design and analysis of ridgelet and curvelet...
TRANSCRIPT
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
15
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
16
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:
17
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 ≡ ∫ .
18
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 → .
19
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
20
• 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
21
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
22
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
23
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
25
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.
30
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