Hölder regularity estimation by Hart Smith and Curvelet

Basic definitionsRegularity estimates

ExamplesDiscussion

Hölder regularity estimation by Hart Smith andCurvelet transforms

Jouni Sampo

Lappeenranta University Of TechnologyDepartment of Mathematics and Physics

Finland

18th September 2007

Jouni Sampo, Lappeenranta University of Technology, Finland


ExamplesDiscussion

This research is done in collaboration with Dr. SongkiatSumetkijakan (Chulalongkorn University, Department ofMathematics, Bangkok, Thailand)



ExamplesDiscussion

Outline

1 Basic definitionsHölder regularitiesVanishing momentsHart Smith TransformContinuous Curvelet Transform

2 Regularity estimatesConditions for kernel functionsUniform regularityPointwise RegularityDirectional Regularity properties

3 Examples

4 Discussion



ExamplesDiscussion

Hölder regularitiesVanishing momentsHart Smith TransformContinuous Curvelet Transform

Uniform and Pointwice Hölder Regularity

Definition

Let α > 0 and α /∈ N. A function f : Rd → R is said to bepointwise Hölder regular with exponent α at u , denoted byf ∈ Cα(u), if there exists a polynomial Pu of degree less than αand a constant Cu such that for all x in a neighborhood of u

|f (x )− Pu (x − u)| ≤ Cu‖x − u‖α. (1)

Let Ω be an open subset of Rd . If (1) holds for all x , u ∈ Ω withCu being a uniform constant independent of u , then we say thatf is uniformly Hölder regular with exponent α on Ω or f ∈ Cα(Ω).



ExamplesDiscussion


Uniform and Pointwise Hölder Exponents

The uniform and pointwise Hölder exponents of f on Ω and at uare defined as

αl(Ω) := supα : f ∈ Cα(Ω)

andαp(u) := supα : f ∈ Cα(u).



ExamplesDiscussion


Local Hölder exponent

Definition

Let (Ii)i∈N be a family of nested open sets in Rd , i.e. Ii+1 ⊂ Ii ,with intersection ∩i Ii = u. The local Hölder exponent of afunction f at u , denoted by αl(u), is

αl(u) = limi→∞

αl(Ii).

In many situations, local and pointwise Hölder exponentscoincide, e.g., if f (x) = |x |γ then αp(0) = αl(0) = γ. However,local Hölder exponents αl(u) is also sensitive to oscillatingbehavior of f near the point u . A simple example is

f (x) = |x |γ sin(

1/ |x |β)

for which αp(0) = γ but αl(0) = γ1+β ,

i.e., αl is influenced by the wild oscillatory behavior of f near 0.Jouni Sampo, Lappeenranta University of Technology, Finland


ExamplesDiscussion


Directional Hölder regularity

Definition

Let v ∈ Rd be a fixed unit vector and u ∈ Rd . A functionf : Rd → R is pointwise Hölder regular with exponent α at u inthe direction v , denoted by f ∈ Cα(u ; v ), if there exist aconstant Cu ,v and a polynomial Pu ,v of degree less than α suchthat

|f (u + λv )− Pu ,v (λ)| ≤ Cu ,v |λ|α

holds for all λ in a neighborhood of 0 ∈ R.If one can choose Cu ,v so that it is independent of u for allu ∈ Ω ⊆ Rd and the inequality holds for all λ ∈ R such thatu + λv ∈ Ω, then we say that f is uniformly Hölder regular withexponent α on Ω in direction v or f ∈ Cα(Ω; v ).



ExamplesDiscussion


Directional Vanishing Moments

Definition

A function f of two variables is said to have an L-orderdirectional vanishing moments along a direction v = (v1, v2)

T

(suppose that v1 6= 0; if v1 = 0 then v2 6= 0 and we can swapthe two dimensions) if∫

Rtnf (t , tv2/v1 − c)dt = 0, ∀c ∈ R, 0 ≤ n ≤ L.

Essentially, the above definition means that any 1-D slicesof the function have vanishing moments of order L.



ExamplesDiscussion


Building Function with Directional Vanishing Moments

At spatial domain the design is challenging if vanishingmoment in many directions are neededIn frequency domain the design is relatively easy

If f (n)(ω1, ω2) vanish at line ω2 = −v1/v2ω1 for alln = 0, . . . , L then f have L-order directional vanishingmoments along a direction v = (v1, v2)

T



ExamplesDiscussion


Building Function with Directional Vanishing Moments

At spatial domain the design is challenging if vanishingmoment in many directions are neededIn frequency domain the design is relatively easy

If f (n)(ω1, ω2) vanish at line ω2 = −v1/v2ω1 for alln = 0, . . . , L then f have L-order directional vanishingmoments along a direction v = (v1, v2)

T



ExamplesDiscussion


Kernel of Hart Smith Transform

For a given ϕ ∈ L2(R2), we define

ϕabθ(x ) = a−34 ϕ(

D 1aR−θ (x − b)

),

for

θ ∈ [0, 2π), b ∈ R2

R−θ is the matrix affecting planar rotation of θ radians inclockwise direction.

