Chapter 2

Design and Analysis of Ridgelet and Curvelet

Transforms

2.1 Ridgelet Transform

In dimensions two and higher, wavelets can efficiently represent

only a small range of the full diversity of interesting behavior. In effect,

wavelets are well adapted for point like phenomena, where in

dimensions greater than one, interesting phenomena can be organized

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along lines, hyperplanes, and other non point like structures, for

which wavelets are poorly adapted.

A recently developed multiresolution analysis is Ridgelet

analysis [41]. It makes available representations by superpositions of

ridge functions or by simple elements that are in some way related to

ridge functions r(a1x1+…+anxn); these are functions of n variables,

constant along hyperplanes a1x1+…+anxn= c; the graph of such a

function in dimension two looks like a ‘ridge’. The terminology ‘Ridge

function’ arose first in tomography, and ridgelet analysis makes use of

a key tomographic concept, the Radon transform [42].

But multiscale ideas as found in the work of Littlewood and Paley

[18] or Calderon and culminating in wavelet theory appear as a crucial

tool. The ridgelet analysis takes the localization notion from wavelet

theory: at all possible locations and orientations, fine scale ridgelets are

concentrated near hyperplanes. Like the wavelets did for point

singularities in dimension one, the ridgelets do for linear singularities in

dimension two – provide an extremely sparse representaion; both either

wavelets or Fourier can not have a similar advantage in representing

linear singularities in dimension two.

2.1.1 Construction of Ridgelet Transform

The continuous ridgelet transform (CRT) in 2R can be defined as

follows [48].

Pick a smooth univariate function ψ : →R R with sufficient

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decay and vanishing mean, ( ) 0.t dtψ =∫ for each a>0, each b∈ R and

each [0,2 )θ π∈ , define the bivariate function 2 2, , :a b θψ →R R by

1/2, , 1 2( ) . ((cos( ) sin( ) ) / ).a b x a x x b aθψ ψ θ θ−= + − (2.1)

This function is constant along “ridges” 1 2cos( ) sin( ) .x x constθ θ+ =

Perpendicular to these ridges it is known as a wavelet; so the name

ridgelet. If an integrable bivariate function ( )f x is given, its ridgelet

coefficients are defined as

, ,( , , ) ( ) ( )f a ba b x f x dxθθ ψℜ = ∫ (2.2)

An idea on ψ guarantees that 2 2ˆ| ( ) | dψ λ λ λ− < ∞∫ , and further that ψ is

normalized such that

2 2ˆ| ( ) | 1dψ λ λ λ− =∫ (2.3)

The reconstruction formula is

2

, , 3 40 0

( ) ( , , ) ( )f a b a

da df x a b x dbπ

θ π

θθ ψ∞ ∞

− ∞= ℜ∫ ∫ ∫ (2.4)

valid for functions that are both square integrable and integrable,

which shows that any function can be written as a superposition of

‘ridge’ functions. This representation is stable as it has a Parseval

relation:

2

2 23 4

0 0| ( ) | | ( , , ) |f a

da df x dx a b dbπ

π

θθ∞ ∞

− ∞= ℜ∫ ∫ ∫ ∫ (2.5)

This approach generalizes to any dimension as follows:

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Given a ψ obeying 2ˆ| ( ) | 1ndψ λ λ λ− =∫ , define , , ( ) (( ) / ) /a b u x u x b a aψ ψ ′= −

and , ,( , , ) ,f a b ua b fθ ψℜ = . Then there is an n-dimensional

reconstruction formula

, , 1( , , ) ( )n f a b u na

daf c a b u x dbduψ += ℜ∫ ∫ ∫ (2.6)

with du the uniform measure on the sphere; and a Parseval’s relation

2 22 1( )

|| || | ( , , ) |n n f nL R a

daf c a b u dbdu+= ℜ∫ ∫ ∫ (2.7)

The continuous ridgelet transform is connected with the Radon

transformation [42]. Put ( , ) ( ) ( )Rf u t f x u x t dxδ ′= −∫ in place of the

integral of f over the hyperplane u x t′ = , then ,( , , ) , ( , )f a ba b u Rf u tψℜ = ,

where , ( ) (( ) / ) /a b t t b a aψ ψ= − is a 1d wavelet. Thus the Ridgelet

transform is the application of a 1d wavelet transform to the pieces of

the Radon transform, where u is constant and t is variable.

For instance, let ‘g’ be the mutilated Gaussian function as

2 21 2( , ) 1 ,1 2 { 0}2

x xg x x ex

− −= >

2x R∈ (2.8)

This is smooth away from that line and discontinuous along the line

2 0x = . The Radon transform of such function can be calculated as

2( )( , ) ( sin / | cos |)tRg t e tθ θ θ−= Φ − , t R∈ , [0,2 ]θ π∈ (2.9)

where 2

( ) u

vv e du

∞−Φ ≡ ∫ .

