# Progressive Geometry Compression. Andrei Khodakovsky Peter Schröder Wim Sweldens

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1 Progressive Geometry Compression Andrei Khodakovsky Peter Schröder Wim Sweldens

2 Motivation Large (up to billions of vertices), finely detailed, arbitrary topology surfaces Difficult manageability of such large amounts of information Computation, Storage, Transmission, Display Strain Progressive Compression is needed

3 What Compression Is All About Accuracy/ bit per vertex Definition of a Geometry Error Measure of the geometric distance between 3D objects New Problems with Respect to Image Compression No direct correspondence between original and compressed surface

4 Algorithm Detailed Mesh Parameters & Connectivity Original Mesh Semi-regular mesh (MAPS) Wavelet Transform Wavelet Coefficients 2-manifold, arbitrary connectivity Smooth global parameterization Semi-regular approximation Progressive transformation and coding of the approximated shapes Geometry Compression Zerotree Coding Entropy Coding Bitrate Coarse Mesh

5 Parameters & Connectivity Geometry-Parameter-Connectivity

6 Geometry Error Parameters & Connectivity For highly detailed, densely sampled meshes, the sample location ( vertices + connectivity) do not contribute in improving the geometric distance, and thus the error Parameter information is thus contained within the surface, and any rate-distortion improving coding should better take advantage from the geometric information only, normal to the surface

7 MAPS algorithm Parameters & Connectivity Multiresolution Adaptive Parametrization of Surfaces based on edge collapsing Semi-regular approximation is achieved on applying triangle quadrisection to the coarse mesh, and taking advantage of the mapping operated by the algorithm between original and coarse mesh.

8 Wavelet Transform-1 Geometry Compression modeling of a complex object of arbitrary topology at multiple levels of detail Replacement of level-n mesh with coarser level- (N-1) mesh + wavelet coefficients Generation of nested meshes Subdivision Rules and Filter-Banks

9 Geometry Compression Wavelet Transform-2 Semi-regular mesh is hierarchically subdivided into a coarser mesh and some details information Reconstruction is achieved from the coarse mesh by hierarchical addition of detail information

10 Wavelet Transform-3 Geometry Compression Filter Bank Algorithm Design of Analysis and Synthesis Filters. Decomposition of a mesh like Our Paper: Synthesis: Analysis: solution [pj,dj]ofsystemforagivenpj+1 P, low-pass reconstruction filter Q, high-pass reconstruction filter small support

11 Wavelet Transform-4 Geometry Compression Decorrelating effect of the Wv Transform Vertex position magnitudes for Venus Wavelet Coefficients Magnitudes

12 Wavelet Coefficients Vector valued x,y,z components pretty correlated, but decorrelation in a local frame Geometry Compression Global Frame Local Frame Coefficients mostly in the normal direction Normal component relevant for geometric information Code each component independently

13 Bitrate Hierarchical Trees Exploit the relationship of wavelet coefficients across bands, and their exponential decay Areas with significant information are similar in shape and location Non significance in a low frequency band for a particular level of accuracy means with high probability non significance of the children nodes. Localizing a zerotree avoids transmitting a large amount of insignificant details with respect to a desired level of accuracy Progressive compression, embedded code

14 Bitrate Zerotree Coding Example of the quality of the coding SPIHT PSNR = db, JPEG PSNR = 31.8 db (quality factor 15%). SAID AND PEARLMAN, A New, Fast, and Efficient Image Codec Based on Set Partitioning in Hierarchical Trees. IEEE Transaction on Circuits and Systems for Video Technology 6, 3 (1996),

15 Zerotree-Howto Bitrate General: let the decoder see just as much coefficients as needed Significance Map: decide which coefficient become significant at a particular bitrate exponentially decreasing threshold Refinement Map: decide which coefficients need to be transmitted, as they became significant in a previous pass Sign Map: additional information for sign coding

16 Bitrate Zerotree-Structure Moves: - wavelet coefficients are coded as 1-dim array - lower frequency bands before higher frequency bands - scan the array and select against a threshold value, which is repeatedly refined - transmit zerotree information, i.e. as soon as a zero is encountered the corresponding coefficient node is treated as non-relevant. - as long as a coefficient remains irrelevant a zerotree symbol will be transmitted for it

17 Bitrate Zerotree-Example

18 Entropy Coding Bitrate Further improvement of the bit budget by eliminating redundancy due to non-uniform distribution Refinement and sign bits are found to be distributed uniformly Significance is a function of bitplane: early bitplanes will contain many insignificant coefficients, which will become significant in later bitplanes

19 Results Results Rate Distortion Curves: Improvement by a factor 4! Relative L 2 error (10-4 ) Bits/vertex

20 Results Live Compression B (e 40) 1528 B (e 12) 4163 B (e 4.7) B (e 0.82)

21 Conclusions Very effective compression algorithm smooth appearance, low (hardware) strain Many details in very early stages of decompression Very fruitful distinction between parameters and geometry Still at the beginning of wavelet 3D model compression...

22 In Fact... Effectiveness of compression inherits very much from a bunch of tested approaches No sound treatment of multiresolution analysis orthogonality, stability... Tentative choice of wavelets Elimination of tangential information (Normal Meshes...)

23 Demo

26 Extra Slides...

27 Wavelet Transform-2 Geometry Compression Ingredients of Multiresolution Analysis (Mallat and Meyer) : Existence of nested linear spaces and of an inner product relative to a subdivision rule. Nested spaces are generated by translations and dilations of a single function, φ(x) Subdivision Rules can be used to define such functions

28 Distance Function Euclidean Distance (L 2 ) d(x,y) between two surfaces X, Y Symmetrize by taking the max of d(x,y) and d(y,x)

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