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Surface Compression with Geometric Bandelets Gabriel Peyré Stéphane Mallat

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Outline Why Discrete Multiscale Geometry? Image-based Surface Processing Geometry in the Wavelet Domain Moving from 2D to 1D The Algorithm in Details Results

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Geometry of Surfaces: Creation Clay modeling Low-poly modeling

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Geometry of Surfaces: Rendering Stoke rendering taken from [Rössl-Kobbelt 01] Line-Art Rendering of 3D Models. Lines of curvatures and highlights

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Geometry of Surfaces: Mesh processing Anisotropic Remeshing [Alliez et Al. 03] Robust Moving Least-squares Fitting with Sharp Features [Fleishman et Al. 05]

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Geometry is Discrete continous discrete multiscale geometry acquisition [Digital Michelangelo Project]

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Geometry is Multiscale? continous discrete multiscale geometry Surface simplification [Garland & Heckbert 97] Normal meshes [Guskov et Al. 00]

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Sharp Geometry is Multiscale! Smoothed Fine Scale Edge extraction is an ill-posed problem. Localization is not needed for compression!

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Geometry 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. [DeRose et Al. 98] Semi-sharp features [DeRose et Al. 98] Subdivision Surfaces in Character Animation

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Outline Why Discrete Multiscale Geometry? Image-based Surface Processing Geometry in the Wavelet Domain Moving from 2D to 1D The Algorithm in Details Results

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[Gu et Al.] Geometry images [Gu et Al.] irregular mesh 2D array of points [r,g,b] = [x,y,z] cutparameterize No connectivity information. Simplify and accelerate hardware rendering. Allows application of image-based compression schemes.

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Our Functional Model 3D model 2D GIM (lit) Uniformly regular areas + Sharp features + Smoothed features

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Outline Why Discrete Multiscale Geometry? Image-based Surface Processing Geometry in the Wavelet Domain Moving from 2D to 1D The Algorithm in Details Results

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Hierarchical Cascad Acquisition (scanner, etc) Wavelet transform Orthogonal dilated filters cascad Proposition: to continue the cascad. Geometric transform ?

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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. HV D

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Outline Why Discrete Multiscale Geometry? Image-based Surface Processing Geometry in the Wavelet Domain Moving from 2D to 1D The Algorithm in Details Results

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Some 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.

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Construction of this Reordering Geometry image 2D Wavelet Transform HV D Choose a direction Project points orthogonally on Report values on 1D axis Resulting 1D signal

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Choosing the square and the direction T -T T + threshold T Too big: direction deviates from geometry How to choose: 1D wavelet transform Too much high coefficients!

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Choosing the Square and the Direction T -T T Bad direction: direction deviates from geometry Still too much high coefficients!

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Choosing the Square and the Direction T -T T Correct direction: direction matches the geometry Nearly no high coefficients!

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Outline Why Discrete Multiscale Geometry? Image-based Surface Processing Geometry in the Wavelet Domain Moving from 2D to 1D The Algorithm in Details Results

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The Algorithm in 10 Steps (1) Geometry Image (2) 2D Wavelet Transf. (3) Dyadic Subdivision (4) Extract Sub-square (5) Sample Geometry (6) Project Points (7) 1D Wavelet Transf. (8) Select Geometry (9) Output Coefficients (10) Build Quadtree Original surface Geometry image

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The Algorithm in 10 Steps (1) Geometry Image (2) 2D Wavelet Tran (3) Dyadic Subdivision (4) Extract Sub-square (5) Sample Geometry (6) Project Points (7) 1D Wavelet Transf. (8) Select Geometry (9) Output Coefficients (10) Build Quadtree Geometry image 2D Wavelet Transform HV D

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The Algorithm in 10 Steps (1) Geometry Image (2) 2D Wavelet Transf. (3) Dyadic Subdivis (4) Extract Sub-square (5) Sample Geometry (6) Project Points (7) 1D Wavelet Transf. (8) Select Geometry (9) Output Coefficients (10) Build Quadtree 2D Wavelet Transform HV D Zoom on D D

