Alignment

1
Alignment

• Introduction

Notes courtesy of Funk et al., SIGGRAPH 2004
2

• Outline
• Challenge
• General Approaches
• Specific Examples

3
Alignment

• Challenge
• The shape of a model does not change when acted
  on by similarity transformations

4
Alignment

• Challenge
• However, the shape descriptors can change if a
  the model is
• Translated
• Scaled
• Rotated
• How do we match shape descriptors across
  transformations that do not change the shape of a
  model?

5

• Outline
• Challenge
• General Approaches
• Specific Examples

6
Alignment

• Approaches
• Given the shape descriptors of two models, find
  the transformation(s) -- translation, scale, and
  rotation -- that minimize the distance between
  the two models
• Exhaustive search
• Closed form solution
• Minimization
• Normalization
• Invariance

7
Alignment

• Aside
• Because translations and rotations preserve
  distances, applying such a transformation to one
  model is equivalent to applying the inverse
  transformation to the other one

8
Alignment

• Aside
• For translations and rotations we can simplify
  the alignment equation

9
Exhaustive Search

• Approach
• Compare the descriptors at all possible
  transformations.
• Find the transformation at which the distance is
  minimal.
• Define the model similarity as the value at the
  minimum.

10
Exhaustive Search

• Approach
• Compare the descriptors at all possible
  transformations.
• Find the transformation at which the distance is
  minimal.
• Define the model similarity as the value at the
  minimum.

Exhaustive search for optimal rotation
11
Exhaustive Search

• Approach
• Compare the descriptors at all possible
  transformations.
• Find the transformation at which the distance is
  minimal.
• Define the model similarity as the value at the
  minimum.

12
Exhaustive Search

• Properties
• Always gives the correct answer
• Needs to be performed at run-time and can be very
  slow to compute
• Computes the measure of similarity for every
  transform. We only need the value at the best one.

13
Closed Form Solution

• Approach
• Explicitly find the transformation(s) that solves
  the equation
• Properties
• Always gives the correct answer
• Only compute the measure of similarity for the
  best transformation.
• A closed form solution does not always exist.
• Often needs to be computed at run-time.

14
Minimization

• Approach
• Coarsely align the models using low frequency
  information.
• Progressively refine the alignment by comparing
  higher frequency components and adjusting the
  alignment.
• Converge to the (locally) optimal alignment.
• Example Light field descriptors

Spherical Extent Function
15
Minimization

• Approach
• Coarsely align the models using low frequency
  information.
• Progressively refine the alignment by comparing
  higher frequency components and adjusting the
  alignment.
• Converge to the (locally) optimal alignment.

Initial Models
Low Frequency
Aligned Models
16
Minimization

• Approach
• Coarsely align the models using low frequency
  information.
• Progressively refine the alignment by comparing
  higher frequency components and adjusting the
  alignment.
• Converge to the (locally) optimal alignment.

Initial Models
Low Frequency
Aligned Models
High Frequency
17
Minimization

• Properties
• Can be applied to any type of transformation
• Needs to be computed at run-time.
• Difficult to do robustly
• Given the low frequency alignment and the
  computed high-frequency alignment, how do you
  combine the two?
• Considerations can include
• Relative size of high and low frequency info
• Distribution of info across the low frequencies
• Speed of oscillation

18
Normalization

• Approach
• Place every model into a canonical coordinate
  frame and assume that two models are optimally
  aligned when each is in its own canonical frame.
• Example COM, Max Radius, PCA

19
Normalization

• Properties
• Can be computed in a pre-processing stage.
• For some transformations this is guaranteed to
  give the optimal alignment.
• For other transformations the approach is only a
  heuristic and may fail.

Failure of PCA-normalization in aligning rotations
20
Invariance

• Approach
• Represent every model by a descriptor that is
  unchanged when the model is transformed by
  discarding information that is transformation
  dependent.

Transformation-invariant descriptor
21
Invariance

• Review
• Is there a general method for addressing these
  basic types of transformations?

Descriptor Translation Scale Rotation
Shape Distributions (D2) -
Extended Gaussian Images - -
Shape Histograms (Shells) - -
Shape Histograms (Sectors) - -
Spherical Parameterizations - -
22
Invariance

• Properties
• Can be computed in a pre-processing stage.
• Works for translation, scale and rotation.
• Gives a more compact representation.
• Tends to discard valuable, discriminating,
  information.

