1 Introduction
Wavelets are used to approximate smooth functions with point singularities. In higher
dimensions wavelets detect the point singularities but do not give information about
the directions where these singularities occur, specially discontinuities along
lines or curves [1]. This problem has
been studied by different class of wavelets like shearlets, which are obtained by
means of dilations, translations and shear parameters [2].
Shearlets have been defined in such a way that they provide not only the point
singularities but also the directions of these singularities. That is, the shearlet
representation is more effective than the wavelet representation for the analysis
and processing of multidimensional data [3].
Wavelets do not detect, with good approximation, the geometry of images specially
with edges. Hence, shearlets have been defined under a similar framework of
wavelets, but with composite dilations. Thus, shearlets have been a very useful tool
to obtain better theoretical results and applications than the ones obtained from
the wavelet theory [1].
In one dimension, for a given admissible function h∈L2(ℝ), the continuous wavelet transform for a given signal f∈L2(ℝ) is defined as [4]:
(Lhf)(a,b)=∫ℝf(x)1|a|h¯(x−ba)dx,
where a≠0 and b∈ℝ.
In this case, h∈L2(ℝ) is admissible if:
0<Ch:=∫ℝ|h^(k)|21|k|dk<∞.
The discrete wavelet transform is obtained by considering discrete values for the
dilation parameter a and the translation parameter b. That is, if a=a0m and b=nb0a0m, with a0>1 and b0>0, are fixed and m,n∈ℤ, the discrete wavelet transform of f∈L2(ℝ) with respect to the admissible function h∈L2(ℝ) is given by [5]:
(Lhf)(a0m,nb0a0m)=∫ℝf(x)1a0mh¯(x−nb0a0ma0m)dx.
The dilation and translation operators Ja0m and Tnb0a0m are defined respectively as:
(Ja0mh)(x)=1a0mh(xa0m),(Tnb0a0mh)(x)=h(x−nb0a0m),
then the discrete wavelet transform can be written as:
(Lhf)(a0m,nb0a0m)=〈f,Tnb0a0mh〉.
Moreover, if (Tnb0a0mJa0mh)(m,n)∈ℤ×ℤ is a tight frame in L2(ℝ), then for all f∈L2(ℝ) there is a positive constant C such that the inversion formula is given by [6]:
f=1C∑m,n∈ℤ〈f,Tnb0a0mJa0mh〉Tnb0a0mJa0mh.
For n dimensions, the dilation parameter a is in ℝ+, and the translation parameter b is in ℝn, where v=(b1,b2,…,bn) with bi>0,i=1,2,…n.
In this case, for a>1 and j=(j1,j2,…,jn) in ℤn, the n×n matrix M(aj) is defined as:
M(aj)=diag(aj1,aj2,…,ajn),
and for b=(b1,b2,…,bn) in ℝ+n, the n×n matrix M(b) is defined as :
M(b)=diag(b1,b2,…,bn).
Now, for a>1 and b=(b1,b2,…,bn) in ℝ+n, the dilation and translation operators are defined respectively as:
(JM(aj)h)(x)=1M(aj)h(M−1(aj)x),
where x∈ℝn, j∈ℤn, and
(TM−1(aj)M(b)kh)(x)=h(x−M−1(aj)M(b)k),
where x∈ℝn and k∈ℤn. Hence, the discrete wavelet transform in n dimensions for f∈L2(ℝn)with respect to a radially symmetric admissible function h∈L2(ℝn), is defined as [7]:
(Lhf)(M−1(aj),M−1(aj)M(b)k)=〈f,TM−1(aj)M(b)kJM(aj)h〉.
In this case, h∈L2(ℝn) is admissible if:
∫0∞|η(k)|21kdk<∞, where h^(y)=η(|y|).
Moreover, if (TM−1(aj)M(b)kJM(aj)h)(j,k)∈ℤn×ℤn is a tight frame in L2(ℝn), then for all f∈L2(ℝn) there is a positive constant C such that [8]:
f=1C∑j,k∈ℤn〈f,TM−1(aj)M(b)kJM(aj)h〉⋅TM−1(aj)M(b)kJM(aj)h.