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An insight into the form of the CRT from this formula is,

remember that the wavelet 2

, , . ( sin / | cos |)ta b e tψ θ θ− Φ − needs to be

evaluated. Effectively, the Gaussian window 2te− makes small

difference. It is of rapid decay and smooth, hence it does little of

interest; consequently the object of real interest is , , ( ( ) )a b s tψ θΦ − ,

where ( ) sin / | cos |s θ θ θ= . Define then ,( , ) , ( )a bW a b tψ= Φ − , which is a

smooth sigmoidal function known as wavelet transform. By the

invariance of the scale the wavelet transform,

1/2, , ( ( ) ) ( ( ) , ( ) ). | ( ) |a b s t W s a s b sψ θ θ θ θ −Φ − = for (0, )θ π∈ and a same

relationship holds for ( , 2 )π π . In short, for a caricature of ( , , )f a b θℜ ,

there exists, which is a simple rescaling of the wavelet transform of Φ

as function of θ , where θ is a function of a and b . This rescaling is

gentle away from / 2θ π= to 3 / 2θ π= and smooth, where it has

singularities.

It is observed that in a certain manner the Continuous ridgelet

transform(CRT) of ‘g’ is sparse. If a wavelet like Meyer is used, the

Continuous ridgelet transform belongs to 3( / )pL da a dbdθ for every

0p > . It is nothing but notifying that the Continuous ridgelet

transform decays rapidly either moving spatially away from 0b = or

{ / 2,3 / 2}θ π π∈ going to fine scales 0a → .

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The formula for the Continuous ridgelet transform of f by

utilizing the Fourier domain where f̂ is denoted as Fourier transform,

, ,1 ˆˆ( , , ) ( ) ( )

2f a ba b f dθθ ψ ξ ξ ξπ

ℜ = ∫ (2.10)

where , ,ˆ ( )a b θψ ξ is denoted in the frequency plane as a distribution

supported on the radial line. Denoting ( , ) ( .cos( ), .sin( ))ξ λ θ λ θ λ θ= ,

eq.2.10 can be written as

1/21 ˆˆ( , , ) ( ) ( ( , ))2

i bf a b a a e f dλθ ψ λ ξ λ θ λ

π

∞−

− ∞ℜ = ∫ (2.11)

From this it can be said that the Continuous ridgelet transform is

obtained by integrating the weighted Fourier transform ,ˆ( ) ( )a bw fξ ξ in

the frequency domain along a radial line, where weight , ( )a bw ξ is given

by 1/2 ˆ ( | |)a aψ ξ times an exponential in i be λ− . The other way it can be

seen that the function of b (with a and θ fixed) , ( ) ( , , )a fb a bθρ θ= ℜ ,

satisfies

1, 1 ,ˆ( ) { ( )}a ab Fθ θρ ρ λ−= (2.12)

where 1F stands for the 1-d Fourier transform, and

1/2,

ˆˆ ˆ( ) ( ) ( ( , ))a a a fθρ λ ψ λ ξ λ θ= λ− ∞ < < ∞ (2.13)

is the restriction of ,0ˆ( ) ( )aw fξ ξ to the radial line. Hence, conceptually,

the CRT at a certain scale a and angle θ can be obtained by the

following steps:

• 2-d Fourier Transform, obtaining ˆ ( )f ξ

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• Radial Windowing, obtaining ,0ˆ( ) ( )aw fξ ξ , say

• 1-d Inverse Fourier Transform along radial line, obtaining , ( )a bθρ

, for all b R∈ .

• 1-d Wavelet Transform for each slice of the above, obtaining

ridgelet coefficients

Based on the above steps the flow graph of CRT is as follows:

Fig. 2.1 Ridgelet Transform flow graph.

2.1.2 Discrete Ridgelet Transform

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For applications to be carried out, it is required to have ridgelets

in discrete domain [43]. Expansions by frames [44] and orthonormal

bases are typical discrete domain representations.

The discrete representations by frames can be formulated as:

To find a method for sampling , ,( , , )j j k j la b θ , so that frame bounds are

obtained, an equivalence exist as

2 2 3, ,

, ,| ( , , ) | | ( , , ) | /f j j k j l f

j k la b a b da a dbdθ θ θℜ ≈ ℜ∑ ∫ ∫ ∫ (2.14)

