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Yves Meyer: restoring the role ofmathematics in signal and image processing

John Rognes

University of Oslo, Norway

May 2017

The Norwegian Academy of Science and Letters has decidedto award the Abel Prize for 2017 to Yves Meyer, École

Normale Supérieure, Paris–Saclay

for his pivotal role in the development of themathematical theory of wavelets.

Yves Meyer (1939-, Abel Prize 2017)

Outline

A biographical sketch

From Fourier to Morlet

Fourier transform

Gabor atoms

Wavelet transform

First synthesis: Wavelet analysis (1984-1985)

Second synthesis: Multiresolution analysis (1986-1988)

Yves Meyer: early years

I 1939: Born in Paris.I 1944: Family exiled

to Tunisia.I High school at Lycée

Carnot de Tunis.

Lycée Carnot

University education

I 1957-1959?: École Normale Supérieure de la rue d’Ulm.I 1960-1963: Military service (Algerian war) as teacher at

Prytanée national militaire.

“Beginning a Ph.D. to avoid being drafted would belike marrying a woman for her money.”

“From teaching in high school I understood that I wasmore happy to share than to possess.”

Prytanée national militaire

Doctoral degree

I 1963-1966: PhD at Strasbourg (unsupervised, formallywith Jean Pierre Kahane).

I Operator theory on Hardy space H1.I Advice from Peter Gabriel:

“Give up classical analysis. Switch to algebraicgeometry (à la Grothendieck). People above 40are completely lost now. Young people can workfreely in this field. In classical analysis you arefighting against the accumulated training andexperience of the old specialists.”

I Meyer’s PhD thesis was soon outdone by Elias Stein.

Jean-Pierre Kahane (1926-) Elias Stein (1931-)

Meyer sets = almost lattices

I 1966-1980: Université Paris-Sud at Orsay.I 1969: Meyer considers “almost lattices” Λ ⊂ Rn such that

Λ− Λ ⊂ Λ + F

where F is a finite set.I A Salem number is an algebraic integer θ > 1 such that

each Galois conjugate θ′ satisfies |θ′| ≤ 1.I If θΛ ⊂ Λ for an almost lattice Λ then θ is a Salem number,

and conversely.I 1976: Rediscovered by Roger Penrose.I Present in Islamic art from ca. 1200 A.D.

Madrasa portal wall in Bukhara, Uzbekistan (rotated)

Ergodic theory

I 1972: Work with Benjamin Weissat Hebrew University, Jerusalem,to prove that Riesz products areBernoulli shifts.

“I was rather ignorant ofergodic theory. [...] Dive intodeep waters and do not bescared to swim against thestream. But only withsomeone who is an expert ofthe field you are entering.You should never do italone!” Benjamin Weiss (1941-)

Calderón program1974: Raphy Coifman at Washington University, St. Louis: “Weshould attack Calderón’s conjectures.”

“At the time I did not know that a singular integraloperator could be. [...] After working day and night fortwo months we were able to prove the boundedness ofthe second commutator.”

Alberto Calderón (1920-1998) Ronald Raphael Coifman (1941-)

Partial differential equations

I 1980-1986: École polytechnique.I Charles Goulaouic: Switch to PDEI Coifman-Meyer theory of paraproducts: “road beyond

pseudodifferential operators”. Led to Jean-Michel Bony’stheory of paradifferential operators

I 1980: Alan McIntosh seeks out Meyer while teaching atOrsay. Insight:

“Calderón’s problem on the boundedness of theCauchy kernel on Lipschitz curves is equivalent toKato’s conjecture on accretive operators.”

I After seven years, Coifman, Meyer and McIntosh proveCalderón’s conjecture.

Alan McIntosh (1942-2016)

A third wayI Antoni Zygmund: “A problem should always be given its

simplest and most concise formulation.”I Nicolas Bourbaki: “A problem should be raised to its most

general formulation before attacking it.”I A third approach: “Translate a problem into the language of

a completely distinct branch of mathematics.”

Antoni Zygmund (1900-1992) Nicolas Bourbaki (1934-)

Applied versus pureI 1984: Jacques Louis Lions (head of French space agency,

CNES): Can we stabilize oscillations in the Spacelab usingsmall rockets? It reduces to “this question” in puremathematics.

I Yves Meyer solved the problem a week later.

Spacelab module in cargo bay

Wavelets

I Autumn 1984: Jean Lascoux (while photocopying):“Yves, I am sure this article will mean somethingto you.”

I Meyer recognized Calderón’s reproducing identity inGrossmann-Morlet’s first paper on wavelets.

