Axial flip invariance and fast exhaustive searching with wavelets
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Axial Flip Invariance and Fast Exhaustive Searching with Wavelets. Matthew Bolitho. 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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Axial Flip Invariance and Fast Exhaustive Searching with Wavelets

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Axial Flip Invariance and Fast Exhaustive Searching with Wavelets

Matthew Bolitho


  • Goals

  • Shape Descriptors

    • Invariance to rigid transformation

  • Wavelets

    • The wavelet transform

    • Haar basis functions

    • Axial ambiguity with wavelets

  • Axial ambiguity Invariance

  • Fast Exhaustive Searching

Wavelet based Shape Descriptor

  • Voxel based descriptor

    • Rasterise model into voxel grid

    • Apply Wavelet Transform

    • Subset of information into feature vectors

    • Compare vectors

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

Project focus

  • Invariance to transformation

  • Efficient matching

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

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


Axial Ambiguity

  • PCA has a problem

  • Eigen-values are only defined up to sign

  • In 3D, flip about x,y,z axes


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

The Wavelet Transform

  • Transforms a function to a new basis: Haar basis functions

  • Invertible

  • Non-Lossy


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

Constant Function

Family i=0

Family i=1

Family i=2


  • Adopt a more convenient indexing scheme




Vector Basis

  • Basis functions can also be represented as a set of orthonormal basis vectors:

  • Wavelet transform of function g is:


  • Given a function

  • Wavelet transform is

  • Aside: given function

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)






Wavelets and Axial Flip

  • Established a mapping for axial flip

    • f0 itself

    • f1  inverse of itself

    • Pairs  inverse of each other


  • 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…

Invariance with pairs

  • For a pair

  • So, a+b and a-b behave like f1 and f0 under axial flip

  • Note: when a+b and a-b are known, a and b can be known – no loss of information; transform invertible



A New Basis

  • Redefine basis with a new mapping S( f )

  • Now all coefficients either map to themselves (+) or their inverse (-) under reflection


  • 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, …

Invariance Example

  • Given g and h from previous example

    Perform wavelet transform:Transform basis:

Invariance Evaluation

  • Advantages

    • Only perform single comparison

  • Disadvantage

    • Discards sign of half the coefficients

       may hurt ability to discriminate

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

Fast Exhaustive searching

  • Distance between g and h, R(g) and h:

    Recall gi , hi : sign according to axial reflection

Fast Exhaustive searching

Recall the mapping of R(gi) gi, thus:

Fast Exhaustive searching

Collect together terms to form:

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

Fast Exhaustive search Example

  • Given g and h from previous examples

    Transform basis:

Fast Exhaustive search Example

Calculate gh+ and gh- from S(W(g)) and S(W(h)):

Calculate norms:

Fast search Evaluation

  • For minimal extra computation, all permutations of flip can be compared

  • No information is discarded

    • c.f. invariance

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

Applying Transforms in 2D

  • Transform rows

Applying Transforms in 2D

  • Transform columns

Exhaustive Searching in 2D

  • In 1D we had gh+ and gh-

  • In 2D we will have gh++, gh+-, gh-+andgh--

  • 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

Cross multiplication

  • gh++, gh+-, gh-+andgh--are determined by cross multiplying the grid

    • + * + = gh++

    • etc

Exhaustive Searching in 2D

In 3D

  • The extension into 3D is similar:

    • 8 flips

    • 8 gh terms

    • 8 ways to combine gh terms


  • Presented a way to overcome PCA alignment ambiguity

    • With minimal extra computation

    • With no loss of useful shape information

Conclusion II

  • PCA still has problems

    • Instability: Small change in PCA alignment can change voxel vote

       Gaussian smoothing can distribute votes better

Future Work

  • Integrate into complete shape descriptor

    • Concise to store

    • Quick to compute

    • Discriminating

    • Robustness

    • etc

  • Actual precision vs. recall results


  • [1] Misha Kazhdan: Alignment slides, October 25 2004

  • [2] Original teapot image from

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