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The Algorithm in 10 Steps (1) Geometry Image (2) 2D Wavelet Transf. (3) Dyadic Subdivis (4) Extract Sub-sq (5) Sample Geometry (6) Project Points (7) 1D Wavelet Transf. (8) Select Geometry (9) Output Coefficients (10) Build Quadtree Zoom on D D Sub-square 2D Wavelet Transform HV D

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The Algorithm in 10 Steps (1) Geometry Image (2) 2D Wavelet Transf. (3) Dyadic Subdivision (4) Extract Sub-square (5) Sample Geometr (6) Project Points (7) 1D Wavelet Transf. (8) Select Geometry (9) Output Coefficients (10) Build Quadtree Sub-squareZoom on D D

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Sub-square The Algorithm in 10 Steps (1) Geometry Image (2) 2D Wavelet Transf. (3) Dyadic Subdivision (4) Extract Sub-square (5) Sample Geometry (6) Project Points (7) 1D Wavelet Transf. (8) Select Geometry (9) Output Coefficients (10) Build Quadtree 1D Signal

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1D Wavelet Transform The Algorithm in 10 Steps (1) Geometry Image (2) 2D Wavelet Transf. (3) Dyadic Subdivision (4) Extract Sub-square (5) Sample Geometry (6) Project Points (7) 1D Wavelet Tran (8) Select Geometry (9) Output Coefficients (10) Build Quadtree 1D Signal

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vs T 1D Wavelet Transform The Algorithm in 10 Steps (1) Geometry Image (2) 2D Wavelet Transf. (3) Dyadic Subdivision (4) Extract Sub-square (5) Sample Geometry (6) Project Points (7) 1D Wavelet Transf. (8) Select Geometry (9) Output Coefficients (10) Build Quadtree 1D Signal T vs T

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The Algorithm in 10 Steps (1) Geometry Image (2) 2D Wavelet Transf. (3) Dyadic Subdivision (4) Extract Sub-square (5) Sample Geometry (6) Project Points (7) 1D Wavelet Transf. (8) Select Geometry (9) Output Coefficie (10) Build Quadtree 1D Signal 1D Wavelet Transform T T vs T Bandelets coefficients Untransformed coefs (For comparison)

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The Algorithm in 10 Steps (1) Geometry Image (2) 2D Wavelet Transf. (3) Dyadic Subdivision (4) Extract Sub-square (5) Sample Geometry (6) Project Points (7) 1D Wavelet Transf. (8) Select Geometry (9) Output Coefficients (10) Build Quadtree Don’t use every dyadic square … Compute an optimal segmentation into squares. Fast pruning algorithm (see paper). 2D Wavelet Transform HV D D Zoom on D

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What does bandelets look like? Transform = decomposition on an orthogonal basis. Basis functions are elongated “bandelets”. The transform adapts itself to the geometry.

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Transform 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).

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Outline Why Discrete Multiscale Geometry? Image-based Surface Processing Geometry in the Wavelet Domain Moving from 2D to 1D The Algorithm in Details Results

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Results: Sharp features Original Wavelets @0.2 bpv Bandelets @0.2 bpv Hausd. PNSR: +2.2dB

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Results: More complex features Original Wavelets @0.2 bpv Bandelets @0.2 bpv Hausd. PNSR: +1.6dB

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Blurred Features Hausd. PNSR: +1.3dB Wavelets @0.2 bpv Wavelets @0.2 bpv Original

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Spherical Geometry Images Wavelets @0.2 bpv Wavelets @0.2 bpv Original Hausd. PNSR: +1.0dB

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Spherical Geometry Images Wavelets @0.2 bpv Wavelets @0.2 bpv Original Hausd. PNSR: +1.25dB

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Conclusion 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.).

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