In this paper, in Section 2 the continuous shearlet transform is studied in two
dimensions and in Section 3 discrete values for the dilation, shear and translation
parameters are taken to obtain the discrete shearlet transform following the same
idea from the definition of the discrete wavelet transform and then, in Sections 4
and 5 a tight frame is considered in L2(ℝ2) to analyze the continuity of a given function f∈L2(ℝ2) under the hypothesis of the fast convergence of its discrete shearlet
transform. In Section 6, an example is given to illustrate the results by means of a
computational experiment, and finally in Section 7 conclusions are presented.
2 Notations and Definitions
First, an overview of the continuous shearlet transform in two dimensions is given.
In this case, the shearlet transform is defined with respect to: dilations, shears
and translations parameters. That is, the following family of operators are
used:
(TbKsJah)(x)=a−34h(A−1(a)S−1(s)(x−b)),
where the matrices A(a) and S(s) are given in the following definition, [9].
Definition 2.1. For a>0 and s in ℝ, let:
A(a)=(a00a) and S(s)=(1s01).
Note that A−1(a)=A(a−1) and S−1(s)=S(−s). Also, A(a1)A(a2)=A(a1a2)>0, where a1,a2>0, and S(s1)S(s2)=S(s1+s2), where s1,s2∈ℝ.
Definition 2.2. For h in L2(ℝ2), the dilation, translation, modulation and shear operators are defined
respectively by:
(Jah)(x)=1det A(a)h(A−1(a)x), where a>0 and x∈ℝ2,
(Tbh)(x)=h(x−b), where x,b∈ℝ2,
(Ech)(x)=e2πic⋅xh(x), where x,c∈ℝ2,
(K2h)(x)=h(S−1(s)x), where s∈ℝ and x∈ℝ2. Besides (KsTh)(x)=h(S−T(s)x).
From the previous definitions the next lemma is obtained directly.
Lemma 2.3.
The operators
Ja, Tb, Ec, Ks
preserve the norm in
L2(ℝ2).
Corollary 2.4.
For
h
in
L2(ℝ2), a>0, s∈ℝ
and
b∈ℝ2, we have:
‖TbKsJah‖2=‖h‖2.
Also, the following results come directly from the definition of the Fourier
transform.
Lemma 2.5.
For the operators
Ja, Tb, Ks, and for
h
in
L1(ℝ2)∩L2(ℝ2):
Jah^=J1ah^, where
a>0,
Tbh^=E−bh^, where
b∈ℝ2,
Ech^=Tch^, where
c∈ℝ2,
Ksh^=K−sTh^, where
s∈ℝ.
In this case, the Fourier transform of
h
is taken as
h^(ξ)=∫ℝ2e−2πiξ⋅xh(x)dx.
The shearlet transform, as well as the wavelet transform, can be defined from the
topological point of view, so that a unitary shearlet group representation can be
used to obtain the inversion formula.
Definition 2.6. Let G={(a,s,b)|a>0,s∈ℝ,b∈ℝ2}. For (a1,s1,b1) and (a2,s2,b2) in G, define:
(a1,s1,b1)⋅(a2,s2,b2)=(a1a2,s1+s2a1,b1+S(s1)A(a1)b2).
Remark 2.7. With this product G becomes a locally compact topological group with identity (1,0,0), where (a,s,b)−1=(1a,−sa,−A−1(a)S−1(s)b) is the inverse of (a,s,b). Moreover, the left Haar measure is d(a,s,b)=1a3dadsdb, and the right Haar measure is dr(a,s,b)=1adadsdb [9]. That is, G is a
non-unimodular group.
Remark 2.8. The shearlet group is isomorphic to the
locally compact group G×ℝ2, where:
G={S(s)A(a)|a>0,s∈ℝ}.
Thus, it is a subgroup of the following group of rotations GL2(ℝ)×ℝ2 with multiplication defined by (M,b)⋅(M′,b′)=(MM′,b+Mb′).
For (a,s,b) in G the three parameter family of operators is defined as:
U(a,s,b)=TbKsJa.
In this case, for h∈L2(ℝ2):
U(a,s,b)h(x)=(TbKsJah)(x)=(KsJah)(x−b), =(Jah)(S−1(s)(s−b)), =a−34h(A−1(a)S−1(s)(x−b)).
Moreover, U is a unitary representation of G acting on L2(ℝ2).
Definition 2.9. A function h in L2(ℝ2) is admissible if:
∫G|〈h,U(a,s,b)h〉|2d(a,s,b)<∞.
