Hölder regularity estimation by Hart Smith and Curvelet ...
Transcript of 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
Basic definitionsRegularity estimates
ExamplesDiscussion
This research is done in collaboration with Dr. SongkiatSumetkijakan (Chulalongkorn University, Department ofMathematics, Bangkok, Thailand)
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
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
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
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).
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Hölder regularitiesVanishing momentsHart Smith TransformContinuous Curvelet Transform
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).
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Hölder regularitiesVanishing momentsHart Smith TransformContinuous Curvelet Transform
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
Basic definitionsRegularity estimates
ExamplesDiscussion
Hölder regularitiesVanishing momentsHart Smith TransformContinuous Curvelet Transform
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 ).
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Hölder regularitiesVanishing momentsHart Smith TransformContinuous Curvelet Transform
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.
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Hölder regularitiesVanishing momentsHart Smith TransformContinuous Curvelet Transform
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
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Hölder regularitiesVanishing momentsHart Smith TransformContinuous Curvelet Transform
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
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Hölder regularitiesVanishing momentsHart Smith TransformContinuous Curvelet Transform
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
)
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Hölder regularitiesVanishing momentsHart Smith TransformContinuous Curvelet Transform
Kernel of Hart Smith Transform
Width and length of essential support of kernel functionϕabθ(x ) are about a and a1/2.
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Hölder regularitiesVanishing momentsHart Smith TransformContinuous Curvelet Transform
Kernel of Hart Smith Transform
Essential support of kernel function ϕabθ(x ) become like aneedle when scale a become smaller.
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Hölder regularitiesVanishing momentsHart Smith TransformContinuous Curvelet Transform
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).
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Hölder regularitiesVanishing momentsHart Smith TransformContinuous Curvelet Transform
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
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Hölder regularitiesVanishing momentsHart Smith TransformContinuous Curvelet Transform
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)
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Hölder regularitiesVanishing momentsHart Smith TransformContinuous Curvelet Transform
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).
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
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.
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Conditions for kernel functionsUniform regularityPointwise RegularityDirectional Regularity properties
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.
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Conditions for kernel functionsUniform regularityPointwise RegularityDirectional Regularity properties
Vanishing moments of kernel function
Frequency support of ϕabθ(x ).
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Conditions for kernel functionsUniform regularityPointwise RegularityDirectional Regularity properties
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.
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Conditions for kernel functionsUniform regularityPointwise RegularityDirectional Regularity properties
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).
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Conditions for kernel functionsUniform regularityPointwise RegularityDirectional Regularity properties
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).
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Conditions for kernel functionsUniform regularityPointwise RegularityDirectional Regularity properties
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).
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Conditions for kernel functionsUniform regularityPointwise RegularityDirectional Regularity properties
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).
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Conditions for kernel functionsUniform regularityPointwise RegularityDirectional Regularity properties
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].
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Conditions for kernel functionsUniform regularityPointwise RegularityDirectional Regularity properties
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).
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
ExamplesDiscussion
Figure: Decay behavior of |〈φa0θ, f 〉| across scales a at various anglesθ for the function f (x ) = e−‖x‖|x1|0.25
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
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
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
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
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
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
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
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
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
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
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
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
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
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
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
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
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
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
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
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
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
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
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
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
Jouni Sampo, Lappeenranta University of Technology, Finland
Basic definitionsRegularity estimates
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
Jouni Sampo, Lappeenranta University of Technology, Finland