To simplify, assume that {1 | | 2}ˆ ( ) 1 ξψ λ ≤ ≤= although the frame result holds

for a large class of ψ ’s as exposed in [44]. Guided by the Little wood-

Paley and the wavelet theories, the scale a and location parameter b

are discretized dyadically, as 0 2 jja a= and , 2 2 j

j kb kπ −= . Following

eq.2.11 the ridgelet coefficients may be written as

1

/2 2 2, 2 | | 2

1 ˆ( , , ) 2 ( ( , ))2 j j

jj if j j ka b e f dλ π

λθ ξ λ θ λ

π +

−− −

≤ ≤ℜ = ∫ (2.15)

and hence, the Plancherel theorem gives

2 2 2, 2 ,012 | | 2

1 ˆ| ( , , ) | | | | ( ( , )) |2f j j k j

k j ja b w f d

λ

θ ξ λ θ λπ +≤ ≤

ℜ =∑ ∫ (2.16)

In short, integrating the square of the Fourier transform along a

dyadic segment is similar to the sum of squares of ridgelet coefficients

across varying spatial location at a fixed scale and angular location.

The angular variable θ when it is discretized results to

evaluating a sampling of such segment-integrals from where the

integral of 2ˆ| ( ) |f ξ over the entire frequency domain requires to be

inferred. This is not possible without support constraints on f , as

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functions f may always be formed with f(x) having slow decay as

| |x → ∞ so that f̂ will become negligible on a collection of disjoint

segments without being identically zero. However, under a support

limitation, so that f is supported inside the unit disk(or any other

compact set), the integrals over the segments can give enough

information to infer 2ˆ| ( ) |f dξ ξ∫ .

Indeed, under a support constraint, the FT ˆ ( )f ξ is bandlimited

function, and over ‘cells’ of appropriate size can only display very

banal behavior. If sampling is carried out once per cell, much of the

behavior of the object can be captured and from those samples size of

the function can be inferred. The solution found by Candes is to

sample with increasing angular resolution at increasingly fine scales,

like , 2 2 jj l lθ π −= .

This strategy gives the equivalence in eq. 2.14. It then follows

that the collection 0

/21 , 2 , ( , , ){2 (2 ( cos( ) sin( ) 2 2 ))}j j j

j l j l j j l kx x kψ θ θ π −≥+ − is a

frame for the unit disk; for any f supported in the disk with finite L2

norm,

22

, , , ,| , |, ,j k l j j k j l

f fa bψ θ⟨ ⟩ ≈∑ (2.17)

The construction generalizes to any dimension n; in two dimensions,

the discretization involved the sampling of angles from the circle, in n

dimensions the sampling of angles from the unit sphere. The angular

variable is also sampled at increasing resolution so that at scale j, the

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discretized set is a net of nearly equispaced points at a distance of

order 2-j , see [44] for details.

The existence of frame bounds implies that there are ‘dual

ridgelets’ , ,j k lψ% so that

, , , ,, ,

, j k l j k lj k l

f f ψ ψ= ⟨ ⟩∑ % (2.18)

and

, , , ,, ,

, j k l j k lj k l

f f ψ ψ= ⟨ ⟩∑ % (2.19)

with equality in an L2 – manner, and so that

22 22, , , ,

, , , ,| , | | , |j k l j k l Lj k l j k l

f f fψ ψ⟨ ⟩ ≈ ⟨ ⟩ ≈∑ ∑% (2.20)

There are no closed form expressions for the structure of dual

ridgelets , ,j k lψ% , only qualitative properties are known.

The discrete representations in orthonormal bases can be

formulated as follows:

Donoho [45] had broaden the idea of ridgelet, to allow the

possibility of systems that obey some frequency/angle localization

properties, and shown that if this broader idea is allowed, then it

becomes possible to have orthonormal ridgelets whose elements can

be specified in closed form. Such a system can be described as

follows:

Let ,( ( ) : , )j k t j kψ ∈ ∈Z Z be an orthonormal basis of Meyer wavelet for

L2(R) [46-49], and let 0

0

0 1, , 0( ( ), 0,....2 -1; ( ), ³ , 0,..., 2 -1)i i

i l i lw l w i i lθ θ= = be an

orhtonormal basis for L2[0,2Π) made of periodized Lemarie scaling

24

functions 0

0,i lw at level 0i and periodized Meyer wavelets 1

,i lw at levels

0i i≥ .(assume particular normalization of these functions).Let ,ˆ ( )j kψ ω

denote the Fourier transform of , ( )j k tψ , and define ridgelets ( )xλρ ,

( , ; , , )j k i lλ ε= as functions of 2x ∈ R utilizing the frequency domain

definition

12

, , , ,ˆ ˆ ˆ( ) | | ( (| |) ( ) ( | |) ( )) / 2j k i l j k i lw wλε ερ ξ ξ ψ ξ θ ψ ξ θ π

−= + − + (2.21)

Here the indices run as follows: ,j k ∈ Z , 10,0,...., 2 1;il i i i j−= − ≥ ≥ .Notice

the restrictions on the range of l and on i .Let Λ denote the set of all

such indices λ [48,49]. It turns out that ( )ρ λ λ ∈ Λ is a complete

orthonormal system for 2 2( )L R .

