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WAVELET LEADER BASED MULTIFRACTAL ANALYSIS
Bruno Lashermes
(1)
,St
´
ephane Jaffard
(2)
, Patrice Abry
(1)
(1) Laboratoire de Physique (2) Laboratoire d’Analyse et de Math
´
ematiques Appliqu
´
ees
CNRS - NS Lyon, France CNRS - Univ. Paris XII, Cr
´
eteil, France
{blasherm,pabry}@ens-lyon.fr jaffard@univ-paris12.fr
ABSTRACT
We introduce a new multifractal formalism based on wavelet lead-
ers and study its properties. Comparing it against previously for-
mulated wavelet coefficient based multifractal formalism, we show
first that this wavelet leader based formalism allows first to obtain
the multifractal spectrum over its entire range, and second that it
does not cease to hold when applied to processes embodying un-
usual chirp-type (or oscillating) singularities (as opposed to the
more common cusp-type ones). We illustrate these results and
properties on four examples of multifractal deterministic functions
or stochastic processes containing a graduation of the major dif-
ficulties. We show that this new multifractal formalism benefits
from excellent theoretical and practical performance. Matlab rou-
tines implementing it are available upon request.
1. MOTIVATION
Scale invariance or scaling is a paradigm that has been largely
used in the past ten years to describe/analyse/model data produced
by a wide variety of applications of very different nature (heart
rhythms, hydrodynamic turbulence, computer network traffic, fi-
nancial markets, ...). After seminal works historically developed
in the field of hydrodynamics turbulence [1, 2, 3], the multifractal
framework became a major concept and tool involved in the prac-
tical analysis of scaling in data.
Following its original formulation in turbulence, the multifractal
formalism has been based on the increments of the studied func-
tion. It is however now well-known that it benefits from significant
theoretical and practical improvements when written with wavelet
coefficients. It is usually computed either from a discrete wavelet
transform (DWT) or from a modulus maxima wavelet transform
(MMWT).
We propose here the formulation of a new multifractal formalism
based on wavelet leaders. We explain how and why it overcomes
two major difficulties partially or unsatisfactorily solved by pre-
vious formulations: It enables to determine the multifractal spec-
trum over its entire range and it works equally well for processes
containing usual cusp-type singularities and for those presenting
less common chirp-type singularities. Its extension to higher di-
mensions is straightforward and it has an extremely low compu-
tational cost. The wavelet leader based multifractal formalism
brings a real enhancement in multifractal analysis both from con-
ceptual or practical viewpoints compared to previous formulations
and should definitely be systematically used.
2. MULTIFRACTAL AND WAVELET COEFFICIENTS
•H
¨
older Exponent and Singularity Spectrum.
Let {X(t)}
t∈R
denote the sample path of the function or stochas-
tic process of interest. Its local regularity is commonly studied
via the notion of pointwise H
¨
older exponent. X(t), which is as-
sumed to be locally bounded, belongs to C
α
(t
0
) at t
0
∈ R, with
α ≥ 0, if there exist a constant C>0 and a polynomial P sat-
isfying deg(P ) <αand such that, in a neighbourhood of t
0
:
|X(t) − P (t − t
0
)|≤C|t− t
0
|
α
. The H
¨
older exponent of X at t
0
is defined as h(t
0
)=sup{α : X ∈ C
α
(t
0
)}. A straightforward
example is given by the cusp-type function X(t)=A+B|t−t
0
|
h
,
whose H
¨
older exponent in t
0
is simply h (when h is not a even in-
teger).
The fluctuations, along time t, of the H
¨
older exponent h are usu-
ally described through the singularity (or multifractal) spectrum,
labelled D(h) and defined as the Hausdorff dimension of the set
of points where the H
¨
older exponent takes the value h. For the
definition of the Hausdorff dimension as well as for further details
on multifractals, the reader is referred to e.g., [4, 5].
•Multifractal Formalism. The determination of the multifractal
spectrum from an observed sample path X(t) is a crucial prac-
tical issue. However, a direct numerical determination based on
the definition of the H
¨
older exponent turns out to be impossible.