0 < a < a0, where a0 is a fixed coarsest scale

D 1a

= diag(

1a , 1√

a

)



ExamplesDiscussion



Width and length of essential support of kernel functionϕabθ(x ) are about a and a1/2.



ExamplesDiscussion



Essential support of kernel function ϕabθ(x ) become like aneedle when scale a become smaller.



ExamplesDiscussion


Reconstruction Formula for Hart Smith Transform

Theorem

There exists a Fourier multiplier M of order 0 so that wheneverf ∈ L2(R2) is a high-frequency function supported in frequencyspace ‖ξ‖ > 2

a0, then

f =

∫ a0

0

∫ 2π

0

∫R2〈ϕabθ, Mf 〉ϕabθ db dθ

daa3 in L2(R2). (2)

The function Mf is defined in the frequency domain by amultiplier formula Mf (ξ) = m(‖ξ‖)f (ξ), where m is a standardFourier multiplier of order 0 (that is, for each k ≥ 0, there is aconstant Ck such that for all t ∈ R,|m(k)(t)| ≤ Ck

(1 + |t |2

)−k/2).



ExamplesDiscussion


Reconstruction Formula for Hart Smith Transform

Because of this and the fact that ϕabθ and Mϕabθ are duals, wecan write reconstruction formula also as

f =

∫ a0

0

∫ 2π

0

∫R2〈Mϕabθ, f 〉ϕabθ db dθ

daa3

=

∫ a0

0

∫ 2π

0

∫R2〈ϕabθ, f 〉Mϕabθ db dθ

daa3 .

Unlike ϕabθ, the dual Mϕabθ do not satisfy true parabolicdilation



ExamplesDiscussion


Curvelets are defined in Fourier Domain

Let W be a positive real-valued function supported inside(12 , 2), called a radial window, and let V be a real-valued

function supported on [−1, 1], called an angular window, forwhich the following admissibility conditions hold:∫ ∞

0W (r)2 dr

r= 1 and

∫ 1

−1V (ω)2 dω = 1. (3)

At each scale a, 0 < a < a0, γa00 is defined by

γa00 (r cos(ω), r sin(ω)) = a34 W (ar) V

(ω/√

a)

for r ≥ 0 and ω ∈ [0, 2π).

For each 0 < a < a0, b ∈ R2, and θ ∈ [0, 2π), a curvelet γabθ isdefined by

γabθ(x ) = γa00 (R−θ (x − b)) , for x ∈ R2. (4)



ExamplesDiscussion


Reconstruction with Curvelets

Theorem

There exists a bandlimited purely radial function Φ such that forall f ∈ L2(R2),

f =

∫R2〈Φb , f 〉Φb db+

∫ a0

0

∫ 2π

0

∫R2〈γabθ, f 〉 γabθ db dθ

daa3 in L2,

(5)where Φb(x ) = Φ(x − b).



ExamplesDiscussion

Conditions for kernel functionsUniform regularityPointwise RegularityDirectional Regularity properties

Extra conditions for kernel functions

For regularity analysis we will need that

Kernel functions have enough directional vanishingmoments

Kernel functions and their derivatives up to desired order(largest α of interest) decay fast enough.



ExamplesDiscussion


Vanishing moments of kernel function

Lemma

There exists C < ∞ (independent of a, b and θ) such thatcurvelet functions γabθ have directional vanishing moments ofany order L < ∞ along all directions v that satisfy|∠(v θ, v )| ≥ Ca1/2. Moreover if there exists finite and strictlypositive constants C1, C′

1 and C2 such thatsupp(ϕ) ⊂ [C1, C′

1]× [−C2, C2], then above is true also forfunctions ϕabθ and Mϕabθ.



ExamplesDiscussion


Vanishing moments of kernel function

Frequency support of ϕabθ(x ).



ExamplesDiscussion


Decay of kernel function

Lemma

Suppose that the windows V and W in the definition of CCTare C∞ and have compact supports. Then for each N = 1, 2, ...there is a constant CN such that

∀x ∈ R2 |∂νγabθ(x )| ≤ CNa−3/4−|ν|

1 + ‖x − b‖2Na,θ

. (6)

Moreover, if ϕ ∈ C∞ and if there exist finite and strictly positiveconstants C1, C′

1, and C2 such thatsupp(ϕ) ⊂ [C1, C′

1]× [−C2, C2], then (6) also holds for functionsϕabθ and Mϕabθ.



