Spectral Compression of Mesh Geometry

(Karni and Gotsman 2000)

Presenter: Eric Lorimer

Overview• Background• Spectral Compression• Evaluation• Recent Work • Future Directions

Background• Mesh geometry compressed

separately from mesh connectivity• Geometry data contains more

information than the connectivity data (15 bpv vs 3 bpv)

• Most techniques are lossless

Background• Standard techniques use

quantization and predictive entropy coding– Quantization: 10-14 bpv visually

indistinguishable from the original (“lossless”)

– Prediction rule• Parallelogram rule

[Touma, Gotsman 1998]

Spectral Compression• Consider now an implicit global

prediction rule: Each vertex is the average of all its neighbors

• Laplacian:– Eigenvalues are “frequencies”– Eigenvectors form orthogonal basis

Spectral Compression

Spectral Compression• Encoder

– Compute eigenvectors of L– Project geometry onto the basis vectors (dot

product) to generate coefficients– Quantize these coefficients and entropy code

them• Decoder

– Compute eigenvectors of L– Unpack coefficients– Sum coefficients * eigenvectors to reproduce

the signals

Spectral Compression• Computing eigenvectors

prohibitively expensive for large matrices

• Partition the mesh– MeTiS partitions mesh into balanced

partitions with minimal edge cuts.– Average submesh ~ 500 vertices

Spectral Compression• Visual Metric• Center: 4.1b/v• Right: TG at 4.1b/v (lossless =

6.5b/v)

Spectral Compression• Connectivity Shapes [Isenburg et

al. 2001]

Evaluation• Pros

– Progressive compression/transmission– Capable of compressing more than

traditional methods• Cons

– Expensive• Eigenvectors computed by decoder• Each mesh requires computing new eigenvectors

– Limited to smooth meshes– Edge effects from partitioning

Recent Work• Fixed spectral basis [Gotsman 2001]

– Don’t compute eigenvector basis vectors for each mesh

– Instead, map mesh to another mesh (e.g. 6-regular mesh) for which you have basis functions

– Good results, but small, expected loss of quality

Fixed Spectral Bases

Future Directions• Wavelets (JPEG2000, MPEG4 still

image coder)• Integration of connectivity and

geometry

References• Z. Karni and C. Gotsman. Spectral Compression

of Mesh Geometry. In Proceedings of SIGGRAPH 2000, pp. 279-286, July 2000.

• M. Ben-Chen and C. Gotsman. On the Optimality of Spectral Compression of Mesh Geometry. To appear in ACM transactions on Graphics 2004

• Z. Karni and C.Gotsman. 3D Mesh Compression Using Fixed Spectral Bases. Proceedings of Graphics Interface, Ottawa, June 2001.

• M. Isenburg., S. Gumhold and C. Gotsman. Connectivity Shapes. Proceedings of Visualization, San Diego, October 2001