In the present form, the system is not visibly related to ridgelets

as defined earlier, but two connections can be exhibited. First, define

a fractionally-differentiated Meyer wavelet:

12

, ,1 ˆ( ) | | ( )

2i t

j k j kt e dωψ ω ψ ω ωπ

∞+

− ∞

= ∫ . Then for 21 2( , )x x x= ∈ R ,

2

, 1 2 ,0

1( ) ( cos sin ) ( )4 j k i lx x x w d

π

λερ ψ θ θ θ θ

π+= +∫ (2.22)

Each , 1 2( cos sin )j k x xψ θ θ+ + is a function known as ridge of 2x ∈ R , i.e., a

function of type 1 2( cos sin )r x xθ θ+ .Therefore λρ is determined by

“averaging" ridge functions with ridge angles θ localized near

, 2 / 2ii l lθ π= .A second relation comes by taking the sampling scheme

underlying ridgelet frames as discussed in previous section . This

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scheme, says that the behavior along line segments must be sampled

and that those segments should be separated in the angular variable

proportional to the scale 2 j− of the wavelet index. The orthonormal

ridgelet system contains elements which are arrainged angularly in

the form; the elements ˆλρ are localized `near' such line segments

because the wavelets , ( )i lwε θ are localized `near' specific points ,i lθ .

The analysis of Orthonormal ridgelet can be seen as a kind of

wavelet analysis in the Radon domain; if ( , )Rf tθ denotes the Radon

transform and if ( , )tλτ θ denotes the function

, , , ,( ) ( ) ( ) ( )) / 2j k i l j k i lt w t wε εψ θ ψ θ π+ ++ − + , the ( : )λτ λ ∈ Λ gives a system of

antipodally symmetrized nonorthogonal tensor wavelets. The ridgelet

coefficients λα ¸ are given by analysis of the Radon transform via

[ , ]Rfλ λα τ= .This means that the ridgelet coefficients contain within

them information about the smoothness in t and θ of the Radon

transform. In particular, if the Radon transform [50,51] exhibits a

certain degree of smoothness, it can be seen that the ridgelet

coefficients exhibit a corresponding rate of decay.

2.1.3 Ridgelet Synthesis

Consider again the Gaussian-windowed halfspace eq.2.8. The

CRT of this object is sparse, which suggests that a discrete ridgelet

26

series can be made which gives a sparse representation of g. This can

be seen in two ways.

It can be shown that there exist constructive and simple

approximations using dual frames (which are not pure ridge

functions) which achieve any desired rate of approximation on

compact sets [41]. Indeed, let A be compact and iψ be a ridgelet-frame

for L2(A). From the exact series

, i ii

g g ψ ψ= ⟨ ⟩∑ % (2.23)

obtain the m-term approximation mg% which keeps only the dual-

ridgelet terms corresponding to the m largest ridgelet coefficients

, ig ψ⟨ ⟩ ; then, the mg% attains the rate

2 ( )r

m rLg g C m−− ≤

A% for any r >0,

provided let ψ is a smooth function whose Fourier transform is

supported away from 0 (like the Meyer wavelet). The result is

generalized to any dimension n and is not restricted to the Gaussian

window. The sparsity of the ridgelet coefficient sequence is the

argument behind this fact; each ridgelet coefficient , ,, ( ,.)j k g j lRψ θ⟨ ⟩

being the 1d wavelet coefficient of the Radon transform ,( ,.)g j lR θ for

fixed θ . From the relation 2

( , ) ( .sin / | cos |)tgR t e tθ θ θ−= Φ − , it is observed

that the coefficients , ,, a bf θψ⟨ ⟩ decay rapidly as θ and/or b move away

from the singularities ( / 2, 0)tθ π= = and ( 3 / 2, 0)tθ π= = .

27

Donoho [41] shows that the orthonormal ridgelet coefficients of

g belong to pl for every p > 0. This means that if an m-term

approximation has been formed by selecting the m terms with the m-

largest coefficients, the reconstruction 1 i i

mm if λ λα ρ

== ∑ has any desired

rate of approximation.

The argument for the orthonormal ridgelet approximation goes

as follows. Because orthonormal ridgelet expansion amounts to a

special wavelet expansion in the Radon domain, the question reduces

to considering the sparsity of the wavelet coefficients of the Radon

transform of g. Now, the Radon transform of g, as indicated above, will

have singularities of order 0 (discontinuities) at ( 0, / 2)t θ π= = and at

( 0, 3 / 2)t θ π= = . Outside of these points the Radon transform is in-

finitely differentiable, uniformly so, away from any neighborhood of

the singularities. If it is `zoomed in' to fine scales one of the

singularities and made a fine change of coordinates, the picture can

be seen is such function as 1/2( , ) | | ( / | |)S u v v u vσ−= for a certain nice

bounded function (.)σ . The wavelet coefficients of such an object are

sparse.

