Surface Compression with Geometric Bandelets Gabriel Peyré Stéphane Mallat.

Surface Compression

with Geometric Bandelets

Gabriel PeyréGabriel Peyré

Stéphane MallatStéphane Mallat

OutlineOutline

• Why Discrete Multiscale Geometry?

• Image-based Surface Processing

• Geometry in the Wavelet Domain

• Moving from 2D to 1D

• The Algorithm in Details

• Results

Geometry of Surfaces: CreationGeometry of Surfaces: Creation

Clay modelingClay modeling Low-poly modelingLow-poly modeling

Geometry of Surfaces: RenderingGeometry of Surfaces: Rendering

Stoke rendering Stoke rendering taken from taken from [Rössl-Kobbelt 01][Rössl-Kobbelt 01]

Line-Art Rendering of 3D Models.Line-Art Rendering of 3D Models.

Lines of curvaturesLines of curvaturesand highlightsand highlights

Geometry of Surfaces:Mesh processingGeometry of Surfaces:Mesh processing

Anisotropic RemeshingAnisotropic Remeshing[Alliez et Al. 03][Alliez et Al. 03]

Robust Moving Least-squares Fitting with Sharp Features [Fleishman et Al. 05][Fleishman et Al. 05]

Geometry is DiscreteGeometry is Discrete

continous

discrete

multiscale

geometry

acquisition

[Digital Michelangelo Project][Digital Michelangelo Project]

Geometry is Multiscale?Geometry is Multiscale?

continous

discrete

multiscale

geometry

Surface simplificationSurface simplification[Garland & Heckbert 97][Garland & Heckbert 97]

Normal meshesNormal meshes[Guskov et Al. 00][Guskov et Al. 00]

Sharp

Geometry is Multiscale!Geometry is Multiscale!

Smoothed

Fine Scale

•Edge extraction is an ill-posed problem.•Localization is not needed for compression!

Geometry is not Defined by Sharp FeaturesGeometry is not Defined by Sharp Features

Edge localization[Ohtake et Al. 04][Ohtake et Al. 04] Ridge-valley lines on

meshes via implicit surface fitting.

Semi-sharp features [DeRose et Al. 98][DeRose et Al. 98] Subdivision

Surfaces in Character Animation

OutlineOutline






• Results

Geometry images [Gu et Al.][Gu et Al.]Geometry images [Gu et Al.][Gu et Al.]

irregular mesh irregular mesh 2D array of points 2D array of pointsirregular mesh irregular mesh 2D array of points 2D array of points

[[rr,,gg,,bb] = [] = [xx,,yy,,zz]]cutcut parameterizeparameterize

•No connectivity information.•Simplify and accelerate hardware rendering.•Allows application of image-based compression schemes.

Our Functional ModelOur Functional Model

3D model 2D GIM (lit)

Uniformly regular areas+ Sharp features+ Smoothed features

OutlineOutline






• Results

Hierarchical CascadHierarchical Cascad

Geometry

2R : f x 2LContinuous

, V nP f f Discrete:

,, j nf Multiscale:

Acquisition (scanner, etc)

Wavelet transform

• Orthogonal dilated filters cascad

• Proposition: to continue the cascad.

Geometric transform?

What is a wavelet transform?What is a wavelet transform?

Original surface

Geometry image Wavelet transform

•Decompose an image at dyadic scales.•3 orientations by scales H/V/D.•Compact representation: few high coefficients.•But… still high coefficients near singularities.

H V

D

OutlineOutline






• Results

Some insights about bandeletsSome insights about bandelets

• Moto: wavelets transform is cool, re-use it!

• Goal: remove the remaining high wavelet coefficients.

• Hope: exploit the anisotropic regularity of the geometry.

• Tool: 2D anisotropy become isotropic in 1D.

Construction of this ReorderingConstruction of this Reordering

Geometry image 2D Wavelet TransformH V

D • Choose a direction

• Project pointsorthogonally on

• Report values on 1D axis

• Resulting 1D signal

Choosing the square and the directionChoosing the square and the direction

T

-T

T

+ threshold T

• Too big: direction deviates from geometry

• How to choose: 1D wavelettransform

• Too much high coefficients!

Choosing the Squareand the DirectionChoosing the Squareand the Direction

T

-T

T

• Bad direction: direction deviates from geometry

• Still too much high coefficients!

