The document describes curve registration through nonparametric testing of Fourier coefficients. It presents a model where curves are represented by their Fourier coefficients with added noise. The hypotheses are that the curves either match up to a shift, or are minimally distant overall after any shift. A generalized likelihood ratio test statistic is developed that minimizes the squared differences between the Fourier coefficients of the two curves over all possible shift values.

1. Curve registration by minimax nonparametric testing
Curve registration by minimax nonparametric testing
Olivier Collier
IMAGINE, Université Paris-Est
CREST, ENSAE
30 janvier 2013

2. Curve registration by minimax nonparametric testing
Collier and Dalalyan. Curve registration by nonparametric
goodness-of-fit testing. Submitted.
Collier. Minimax hypothesis testing for curve registration,
Electron. J. Statist., 6:1129–1154, 2012.

3. Curve registration by minimax nonparametric testing
Introduction
1 Introduction
2 Model
3 Generalized likelihood ratio
4 Wilks’ phenomenon
5 Minimax considerations
6 Upper bound
7 Lower bound
8 Adaptation

4. Curve registration by minimax nonparametric testing
Introduction
Problem:

5. Curve registration by minimax nonparametric testing
Introduction
Problem:

6. Curve registration by minimax nonparametric testing
Introduction
Problem:
Question: How to detect matches ?

7. Curve registration by minimax nonparametric testing
Introduction
Solution:

8. Curve registration by minimax nonparametric testing
Introduction
Solution:
keypoint ⇒ descriptor

9. Curve registration by minimax nonparametric testing
Introduction
Solution:
keypoint ⇒ descriptor
We say two points match if they have the same descriptor.

10. Curve registration by minimax nonparametric testing
Introduction
Example: Descriptors of the main orientation of the local gradient

11. Curve registration by minimax nonparametric testing
Introduction
The descriptors should be:

12. Curve registration by minimax nonparametric testing
Introduction
The descriptors should be:
discriminating enough,

13. Curve registration by minimax nonparametric testing
Introduction
The descriptors should be:
discriminating enough,
invariant for some basic transformations.

14. Curve registration by minimax nonparametric testing
Introduction
Transformations to consider:

15. Curve registration by minimax nonparametric testing
Introduction
Transformations to consider:
translation,

16. Curve registration by minimax nonparametric testing
Introduction
Transformations to consider:
translation,
rotation,

17. Curve registration by minimax nonparametric testing
Introduction
Transformations to consider:
translation,
rotation,
scale change...

18. Curve registration by minimax nonparametric testing
Introduction
Famous example: SIFT

19. Curve registration by minimax nonparametric testing
Introduction
Famous example: SIFT
Compute the histogram of the local gradient with relation to
the angle.
−→

20. Curve registration by minimax nonparametric testing
Introduction
Invariance by translation: the position of the keypoint is not
taken into account.

21. Curve registration by minimax nonparametric testing
Introduction
Invariance by translation: the position of the keypoint is not
taken into account.
Invariance by rotation: the histogram is centered on the main
gradient orientation.
−→

22. Curve registration by minimax nonparametric testing
Introduction
New approach:

23. Curve registration by minimax nonparametric testing
Introduction
New approach:
Use the non-centered histogram, to avoid the computation of
the main orientation.

24. Curve registration by minimax nonparametric testing
Introduction
New approach:
Use the non-centered histogram, to avoid the computation of
the main orientation.
But a rotation of the image yields a translation of the
non-centered histogram.

25. Curve registration by minimax nonparametric testing
Introduction
New approach:
Use the non-centered histogram, to avoid the computation of
the main orientation.
But a rotation of the image yields a translation of the
non-centered histogram.
⇒ New matching criterion: Two keypoints match if their
descriptors are shifted from each other.

26. Curve registration by minimax nonparametric testing
Introduction
Consequence: We want to detect when
∃ τ ∈ [0, 2π], f (t) = g (t + τ ) ?

27. Curve registration by minimax nonparametric testing
Model
1 Introduction
2 Model
3 Generalized likelihood ratio
4 Wilks’ phenomenon
5 Minimax considerations
6 Upper bound
7 Lower bound
8 Adaptation

28. Curve registration by minimax nonparametric testing
Model
We state the model using Fourier coefficients:

29. Curve registration by minimax nonparametric testing
Model
We state the model using Fourier coefficients:
Xi = ci + σξi ,
Xi# = ci# + σξi# .

30. Curve registration by minimax nonparametric testing
Model
Xi = ci + σξi
Xi# = ci# + σξi#

31. Curve registration by minimax nonparametric testing
Model
Xi = ci + σξi
Xi# = ci# + σξi#
∗
H0 : ∃τ ∗ ∈ [0, 2π], ∀j ≥ 1, cj# = e ijτ cj ,

32. Curve registration by minimax nonparametric testing
Model
Xi = ci + σξi
Xi# = ci# + σξi#
∗
H0 : ∃τ ∗ ∈ [0, 2π], ∀j ≥ 1, cj# = e ijτ cj ,
H1 : d (c, c # ) minτ ∈[0,2π] +∞
j=1 |cj − e −ijτ cj# |2 ≥ ρ.