Lemma 2.10.
Suppose that
f, h
are in
L2(ℝ2), then:
∫G|〈f,U(a,s,b)h〉|2d(a,s,b)=Ch‖f‖2,
where
Ch:=∫ℝ2|h^(k1,k2)|21k12dk1dk2.
Proof. See [9].
Remark 2.11. From Lemma 2.10, we have that h in L2(ℝ2) is admissible if and only if:
Ch:=∫ℝ2|h^(k1,k2)|21k12dk1dk2<∞.
(2.1)
Definition 2.12. Let h be an admissible function h in L2(ℝ2), and let (a,s,b) be in G. The continuous shearlet transform with respect to h is defined as the map:
Sh(a,s,b):L2(ℝ2,dx)→L2(G,d(a,s,b)),
such that for f in L2(ℝ2):
(Shf)(a,s,b)=〈f,U(a,s,b)h〉=〈f,TbKsJah〉.
That is:
(Shf)(a,s,b)=∫ℝ2f(x)1det A(a)h¯(A−1(a)S−1(s)(x−b))dx.
The continuous shearlet transform can be expressed as convolution, as the following
remark states.
Remark 2.13. Let h be admissible in L2(ℝ2). Then for f∈L2(ℝ2) and (a,s,b)∈G:
(Shf)(a,s,b)=[(KsJah¯)~∗f](b),
(2.2)
where the symbol ~ means h~(x)=h(−x).
The next result corresponds to the inverse shearlet transform, [9].
Lemma 2.14.
If
f, h
are in
L2(ℝ2), and
(a,s,b)∈G, then:
f=1Ch∫G(Shf)(a,s,b)U(a,s,b)h(a,s,b),
where the convergence is in the weak sense.
Remark 2.15. In the case of band limited shearlets,
that is when supp h^ is compact, the function h∈L2(ℝ2) is taken as:
h^(w)=h^(w1,w2)=h^1(w1)h^2(w2w1),
where w=(w1,w2)∈ℝ2^, with w1≠0, and where h1 is a continuous wavelet, h^1∈C∞(ℝ), and supp h^1⊆[−2,−12] and where h2 is such that h^2∈C∞(ℝ) and supp h^2⊆[−1,1]. This generating function was used in [10] to show that the continuous shearlet transform resolves
the wave front set.
Moreover, this function satisfies the admissibility condition given in (2.1). That is h∈L2(ℝ2) is admissible if h^(w)=h^(w1,w2)=h^1(w1)h^2(w2w1), with w1≠0 and h1∈L2(ℝ) satisfies ∫|h^1(aξ)|2daa=1, for a.e. ξ∈ℝ, and ‖h2‖2=1. [10].
3 Discrete Shearlet Transform
To define the discrete shearlet transform discrete values for the dilation, shear and
translation parameters are considered. In this paper this transform is applied to
analyze the singularities of functions in L2(ℝ2) by means of the decay of the discrete shearlet transform. For this
purpose, similar matrices are considered like the ones given in Definition 2.1 with a=4, and s=−1.
Definition 3.1. Consider the following four matrices:
A1=(1220012),B1=(1−101),A2=(1200122),B2=(10−11).
Then for j,k∈ℤ:
A1j=(122j0012j),B1k=(1−k01),A2j=(12j00122j),B2k=(10−k1).
Moreover:
A1−j=(22j002j),B1−k=(1k01),A2−j=(2j0022j),B2−k=(10k1).
Definition 3.2. Consider the group G of the form:
G={(M,z):M∈GL2(ℝ),z∈ℝ2},
where M is the set of matrices of size 2×2 of the form:
M=Md(jk)=AdjBdk,
where j∈ℤ, k∈ℤ, and d=1,2.
The group law in G is given by (M,z)⋅(M′,z′)=(MM′,Mz′+z), where the inverse is (M,z)−1=(M−1,−M−1z), and the identity is (I,0).
Remark 3.3. Note that for d=1:
M1(jk)=A1jB1k=(122j00122j)(1−k01)=(122j−k22j012j),
and for d=2:
M2(jk)=A2jB2k=(12j00122j)(10−k1)=(12j0−k22j122j).
Hence, note that for d=1,2:
det Md(jk)=det Adj=123j.
