【免费】快速离散剪切波变换教程论文

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Hauser_S_2012_PREPRINT_fast_fst-shearlets-tutorial.zip （1个子文件）

Hauser_S_2012_PREPRINT_fast_fst-shearlets-tutorial.pdf 973KB

Fast Finite Shearlet Transform: a tutorial

S¨oren H¨auser

∗

February 9, 2012

Contents

1 Introduction 1

2 Shearlet transform 3

2.1 Some functions and their properties . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 The continuous shearlet transform . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Shearlets on the cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Scaling function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Computation of the shearlet transform 13

3.1 Finite discrete shearlets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 A discrete shearlet frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Inversion of the shearlet transform . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Smooth shearlets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.5 Implementation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.1 Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.2 Computation of spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6 Short documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.7 Download & Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.8 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.9 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1 Introduction

In recent years, much eﬀort has been spent to design directional representation systems for

images such as curvelets [1], ridgelets [2] and shearlets [10] and corresponding transforms

(this list is not complete). Among these transforms, the shearlet transform stands out since

it stems from a square-integrable group representation [4] and has the corresponding useful

∗

University of Kaiserslautern, Dept. of Mathematics, Kaiserslautern, Germany, haeuser@mathematik.uni-kl.de

arXiv:1202.1773v1 [math.NA] 8 Feb 2012

mathematical properties. Moreover, similarly as wavelets are related to Besov spaces via atomic

decompositions, shearlets correspond to certain function spaces, the so-called shearlet coorbit

spaces [5]. In addition shearlets provide an optimally sparse approximation in the class of

piecewise smooth functions with C

singularity curves, i.e.,

kf − f

≤ CN

−2

(log N)

as N → ∞,

where f

is the nonlinear shearlet approximation of a function f from this class obtained by

taking the N largest shearlet coeﬃcients in absolute value.

Shearlets have been applied to a wide ﬁeld of image processing tasks, see, e.g., [7, 11, 12, 17].

In [9] the authors showed how the directional information encoded by the shearlet transform

can be used in image segmentation. Fig 1 illustrates the directional information in the shearlet

coeﬃcients. To this end, we introduced a simple discrete shearlet transform which translates

the shearlets over the full grid at each scale and for each direction. Using the FFT this

transform can be still realized in a fast way. This tutorial explains the details behind the

Matlab-implementation of the transform and shows how to apply the transform. The software

is available for free under the GPL-license at

http://www.mathematik.uni-kl.de/

haeuser/FFST/

In analogy with other transforms we named the software FFST – Fast Finite Shearlet Transform.

The package provides a fast implementation of the ﬁnite (discrete) shearlet transform.

(a) Forms with diﬀerent edge ori-

entations

(b) Shearlet coeﬃcients

for a =

and s = −1

for a =

for all s

Figure 1: Shearlet coeﬃcients can detect edges with diﬀerent orientations.

This tutorial is organized as follows: In Section 2 we introduce the continuous shearlet trans-

form and prove the properties of the involved shearlets. We follow in Section 3 the path via

the continuous shearlet transform, its counterpart on cones and ﬁnally its discretization on the

full grid to obtain the translation invariant discrete shearlet transform. This is diﬀerent to

other implementations as, e.g., in ShearLab

, see [16]. Our discrete shearlet transform can be

eﬃciently computed by the fast Fourier transform (FFT). The discrete shearlets constitute a

Parseval frame of the ﬁnite Euclidean space such that the inversion of the shearlet transform

http://www.shearlab.org

can be simply done by applying the adjoint transform. The second part of the section covers

the implementation and installation details and provides some performance measures.

2 Shearlet transform

In this section we combine some mostly well-known results from diﬀerent authors. To make this

paper self-contained and to obtain a complete documentation we also include the proofs. The

functions where taken from [15, 14]. The construction of the shearlets is based on ideas from

[12] and [13]. The shearlet transform and the concept of shearlets n the cone was introduced

in [6].

2.1 Some functions and their properties

To deﬁne usable shearlets we need functions with special properties. We begin with deﬁning

these functions and prove their necessary properties. The results will be used later. The

following results are taken basically from [12] and [13].

We start by deﬁning an auxiliary function v : R → R as

v(x) :=











0 for x < 0

35x

− 84x

+ 70x

− 20x

for 0 ≤ x ≤ 1

1 for x > 1.

(1)

This function was proposed by Y. Meyer in [15, 14]. Other choices of v are possible, in [16]

the simpler function

ev(x) =











0 for x < 0

for 0 ≤ x ≤

1 − 2(1 − x)

for

≤ x ≤ 1

1 for x > 1

was chosen. As we will see the useful properties of v for our purposes are its symmetry around

(

) and the values at 0 and 1 with increase in between. A plot of v is shown in Fig. 2(a).