This is first because by definition the H
¨
older exponent of a mul-
tifractal path varies widely from one time to another and second
because actual empirical data come with practical limitations such
as discrete time sampling and finite resolution. To overcome such
difficulties, Parisi and Frisch, in a seminal work in turbulence [2],
proposed to overcome such difficulties by introducing a multifrac-
tal formalism: it consists in reaching the multifractal spectrum
through auxiliary easily computable quantities: the structure func-
tions (see below). Their original proposition was based on incre-
ments X(t + τ ) − X(t) of X(t), they are now advantageously
replaced by wavelet coefficients d
X
(j, k).
•Wavelet coefficients and H
¨
older Regularity. It has long been
recognised that wavelet coefficients constitute ideal quantities to
study the regularity of a path (see e.g., [6, 4, 5]). Let us briefly
recall how and why. Let ψ
0
(t) denote a reference pattern with
fast exponential decay and called the mother-wavelet.Itisalso
characterised by a strictly positive integer N ≥ 1, called its num-
ber of vanishing moments,defined as ∀k =0, 1,...,N − 1,
R
R
t
k
ψ
0
(t)dt ≡ 0 and
R
R
t
N
ψ
0
(t)dt =0.Let{ψ
j,k
(t)=
2
−j
ψ
0
(2
−j
t − k),j ∈ N,k ∈ N} denote templates of ψ
0
di-
lated to scales 2
j
and translated to time positions 2
j
k. The discrete
wavelet transform (DWT) of X is defined through its coefficients:
d
X
(j, k)=
Z
R
X(t)2
−j
ψ
0
(2
−j
t − k) dt. (1)
For further details on the wavelet transforms, the reader is referred
to e.g., [7]. It has been shown that on condition that N>h,
when X ∈ C
α
(t
0
) then there exists a constant C>0 such that
IV - 1610-7803-8874-7/05/$20.00 ©2005 IEEE ICASSP 2005
➠
➡
|d
X
(j, k)|≤C2
jh
(1 + |2
−j
t
0
− k|
h
). Loosely speaking, it is
commonly read as the fact that when X has H
¨
older exponent h
at t
0
=2
j
k, the corresponding wavelet coefficients d
X
(j, k) are
of the order of magnitude |d
X
(j, k)|∼2
jh
. This is precisely the
case of the cusp like function mentioned above. Further results re-
lating the decrease along scales of wavelet coefficients and H
¨
older
exponent can be found in e.g., [8].
•Wavelet Coefficient based Multifractal Formalism.
Wavelet coefficient based structure functions and scaling expo-
nents are defined as:
S
d
(q, j)=
1
n
j
n
j
X
k=1
|d
X
(j, k)|
q
, (2)
ζ
d
(q)=liminf
j→0
„
log
2
S
d
(q, j)
j
«
, (3)
where n
j
is the number of available d
X
(j, k) at octave j: n
j
n
0
2
−j
.Bydefinition of the multifractal spectrum, there are about
(2
j
)
−D(h)
points with H
¨
older exponent h, hence with wavelet co-
efficients of the order |d
X
(j, k)|(2
j
)
h
. They contribute to
S
d
(q, j) as ∼ 2
j
(2
j
)
qh
(2
j
)
−D(h)
=(2
j
)
1+qh−D(h)
. Therefore,
S
d
(q, j) will behave as ∼ c
q
(2
j
)
ζ
d
(q)
and a standard steepest de-
scent argument yields a Legendre transform relationship between
the multifractal spectrum D(h) and the scaling exponents ζ
d
(q):
ζ
d
(q)=inf
h
(1 + qh − D(h)). The Wavelet Coefficient based
Multifractal Formalism (hereafter WCMF) is standardly said to
hold when the following equality is valid:
D(h)=inf
q=0
(1 + qh − ζ
d
(q)) . (4)
•Limitations. However, the wavelet coefficient based multifractal
formalism suffers from two major drawbacks, illustrated in Sec-
tion 4 below.
First, by definition, wavelet decompositions necessarily yield a
large number of close to 0 coefficients. This implies that the com-
putation of S
d
(q, j) for negative qs will be numerically instable.
In a stochastic framework, it would translate into the fact that the
wavelet coefficients are random variables with a strictly positive
probability density function at the origin and hence infinite mo-
ments of order q<−1. In both cases, it implies practically that
wavelet coefficient based structure functions cannot be used for
q<−1. The Legendre transform indicates that it will prevent us
from measuring the, roughly speaking, right part of D(h) (i.e., the
part of D such that h>h
0
, where h
0
corresponds to the value of
h where inf
q
(1 + qh − ζ
d
(q)) is the largest).