(3.1)
Definition 3.4. For h in L2(ℝ2), and each d=1,2, define the dilation and translation operators respectively by:
(JMd(jk)h)(x)=1det Md(jk)h(Md−1(jk)x),
where x∈ℝ2 and j,k∈ℤ.
(TMd(jk)lh)=h(x−Md(jk)l), where x∈ℝ2, j,k∈ℤ, and l∈ℤ2.
Remark 3.5. For d=1,2, the operators JMd(jk) and TMd(jk)l preserve the norm in L2(ℝ2). Moreover, the adjoints are their inverses respectively.
Definition 3.6.
h∈L2(ℝ2) is admissible [9], if:
0<Ch:=∫ℝ∫ℝ|h^(y1,y2)|21y12dy1dy2<∞.
Following [11], the discrete shearlet
transform is defined as:
Definition 3.7. Let h be an admissible function in L2(ℝ2), and for each d=1,2, let (Md(jk),Md(jk)l) in GL2(ℝ)×ℝ2. Then the discrete shearlet transform of f in L2(ℝ2), with respect to h is defined as:
(Dhf)(M1(jk),M2(jk),M1(jk)l,M2(jk)l)=∑d=12〈f,TMd(jk)lJMd(jk)h〉.
Definition 3.8. The family of functions:
{TMd(jk)lJMd(jk)h},
in L2(ℝ2) with j,k∈ℤ, l∈ℤ2 and where d=1,2, j≥0 is a tight frame [11], if
there is a positive constant C such that for any f∈L2(ℝ2):
C‖f‖22=∑j,k,l,d|〈f,TMd(jk)lJMd(jk)h〉|2.
Theorem 3.9.
For any
f
in
L2(ℝ2)
and for a given admissible function
h∈L2(ℝ2), if
TMd(jk)lJMd(jk)h
is a tight frame, then there is a constant
C>0
such that:
f=1C∑j,k,l,d〈f,TMd(jk)lJMd(jk)h〉TMd(jk)lJMd(jk)h,
(3.2)
where the convergence is in the weak sense, and where
j,k∈ℤ, l∈ℤ2, and
d=1,2. [11].
Remark 3.10. According to Definitions 3.4 and 3.7,
the discrete shearlet transform can be written as:
(Dhf)(M1(jk),M2(jk),M1(jk)l,M2(jk)l)=∑d=12∫ℝ2f(x)(TMd(jk)lJMd(jk)h¯)(x)dx,∑d=12∫ℝ2f(x)(JMd(jk)h¯)(x−Md(jk)l)dx,∑d=12∫ℝ2f(x)1det Md(jk)h¯(Md−1(jk)x−l)dx.
(3.3)
That is:
(Dhf)(M1(jk),M2(jk),M1(jk)l,M2(jk)l)=∑d=12∫ℝ2f(x)1det Adjh¯(Bd−kAd−jx−l)dx,=∫ℝ2f(x)1det A1jh¯(B1−kA1−jx−l)dx,+∫ℝ2f(x)1det A2jh¯(B2−kA2−jx−l)dx,=∫ℝ2f(x)232jh¯(22jx1+k2jx2−l1,2jx2−l2)dx,+∫ℝ2f(x)232jh¯(2jx1−l1,22jx2+k2jx1−l2)dx.
Lemma 3.11.
The discrete shearlet transform can be expressed as a convolution. That
is:
(Dhf)(M1(jk),M2(jk),M1(jk)l,M2(jk)l)=∑d=12[(JMd(jk)h¯)~∗f](Md(jk)l),
where
~
means
ϕ~(x)=ϕ(−x).
Proof. From (3.3):
∑d=12[(JMd(jk)h)~∗f](Md(jk)l),=∑d=12∫ℝ2(JMd(jk)h¯)~(Md(jk)l−x)f(x)dx,=∑d=12∫ℝ2(JMd(jk)h¯)(x−Md(jk)l)f(x)dx,=∑d=12∫ℝ21det Adjh¯(Md(jk)−1x−l)f(x)dx,(Dhf)(M1(jk),M2(jk),M1(jk)l,M2(jk)l).
4 Partial Result
Lemma 4.1.
Suppose
h
be in
C0(ℝ2)
is an admissible function not identically zero, such that
∫ℝh(x)dx=0. Consider
f
in
L2(ℝ2), and for each
d=1,2, let:
(Ph(d)f)(Md(jk),Md(jk)l):=1det Md(jk)〈f,TMd(jk)lJMd(jk)h〉.
