Hölder regularity estimation by Hart Smith and Curvelet transforms

Transcription

1 Hölder regularity estimation by Hart Smith and Curvelet transforms Jouni Sampo Lappeenranta University Of Technology Department of Mathematics and Physics Finland 18th September 2007

2 This research is done in collaboration with Dr. Songkiat Sumetkijakan (Chulalongkorn University, Department of Mathematics, Bangkok, Thailand)

3 Outline Basic definitions 1 Basic definitions Hölder regularities Vanishing moments Hart Smith Transform Continuous Curvelet Transform 2 Conditions for kernel functions Uniform regularity Pointwise Regularity Directional Regularity properties 3 4

4 Hölder regularities Vanishing moments Hart Smith Transform Continuous Curvelet Transform Uniform and Pointwice Hölder Regularity Definition Let α > 0 and α / N. A function f : R d R is said to be pointwise Hölder regular with exponent α at u, denoted by f C α (u), if there exists a polynomial P u of degree less than α and a constant C u such that for all x in a neighborhood of u f (x) P u (x u) C u x u α. (1) Let Ω be an open subset of R d. If (1) holds for all x, u Ω with C u being a uniform constant independent of u, then we say that f is uniformly Hölder regular with exponent α on Ω or f C α (Ω).

5 Hölder regularities Vanishing moments Hart Smith Transform Continuous Curvelet Transform Uniform and Pointwise Hölder Exponents The uniform and pointwise Hölder exponents of f on Ω and at u are defined as α l (Ω) := sup{α: f C α (Ω)} and α p (u) := sup{α: f C α (u)}.

6 Local Hölder exponent Hölder regularities Vanishing moments Hart Smith Transform Continuous Curvelet Transform Definition Let (I i ) i N be a family of nested open sets in R d, i.e. I i+1 I i, with intersection i I i = {u}. The local Hölder exponent of a function f at u, denoted by α l (u), is α l (u) = lim i α l (I i ). In many situations, local and pointwise Hölder exponents coincide, e.g., if f (x) = x γ then α p (0) = α l (0) = γ. However, local Hölder exponents α l (u) is also sensitive to oscillating behavior of f near ( the point u. A simple example is f (x) = x γ sin 1/ x β) for which α p (0) = γ but α l (0) = γ 1+β, i.e., α l is influenced by the wild oscillatory behavior of f near 0.

7 Directional Hölder regularity Hölder regularities Vanishing moments Hart Smith Transform Continuous Curvelet Transform Definition Let v R d be a fixed unit vector and u R d. A function f : R d R is pointwise Hölder regular with exponent α at u in the direction v, denoted by f C α (u; v), if there exist a constant C u,v and a polynomial P u,v of degree less than α such that f (u + λv) P u,v (λ) C u,v λ α holds for all λ in a neighborhood of 0 R. If one can choose C u,v so that it is independent of u for all u Ω R d and the inequality holds for all λ R such that u + λv Ω, then we say that f is uniformly Hölder regular with exponent α on Ω in direction v or f C α (Ω; v).

8 Directional Vanishing Moments Hölder regularities Vanishing moments Hart Smith Transform Continuous Curvelet Transform Definition A function f of two variables is said to have an L-order directional vanishing moments along a direction v = (v 1, v 2 ) T (suppose that v 1 0; if v 1 = 0 then v 2 0 and we can swap the two dimensions) if t n f (t, tv 2 /v 1 c)dt = 0, c R, 0 n L. R Essentially, the above definition means that any 1-D slices of the function have vanishing moments of order L.

9 Hölder regularities Vanishing moments Hart Smith Transform Continuous Curvelet Transform Building Function with Directional Vanishing Moments At spatial domain the design is challenging if vanishing moment in many directions are needed In frequency domain the design is relatively easy If ˆf (n) (ω 1, ω 2 ) vanish at line ω 2 = v 1 /v 2 ω 1 for all n = 0,..., L then f have L-order directional vanishing moments along a direction v = (v 1, v 2 ) T

10 Hölder regularities Vanishing moments Hart Smith Transform Continuous Curvelet Transform Building Function with Directional Vanishing Moments At spatial domain the design is challenging if vanishing moment in many directions are needed In frequency domain the design is relatively easy If ˆf (n) (ω 1, ω 2 ) vanish at line ω 2 = v 1 /v 2 ω 1 for all n = 0,..., L then f have L-order directional vanishing moments along a direction v = (v 1, v 2 ) T

11 Kernel of Hart Smith Transform Hölder regularities Vanishing moments Hart Smith Transform Continuous Curvelet Transform For a given ϕ L 2 (R 2 ), we define ) ϕ abθ (x) = a 3 4 ϕ (D 1 R θ (x b), a for θ [0, 2π), b R 2 R θ is the matrix affecting planar rotation of θ radians in clockwise direction. 0 < a < a 0, where a 0 is a fixed coarsest scale ( ) = diag 1 a, a 1 D 1 a

12 Kernel of Hart Smith Transform Hölder regularities Vanishing moments Hart Smith Transform Continuous Curvelet Transform Width and length of essential support of kernel function ϕ abθ (x) are about a and a 1/2.