Second and foremost, while valid for cusp-type singularities, the
wavelet coefficient based multifractal formalism fails to hold for
instance when the analysed X(t) embodies oscillating (or chirp-
type) singularities, i.e., when X(t) is of the form: X(t)=|t −
t
0
|
α
sin
`
1/|t − t
0
|
β
´
.
3. MULTIFRACTAL AND WAVELET LEADERS
•Wavelet Leaders. To overcome those two drawbacks, it has re-
cently been suggested that the relevant quantities the multifrac-
tal formalism should be based on are not wavelet coefficients but
wavelet leaders [9].
From now on, let us assume that ψ
0
is a compact support mother
wavelet and that the {2
−j/2
ψ
0
(2
−j
t − k),j ∈ N,k ∈ N} form
an orthonormal basis. Let us define an alternative indexing for
the dyadic intervals λ (= λ
(j,k)
)=
ˆ
k2
j
, (k +1)2
j
´
, so that
d
λ
≡ d
X
(j, k). Finally, let 3λ denote the union of the interval λ
and its 2 adjacent dyadic intervals: 3λ
j,k
= λ
j,k−1
∪λ
j,k
∪λ
j,k+1
.
One has |d
λ
|≤
R
|X(t))||ψ
j,k
(t)|dt ≤ C ψ
0
L
1
X
L
∞
as
soon as X ∈ L
∞
, and the quantities,
L
X
(j, k) ≡ L
λ
=sup
λ
⊂3λ
|d
λ
| (5)
are thus finite. They are referred to as wavelet leaders.Inall
generality, we now have that X ∈ C
α
(t =2
j
k) is equivalent to
the fact that the wavelet leaders L
X
(j, k) decay as power laws of
the scales ∼ 2
jh
(up to a logarithmic correction), cf. [4].
•Multifractal Formalism. Let S
L
(q, j) denote the wavelet leader
based structure functions and ζ
L
(q) the corresponding scaling ex-
ponents:
S
L
(q, j)=
1
n
j
n
j
X
k=1
|L
X
(j, k)|
q
, (6)
ζ
L
(q)=liminf
j→0
„
log
2
S
L
(q, j)
j
«
. (7)
Arguments similar to those of the previous section yield
ζ
L
(q)=inf
h
(1 + qh − D(h)) . (8)
This leads to a Wavelet Leader based Multifractal Formalism (here-
after WLMF), stated as:
D(h)=inf
q=0
(1 + qh − ζ
L
(q)) . (9)
It is mathematically proven that inf
q=0
(1 + qh − ζ
L
(q)) actsasa
sharp upper bound for D(h) for all functions or processes (on con-
dition that they satisfy some mild uniform regularity conditions)
[9]. This is in contrast with the WCMF for which much weaker
mathematical results hold.
•WCMF vs WLMF. As opposed to the WCMF, the WLMF over-
comes the two drawbacks mentioned above: It holds both for pos-
itive and negative qs and when the function or process under study
embodies chirp-type singularities and enables to obtain in any case
the whole range of the multifractal spectrum [9]. This will be illus-
trated in Section 4. Furthermore, tracking, for q>0, the discrep-
ancies between ζ
d
(q) and ζ
L
(q) or between their Legendre trans-
forms is likely to enable us to detect whether the process under
study contains chirp-like singularities or simply cusp-like singu-
larities.
However, numerically deriving the ζ
L
(q)s requires the knowledge
of wavelet coefficients on a wider range of scales than that nec-
essary to get the ζ
d
(q)s: indeed, in order to be meaningful, the
computation of L
λ
at a given scale requires that of the wavelet co-
efficients d
λ
over several scales below.
•Higher dimensions. For sake of simplicity, the presentation was
proposed here for 1D processes or functions. However, the wavelet
leader multifractal formalism can be straightforwardly extended to
arbitrary higher dimensions d ≥ 1, {X(t)}
t∈R
d
, simply by adapt-
ing the definitions of the wavelet coefficients and leaders as well as
that of the Legendre transform, inf
q
(d+hq −ζ
L
(q)). An example
in the case d =2will be given in Section 4.