2.1.4 Multiscale Ridgelets

Think of orthoridgelets as objects which have a “length” of about

1 and a “width” which can be arbitrarily fine. The multiscale ridgelet

28

system renormalizes and transports such objects, so that one has a

system of elements at all lengths and all finer widths.

The construction begins with a smooth partition of energy

function 1 2( , ) 0,w x x ≥ 20 ([ 1,1] )w C ∞∈ − obeying

1 2

21 1 2 2,

( , ) 1k k

w x k x k− − ≡∑ .

Define a transport operator, so that with index Q indicating a dyadic

square 1 2( , , )Q s k k= of the form 1 1 2 2[ / 2 , ( 1) / 2 ) [ / 2 , ( 1) / 2 )s s s sk k k k+ × + , by

1 2 1 1 2 2( )( , ) (2 , 2 )s sQT f x x f x k x k= − − . The multiscale ridgelet [52] with index

( , )Qµ λ= is then

2 ( ).sQT wµ λψ ρ= (2.24)

In short, one transports the normalized, windowed orthoridgelet.

Letting Qs denote the dyadic squares of side 2 s− , It can be

defined the subcollection of monoscale ridgelets at scale s:

{( , ) : Q , }s sM Q Qλ λ= ∈ ∈ Λ .

It is immediate from the orthonormality of the ridgelets that

each system of monoscale ridgelets makes tight frame, in particular

obeying the Parseval relation

22 2, || ||

s

LM

f fµµ

ψ∈

⟨ ⟩ =∑ (2.25)

It follows that the dictionary of multiscale ridgelets at all scales,

indexed by

1s

sM M

≥= U

is not frameable, as there is energy blow-up,

29

2,M

fµµ

ψ∈

⟨ ⟩ = ∞∑ (2.26)

The multiscale ridgelets dictionary is simply too massive to form a

good analyzing set. It lacks interscale orthogonality; ( , )Qψ λ is not

typically orthogonal to ( , )Qψ λ′ ′ if Q and Q′ are squares at different

scales and overlapping locations. In analyzing a function using this

dictionary, the repeated interactions with all different scales causes

energy blow-up as in eq.2.26.

The construction of curvelets solves this problem by in effect

disallowing the full richness of the multiscale ridgelets dictionary.

Instead of allowing all different combinations of “lengths” and

“widths,” it will allow only those where 2width length≈ .

2.2 Curvelet Transform

2.2.1 Construction of Curvelets

Curvelets provide optimally sparse representation of otherwise

smooth objects. The curvelet transform [1,2,52] is derived by

combining various notions:

1. Ridgelets: a method of analysis suitable for objects with

discontinuities across straight lines [53] as described in

previous section.

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2. Multiscale Ridgelets: a pyramid of windowed ridgelets,

renormalized and transported to a wide range of scales and

locations as described in previous section.

3. Bandpass Filtering (Subband filtering): a method of separating

an object out into a series of disjoint scales. The remedy to the

“energy blow-up” as in eq.2.26 is to decompose f into subbands

using standard filterbank ideas. Then one specific monoscale

dictionary sM to analyze one specific (and specially chosen)

subband. Defined coronae of frequencies 2 2 2| | [2 , 2 ]s sξ +∈ , and

subband filters sD extracting components of f in the indicated

subbands; a filter oP deals with frequencies |ξ|≤1. The filters

decompose the energy exactly into subbands:

2 2 2

2 2 2.o s

sf P f D f= + ∑ (2.27)

The construction of such operators is standard [54]; the

coronization oriented around powers 22s is nonstandard and

essential here. Explicitly, a sequence of filters oΦ and

4 22 2 (2 .), 0,1, 2,...,s ss sψ ψ= = with the following properties: oΦ is a

lowpass filter concentrated near frequencies | | 1ξ ≤ ; 2sψ is

bandpass, concentrated near 2 2 2| | [2 , 2 ]s sξ +∈ and there is

2 22

0

ˆ ˆ( ) (2 ) 1so

sξ ψ ξ−

≥

Φ + =∑ ξ∀ (2.28)

Hence, sD is simply the convolution operator 2 *s sD f fψ= .

31

The curvelet transform can be described as a combination of

reversible transformations as follows:

• 2D Wavelet transform

• Smooth partitioning

• Ridgelet transform

• Radon transform

• 1D Wavelet transform

The flow graph is as shown in Fig. 2.2

Fig. 2.2 Flow graph of curvelet transform

Assembling the above ingredients, the definition of the curvelet

transform can be sketched as:

32

let M ′ consist of M merged with the collection of integral triples

1 2( , , , )s k k e where 0, {0,1},s e≤ ∈ indexing all dyadic squares in the plane

of side 2s> 1. The curvelet transform is a map 2 2 2( ) ( ),L l M ′R a yielding

curvelet coefficients ( : ).Mµα µ ′∈ These come in two types.

