are visted in random

ordering– Local coordinate frames– Neighborhood construction

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Local Coordinate Frames

• Assigned at coarsest level. Finer levels interpolate from higher levels.

• Random works well for isotropic textures.• Relaxation produces n-way symmetry:

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Local Coordinate Frames

Isotropictextures

Anisotropic textures

Random Spherical Mapping Relaxed

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Neighborhood Construction

• Local flattening of the mesh and resampling.• Orthographically project neighboring triangles

of a point p into its local plane.• Grow region until neighborhood template is

fully covered.

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Neighborhood Construction

• For lower-resolution neighborhood, project parent face intersected by ray from p in the direction of the normal.

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Results

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References

[GSS99]

[LMH00]

[Turk91]

[Turk01]

[WL01]

I. Guskov, W. Sweldens and P. Schröder, Multiresolution Signal Processing for Meshes, Proceedings of SIGGRAPH ’99, pp. 325-334, 1999.

A. Lee, H. Morenton and H. Hoppe. Displaced Subdivision Surfaces, Proceedings of SIGGRAPH 2000, pp.84-97, 2000.

G. Turk, Generating Textures for Arbitrary Surfaces Using Reaction-Diffusion, Proceedings of SIGGRAPH ’91, pp 289-298, 1991.

G. Turk, Texture Synthesis on Surfaces, Proceedings of SIGGRAPH ’01, pp.347-354, 2001.

L. Wei and M. Levoy. Texture Synthesis over Arbitrary Manifold Surfaces, Proceedings of SIGGRAPH ’01, pp. 355-360, 2001.