Partial Shape Matching

1
Partial Shape Matching
2

• Outline
• Motivation
• Sum of Squared Distances

3
Representation Theory

• Motivation
• We have seen a large number of different shape
  descriptors
• Shape Distributions
• Extended Gaussian Images
• Shape Histograms
• Gaussian EDT
• Wavelets
• Spherical Parameterizations
• Spherical Extent Functions
• Light Field Descriptors
• Shock Graphs
• Reeb Graphs

4
Representation Theory

• Challenge
• Partial shape matching problem
• Given a part of a model S and a whole model M
  determine if the part is a subset of the whole
  S?M.

S
M
5
Representation Theory

• Difficulty
• For whole object matching, we would associate a
  shape descriptor vM to every model M and would
  define the measure of similarity between models M
  and N as the distance between their descriptors
• D(M,N)vM-vN

6
Representation Theory

• Difficulty
• For partial object matching, we cannot use the
  same approach
• Vector differences are symmetric but subset
  matching is not
• If S?M, then we would like to have vS?vM
  andwhich means that we cannot use difference
  norms to measure similarity.

7
Representation Theory

• Motivation
• We have seen a number of different ways for
  addressing the alignment problem
• Center of Mass Normalization
• Scale Normalization
• PCA Alignment
• Translation Invariance
• Rotation Invariance

8
Representation Theory

• Motivation
• Most of these methods will change give very
  different descriptors if only part of the model
  is given.
• Center of mass, variance, and principal axes of a
  part of the model will not be the same as those
  of the whole.

9
Representation Theory

• Motivation
• Most of these methods will change give very
  different descriptors if only part of the model
  is given.
• Changing the values of a function will change the
  (non-constant) frequency distribution in
  non-trivial ways.

10

• Outline
• Motivation
• Sum of Squared Distances

11
Representation Theory

• Goal
• Design a new paradigm for shape matching that
  associates a simple structure to each shape M?vM
  such that if S?M, then
• vS?vM (unless SM)
• but D(S,M)0
• That is, we would like to define a measure of
  similarity that answers the questionHow close
  is S to being a subset of M?

12
Representation Theory

• Key Idea
• Instead of using the norm of the difference, use
  the dot product
• Then, S is a subset of M if is orthogonal to
  .
• To do this, we have to define different
  descriptors for a model depending on whether it
  is the target or the query.

13
Representation Theory

• Implementation
• For a model M, represent the model by two
  different 3D function

M
RasterM
EDTM
14
Representation Theory

• Implementation
• Then RasterM is non-zero only on the boundary
  points of the model, and EDTM is non-zero
  everywhere else. Consequently we haveand
  hence

M
RasterM
EDTM
15
Representation Theory

• Implementation
• Moreover, if S?M, then we still haveso that

S
RasterS
M
EDTM
16
Representation Theory

• What is the value of D(S,M)?

17
Representation Theory

• What is the value of D(S,M)?

18
Representation Theory

• What is the value of D(S,M)?
• Since RasterS is equal to 1 for points that lie
  on S and equal to 0 everywhere else

19
Representation Theory

• What is the value of D(S,M)?
• So that distance between S and M is equal to the
  sum of squared distances from points on S to the
  nearest point M.

S
M
20
Representation Theory

• What is the value of D(S,M)?
• Note that if we rasterize the models into an
  nxnxn voxel grid, then a brute force computation
  would compute the sum of the distances for each
  of O(n2) on the query by testing against each of
  O(n2) points on the target for the minimum
  distance, giving a total running time of O(n4).
  By pre-computing the EDT, we reduce the
  computation to O(n2) operations.

21
Representation Theory

• Advantages
• Model similarity is defined in terms of the
  dot-product
• We can still use SVD for efficiency and
  compression (since rotations do not change the
  dot product)
• We can still use fast correlation methods
  (translation, rotation, axial flip) but now we
  want to find the transformation minimizing the
  correlation.

22
Representation Theory

• Advantages
• We can use a symmetric version of this for whole
  object matching.

23
Representation Theory

• Advantages
• We can perform importance matching by assigning a
  value larger than 1 to sub-regions of the
  rasterization.

24
Representation Theory

• Disadvantage
• Aside from using fast Fourier /
  Spherical-Harmonic / Wigner-D transforms, we
  still have no good way to address the alignment
  problem.