13 Kernel of Hart Smith Transform Hölder regularities Vanishing moments Hart Smith Transform Continuous Curvelet Transform Essential support of kernel function ϕ abθ (x) become like a needle when scale a become smaller.

14 Hölder regularities Vanishing moments Hart Smith Transform Continuous Curvelet Transform Reconstruction Formula for Hart Smith Transform Theorem There exists a Fourier multiplier M of order 0 so that whenever f L 2 (R 2 ) is a high-frequency function supported in frequency space ξ > 2 a 0, then f = a0 2π 0 0 R 2 ϕ abθ, Mf ϕ abθ db dθ da a 3 in L 2 (R 2 ). (2) The function Mf is defined in the frequency domain by a multiplier formula Mf (ξ) = m( ξ )ˆf (ξ), where m is a standard Fourier multiplier of order 0 (that is, for each k 0, there is a constant C k such that for all t R, m (k) (t) C k ( 1 + t 2 ) k/2 ).

15 Hölder regularities Vanishing moments Hart Smith Transform Continuous Curvelet Transform Reconstruction Formula for Hart Smith Transform Because of this and the fact that ϕ abθ and Mϕ abθ are duals, we can write reconstruction formula also as a0 2π f = Mϕ abθ, f ϕ abθ db dθ da 0 0 R 2 a 3 a0 2π = ϕ abθ, f Mϕ abθ db dθ da 0 0 R 2 a 3. Unlike ϕ abθ, the dual Mϕ abθ do not satisfy true parabolic dilation

16 Hölder regularities Vanishing moments Hart Smith Transform Continuous Curvelet Transform Curvelets are defined in Fourier Domain ( Let W be a positive real-valued function supported inside 1 2, 2), called a radial window, and let V be a real-valued function supported on [ 1, 1], called an angular window, for which the following admissibility conditions hold: 0 W (r) 2 dr r = 1 and 1 At each scale a, 0 < a < a 0, γ a00 is defined by γ a00 (r cos(ω), r sin(ω)) = a 3 4 W (ar) V ( ω/ a ) 1 V (ω) 2 dω = 1. (3) for r 0 and ω [0, 2 For each 0 < a < a 0, b R 2, and θ [0, 2π), a curvelet γ abθ is defined by γ abθ (x) = γ a00 (R θ (x b)), for x R 2. (4)

17 Reconstruction with Curvelets Hölder regularities Vanishing moments Hart Smith Transform Continuous Curvelet Transform Theorem There exists a bandlimited purely radial function Φ such that for all f L 2 (R 2 ), f = Φ b, f Φ b db+ R 2 where Φ b (x) = Φ(x b). a0 2π 0 0 R 2 γ abθ, f γ abθ db dθ da a 3 in L 2, (5)

18 Conditions for kernel functions Uniform regularity Pointwise Regularity Directional Regularity properties Extra conditions for kernel functions For regularity analysis we will need that Kernel functions have enough directional vanishing moments Kernel functions and their derivatives up to desired order (largest α of interest) decay fast enough.

19 Conditions for kernel functions Uniform regularity Pointwise Regularity Directional Regularity properties Vanishing moments of kernel function Lemma There exists C < (independent of a, b and θ) such that curvelet functions γ abθ have directional vanishing moments of any order L < along all directions v that satisfy (v θ, v) Ca 1/2. Moreover if there exists finite and strictly positive constants C 1, C 1 and C 2 such that supp( ˆϕ) [C 1, C 1 ] [ C 2, C 2 ], then above is true also for functions ϕ abθ and Mϕ abθ.

20 Conditions for kernel functions Uniform regularity Pointwise Regularity Directional Regularity properties Vanishing moments of kernel function Frequency support of ϕ abθ (x).

21 Decay of kernel function Conditions for kernel functions Uniform regularity Pointwise Regularity Directional Regularity properties Lemma Suppose that the windows V and W in the definition of CCT are C and have compact supports. Then for each N = 1, 2,... there is a constant C N such that x R 2 ν γ abθ (x) C Na 3/4 ν 1 + x b 2N. (6) a,θ Moreover, if ˆϕ C and if there exist finite and strictly positive constants C 1, C 1, and C 2 such that supp( ˆϕ) [C 1, C 1 ] [ C 2, C 2 ], then (6) also holds for functions ϕ abθ and Mϕ abθ.