Choosing the Squareand the DirectionChoosing the Squareand the Direction

T

-T

T

• Correct direction: direction matches the geometry

• Nearly no high coefficients!

OutlineOutline






• Results

The Algorithm in 10 StepsThe Algorithm in 10 Steps

(1) Geometry Image(1) Geometry Image

(2) 2D Wavelet Transf.

(3) Dyadic Subdivision

(4) Extract Sub-square

(5) Sample Geometry

(6) Project Points


(8) Select Geometry

(9) Output Coefficients

(10) Build Quadtree

Original surfaceGeometry image


(1) Geometry Image

(2) 2D Wavelet Tran(2) 2D Wavelet Tran



(5) Sample Geometry

(6) Project Points


(8) Select Geometry


(10) Build Quadtree

Geometry image 2D Wavelet Transform

H V

D


(1) Geometry Image


(3) Dyadic Subdivis(3) Dyadic Subdivis


(5) Sample Geometry

(6) Project Points


(8) Select Geometry


(10) Build Quadtree

2D Wavelet Transform

H V

D

Zoom on D

D


(1) Geometry Image


(3) Dyadic Subdivis

(4) Extract Sub-sq(4) Extract Sub-sq

(5) Sample Geometry

(6) Project Points


(8) Select Geometry


(10) Build Quadtree

Zoom on D

D

Sub-square2D Wavelet Transform

H V

D


(1) Geometry Image




(5) Sample Geometr(5) Sample Geometr

(6) Project Points


(8) Select Geometry


(10) Build Quadtree

Sub-squareZoom on D

D

Sub-square


(1) Geometry Image




(5) Sample Geometry

(6) Project Points(6) Project Points


(8) Select Geometry


(10) Build Quadtree

1D Signal



(1) Geometry Image




(5) Sample Geometry

(6) Project Points

(7) 1D Wavelet Tran(7) 1D Wavelet Tran

(8) Select Geometry


(10) Build Quadtree

1D Signal

vs

T



(1) Geometry Image




(5) Sample Geometry

(6) Project Points


(8) Select Geometry(8) Select Geometry


(10) Build Quadtree

1D Signal

T

vs

T

vs


(1) Geometry Image




(5) Sample Geometry

(6) Project Points


(8) Select Geometry

(9) Output Coefficie(9) Output Coefficie

(10) Build Quadtree

1D Signal 1D Wavelet Transform

T

T

vs

T

Bandelets coefficientsUntransformed coefs(For comparison)


(1) Geometry Image




(5) Sample Geometry

(6) Project Points


(8) Select Geometry


(10) Build Quadtree(10) Build Quadtree

• Don’t use every dyadic square …

• Compute an optimal segmentation into squares.

• Fast pruning algorithm (see paper).


H V

DD

Zoom on D

What does bandelets look like?What does bandelets look like?

• Transform = decomposition on an orthogonal basis.

• Basis functions are elongated “bandelets”.

• The transform adapts itself to the geometry.

Transform Coding in a Bandelet BasisTransform Coding in a Bandelet Basis

• Bandelet coefficients are quantized and entropy coded.

• Quadtree segmentation and geometry is coded.

• Possibility to use more advanced image coders (e.g. JPEG2000).

OutlineOutline






• Results

Results:Sharp featuresResults:Sharp features

Original [email protected] bpv

[email protected] bpv

Hausd. PNSR: +2.2dB

Results:More complex featuresResults:More complex features

Original [email protected] bpv


Hausd. PNSR: +1.6dB

Blurred FeaturesBlurred Features

Hausd. PNSR: +1.3dB



Original

Spherical Geometry ImagesSpherical Geometry Images



Original

Hausd. PNSR: +1.0dB

Spherical Geometry ImagesSpherical Geometry Images



Original

Hausd. PNSR: +1.25dB

ConclusionConclusion

• Approach: re-use wavelet expansion.

• Contribution: bring geometry into the multiscale framework.

• Results: improvement over wavelets even for blurred features.

• Extension: other maps (normals, BRDF, etc.) and other processings (denoising, deblurring, etc.).

Surface Compression with Geometric Bandelets Gabriel Peyré Stéphane Mallat.

Documents

d wavelet

hv d slide

sharp geometry

d signal slide

geometry of surfaces

reordering geometry

d axis

d anisotropy