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Spectral bases for 3D shapes representation Spectral bases – reminder Last week we spoke of Fourier analysis 1D sine/cosine bases for signals: Spectral bases – reminder Last week we spoke of Fourier analysis 2D bases for 2D signals (images) How about 3D shapes?

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Presentation Transcript
spectral bases reminder
Spectral bases – reminder
  • Last week we spoke of Fourier analysis
    • 1D sine/cosine bases for signals:
spectral bases reminder3
Spectral bases – reminder
  • Last week we spoke of Fourier analysis
    • 2D bases for 2D signals (images)
how about 3d shapes
How about 3D shapes?
  • Problem: general 3D shapes cannot be described as height functions

Height function, regularly sampled above a 2D domain

General 3D shapes

irregular meshes
Irregular meshes
  • In graphics, shapes are mostly represented by triangle meshes
irregular meshes6
Irregular meshes
  • In graphics, shapes are mostly represented by triangle meshes
  • Geometry: Vertex coordinates

(x1, y1, z1)

(x2, y2, z2)

. . .

(xn, yn, zn)

  • Connectivity: List of triangles

(i1, j1, k1)

(i2, j2, k2)

. . .

(im, jm, km)

irregular meshes7
Irregular meshes
  • In graphics, shapes are mostly represented by triangle meshes
    • The connectivity (the graph) can be very efficiently encoded
      • About 2 bits per vertex only
    • The geometry (x,y,z) is heavy!
      • When stored naively, at least 12 bits per coordinate are needed, i.e. 36 bits per vertex
how to define efficient bases
How to define efficient bases?
  • Extension of the 2D DCT basis to a general (irregular) mesh

DCT

???

basis shapes
Basis shapes
  • We need a collection of basis functions
    • First basis functions will be very smooth, slowly-varying
    • Last basis functions will be high-frequency, oscillating
  • We will represent our shape (mesh geometry) as a linear combination of the basis functions
  • We will induce the basis functions from the mesh connectivity!
    • We assume that the connectivity is known to the decoder side (it is transferred first)
the mesh laplacian operator
The Mesh Laplacian operator
  • Measures the local smoothness at each mesh vertex
laplacian matrix
Laplacian matrix
  • L is constructed based on the connectivity alone
  • So, if the decoder got the connectivity first, it can construct L
spectral bases
Spectral bases
  • L is a symmetric nn matrix
  • Perform spectral analysis of L!

T

1

L

2

=

n

Basis vectors

Frequencies,

sorted in ascending

order

how to represent our mesh geometry in the spectral basis
How to represent our mesh geometry in the spectral basis?
  • X, Y, Z Rn. Decompose in the spectral basis:
how to represent our mesh geometry in the spectral basis15
How to represent our mesh geometry in the spectral basis?
  • Decompose the mesh geometry in the spectral basis:

The first components are low-frequency

The last components are high-frequency

the spectral basis
The spectral basis
  • First functions are smooth and slow, last oscillate a lot

chain connectivity

horse connectivity

2nd basis function

10th basis function

100th basis function

spectral basis of L =

the DCT basis

why this works
Why this works?
  • Low-frequency basis vectors are smooth and slowly-varying:
    • The first basis vector:
    • It is the smoothest possible function (Laplacian is constant zero on it)
why this works18
Why this works?
  • Low-frequency basis vectors are smooth and slowly-varying:
    • The first basis vector:
    • Lb1 = 0 because all the rows of L sum to zero
why this works19
Why this works?
  • Low-frequency basis vectors are smooth and slowly-varying:
    • The following basis vectors:

The frequency is small  the Laplacian operator gives small values 

the shape of the function is smooth

why this works20
Why this works?
  • High-frequency basis vectors are non-smooth, oscillating:

The frequency is large  the Laplacian operator gives large values 

the shape of the function is less smooth!

geometry compression
Geometry compression

As in JPEG compression,

drop the high-frequency components!

geometry compression22
Geometry compression

Increasing number of participating

basis vectors.

The less coefficients are dropped, the less shape information is lost

technical algorithm
Technical algorithm
  • The encoding side:
    • Compute the spectral bases from mesh connectivity
    • Represent the shape geometry in the spectral basis and decide how many coeffs. to leave (K)
    • Store the connectivity and the K non-zero coefficients
  • The decoding side:
    • Compute the first K spectral bases from the connectivity
    • Combine them using the K received coefficients and get the shape
challenges and trade offs
Challenges and trade-offs
  • Performing spectral decomposition of a large matrix (n>1000) is prohibitively expensive (O(n3))
    • Today’s meshes come with 50,000 and more vertices
    • We don’t want the decompressor to work forever!
  • Solution: like in JPEG, work on small blocks:
challenges and trade offs25
Challenges and trade-offs
  • As in JPEG, blocking artifacts appear for low bit rates, because the compression is lossy (too much information is lost)
further info
Further info
  • “Spectral Mesh Compression”, Zachi Karni and Craig Gotsman, ACM SIGGRAPH 2000

http://www.mpi-inf.mpg.de/~karni/publications/spectral.pdf

  • “3D Mesh compression using fixed spectral bases”, Z. Karni and C. Gotsman, Graphics Interface, 2001

http://www.mpi-inf.mpg.de/~karni/publications/fixed.pdf