22 Conditions for kernel functions Uniform regularity Pointwise Regularity Directional Regularity properties Necessary Condition for Uniform Regularity Theorem If a bounded function f C α (R 2 ), then there exist a constant C and a fixed coarsest scale a 0 for which φ abθ, f Ca α+ 3 4 for all 0 < a < a 0, b R 2, and θ [0, 2π).

23 Conditions for kernel functions Uniform regularity Pointwise Regularity Directional Regularity properties Sufficient Condition for Uniform Regularity Theorem Let f L 2 (R 2 ) and α > 0 a non-integer. If there is a constant C < such that φ abθ, f Ca α+ 5 4, for all 0 < a < a 0, b R 2, and θ [0, 2π), then f C α (R 2 ).

24 Conditions for kernel functions Uniform regularity Pointwise Regularity Directional Regularity properties Necessary Condition for Pointwise Regularity Theorem If a bounded function f C α (u) then there exists C < such that ( φ abθ, f Ca α b u α) a 1/2 (7) for all 0 < a < a 0, b R 2, and θ [0, 2π).

25 Conditions for kernel functions Uniform regularity Pointwise Regularity Directional Regularity properties Sufficient Condition for Pointwise Regularity Theorem Let f L 2 (R 2 ) and α be a non-integer positive number. If there exist C < and α < 2α such that ( φ abθ, f Ca α b u α ) a 1/2, (8) for all 0 < a < a 0, b R 2, and θ [0, 2π), then f C α (u).

26 Conditions for kernel functions Uniform regularity Pointwise Regularity Directional Regularity properties Necessary Conditions for direction of Singularity Line Now we consider situation that background is sufficiently smooth, i.e. regularity to one direction is higher. Theorem Let f be bounded with local Hölder exponent α (0, 1] at point u and f C 2α+1+ε (R 2, v θ0 ) for some θ 0 [0, 2π) with any fixed ε > 0. Then there exist α [α ε, α] and C < such that for a > 0 and b R 2, Ca α+ 5 4, ( if θ / θ 0 + Ca 1/2 [ 1, 1], φ abθ, f Ca α b u α ), if θ θ 0 + Ca 1/2 [ 1, 1]. a

27 Conditions for kernel functions Uniform regularity Pointwise Regularity Directional Regularity properties Sufficient Conditions for direction of Singularity Line Theorem Let f L 2 (R 2 ), u R 2, and assume that α > 0 is not an integer. If there exist α < 2α, θ 0 [0, 2π], and C < such that ( Ca α b u α ) a 1/2, if θ / θ 0 + Ca 1/2 [ 1, 1] φ abθ, f ( Ca α b u α ) a 1/2, if θ θ 0 + Ca 1/2 [ 1, 1] for all 0 < a < a 0, b R 2, and θ [0, 2π), then f C α (u).

28 Figure: Decay behavior of φ a0θ, f across scales a at various angles θ for the function f (x) = e x x

29 Figure: Estimation errors of the Hölder exponents by α e (s, θ) at scales 2 s and angles θ for the function f (x) = e x x

30 Generalizations Basic definitions Assumption α < 1 can be removed from all theorems Everything holds also for discrete curvelet transform Assumptions can be relaxed to hold only at ball of radius ε. Assumptions about Fourier properties of kernel functions can be relaxed Real valued kernel functions if support include reflection respect origin Compact support on Fourier domain may not be necessary Theorems could work for contourlets also With some extra assumptions of background regularity Necessary and sufficient conditions would be the same up to ε. Similar estimates for C 2 curve (now only for line singularity) Generalization from R 2 to R d

31 Generalizations Basic definitions Assumption α < 1 can be removed from all theorems Everything holds also for discrete curvelet transform Assumptions can be relaxed to hold only at ball of radius ε. Assumptions about Fourier properties of kernel functions can be relaxed Real valued kernel functions if support include reflection respect origin Compact support on Fourier domain may not be necessary Theorems could work for contourlets also With some extra assumptions of background regularity Necessary and sufficient conditions would be the same up to ε. Similar estimates for C 2 curve (now only for line singularity) Generalization from R 2 to R d

32 Generalizations Basic definitions Assumption α < 1 can be removed from all theorems Everything holds also for discrete curvelet transform Assumptions can be relaxed to hold only at ball of radius ε. Assumptions about Fourier properties of kernel functions can be relaxed Real valued kernel functions if support include reflection respect origin Compact support on Fourier domain may not be necessary Theorems could work for contourlets also With some extra assumptions of background regularity Necessary and sufficient conditions would be the same up to ε. Similar estimates for C 2 curve (now only for line singularity) Generalization from R 2 to R d