•Modulus Maxima of the Wavelet Transform. The wavelet leader
approach is highly reminiscent of the Modulus Maxima of the
Wavelet Transform technique (MMWT) initially introduced by S.
Mallat [7] and developed in the context of multifractal analysis by
Arneodo et al.[6]. It is based on the continuous wavelet transform
(CWT) defined in the upper half plane {(a, t): a>0,t∈ R}:
T
X
(a, t)=
1
a
Z
X(u)ψ
„
t − u
a
«
du. (10)
IV - 162
➡
➡
It consists in finding local maxima of the functions t →|T
X
(a, t)|
along time t for each given scale a and in chaining maxima along
scales at given time positions. Structure functions are then based
on this skeleton. This MMWT based multifractal formalism is
known to solve the q<0 issue [6, 10], it has also been shown
to work on examples containing oscillating singularities [10]. The
main difference between the leader and the modulus maxima ap-
proaches lies in the fact that in this latter method, the spacing be-
tween local maxima need not be of the order of magnitude of the
scale a or even be regularly spaced. Therefore, the MMWT scal-
ing exponents can differ from those obtained with leaders (see [4]
where counterexamples are constructed). It follows that no mathe-
matical result such as the one in Eq. (9) is expected to hold for the
MMWT method.
On a more practical side, the computation of the MMWT involves
that of a CWT plus maxima tracking and chaining operations. This
results in a high computational cost. The WLMF can be imple-
mented using the fast pyramidal algorithm underlying the DWT
and has thus a significantly lower computational cost. Further-
more, as already mentioned, the wavelet leader approach can be
easily theoretically and practically generalised to higher dimen-
sions. This is far less the case for the MMWT method.
4. ILLUSTRATIONS AND EXAMPLES
•Methodology. The goal of this section is to illustrate the wavelet
coefficient and wavelet leader multifractal formalisms, on 4 well-
chosen reference examples. They consist of both deterministic
functions and stochastic processes and contain either cusp-like or
chirp-like singularities. Their multifractal spectra D(h) are known
theoretically and so are their ζ
L
(q) through Eq. (7) or Eq. (8).
The DWT is computed using least asymmetric compact support
orthonormal Daubechies wavelets with N =3[7]. The scaling
exponents and multifractal spectra obtained from the WCMF and
WLMF will be denoted
ˆ
ζ
d
(q),
ˆ
ζ
L
(q),
ˆ
D
d
(h) and
ˆ
D
L
(h) respec-
tively. The
ˆ
ζ
d
(q) and
ˆ
ζ
L
(q) are computed via linear regressions in
log of structure functions versus log of scales diagrams, as thor-
oughly described in [11, 12], a standard Legendre transform al-
gorithm is then used to get
ˆ
D
d
(h) and
ˆ
D
L
(h). The orders qs
are chosen in a range that avoid the occurrence of the so-called
linearisation effect that necessarily takes place in any scaling ex-
ponent estimation procedure (cf. [11, 12]). Results reported and
compared in Fig. 1 and 2, are obtained on sample paths of duration
nbpoints and, for the case of stochastic processes, by averaging
over nbreal = 1000 replications to ensure statistical convergence
of the measures. Matlab routines implementing both the process
synthesis procedures and the WCMF and WLMF analysis proce-
dures were developped by ourseleves and are available upon re-
quest.
•Fractional Brownian Motion (FBM). Our first example consists
of the FBM with self-similarity parameter H (see e.g., [5]). FBM
is a stochastic process that is almost surely everywhere singular
with constant H
¨
older exponent h(t)=H, ∀t ∈ R. Therefore,
its D(h) reduces to a single point (h = H, D =1)(ζ
L
(q)=
qH,∀q ∈ R). Moreover, it contains only cusp-like singularities.
Fig. 1 (first row) shows that the
ˆ
ζ
d
(q) depart from qH when q<
−1, the corresponding Legendre transform yields the inaccurate
ˆ
D
d
(h)=1−h+H, h ∈ [H, H +1]. The WCMF hence produces
an incorrect determination of D(h) while the WLMF results in a
perfect one (here, nbpoints =2
20
).
•Canonical Mandelbrot’s Cascade (CMC). CMC constitute the
historical archetype for stochastic multifractal processes [3, 1].