At coarse scales there are wavelet coefficients:

1 2, , , , ,s k k e oW P fµα = ⟨ ⟩ 1 2( , , ) \ ,s k k M Mµ ′= ∈ (2.29)

where each 1 2, , ,s k k eW is a Meyer wavelet, while at fine scale there are

multiscale ridgelet coefficients of the bandpass filtered object:

, ,sD fµ µα ψ= ⟨ ⟩ , 1, 2,.....sM sµ ∈ = (2.30)

Note well that for s>0, each coefficient associated to scale 2 s− derives

from the subband filtered version of sf D− − and not from f .

2.2.2 Analysis

The curvelet decomposition [2] can be stated in the following

form:

• Subband Decomposition: The object f is filtered into subbands:

1 2( , , ,....)of P f f f∆ ∆a with the property that the passband filter

s∆ is concentrated near frequencies 2 2 2[2 ,2 ],s s+ e.g.

2 *s s f∆ = Ψ , 22ˆ ˆ( ) (2 ).ssψ ξ ψ ξ−= (2.31)

• Smooth Partitioning: Each subband is smoothly windowed into

“squares” of an appropriate scale:

33

Q( ) sQs Q sf w f ∈∆ ∆a (2.32)

Where Q denote dyadic square

1 1 2 2[ / 2 , ( 1) / 2 ) [ / 2 , ( 1) / 2 )s s s sQ k k k k= + × + (2.33)

and let Q be the collection of all such dyadic squares. The notation

Qs will correspond to all dyadic squares of scale s. Let Qw be a

window centered nearQ , obtained after dilation and translation of

a single w , such that the 2Qw ’s, QsQ ∈ , make up a partition of

unity.

• Renormalization: Each resulting square is renormalized to unit

scale

1( ) ( ),Q Q Q sg T w f−= ∆ QsQ ∈ . (2.34)

Here multiscale ridgelets are defined by ,{ : , Q , }Q o ss s Qλρ λ≥ ∈ ∈ Λ

, ,Q Q Qw Tλ λρ ρ= (2.35)

Where

1 1 2 22 (2 ,2 ).s s sQT f f x k x k= − −

The multiscale ridgelet system renormalizes and transports the

ridgelet basis, so that one has a system of elements at all lengths

and all finer widths.

• Ridgelet Analysis: Each square is analyzed in the orthonormal

ridgelet system. This is a system of basis elements λρ making

an orthobasis for 2 2( ) :L R

, ,Qgµ λα ρ= ⟨ ⟩ ( , ).Qµ λ= (2.36)

34

Fig. 2.3 illustrates an overview of the arrangement of the curvelet

transform.

For an understanding of why the procedure might be organized

as it is, consider Fig. 2.4.

Consider an object f which exhibits an edge. Upon subband

filtering, each resulting fine-scale subband output s f∆ will contain a

map of the edge in f , thickened out to a width 22 s− according to the

scale of the subband filter operator. This gives the subband the

appearance of a collection of smooth ridges. When it is smoothly

partitioned each subband into `squares', it can be seen either an

`empty square'- if the square does not intersect the edge - or a ridge

fragment. Moreover, the ridge fragments are nearly straight at fine

scales, because the edge is nearly straight at fine scales. Such nearly

straight ridge fragments are precisely the desired input for the ridgelet

transform.

Curvelets are then given by

, ,s Qµ λγ ρ= ∆ ( , Q ).sQµ λ= ∈ Λ ∈ (2.37)

It follows from both the definition of the ridgelets[53] and of the

multiscale ridgelets that ,ˆQ λρ is supported near the corona 1 1[2 , 2 ].j j s+ + +

Therefore, the only multiscale ridgelets that survive to the passband

filtering are those sλ such that the scale j satisfies

2 .j s s+ ≈ (2.38)

35

Fig. 2.3 Overview of organization of the curvelet transform.

Fig. 2.4 Spatial decomposition of a single subband.

36

Fig. 2.5 Big Mac image and stages of curvelet analysis.

The above observation is an important key for having thorough

knowledge about the main properties of the curvelets which will be

described in the coming section.

Fig. 2.5 shows the operation of curvelet transform on a digital

image, where the size of the image is 256 x 256 pixels. The main three

stages of the curvelet transform are clearly described.

2.2.3 Synthesis

The procedural definition of the reconstruction algorithm is

• Ridgelet Synthesis: Each ‘square’ is reconstructed from the

orthonormal ridgelet system.

