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  1. Axial Flip Invariance and Fast Exhaustive Searching with Wavelets Matthew Bolitho

  2. 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

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

  4. 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

  5. Project focus • Invariance to transformation • Efficient matching

  6. 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

  7. 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 [1]

  8. Axial Ambiguity • PCA has a problem • Eigen-values are only defined up to sign • In 3D, flip about x,y,z axes [1]

  9. 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

  10. The Wavelet Transform • Transforms a function to a new basis: Haar basis functions • Invertible • Non-Lossy [2]

  11. 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

  12. Constant Function

  13. Family i=0

  14. Family i=1

  15. Family i=2

  16. Nomenclature • Adopt a more convenient indexing scheme i=0 i=1 i=2

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

  18. Example • Given a function • Wavelet transform is • Aside: given function

  19. 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)

  20. Observation

  21. Observation

  22. Observation

  23. Observation

  24. Observation

  25. Wavelets and Axial Flip • Established a mapping for axial flip • f0 itself • f1  inverse of itself • Pairs  inverse of each other

  26. 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…

  27. 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

  28. Observation

  29. Observation

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

  31. 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, …

  32. Invariance Example • Given g and h from previous example Perform wavelet transform:Transform basis:

  33. Invariance Evaluation • Advantages • Only perform single comparison • Disadvantage • Discards sign of half the coefficients  may hurt ability to discriminate

  34. 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

  35. Fast Exhaustive searching • Distance between g and h, R(g) and h: Recall gi , hi : sign according to axial reflection

  36. Fast Exhaustive searching Recall the mapping of R(gi) gi, thus:

  37. Fast Exhaustive searching Collect together terms to form:

  38. 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

  39. Fast Exhaustive search Example • Given g and h from previous examples Transform basis:

  40. Fast Exhaustive search Example Calculate gh+ and gh- from S(W(g)) and S(W(h)): Calculate norms:

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

  42. 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

  43. Applying Transforms in 2D • Transform rows

  44. Applying Transforms in 2D • Transform columns

  45. 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

  46. Cross multiplication • gh++, gh+-, gh-+andgh--are determined by cross multiplying the grid • + * + = gh++ • etc

  47. Exhaustive Searching in 2D

  48. In 3D • The extension into 3D is similar: • 8 flips • 8 gh terms • 8 ways to combine gh terms

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

  50. Conclusion II • PCA still has problems • Instability: Small change in PCA alignment can change voxel vote  Gaussian smoothing can distribute votes better