33. Curve registration by minimax nonparametric testing
Model
Hypotheses:

34. Curve registration by minimax nonparametric testing
Model
Hypotheses:
The noise level σ is known.

35. Curve registration by minimax nonparametric testing
Model
Hypotheses:
The noise level σ is known.
+∞ 2k 2
c, c # ∈ Fk,L {u, j=1 j |uj | ≤ L}.

36. Curve registration by minimax nonparametric testing
Model
Hypotheses:
The noise level σ is known.
+∞ 2k 2
c, c # ∈ Fk,L {u, j=1 j |uj | ≤ L}.
The regularity parameters k, L are known.

37. Curve registration by minimax nonparametric testing
Generalized likelihood ratio
1 Introduction
2 Model
3 Generalized likelihood ratio
4 Wilks’ phenomenon
5 Minimax considerations
6 Upper bound
7 Lower bound
8 Adaptation

38. Curve registration by minimax nonparametric testing
Generalized likelihood ratio
Standard likelihood ratio:

39. Curve registration by minimax nonparametric testing
Generalized likelihood ratio
Standard likelihood ratio:
The minimization of the negative log-likelihood in the general
set-up leads to

40. Curve registration by minimax nonparametric testing
Generalized likelihood ratio
Standard likelihood ratio:
The minimization of the negative log-likelihood in the general
set-up leads to
1 2
min X −u + λj j 2k |uj |2 + . . . ,
u 2σ 2
j≥1
where the λj are the Lagrange multipliers.

41. Curve registration by minimax nonparametric testing
Generalized likelihood ratio
Generalized likelihood ratio:
We replace

42. Curve registration by minimax nonparametric testing
Generalized likelihood ratio
Generalized likelihood ratio:
We replace
1 2
min X −u + λj j 2k |uj |2
u 2σ 2
j≥1

43. Curve registration by minimax nonparametric testing
Generalized likelihood ratio
Generalized likelihood ratio:
We replace
1 2
min X −u + λj j 2k |uj |2
u 2σ 2
j≥1
by
1 2
min X −u + ωj |uj |2 .
u 2σ 2
j≥1

44. Curve registration by minimax nonparametric testing
Generalized likelihood ratio
So we get the test statistic:
1
Tσ = min νj |Xj − e −ijτ Xj# |2 .
σ 2 τ ∈[0,2π]
j≥1
with νj = 1/(1 + ωj ).

45. Curve registration by minimax nonparametric testing
Generalized likelihood ratio
Possible weights:
νj = 1j≤N , (projection weights),

46. Curve registration by minimax nonparametric testing
Generalized likelihood ratio
Possible weights:
νj = 1j≤N , (projection weights),
−1
j
νj = 1 + N 1j≤N , (Tikhonov weights),

47. Curve registration by minimax nonparametric testing
Generalized likelihood ratio
Possible weights:
νj = 1j≤N , (projection weights),
−1
j
νj = 1 + N 1j≤N , (Tikhonov weights),
j
νj = 1 − N , (Pinsker weights).
+

48. Curve registration by minimax nonparametric testing
Wilks’ phenomenon
1 Introduction
2 Model
3 Generalized likelihood ratio
4 Wilks’ phenomenon
5 Minimax considerations
6 Upper bound
7 Lower bound
8 Adaptation

49. Curve registration by minimax nonparametric testing
Wilks’ phenomenon
Theorem
We assume that
|c1 | > 0 and c ∈ F1,L ,

50. Curve registration by minimax nonparametric testing
Wilks’ phenomenon
Theorem
We assume that
|c1 | > 0 and c ∈ F1,L ,
ν satisfies reasonable assumptions, i.e., ν 2 ≈ Nσ with
2
5/2
σ 2 Nσ log(Nσ ) → 0.

51. Curve registration by minimax nonparametric testing
Wilks’ phenomenon
Theorem
We assume that
|c1 | > 0 and c ∈ F1,L ,
ν satisfies reasonable assumptions, i.e., ν 2 ≈ Nσ with
2
5/2
σ 2 Nσ log(Nσ ) → 0.
Then, under H0 ,
Tσ − 4 ν 1 L
− N (0, 1).
→
4 ν 2

52. Curve registration by minimax nonparametric testing
Wilks’ phenomenon
Theorem (Wilks’ phenomenon)
The generalized likelihood ratio test
1{Tσ ≥4 ν 1 +4 ν 2 qα }
is asymptotically of level α and does not depend on the nuisance
parameters.