They are built on an iterative split/multiply cascade scheme (see
e.g., [5, 3, 1]). They contain only cusp-like singularities and are
everywhere singular with bell-shaped D(h).Wechosehereafrac-
tionally integrated 2D cascade, (this is why its D(h) ranges from
0 to 2) with log-normal characteristics, i.e., D(h)=2− (h −
H)
2
/(2σ
2
) and hence ζ
L
(q)=qH − σ
2
q
2
/2. The 2D frac-
tional integration (cf. [9]) of parameter 1 ensures that the anal-
ysed process contains singularities with h>0 only. Again, the
WCMF yields an incorrect determination of the scaling exponents
for q<−1 and of D(h) for its upper (or right) part while the
WLMF produces a very relevant one (cf. Fig. 1, second row, here
nbpoints =2
10
×2
10
). Note that the MMWT would also produce
a correct determination of D(h) at the prices of significant con-
ceptual difficulties (for the 2D extension) and of a huge increase
of the computational cost (2D CWT plus local maxima detection
and chaining).
•Riemann’s Function. It is a deterministic function, defined as
X(t)=
P
n∈N
sin(2πn
2
t)
n
2
, providing us with a reference example
for chirp-like singularities. Its D(h) consists of D(h)=4(h −
0.5),h ∈ [0.5, 0.75] plus an isolated point (h =1.5,D =0).
This latter point constitute a signature of the existence of chirp-
like singularities. Fig. 1, third row, shows that the WCMF and
the WLMF coincide on the linear part of the spectrum but that
the WCMF totally misses the isolated point while the WCMF cor-
rectly recovers it. Equivalently, this corresponds to the discrepancy
between
ˆ
ζ
d
(q) and
ˆ
ζ
L
(q) for q<0 (here, nbpoints =2
20
,the
summation is practically truncated to 100 terms).
•Random Wavelet Series (RWS). RWS are multifractal stochas-
tic processes that were very recently introduced in [13] and that
are not based on an iterative multiplicative cascade. They are ev-
erywhere singular and contain chirp-like singularities (note that
this cannot be reached with iterative multiplicative constructions).
We chose here RWS such that their D(h) consists of a piece of
parabola together with a linear part (see solid line in Fig. 1, bot-
tom row, left). The fact that D(h) departs from the parabola when
approaching its abruptly ending point at its maximum D =1con-
stitutes a signature for the chirp-type nature of the singularities in
RWS. Fig. 1, bottom row shows that the WCMF creates a right
part of D(h) that does not actually exists (again, it corresponds to
the discrepancy between
ˆ
ζ
d
(q) and
ˆ
ζ
L
(q) for q<0) while
ˆ
D
L
(h)
satisfactorily recovers the theoretical D(h). Notably, the ending
point of D(h) is clearly captured by
ˆ
D
L
(h) while totally missed
by
ˆ
D
d
(h) that continues toward its right (here, nbpoints =2
15
).
Fig. 2 shows an important difference between the CMC example
(cusp-type singularities) and the RWS one (chirp-type singulari-
ties). For CMC, the left parts of
ˆ
D
L
(h) and
ˆ
D
d
(h) (i.e., the parts
obtained from the positive qsoftheζ
L
(q) and ζ
d
(q) via the Leg-
endre transform) are close. For RWS, they significantly differ, in
particular
ˆ
D
d
(h) clearly misses the linear part of D(h) and the
end point at D =1while
ˆ
D
L
(h) does not. Such a discrepancy
between the left parts of
ˆ
D
L
(h) and
ˆ
D
d
(h) may therefore be used
to assess the presence of chirp-type singularities in the analysed
data.
5. CONCLUSION AND PERSPECTIVE
We gave here the definition of a new multifractal formalism based
on the leaders of the wavelet coefficients of a discrete wavelet
transform. It enables a precise determination of the multifractal
IV - 163
➡
➡
spectrum over its full range for very large classes of processes, in-
cluding those containing oscillating (or chirp-like) singularities. It
has a very low computational cost and can be easily implemented
for processes in any dimension. Compared to other previously ex-
isting multifractal formalisms, it benefits from more accurate the-
oretical results and brings a real enhacement from a practical point
of view. Its ability to detect the existence of chirp-like singulari-
ties is under current investigation. The statistical performance of
estimation procedures, based on wavelet leaders, for the scaling
exponents and multifractal spectra of stochastic processes will be
further studied. The occurence of the linearisation effect [11, 12]
will specifically be investigated. Applications to the analysis of
data coming from hydrodynamic turbulence and internet trafficare
considered. Matlab routines implementing practically this wavelet
leader multifractal formalism are available upon request.