If
f
is continuous in a neighborhood of
x=0∈ℝ2, then for each
d=1,2, and any
j,k∈Z
and any
l∈ℤ2:
lim(Md(jk),Md(jk)l)→(0,0)(Ph(d)f)(Md(jk),Md(jk)l)=0.
Proof. Note that if j↛+∞, then from (3.1), we have det Md(jk)↛0. Hence, from (2.2), and
for any d=1,2, the function Ph(d)f is continuous for any (Md(jk),Md(jk)l)∈GL2(ℝ)×ℝ2.
Consider now the case when j→+∞, then det Md(jk)→0. Thus, by hypothesis suppose that f is continuous in a neighborhood of x=0 containing the closed ball BR(0)¯, where R>0, and take Md(jk)l in the open ball BR2(0).
Now, since h∈C0(ℝ2) there is L>0 such that supp h⊂BL(0). Then since the adjoints of the operators TMd(jk)l,JMd(jk) are their inverses respectively (Remark 3.5), then for each d=1,2:
(Ph(d)f)(Md(jk),Md(jk)l)=1det Md(jk)〈f,TMd(jk)lJMd(jk)h〉,1det Md(jk)〈JMd(jk)−1TMd(jk)l−1f,h〉,=1det Md(jk)∫BL(0)(JMd(jk)−1TMd(jk)l−1f)(x)h¯(x)dx,=∫BL(0)TMd(jk)l−1f(Md(jk)x)h¯(x)dx,=∫BL(0)f(Md(jk)+Md(jk)l)h¯(x)dx.
Since ∫ℝh(x)dx=0, and f is continuous near 0, it follows that for any j,k∈ℤ and any l∈ℤ2:
lim(Md(jk),Md(jk)l)→(0,0)(Ph(d)f)(Md(jk),Md(jk)l)=f(0)∫BL(0)h¯(x)dx=f(0)⋅0=0.
5 Main Result
Theorem 5.1.
Suppose
h
be in
C0(ℝ2)
is an admissible function not identically zero, such that
∫ℝh(x)dx=0. For
d=1,2
suppose that
TMd(jk)lJMd(jk)h
is a tight frame. Consider
f
in
L2(ℝ2)
and
(Md(jk),z)∈GL2(ℝ)×ℝ2.
If for each
d=1,2:
lim(Md(jk),zd)→(0,z′d)(Ph(d)f)(Md(jk),zd),
exists for each
k
in
[−Q,Q]
for some positive
Q∈ℤ
and any
z′d
in an open neighborhood of
x=0∈ℝ2, then
f
in
L2(ℝ2)
is continuous in a neighborhood of
x=0∈ℝ2.
Proof of Theorem 5.1. Suppose that for each d=1,2:
lim(Md(jk),zd)→(0,z′d)(Ph(d)f)(Md(jk),zd):=Fd(0,z′d),
(5.1)
exists for each k in [−Q,Q] and any z′d in an open neighborhood containing the closed ball BR(0)¯, with R>0.
Now for fixed x in the open ball BR(0) and y∈ℝ2, for each d=1,2 let:
ℐd(Md(jk),x,y),={h(−y)(Ph(d)f)(Md(jk),x+Md(jk)y),if j↛−∞, andh(−y)Fd(Md(jk),x) if j→−∞.
(5.2)
Note that for such x, the function ℐd is well-defined for all j∈ℤ, k∈[−Q,Q], and y∈ℝ2.
Then we have the following three Claims.
Claim 1. For each d=1,2, the function ℐd is continuous in GL2(ℝ)×BR(0)¯×ℝ2.
Proof. See Appendix
Claim 2. For each d=1,2 and fixed x∈BR(0)¯, the triple series:
∑j∈ℤ∑k∈ℤ∑l∈ℤ2ℐd(Md(jk),x,l−Md−1(jk)x),
converges uniformly on BR(0)¯.
Proof. See Appendix
Claim 3. For each d=1,2 and x∈BR(0)¯, the function:
Wd(x):=∑j∈ℤ∑k∈ℤ∑l∈ℤ2ℐd(Md(jk),x,l−Md−1(jk)x),
is continuous at x=0.