Grossmann-Morlet, SIAM J. MATH. ANAL., 1984)

16 precursors to wavelets, I

I Haar basis (1909)I Franklin orthonormal system (1927)I Littlewood-Paley theory (1930s)I Calderón’s reproducing identity (1960)I Atomic decompositions (1972)I Calderón-Zygmund theoryI Geometry of Banach spaces (Strömberg 1981)I Time frequency atoms in speech signal processing (Gabor

16 precursors to wavelets, II

I Subband coding (Croisier, Esteban, Galand 1975)I Pyramid algorithms in image processing (Burt, Adelson

1982)I Zero-crossings in human vision (Marr 1982)I Spline approximationsI Multipole algorithm (Rokhlin 1985)I Refinement schemes in computer graphicsI Coherent states in quantum mechanicsI Renormalization in quantum field theory

Navier-Stokes

I 1985-1995: Ceremade, Université Paris-Dauphine.I Proved div-curl-lemma with Pierre Louis Lions: If

B,E ∈ L2(Rn), div E = 0, curl B = 0 then E · B ∈ H1(Rn).“No, it cannot be true; otherwise I would haveknown it.”

I Paul Federbush’s paper “Navier and Stokes meet thewavelet.” Jacques Louis Lions puzzled and irritated by title:

“What is you opinion?”

I Yves Meyer, Marco Cannone* and Fabrice Planchon*:Prove existence of global Kato solution to Navier-Stokeswhen initial conditions are oscillating.

Pierre Louis Lions (1956-) Jacques Louis Lions (1928-2001)

Marco Cannone* (1966-) Fabrice Planchon* (?-)

CNRS, CNAM, CachanI 1995-1999: Directeur de recherche au CNRS.I 2000: Conservatoire national des arts et métiers.I 1999-2003: École Normale Supérieure de Cachan.

Yves Meyer at CNAM (2000)

Gauss PrizeI 2003-: Professor emeritus of ENS Cachan, now ENS

Paris-Saclay.I 2010: Awarded Gauss prize at Hyderabad ICM.I At least 51 PhD students and 138 descendants.

Yves Meyer receives Gauss prize (2010)

Outline

Fourier transform

Gabor atoms

Wavelet transform

Outline

Fourier transform

Gabor atoms

Wavelet transform

Joseph Fourier (1768-1830)

Extended version of Fourier’s 1807 Memoir

Heat equation:∂u∂t

=∂2u∂x2

Fourier’s claim

Every 2π-periodic function f (t) can be represented by sum

a0 +∞∑

(an cos nt + bn sin nt)

∫ 2π

0f (t) dt

an =1π

∫ 2π

0f (t) cos nt dt

bn =1π

∫ 2π

0f (t) sin nt dt .

Fourier series

I For 2π-periodic f (t) Fourier coefficients are

∫ 2π

0f (t)e−int dt for n ∈ Z

I Fourier series is

f (t) ?=

fn eint

I Functions {eint}n are orthonormal for inner product

〈f ,g〉 =1

∫ 2π

0f (t)g(t) dt .

1873: Paul du Bois-Reymond (1831-1889) constructed a continuousfunction whose Fourier series diverges at one point

1903: Lipót Fejér (1880-1959) proved Cesàro sum convergence forFourier series of any continuous function

Henri Lebesgue (1875-1941) established convergence in norm forFourier series of any L2-function on [0,2π]

1923/1926: Andrey Kolmogorov (1903-1987) constructed anL1-function whose Fourier series diverges (almost) everywhere

1966: Lennart Carleson (1928-, Abel Prize 2006) proved that Fourierseries of an L2-function converges almost everywhere

Time-invariant linear operators

I Consider functions g : u 7→ g(u) for u ∈ R.I Time-invariant linear operators L : g 7→ Lg given by

convolution products

Lg(u) = (f ∗ g)(u) =

∫ +∞

−∞f (t)g(u − t) dt .

I Here f = Lδ is impulse response of L to the Dirac δ.

Fourier transform

I g(t) = eiωt is an eigenfunction/eigenvector of L

Lg(u) =

∫ +∞

−∞f (t)eiω(u−t) dt = f (ω)g(u) .

I Eigenvalue

f (ω) =

∫ +∞

−∞f (t)e−iωt dt

is the Fourier transform of f at ω ∈ R.

I The value f (ω) is large when f (t) is similar to eiωt for t in aset of large measure.

Inverse Fourier transform

I Fourier reconstruction formula

f (t) ?=

∫ +∞

−∞f (ω)eiωt dω .

I When is f (t) well approximated by weighted sums orintegrals of exponential functions {eiωt}ω?

I When do few ω suffice?

Regularity and decay

A function f (t) is many times differentiable if f (ω) tends quicklyto zero as |ω| grows.

Theorem

Let r ≥ 0. If ∫ +∞

−∞|f (ω)|(1 + |ω|r ) dω <∞

then f (t) is r times continuously differentiable.

