Axial Flip Invariance and Fast Exhaustive Searching with Wavelets

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Axial Flip Invariance and Fast Exhaustive
Searching with Wavelets

• Matthew Bolitho

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Outline

• Goals
• Shape Descriptors
• Invariance to rigid transformation
• Wavelets
• The wavelet transform
• Haar basis functions
• Axial ambiguity with wavelets
• Axial ambiguity Invariance
• Fast Exhaustive Searching

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Wavelet based Shape Descriptor

• Voxel based descriptor
• Rasterise model into voxel grid
• Apply Wavelet Transform
• Subset of information into feature vectors
• Compare vectors

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Shape Descriptor Goals

• Concise to store
• Quick to compute
• Efficient to match
• Discriminating
• Invariant to transformations
• Invariant to deformations
• Insensitive to noise
• Insensitive to topology
• Robust to degeneracies

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Project focus

• Invariance to transformation
• Efficient matching

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Scale, Translation, Rotation Invariance

• Invariance through normalisation
• Scale scale voxel grid such that is just fits
  the whole model
• Translation set the origin of voxel grid to be
  model center of mass
• Rotation Principal Component Analysis

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Principal Component Analysis

• Align model to a canonical frame
• Calculate variance of points
• Eigen-values of covariance matrix map to (x,y,z)
  axes in order of size

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Axial Ambiguity

• PCA has a problem
• Eigen-values are only defined up to sign
• In 3D, flip about x,y,z axes

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Resolving the Ambiguity

• Exhaustive search approach
• Compare all possible alignments (8 in 3D)
• Select alignment with minimal distance as best
  match
• An invariant approach make comparison invariant
  to axial flip

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The Wavelet Transform

• Transforms a function to a new basis Haar basis
  functions
• Invertible
• Non-Lossy

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Haar Basis Functions

• Family of step functions
• i specifies frequency family
• j indexes family
• Orthogonal
• Orthonormal when scaled by
• Fast to compute
• Compute in-place

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Constant Function
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Family i0
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Family i1
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Family i2
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Nomenclature

• Adopt a more convenient indexing scheme

i0
i1
i2
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Vector Basis

• Basis functions can also be represented as a set
  of orthonormal basis vectors
• Wavelet transform of function g is

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Example

• Given a function
• Wavelet transform is
• Aside given function

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Resolving Axial Ambiguity

• Exploit wavelets to get
• Axial Flip Invariance
• Make Wavelet Transform invariant to axial flip
• Fast Exhaustive Search
• Reduce the complexity of exhaustively testing all
  permutations of flip (recall 8 in 3D)

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Observation
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Observation
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Observation
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Observation
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Observation
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Wavelets and Axial Flip

• Established a mapping for axial flip
• f0 ? itself
• f1 ? inverse of itself
• Pairs ? inverse of each other

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Invariance

• Goal Discard information that determines flip
• Goal Not loose too much information
• Use mapping to make wavelet transform invariant
  to flip
• f0 is already invariant
• f1 is invariant
• Pairs are not, yet

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Invariance with pairs

• For a pair
• So, ab and a-b behave like f1 and f0 under axial
  flip
• Note when ab and a-b are known, a and b can be
  known no loss of information transform
  invertible

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Observation
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Observation
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A New Basis

• Redefine basis with a new mapping S( f )
• Now all coefficients either map to themselves ()
  or their inverse (-) under reflection

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Invariance

• New basis defines reflections with change in sign
  of half the coefficients
• Invariance
• Store f0, f3, f6, f7
• Store absolute value of f1, f2, f4, f5,

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Invariance Example

• Given g and h from previous example
• Perform wavelet transformTransform basis

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Invariance Evaluation

• Advantages
• Only perform single comparison
• Disadvantage
• Discards sign of half the coefficients
• ? may hurt ability to discriminate

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Exhaustive Searching

• Rather than making comparison invariant, perform
  it a number of times
• R is the set of all possible axial reflections
• Good Idea If possible reduce this comparison
  cost
• ? fast exhaustive searching

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Fast Exhaustive searching

• Distance between g and h, R(g) and h
• Recall gi , hi sign according to axial
  reflection

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Fast Exhaustive searching
Recall the mapping of R(gi) ? gi, thus
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Fast Exhaustive searching
Collect together terms to form
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Fast Exhaustive searching

• Now, we can express andonly in terms
  of gi and hi
• We can calculate both from the decomposition of
  the first, with minimal extra computation

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Fast Exhaustive search Example

• Given g and h from previous examples
• Transform basis

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Fast Exhaustive search Example

• Calculate gh and gh- from S(W(g)) and
  S(W(h))
• Calculate norms

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Fast search Evaluation

• For minimal extra computation, all permutations
  of flip can be compared
• No information is discarded
• c.f. invariance

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Higher Dimensions

• Both invariance and fast exhaustive search apply
  to higher dimensions
• As dimensionality increases, invariance needs to
  discard more and more information
• In 2D, 4 flips
• In 3D, 8 flips

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Applying Transforms in 2D

• Transform rows

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Applying Transforms in 2D

• Transform columns

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Exhaustive Searching in 2D

• In 1D we had gh and gh-
• In 2D we will have gh, gh-, gh- and gh--
• By applying both W(g) and S(g) in rows then
  columns, the 2D flip problem is reduced to two 1D
  flip problems
• This makes the cross multiplication easier

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Cross multiplication

• gh, gh-, gh- and gh--are determined by cross
  multiplying the grid
• gh
• etc

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Exhaustive Searching in 2D
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In 3D

• The extension into 3D is similar
• 8 flips
• 8 gh terms
• 8 ways to combine gh terms

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Conclusion

• Presented a way to overcome PCA alignment
  ambiguity
• With minimal extra computation
• With no loss of useful shape information

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Conclusion II

• PCA still has problems
• Instability Small change in PCA alignment can
  change voxel vote
• ? Gaussian smoothing can distribute votes better

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Future Work

• Integrate into complete shape descriptor
• Concise to store
• Quick to compute
• Discriminating
• Robustness
• etc
• Actual precision vs. recall results

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References

• 1 Misha Kazhdan Alignment slides, October 25
  2004
• 2 Original teapot image from http//www.plunk.or
  g/grantham/public/graphics.html