Proof. See Appendix
Back to the proof of Theorem 5.1, for any integer r≥0, any x∈ℝ2, and for each d=1,2, define:
Ud,r(x):=∑j=−rr∑k∈ℤ∑l∈ℤ2〈f,TMd(jk)lJMd(jk)h〉⋅1det Md(jk)h(Md(jk)x−l).
Then by Claim 3, for each d=1,2 and for any: x∈BR(0)¯,
limr→∞Ud,r(x)=Wd(x).
That is, for each d=1,2, it follows that Ud,r→Wd pointwise on BR(0) as r→∞. Hence, U1,r+U2,r→W1+W2 pointwise on BR(0) as r→∞.
On the other hand from (3.2), U1,r+U2,r→Cf weakly in L2(M22×ℝ2). Hence, f=1C(W1+W2) almost everywhere, and due to Claim 3, since W1 and W2 are continuous at x=0, then f is continuous at x=0.
This completes the proof of Theorem 5.1.
6 Experiments
To illustrate the main results given in Lemma 4.1 and Theorem 5.1, grayscale images
with width W and height H were processed. Grayscale pixel values are in [0,255] where 0 means a black pixel and 255 means a white pixel. Note that, although f∈L2(ℝ2), pixel values are integer and the energy of the image is given by the
sum of the square values of the pixels.
An image IMG with W=H=128 was built from:
f(x1,x2)=255∗e−(x1−64)2+(x2−64)2512,
by getting integer values. Since IMG represents a continuous gaussian function, a
cross section was produced with zero values for x1<x2 to get image IMG1 with two continuous sections, as is shown in Figure 1.
In a similar way, image IMG2 was built from IMG with zero values for x1<64(1+sin(πx2/64)). Figure 2 shows a top view
projection for IMG2 (left) as grayscale heat map and a colored angular projection
(right).
In both cases, for IMG1 and IMG2, the insertion of zero values aims to define black
regions and discontinuities to be studied. Shearlab [12] software was used to get the shearlet coefficients for
IMG1 and IMG2. For a single shearlet scale, N=9 decomposition bands are obtained for each image. To illustrate how the
shearlet transform coefficients tends to zero in continuous sections, shearlet
coefficients were scaled to [0,255] to appreciate a 128×128 grayscale heat map where dark pixels mean close to zero values (shearlet
transform converges) and discontinuities (where shearlet transform does not
converge) are perceived as non-black pixels (”clear” pixels that tends to
white).
Note that by translation to any point x∈ℝ2, and not only at (0,0), the results described in Lemma 4.1 and Theorem 5.1 show that:
If we take a point in the black zone of the shearlet transform (where it
tends/converges to zero) then by Theorem 5.1 the function is
continuous.
If we take a point in the white region of the shearlet transform (the
shearlet coefficients takes large values, meaning that it does not
converge) then by Lemma 4.1 the function is not continuous.
For IMG1, two illustrative shearlet images were chosen as high-frequency bands and
they are shown in Figure 3 where clear pixels
follow the line corresponding to x1=x2 as it was expected from Figure
1.
For IMG2, eight shearlet images were chosen (non-low pass frequency bands) and they
are shown in Figure 4 where there are clear
pixels along x1=64(1+sin(πx2/64)) by detecting the discontinuity.
Note that subfigures of Figure 4 have distinct
directionality of shearlets and non-dark pixels line up to 8 different
directions.
To expose the directionality of shearlets an image IMG3 with zero values except at
(63, 63) with a 255 value was generated to simulate a pulse embedded in 128×128 pixels. The corresponding images built from the shearlet coefficients
are shown in Figure 5. Note the directionality
change counterclockwise when reading subfigures from left to right and top to bottom
in Figure 5.
Additionally, the 2D discrete shearlet transform was applied to the ”Barbara” image
(see Figure 6) and from the shearlet
coefficients 8 subfigures were generated (see Figure
7) where it is possible to appreciate pixel patterns that match the
different directions illustrated in Figure
5.
7 Conclusions
There are several works about application of shearlets, in particular in two
dimensions for images with edges.
This manuscript aims to support these applications, once it was shown that it is
possible to study the continuity of a function in two dimensions through the
convergence of the 2D discrete shearlet transform. Close to zero shearlet
coefficients are associated to continuous sections in images, whereas high values of
shearlet coefficients reveal the edges.