33 Generalizations Basic definitions Assumption α < 1 can be removed from all theorems Everything holds also for discrete curvelet transform Assumptions can be relaxed to hold only at ball of radius ε. Assumptions about Fourier properties of kernel functions can be relaxed Real valued kernel functions if support include reflection respect origin Compact support on Fourier domain may not be necessary Theorems could work for contourlets also With some extra assumptions of background regularity Necessary and sufficient conditions would be the same up to ε. Similar estimates for C 2 curve (now only for line singularity) Generalization from R 2 to R d

34 Generalizations Basic definitions Assumption α < 1 can be removed from all theorems Everything holds also for discrete curvelet transform Assumptions can be relaxed to hold only at ball of radius ε. Assumptions about Fourier properties of kernel functions can be relaxed Real valued kernel functions if support include reflection respect origin Compact support on Fourier domain may not be necessary Theorems could work for contourlets also With some extra assumptions of background regularity Necessary and sufficient conditions would be the same up to ε. Similar estimates for C 2 curve (now only for line singularity) Generalization from R 2 to R d

35 Generalizations Basic definitions Assumption α < 1 can be removed from all theorems Everything holds also for discrete curvelet transform Assumptions can be relaxed to hold only at ball of radius ε. Assumptions about Fourier properties of kernel functions can be relaxed Real valued kernel functions if support include reflection respect origin Compact support on Fourier domain may not be necessary Theorems could work for contourlets also With some extra assumptions of background regularity Necessary and sufficient conditions would be the same up to ε. Similar estimates for C 2 curve (now only for line singularity) Generalization from R 2 to R d

36 Generalizations Basic definitions Assumption α < 1 can be removed from all theorems Everything holds also for discrete curvelet transform Assumptions can be relaxed to hold only at ball of radius ε. Assumptions about Fourier properties of kernel functions can be relaxed Real valued kernel functions if support include reflection respect origin Compact support on Fourier domain may not be necessary Theorems could work for contourlets also With some extra assumptions of background regularity Necessary and sufficient conditions would be the same up to ε. Similar estimates for C 2 curve (now only for line singularity) Generalization from R 2 to R d

37 Generalizations Basic definitions Assumption α < 1 can be removed from all theorems Everything holds also for discrete curvelet transform Assumptions can be relaxed to hold only at ball of radius ε. Assumptions about Fourier properties of kernel functions can be relaxed Real valued kernel functions if support include reflection respect origin Compact support on Fourier domain may not be necessary Theorems could work for contourlets also With some extra assumptions of background regularity Necessary and sufficient conditions would be the same up to ε. Similar estimates for C 2 curve (now only for line singularity) Generalization from R 2 to R d

38 Generalizations Basic definitions Assumption α < 1 can be removed from all theorems Everything holds also for discrete curvelet transform Assumptions can be relaxed to hold only at ball of radius ε. Assumptions about Fourier properties of kernel functions can be relaxed Real valued kernel functions if support include reflection respect origin Compact support on Fourier domain may not be necessary Theorems could work for contourlets also With some extra assumptions of background regularity Necessary and sufficient conditions would be the same up to ε. Similar estimates for C 2 curve (now only for line singularity) Generalization from R 2 to R d

39 Generalizations Basic definitions Assumption α < 1 can be removed from all theorems Everything holds also for discrete curvelet transform Assumptions can be relaxed to hold only at ball of radius ε. Assumptions about Fourier properties of kernel functions can be relaxed Real valued kernel functions if support include reflection respect origin Compact support on Fourier domain may not be necessary Theorems could work for contourlets also With some extra assumptions of background regularity Necessary and sufficient conditions would be the same up to ε. Similar estimates for C 2 curve (now only for line singularity) Generalization from R 2 to R d

40 Generalizations Basic definitions Assumption α < 1 can be removed from all theorems Everything holds also for discrete curvelet transform Assumptions can be relaxed to hold only at ball of radius ε. Assumptions about Fourier properties of kernel functions can be relaxed Real valued kernel functions if support include reflection respect origin Compact support on Fourier domain may not be necessary Theorems could work for contourlets also With some extra assumptions of background regularity Necessary and sufficient conditions would be the same up to ε. Similar estimates for C 2 curve (now only for line singularity) Generalization from R 2 to R d

41 Generalizations Basic definitions Assumption α < 1 can be removed from all theorems Everything holds also for discrete curvelet transform Assumptions can be relaxed to hold only at ball of radius ε. Assumptions about Fourier properties of kernel functions can be relaxed Real valued kernel functions if support include reflection respect origin Compact support on Fourier domain may not be necessary Theorems could work for contourlets also With some extra assumptions of background regularity Necessary and sufficient conditions would be the same up to ε. Similar estimates for C 2 curve (now only for line singularity) Generalization from R 2 to R d