Next we deﬁne the function b : R → R with

b(ω) :=











sin(

v(|ω| − 1)) for 1 ≤ |ω| ≤ 2

cos(

|ω| −1)) for 2 < |ω| ≤ 4

0 otherwise,

(2)

where b is symmetric, positive, real and supp b = [−4, −1] ∪ [1, 4]. We further have that

b(±2) = 1. A plot of b is shown in Fig. 2(b).

Because of the symmetry we restrict ourselves in the following analysis to the case ω > 0. Let

:= b(2

−j

·), j ∈ N

, thus, supp b

= 2

[1, 4] = [2

, 2

j+2

] and b

j+1

) = 1. Observe that b

increasing for ω ∈ [2

, 2

j+1

] and decreasing for ω ∈ [2

j+1

, 2

j+2

]. Obviously all these properties

−0.5 0 0.5 1 1.5

0.2

0.4

0.6

0.8

(a) v(x)

−5 −4 −3 −2 −1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

(b) solid: b(ω), dashed: b(2ω)

Figure 2: The two auxiliary functions v in (1) and b in (2)

carry over to b

. These facts are illustrated in the following diagram where % stands for the

increasing function and & for decreasing function.

ω 2

j+1

j+2

j+3

0 % 1 & 0

j+1

0 % 1 & 0

For j

6= j

the overlap between the support of b

and b

is empty except for |j

−j

| = 1. Thus,

for b

and b

j+1

we have that supp b

∩supp b

j+1

= [2

j+1

, 2

j+2

]. In this interval b

is decreasing

with b

= cos

(

−j

|ω|−1)) and b

j+1

is increasing with b

j+1

= sin

(

v(2

−(j+1)

|ω|−1)). We

get for their sum in this interval

(ω) + b

j+1

(ω) = cos



v(2

−j−1

|ω| −1)



+ sin



v(2

−j−1

|ω| −1)



= 1.

Hence, we can summarize

+ b

j+1

)(ω) =











for ω < 2

j+1

1 for 2

j+1

≤ ω ≤ 2

j+2

j+1

for ω > 2

j+2

Consequently, we have the following lemma

Lemma 2.1. For b

deﬁned as above, the relations

∞

j=−1

(ω) =

∞

j=−1

−j

ω) = 1 for |ω| ≥ 1

and

∞

j=−1

(ω) =











0 for |ω| ≤

sin



v(2ω − 1)



for

< |ω| < 1

1 for |ω| ≥ 1

(3)

hold true.

Proof. In each interval [2

j+1

, 2

j+2

] only b

and b

j+1

, j ≥ −1, are not equal to zero. Thus, it is

suﬃcient to prove that b

+ b

j+1

≡ 1 in this interval. We get that

+ b

j+1

)(ω) = b

−j

|{z}

∈ 2

−j

j+1

, 2

j+2

] = [2, 4]

) + b

−j−1

| {z }

∈ 2

−j−1

j+1

, 2

j+2

] = [1, 2]

)

= cos



· 2

−j

ω − 1)



+ sin



v(2

−j−1

ω − 1)



= cos



v(2

−j−1

ω − 1)



+ sin



v(2

−j−1

ω − 1)



= 1.

The second relation follows by straightforward computation.

Recall that the Fourier transform F: L

) → L

) and the inverse transform are deﬁned

Ff(ω) =

f(ω):=

f(t)e

−2πihω,ti

dt,

−1

f(ω) = f (t) =

f(ω)e

2πihω,ti

dω.

Now we deﬁne the function ψ

: R → R via its Fourier transform as

(ω) :=

(2ω) + b

(ω). (4)

Fig. 3(a) shows the function. The following theorem states an important property of ψ

Theorem 2.2. The above deﬁned function

has supp

= [−4, −

] ∪ [

, 4] and fulﬁlls

j≥0

−2j

ω)|

= 1 for |ω| > 1.

Proof. The assumption on the support follows from the deﬁnition of b. For the sum we have

j≥0

−2j

ω)|

∞

j=0

(2 · 2

−2j

ω) + b

−2j

ω)

∞

j=0

−2j+1

ω) + b

−2j

ω),

where −2j + 1 ∈ {+1, −1, −3, . . .} (odd) and −2j ∈ {0, −2, −4, . . .} (even). Thus, by Lemma

2.1, we get

j≥0

−2j

ω)|

∞

j=−1

−j

ω)

= 1.

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