Both properties of detecting discontinuities and providing directionality inside
images, make the discrete shearlet transform an interesting and useful tool in
applications such as image classification and opens the possibility to extend these
results to higher dimensions.
References
1.
1. Easley,
G., Labate,
D., & Lim,
W.Q. (2008).
Sparse directional image representations using the discrete shearlet transform.
Appl. Comput. Harmon. Anal., Vol. 25, No.
1, pp. 25–46. DOI: 10.1016/j.acha.2007.09.003.
[ Links ]
2.
2. Labate,
D., Lim,
W.Q., Kutyniok,
G., & Weiss,
G. (2005). Sparse
multidimensional representation using shearlets. Proc.
SPIE Wavelets XI, Vol. 5914, pp. 254–262.
DOI:10.1117/12.613494.
[ Links ]
3.
3. Guo,
K., Kutyniok,
G., & Labate,
D. (2006). Sparse
multidimensional representations using anisotropic dilation and shear operators.
Wavelets and Splines
Chen, G.,
Lai, M., Eds.,
Nashboro Press: Brentwood.
[ Links ]
4.
4. Navarro,
J. (1998).
Wavelet transforms and singularities of L2-functions
in Rn. Revista Colombiana de
Matemáticas, Vol. 32, pp. 93–99.
[ Links ]
5.
5. Navarro,
J., & Herrera,
O. (2012).
Convergence of the discrete wavelet transform. Int. J. Wavelets
Multi., Vol. 10, No. 6, pp. 1–12.
[ Links ]
6.
6. Daubechies,
I. (1992).
Ten lectures on wavelets. SIAM: Philadelphia.
[ Links ]
7.
7. Navarro,
J., & Herrera,
O. (2016). Decay
properties of the discrete wavelet transform in n dimensions with independent
dilation parameters. J. Inequal. Appl., pp.
1–19.
[ Links ]
8.
8. Kugarajah,
T., & Qinghua,
Z. (1995).
Multidimensional wavelet frames. IEEE Trans. Neural
Netw., Vol. 6, No. 6, pp.
1552–1556.
[ Links ]
9.
9. Dahlke,
S., Kutyniok,
G., Maass,
P., Sagiv,
C., Stark,
H.G., & Teschke,
G. (2008). The
uncertainty principle associated with the continuous shearlet transform.
Int. J. Wavelets Multi., Vol. 6, No. 2,
pp. 157–181.
[ Links ]
10.
10. Kutyniok,
G., & Labate,
D. (2009).
Resolution of the wavefront set using continuous shearlets. Trans. Amer.
Math. Soc., Vol. 36, No. 5, pp.
2719–2754.
[ Links ]
11.
11. Goossens,
B., Aelterman,
J., Luong,
H., Pizurica,
A., & Philips,
W. (2009).
Efficient design of a low redundant discrete shearlet transform.
LNLA:2009 Int. Workshop on Local and Non-Local approximation in
Image Proccesing, pp. 112–124.
[ Links ]
12.
12. Kutyniok,
G., Lim,
W.Q., & Reisenhofer,
R. (2016).
ShearLab 3D: Faithful digital shearlet transforms based on compactly supported
shearlets. ACM Trans. Math. Software, Vol. 42,
No. 5.
[ Links ]
8 Appendix A
Proof of Claim 1. If j↛−∞, then from (5.2):
ℐd(Md(jk),x,y),=h(−y)(Ph(d)f)(Md(jk),x+Md(jk)y),=h(−y)1det Md(jk)〈f,Tx+Md(jk)yJMd(jk)h〉,=h(−y)1det Md(jk),⋅ [(JMd(jk)h)~∗f](x+Md(jk)y).
Since h∈C0(ℝ2) and f∈L2(ℝ2), then the convolution (JMd(jk)h)~∗f is a continuous function. Then for each d=1,2, the function ℐd is continuous in GL(ℝ)×BR(0)¯×ℝ2.
In this case, the Frobenious norm is taken in GL(ℝ), where ‖A‖=〈A,A〉12.
Now if j→−∞, then from (5.1), for
any (0,x1,y1)∈GL(ℝ)×BR(0)¯×ℝ2:
lim(Md(jk),x,y)→(0,x1,y1)ℐd(Md(jk),x,y),=lim(Md(jk),x,y)→(0,x1,y1)h(−y), ⋅ (Ph(d)f)(Md(jk),x+Md(jk)y),=h(−y1)lim(Md(jk),zd)→(0,x1)(Ph(d)f)(Md(jk),zd),=h(−y1)Fd(0,x1)=ℐd(0,x1,y1).