.....No Invariance ..Rotation Translation
Rotation ...Rotation
23

• Outline
• Challenge
• General Approaches
• Specific Examples
• Normalization PCA
• Closed Form Solution Ordered Point Sets

24
PCA Alignment

• Treat a surface as a collection of points and
  define the variance function

25
PCA Alignment

• Define the covariance matrix M
• Find the eigen-values and align so that the
  eigen-values map to the x-, y-, and z-axes

26
PCA Alignment

• Limitations
• Eigen-values are only defined up to sign!PCA
  alignment is only well-defined up to axial flips
  about the x-, y-, and z-axes.

27
PCA Alignment

• Limitations
• Assumes that the eigen-values are distinct and
  therefore the eigen-vectors are well-defined (up
  to sign).
• This is not always true and can make PCA
  alignment unstable.

28

• Outline
• Challenge
• General Approaches
• Specific Examples
• Normalization PCA
• Closed Form Solution Ordered Point Sets

29
Ordered Point Sets

• Challenge
• Given ordered point sets Pp1,,pn,
  Qq1,,qn, find the rotation/reflection R
  minimizing the sum of squared differences

q4
q4
p2
R(q2)
p1
p3
R(q1)
R(q3)
R
q5
q5
q3
q3
q6
q6
q2
q2
p4
p6
R(q6)
R(q4)
p5
R(q5)
q1
q1
30
Review

• Vector dot-product
• If v (v1,,vn) and w(w1,,wn) are two
  n-dimensional vectors the dot-product of v with w
  is the sum of the product of the coefficients

31
Review

• Trace
• The trace of a nxn matrix M is the sum of the
  diagonal entries of M
• Properties

32
Review

• Trace
• If M is any nxn matrix and D is a diagonal nxn
  matrix, then the trace of MD is the sum of the
  products of the diagonal entries.

M
D
33
Review

• Matrix multiplication
• If M and N are two then the (i,j)th entry of the
  matrix MN is the dot-product of the jth row
  vector of M with the ith column vector of N.

M
MN
N
jth row
ith column
(i,j)th entry
34
Review

• Matrix dot-product
• If M and N are two mxn matrices then the ith
  diagonal entry of the matrix MtN is the
  dot-product of the ith column vector of M with
  the ith column vector of N.

m
n
Mt
MtN
N
n
m
ith row
ith column
ith diagonal entry
35
Review

• Matrix dot-product
• We define the dot-product of two mxn matrices, M
  and N, to be the trace of the matrix product
• (the sum of the dot-products of the column
  vectors).

36
Review

• SVD Factorization
• If M is an mxm matrix, then M can be factored as
  the product
• where D is a diagonal mxm matrix with
  non-negative entries and U and V are orthonormal
  (i.e. rotation/reflection) mxm matrices.

37
Trace Maximization

• Claim
• If M is an mxm matrices, whose SVD factorization
  is
• then the orthonormal transformation RVUt is the
  orthonormal transformation maximizing the trace

38
Trace Maximization

• Proof
• We can rewrite the trace equation as
• If we set R0 to be the rotation R0VtRU, we get

39
Trace Maximization

• Proof
• Since R0 is a rotation, each of its entries can
  have value no larger than one.
• Since D is diagonal, the value of the trace is
  the product of the diagonal elements of D and R0.

40
Trace Maximization

• Proof
• To maximize the trace we want R0 to be maximal on
  the diagonal (i.e. have only 1s).
• Thus, R0 is the identity matrix and we have
• So that the rotation/reflection R that maximizes
  Trace(RM) is

41
Ordered Point Sets

• Challenge
• Given ordered point sets Pp1,,pn,
  Qq1,,qn, find the rotation/reflection R
  minimizing the sum of squared differences

42
Ordered Point Sets

• Solution
• Use the fact that we can express the difference
• to rewrite the equation as

43
Ordered Point Sets

• Solution
• Use the fact that rotations preserves lengths
• to rewrite the equation as

44
Ordered Point Sets

• Solution
• Since the value
• Does not depend on the choice of rotation R,
  minimizing the sum of squared distances is
  equivalent to maximizing the sum of dot-products

45
Ordered Point Sets

• Solution
• If we let MP (respectively MQ) be the 3xn matrix
  whose columns are the points pi (respectively qi)
  then we can rewrite the sum of the vector
  dot-products as the matrix dot product
• and we can find the maximizing rotation/reflection
  R by trace maximization.