37

( , )Qg Q λλ

α λ ρ= ∑ (2.39)

• Renormalization: Each ‘square’ resulting in the previous stage is

renormalized to its own proper square

( ) ,Q Q Qh T g= QsQ ∈ (2.40)

• Smooth Integration: Reverse the windowing dissection to each of

the windows reconstructed in the previous stage of the

algorithm.

. .s

s Q QQ

f w h∈

∆ = ∑¤

(2.41)

• Subband Recomposition: Undo the bank of subband filters,

using the reproducing formula:

0

( ) ( ).o o s ss

f P P f f>

= + ∆ ∆∑ (2.42)

Fig. 2.6 shows the different stages of the reconstruction of an

image from the datum of its curvelet coefficients.

Fig. 2.6 Curvelet coefficients and stages of curvelet synthesis.

38

2.2.4 Properties of Curvelets

The curvelet tight frame for 2 2( )L R is a collection of analyzing

elements 1 2( , )x xµ µγ γ= indexed by tuples Mµ ′∈ to be described below.

It has the following key properties:

• Transform Definition or Existence of Coefficient

Representaters(Frame Elements):

, ,fµ µα γ≡ ⟨ ⟩ .Mµ ′∈ (2.43)

• Tight Frame or Parseval relation:

2 22|| || | | .

Mf µ

µα

′∈= ∑ (2.44)

• 2L Reconstruction Formula:

, .M

f f µ µµ

γ γ′∈

= ⟨ ⟩∑ (2.45)

• Formula for Frame Elements:

,sµ µγ ψ= ∆ Q .sµ ∈ (2.46)

With an equality holding in a 2L sense. Both equalities in

eqs.2.44 and 2.45 are having practical relevance considerably: an

object can be expanded into a curvelet series in a stable and very

concrete fashion. Although the Parseval relation apart from the

decomposition formula are reminiscent of an orthonormal

decomposition, it is emphasized that the curvelet system is

redundant, and hence, not orthonormal.

39

Along with the tight frame property, the curvelet transform

exhibit an interesting structure that sets it apart from existing image

representations:

• The curvelet transform exhibits a new kind of pyramid

structure.

• Curvelet frame elements exhibit new scaling laws.

• Curvelets provide an efficient representation of images with

edges.

2.2.5 Pyramid Structure

Fig. 2.5 highlighted the three stages of the curvelet transform.

The first two stages, called, the decomposition into subbands and the

spatial localization of each subband are well known although the

coronization [22s , 22s+2] associated with the bandpass filters is

nonstandard. In a well known wavelet pyramid, the coronization

would, instead, be of the form [2s , 2s+1]. Keeping this important

difference in mind, the curvelet transform might be described as a

sequence of two pyramids:

• A first pyramid, indexed by Q whose range is recalled to be the

set of all dyadic squares, which localizes the image both in

space and frequency;

• A second pyramid, namely, the ridgelet pyramid which analyzes

each renormalized block of image data that obey spatial and

40

frequency localization properties (they were denoted by Qg in the

previous section) using directional and anisotropic elements.

First, although the interpretation of the scale may appear somewhat

ambiguous, the transform displays the well known ideas of

• dyadic scale, and

• dyadic location.

Second, the ridgelet adds an originality to the pyramid: the latter

transform introduces two new ingredients, namely,

• direction, and

• microlocation.

There are two levels of location corresponding to the curvelet

transform: a coarse level that corresponds to the dyadic square Q , an

approximate location of a curvelet is given by this level; and a finer

one, the location of the curvelet relative to this dyadic square is given

by this level, a piece of information given by the parameter / 2 jk .The

word ‘microlocation’ refers to this finer level.

2.2.6 Scaling Laws

In the curvelet pyramid, define the scale as being the side length

of the dyadic square Q indexing a curvelet element. With this

definition and eq.2.37, there exist that the scale of a curvelet is

roughly equal to its length. In other words,

( ) 2 .slength µγ −≈ (2.47)

41

The following anisotropy scaling relation is key to the construction;

curvelets have support obeying the scaling law

2width length≈ (2.48)

Hence, the anisotropy is increasing with decreasing scales according

to a quadratic power law. This property seems to be new in the

literature of computational harmonic analysis. Fig. 2.7 shows a few

curvelets at various scales.

Fig. 2.7 Showing few curvelets at various scales. The left corresponds

to a scale s = 2, and the right corresponds to a scale s = 3.

The quadratic scaling relation is followed from the previous

observation i.e. in eq. 2.38 . A curvelet is supported near the dyadic

square Q and hence, its length is effectively as that of Q , i.e. 2 s− . eq.

2.38 provides its approximate width, the width of ,Q λρ is about

22 2s j s− − −≈ , which justifies eq. 2.48.