Therefore, for each d=1,2, the function ℐd is continuous in GL(ℝ)×BR(0)¯×ℝ2.
Proof of Claim 2. Note that if j↛∞, and if for each d=1,2, take −y=−l+Md−1(jk)x, then from (5.2):
ℐd(Md(jk),x,l−Md−1(jk)x),=h(−l+Md−1(jk)x)1det Md(jk), ⋅ 〈f,TMd(jk)lJMd(jk)h〉.
Then, from (3.1):
|ℐd(Md(jk),x,l−Md−1(jk)x)|,≤‖f‖2‖h‖2232j|h(−l+Md−1(jk)x)|.
Now for a given positive integer V, define:
Gd(Md(jk),l−Md−1(jk)x),:={|ℐd(Md(jk),x,l−Md−1(jk)x)|, if j∈[−V,V],‖f‖2‖h‖2232j|h(−l+Md−1(jk)x)|, if j∉[−V,V].
(8.1)
Hence:
|ℐd(Md(jk),x,l−Md−1(jk)x)|,≤Gd(Md(jk),l−Md−1(jk)x),
for all (Md(jk),l−Md−1(jk)x)∈GL2(ℝ)×ℝ2.
On the other hand, since h∈C0(ℝ2) there is L>0 such that supp h⊂BL(0)¯. So, there is a positive integer N>L so that h(−l+Md−1(jk)x)=0 for l=(l1,l2)∈ℤ2 with li>N for i=1,2. Hence, it can be considered the series over li∈ℤ only from −N to N for i=1,2, and since k∈[−Q,Q], then:
∑j∈ℤ∑k∈ℤ∑l∈ℤ2|Gd(Md(jk),l−Md−1(jk)x)|,=(∑j=−∞−V−1+∑j=−VV+∑j=V+1∞)∑k=−QQ(∑l1=−NN∑l2=−NN),|Gd(Md(jk),l−Md−1(jk)x)|.
Due to the fact that j↛−∞, then from (8.1):
∑j∈ℤ∑k∈ℤ∑l∈ℤ2|Gd(Md(jk),l−Md−1(jk)x)|,=∑j=−VV∑k=−QQ(∑l1=−NN∑l2=−NN),|ℐd(Md(jk),x,l−Md−1(jk)x)|+∑j=V+1∞∑k=−QQ(∑l1=−NN∑l2=−NN),2−32j‖f‖2‖h‖2|h(−l+Md−1(jk)x)|.
(8.2)
Note that since from Claim 1, the function ℐd is continuous, it follows that the series in the first term of (8.2) converges.
On the other hand, if:
S=Sup |h(−l+Md−1(jk)x)|, it follows that the series in the second term of (8.2) converges to:
S‖f‖2‖h‖2(2N+1)2(2Q+1)(∑j=V+1∞2−32j).
Thus, for d=1,2:
∑j∈ℤ∑k∈ℤ∑l∈ℤ2|Gd(Md(jk),l−Md−1(jk)x)|,
converges.
Hence, for fixed x∈BR(0)¯, the triple series:
∑j∈ℤ∑k∈ℤ∑l∈ℤ2ℐd(Md(jk),x,l−Md−1(jk)x),
converges uniformly.
Proof of Claim 3. By Claim 1, the function ℐd is continuous on GL2(ℝ)×BR(0)¯×ℝ2, and by Claim 2, the triple series:
∑j∈ℤ∑k∈ℤ∑l∈ℤ2ℐd(Md(jk),x,l−Md−1(jk)x),
converges uniformly on BR(0)¯.
Then in particular for x=0, the triple series:
∑j∈ℤ∑k∈ℤ∑l∈ℤ2ℐd(Md(jk),0,l),
converges absolutely and uniformly on BR(0)¯.
Hence, for each d=1,2:
limx→0 Wd(x)=∑j∈ℤ∑k∈ℤ∑l∈ℤ2(limx→0ℐd(Md(jk),x,l−Md−1(jk)x)),=∑j∈ℤ∑k∈ℤ∑l∈ℤ2ℐd(Md(jk),0,l)=Wd(0).
This proves Claim 3.
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