53. Curve registration by minimax nonparametric testing
Wilks’ phenomenon
Theorem
We assume that ν satisfies reasonable conditions and σ 4 Nσ → 0,
then, under H1 ,
P
Tσ − +∞.
→

54. Curve registration by minimax nonparametric testing
Minimax considerations
1 Introduction
2 Model
3 Generalized likelihood ratio
4 Wilks’ phenomenon
5 Minimax considerations
6 Upper bound
7 Lower bound
8 Adaptation

55. Curve registration by minimax nonparametric testing
Minimax considerations
We consider the sets

56. Curve registration by minimax nonparametric testing
Minimax considerations
We consider the sets

 Θ0 = {(c, c # ) ∈ Fs,L | d (c, c # ) = 0},


Θ1 = {(c, c # ) ∈ Fs,L | d (c, c # ) ≥ C ρσ },


57. Curve registration by minimax nonparametric testing
Minimax considerations
and the errors (for a test ψ)

 α(ψ, Θ0 ) = supΘ0 P(c,c # ) (ψ = 1),


β(ψ, Θ1 ) = supΘ1 P(c,c # ) (ψ = 0).


58. Curve registration by minimax nonparametric testing
Minimax considerations
Problem:
What is the smallest rate ρσ allowing to consistently decide
between Θ0 and Θ1 ?

59. Curve registration by minimax nonparametric testing
Upper bound
1 Introduction
2 Model
3 Generalized likelihood ratio
4 Wilks’ phenomenon
5 Minimax considerations
6 Upper bound
7 Lower bound
8 Adaptation

60. Curve registration by minimax nonparametric testing
Upper bound
We consider the tests
ψ(N, q) = 1{λσ (N)>q}
where
N √
1
λσ (N) = √ min |Xj − e −ijτ Xj# |2 − N.
4σ 2 N τ
j=1

61. Curve registration by minimax nonparametric testing
Upper bound
Theorem (Upper bound)
Let α be in (0, 1). Define ψσ,α = ψ(Nσ , qα ) with
 −1/s
 Nσ = [cs,L ρσ ],
 √ 2/4s+1
cs,L = 4sL2 4s + 1

qα the quantile of order 1 − α of the N (0, 1) distribution.

−2s 256cs,L 2s/4s+1
If C 2 > 4L2 cs,L + 4s+1 and ρσ = σ 2 log σ −1 , then
lim supσ→0 α(ψσ,α , Θ0 ) ≤ α
limσ→0 β(ψσ,α , Θ1 ) = 0.

62. Curve registration by minimax nonparametric testing
Lower bound
1 Introduction
2 Model
3 Generalized likelihood ratio
4 Wilks’ phenomenon
5 Minimax considerations
6 Upper bound
7 Lower bound
8 Adaptation

63. Curve registration by minimax nonparametric testing
Lower bound
Consider the general model
Xi = ci + σξi ,
Xi# = ci# + σξi# .
where we want to test
(c, c # ) ∈ Θ0
against (c, c # ) ∈ Θ1 .

64. Curve registration by minimax nonparametric testing
Lower bound
If c # = 0, we get the simpler model
Xi = ci + σξi ,
where we want to test
c ∈ Θclass = {0}
0
against c ∈ Θclass = {c ∈ Fs,L , c
1 2 ≥ C ρσ }.

65. Curve registration by minimax nonparametric testing
Lower bound
Theorem (Lower bound)
If ρσ σ 4s/4s+1 then consistent testing of H0 against H1 is
impossible.

66. Curve registration by minimax nonparametric testing
Adaptation
1 Introduction
2 Model
3 Generalized likelihood ratio
4 Wilks’ phenomenon
5 Minimax considerations
6 Upper bound
7 Lower bound
8 Adaptation

67. Curve registration by minimax nonparametric testing
Adaptation
Problem:
How to obtain similar performances when you do not know the
regularity parameter s and the radius L ?

68. Curve registration by minimax nonparametric testing
Adaptation
Assume that
(s, L) ∈ [s1 , s2 ] × [L1 , L2 ].

69. Curve registration by minimax nonparametric testing
Adaptation
Assume that
(s, L) ∈ [s1 , s2 ] × [L1 , L2 ].
We define

 Σ = {s = s1 + j

 log σ −1
| s1 ≤ s ≤ s2 },

N = (σ 2 log σ −1 )−2/4s+1 | s ∈ Σ .


70. Curve registration by minimax nonparametric testing
Adaptation
We denote
 √
 λσ (N) =

 4σ 2
1
√
N
minτ N
j=1 |Xj − e −ijτ Xj# |2 − N,
 ˜ √

 ψσ = maxN∈N 1 .
λ σ (N)> 2 log log σ −1

71. Curve registration by minimax nonparametric testing
Adaptation
Theorem
If ρσ (s) = (σ 2 log σ −1 )2s/4s+1 , then ∃C > 0 with

˜
 α(ψσ , Θ0 ) → 0,

sups,L β(ψσ , Θs,L ) → 0.
˜


1

72. Curve registration by minimax nonparametric testing
Adaptation
Thank you !