ExamplesDiscussion


Necessary Condition for Uniform Regularity

Theorem

If a bounded function f ∈ Cα(R2), then there exist a constant Cand a fixed coarsest scale a0 for which

|〈φabθ, f 〉| ≤ Caα+ 34

for all 0 < a < a0, b ∈ R2, and θ ∈ [0, 2π).



ExamplesDiscussion


Sufficient Condition for Uniform Regularity

Theorem

Let f ∈ L2(R2) and α > 0 a non-integer. If there is a constantC < ∞ such that

|〈φabθ, f 〉| ≤ Caα+ 54 ,

for all 0 < a < a0, b ∈ R2, and θ ∈ [0, 2π), then f ∈ Cα(R2).



ExamplesDiscussion


Necessary Condition for Pointwise Regularity

Theorem

If a bounded function f ∈ Cα(u) then there exists C < ∞ suchthat

|〈φabθ, f 〉| ≤ Caα2 + 3

4

(1 +

∥∥∥∥b − ua1/2

∥∥∥∥α)(7)

for all 0 < a < a0, b ∈ R2, and θ ∈ [0, 2π).



ExamplesDiscussion


Sufficient Condition for Pointwise Regularity

Theorem

Let f ∈ L2(R2) and α be a non-integer positive number. If thereexist C < ∞ and α′ < 2α such that

|〈φabθ, f 〉| ≤ Caα+ 54

(1 +

∥∥∥∥b − ua1/2

∥∥∥∥α′), (8)

for all 0 < a < a0, b ∈ R2, and θ ∈ [0, 2π), then f ∈ Cα(u).



ExamplesDiscussion


Necessary Conditions for direction of Singularity Line

Now we consider situation that background is sufficientlysmooth, i.e. regularity to one direction is higher.

Theorem

Let f be bounded with local Hölder exponent α ∈ (0, 1] at pointu and f ∈ C2α+1+ε(R2, v θ0) for some θ0 ∈ [0, 2π) with any fixedε > 0. Then there exist α′ ∈ [α− ε, α] and C < ∞ such that fora > 0 and b ∈ R2,

|〈φabθ, f 〉| ≤

Caα+ 5

4 , if θ /∈ θ0 + Ca1/2[−1, 1],

Caα′+ 34

(1 +

∥∥∥∥b − ua

∥∥∥∥α′), if θ ∈ θ0 + Ca1/2[−1, 1].



ExamplesDiscussion


Sufficient Conditions for direction of Singularity Line

Theorem

Let f ∈ L2(R2), u ∈ R2, and assume that α > 0 is not aninteger. If there exist α′ < 2α, θ0 ∈ [0, 2π], and C < ∞ such that

|〈φabθ, f 〉| ≤

Caα+ 5

4

(1 +

∥∥∥∥b − ua1/2

∥∥∥∥α′), if θ /∈ θ0 + Ca1/2[−1, 1]

Caα+ 34

(1 +

∥∥∥∥b − ua1/2

∥∥∥∥α′), if θ ∈ θ0 + Ca1/2[−1, 1]

for all 0 < a < a0, b ∈ R2, and θ ∈ [0, 2π), then f ∈ Cα(u).



ExamplesDiscussion

Figure: Decay behavior of |〈φa0θ, f 〉| across scales a at various anglesθ for the function f (x ) = e−‖x‖|x1|0.25



ExamplesDiscussion

Figure: Estimation errors of the Hölder exponents by αe(s, θ) atscales 2−s and angles θ for the function f (x ) = e−‖x‖|x1|0.25



ExamplesDiscussion

Generalizations

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 functionscan be relaxed

Real valued kernel functions if support include reflectionrespect originCompact support on Fourier domain may not be necessaryTheorems could work for contourlets also

With some extra assumptions of background regularityNecessary and sufficient conditions would be the same upto ε.Similar estimates for C2 curve (now only for line singularity)

Generalization from R2 to Rd



ExamplesDiscussion

Generalizations









ExamplesDiscussion

Generalizations









ExamplesDiscussion

Generalizations









ExamplesDiscussion

Generalizations









ExamplesDiscussion

Generalizations









ExamplesDiscussion

Generalizations









ExamplesDiscussion

Generalizations









ExamplesDiscussion

Generalizations









ExamplesDiscussion

Generalizations









ExamplesDiscussion

Generalizations









ExamplesDiscussion

Generalizations








Documents

Hölder regularity estimation by Hart Smith and Curvelet