Axial Flip Invariance and Fast Exhaustive Searching with Wavelets

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