6. REFERENCES
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cascades: Divergence of high moments and dimension of the
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[2] G. Parisi and U. Frisch, “On the singularity structure of
fully developed turbulence, appendix to fully developed tur-
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mer school Phys. Enrico Fermi, North Holland, 1985.
[3] A.M. Yaglom, “Effect of fluctuations in energy dissipation
rate on the form of turbulence characteristics in the inertial
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[4] S. Jaffard, “Multifractal formalism for functions,” S.I.A.M.
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[5] R. H. Riedi, “Multifractal processes,” in: “Theory and ap-
plications of long range dependence ”, eds. Doukhan, Op-
penheim and Taqqu, pp. 625–716, 2003.
[6] A. Arneodo, E. Bacry, and J.F. Muzy, “The thermodynamics
of fractals revisited with wavelets,” Physica A, vol. 213, pp.
232–275, 1995.
[7] S. Mallat, A Wavelet Tour of Signal Processing, Academic
Press, San Diego, CA, 1998.
[8] S. Jaffard, “Exposants de h
¨
older en des points donn
´
es et
coefficients d’ondelettes,” C. R. Acad. Sci. S
´
er. I Math., vol.
308, pp. 79–81, 1989.
[9] S. Jaffard, “Wavelet tecnhiques in multifractal analysis,” to
appear in: ”Fractal Geometry and Applications: A Jubilee
of Benoit Mandelbrot”, eds. M. Lapidus et M. van Franken-
huysen, Proc. of Symp. in Pure Mathematics, 2004.
[10] A. Arneodo, E. Bacry, S. Jaffard, and J.F. Muzy, “Oscillating
singularities on cantor sets: a grand-canonical multifractal
formalism,” J. Stat. Phys., vol. 87(1–2), pp. 179–209, 1997.
[11] B. Lashermes, P. Abry, and P. Chainais, “New insights on the
estimation of scaling exponents,” Int. J. of Wavelets, Mul-
tiresolution and Information Processing, to appear, 2004.
[12] B. Lashermes, P. Abry, and P. Chainais, “Scaling exponents
estimation for multiscaling processes,” in ICASSP 2004,
Montr
´
eal, Canada, 2004.
[13] J.-M. Aubry and S. Jaffard, “Random wavelet series,”
Comm. Math. Phys., vol. 227, pp. 483–514, 2002.
−10 −5 0 5 10
−20
−15
−10
−5
0
5
10
0 0.5 1 1.5 2
0
0.5
1
−4 −2 0 2 4
−6
−3
0
3
0 1 2 3
0
1
2
ï10 ï5 0 5 10
ï20
ï10
0
10
0 0.5 1 1.5 2
0
0.5
1
−4 −2 0 2 4
−6
−4
−2
0
2
0 0.5 1
0
0.5
1
Fig. 1. Illustration of the MultiFractal Formalisms. From top
to bottom, Fractional Brownian Motion (FBM), Canonical Man-
delbrot’s Cascade (CMC), Riemann’s function, Random Wavelet
Series (RWS); left column: scaling exponents ζ
L
(q),
ˆ
ζ
d
(q),
ˆ
ζ
L
(q),
right column: multifractal spectra D(h),
ˆ
D
d
(h),
ˆ
D
L
(h); solid
line and isolated points: theory, (+) and (o) wavelet coefficient
and wavelet leader based multifractal spectra, respectively.
1 1.5
1.8
2
0.4 0.6
0.8
1
Fig. 2. Cusp-like vs Chirp-like singularities. Left, for a CMC
(with cusp-like singularities only), both
ˆ
D
d
(h) and
ˆ
D
L
(h) are
equally satisfactorily close to D(h). Right, for a RWS (with chirp-
like singularities), significant discrepancies between
ˆ
D
d
(h) and
ˆ
D
L
(h) can be observed (with
ˆ
D
L
much closer to D) and can be
interpreted as a way to detect chirp-type singularities.
IV - 164
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