42

This same principle, namely, eq. 2.38 provides two additional

scaling relations. Because ,j s≈ it is followed from the definition of a

curvelet coefficient

,Qgµ λα ρ= ⟨ ⟩

(where Qg is given by eq. 2.34) that

• the number of directions is about proportional to the inverse of

the scale, and

• the number of microlocations is about proportional to the

inverse of the scale.

The other way to explain the scaling relations that operate within the

curvelet transform is to test the transition from one scale to the next

finer scale, i.e. from 2 s− to 12 .s− − Each refinement of scale

• Doubles the spatial resolution; that is, the size of the dyadic

squares in the pyramid is reduced by a factor of two (much like

wavelet pyramids).

• Doubles the angular resolution; that is, the number of

directions of the anisotropic analyzing elements is increased by

a factor of two.

This combination is very much interested and new.

43

2.2.7 Fast Discrete Curvelet Transforms

The curvelet transform is mathematically valid, and a very

promising potential in traditional application areas for wavelet-like

ideas such as image processing, data analysis, scientific computing

and communications. To realize this potential though, and deploy this

technology to a wide range of problems, one would need a fast and

accurate discrete curvelet transform [55] operating on digital data.

The Fast Discrete Curvelet Transform (FDCT) is implemented via

USFFT and Wrapping , the details about implementations are

available in [16 ]. The algorithms are as follows:

These digital transformations are linear and take as input Cartesian

arrays of the form 1 2[ , ]f t t , 1 20 ,t t n≤ < , which allows to think of the

output as a collection of coefficients ( , , )Dc j l k obtained by the digital

analog to eq. 2.43

1 2

1 2 , , 1 20 ,

( , , ) : [ , ] [ , ],D Dj l k

t t nc j l k f t t t tϕ

≤ <

= ∑ (2.49)

where each , ,Dj l kϕ is a digital curvelet waveform (the superscript

D stands for “digital”).

The first implementation which is referred to as the FDCT via

USFFT, and whose architecture is as follows:

1. Apply the 2D FFT and obtain Fourier samples

1 2 1 2ˆ[ , ], / 2 , / 2.f n n n n n n− ≤ <

44

2. For each scale/angle pair ( , )j l , resample (or interpolate)

1 2ˆ[ , ]f n n to obtain sampled values 1 2 1

ˆ[ , tan ]lf n n n θ− for 1 2( , ) jn n P∈ .

3. Multiply the interpolated (or sheared) object f̂ with the

parabolic window jU% , effectively localizing f̂ near the

parallelogram with orientation lθ , and obtain

, 1 2 1 2 1 1 2ˆ[ , ] [ , tan ] [ , ]j l l jf n n f n n n U n nθ= −% % .

4. Apply the inverse 2D FFT to each ,j lf% , hence collecting the

discrete coefficients ( , , )Dc j l k .

For practical purposes it takes 2( log )O n n flops for computation, and

requires 2( )O n storage, where 2n is the number of pixels.

The second implementation which is referred to as the FDCT via

wrapping, and whose architecture is as follows:

1. Apply the 2D FFT and obtain Fourier samples

1 2 1 2ˆ[ , ], / 2 , / 2.f n n n n n n− ≤ <

2. For each scale j and angle l , form the product

, 1 2 1 2ˆ[ , ] [ , ].j lU n n f n n%

3. Wrap this product around the origin and obtain

, 1 2 , 1 2ˆ[ , ] ( )[ , ],j l j lf n n W U f n n=% %

where the range for 1n and 2n is now 1 1,0 jn L≤ < and 2 2,0 jn L≤ <

(for θ in the range ( / 4, / 4)π π− ).

45

4. Apply the inverse 2D FFT to each ,j lf% , hence collecting the

discrete coefficients ( , , )Dc j l k .

This algorithm has computational complexity 2( log )O n n and in

practice, its computational cost does not exceed that of 6 to 10 two-

dimensional FFTs.

Both the forward transform algorithms specified here are

invertible .

2.2.8 Analysis of Curvelet Coefficients

To analyze the curvelet coefficients, they have been displayed for

various images as shown in Figs. 2.8 to 2.10, where the display is as

follows:

1. The low frequency (coarse scale) coefficients are stored at the

center of the display.

2. The Cartesian concentric coronae show the coefficients at

different scales; the outer coronae correspond to higher

frequencies.

3. There are four strips associated to each corona, corresponding

to the four cardinal points; these are further subdivided in

angular panels.

4. Each panel represent coefficients at a specified scale and along

the orientation suggested by the position of the panel.

46

(a) (b)

Fig. 2.8 Curvelet coefficients analysis (a) Original image 1 (b) it’s

coefficients display.

(a) (b)

Fig. 2.9 Curvelet coefficients analysis (a) Original image 2 (b) it’s

coefficients display.

(a) (b)

Fig. 2.10 Curvelet coefficients analysis (a) Original image 3 (b) it’s

coefficients display.

47