Lack of localization

I Decay of f depends on worst singular behavior of f .I Indicator function

f (t) =

{1 for −1 ≤ t ≤ 1,0 otherwise

has Fourier transform f (ω) = 2 sin(ω)/ω, with slow decay.

ωf (ω)

Outline

Fourier transform

Gabor atoms

Wavelet transform

Localization in time and frequency

I Exponential eiωt is localized in frequency, but not in time.I Delaying f (t) by t0 is not seen by |f (ω)|.I To study transient, time-dependent phenomena, it could be

better to replace functions eiωt with functions g(t) that arelocalized in both time and frequency.

I Can both g(t) and g(ω) have small support, or decayquickly?

Heisenberg’s uncertainty principle, I

I Suppose ‖g‖2 =∫ +∞−∞ |g(t)|2 dt = 1, so that |g(t)|2 is a

probability density on R.I Plancherel: ‖g‖2 =

∫ +∞−∞ |g(ω)|2 dω = 2π.

I Mean value of t

∫ +∞

−∞t |g(t)|2 dt .

I Variance around µt

σ2t =

∫ +∞

−∞(t − µt )

2|g(t)|2 dt .

Heisenberg’s uncertainty principle, II

I Mean value of ω

µω =1

∫ +∞

−∞ω|g(ω)|2 dω .

I Variance around µω

σ2ω =

∫ +∞

−∞(ω − µω)2|g(ω)|2 dω .

Theorem (Heisenberg (1927))

σt · σω ≥12.

I Theoretical limit to combined localization of time andfrequency.

Werner Heisenberg (1901-1976, Nobel prize in physics 1932)

Gabor atoms

Minimum σt · σω = 1/2 only for Gabor atoms

g(t) = ae−bt2

with a,b ∈ C, and their twists

gt0,ω0(t) = g(t − t0) · eiω0t

obtained by translating by t0 in time and by ω0 in frequency.

u = Re g(t)

ω0 = 10

ω0 = 5

Dennis Gabor (1900-1979, Nobel prize in physics 1971)

Time-frequency supportCorrelation of f (t) with g(t),

〈f ,g〉 =

∫ +∞

−∞f (t)g(t) dt =

∫ +∞

−∞f (ω)g(ω) dω ,

depends on f and f at (t , ω) where g and g are not neglible.

Heisenberg box

ω0 σω

|g(t)|

|g(ω)|

Windowed Fourier transform

I Gabor (1946) showed that decomposition of audio signalsas a sum or integral of twists

gt0,ω0(t) = g(t − t0) · eiω0t

of atoms g(t) = ae−bt2(which he called logons) is closely

related to our perception of sounds.I Conjectured that a time-frequency dictionary of logons

gt0,ω0(t) for

t0 = k∆t and ω0 = `∆ω ,

integer multiples of fixed ∆t and ∆ω, could give anorthonormal basis for L2(R).

Balian-Low theoremI Such a basis would tile time-frequency plane by congruent

translates of Heisenberg box of g(t) by integer multiples of∆t and ∆ω.

I Roger Balian (1981) and Francis Low provedindependently that an orthonormal basis cannot beobtained this way, for any smooth, localized “window” g(t).

t0 −∆t t0 t0 + ∆t

ω0 + ∆ω

ω0 −∆ω

Roger Balian (1933-) Francis Low (1921-2007)

Outline

Fourier transform

Gabor atoms

Wavelet transform

Jean Morlet (1931-2007)

Jean Morlet, I

I 1970s: Worked as a geophysicist for French oil companyElf-Aquitaine.

I Processed backscattered seismic signals to obtaininformation about geological layers.

I Found that at high frequencies windowed Fourier transform(twisted Gabor atoms = logons) had too long duration toresolve thin borders between layers.

I 1981: Proposed to use dilations by m and translations by c

ψ(t − c

of a constant shape ψ(t) = Re g(t).

Jean Morlet, II

I Adopted Balian’s term “ondelette/wavelet” for this shape.I Turn from time-frequency analysis to time-scale analysis.I Elf: “If it were true [that this can work], it would be known.”I Balian directed Morlet to Alexandre Grossmann in

Marseille.

ψ(2t)

Alexandre Grossmann (1930-)

Continuous wavelet transform (CWT)

I Normalizeψ(m,c)(t) =

1√mψ(

t − cm

I Inner product

Wf (m, c) = 〈f , ψ(m,c)〉 =

∫ +∞

−∞f (t)ψ(m,c)(t) dt

for (m, c) ∈ (0,∞)× R defines continuous wavelettransform Wf of f .

I Wf (m, c) is large if f is similar to ψ at scale m near time c.

Grossmann-Morlet reconstruction formula

Theorem (Grossmann-Morlet (1984))

f (t) =

∫∫Wf (m, c) ψ(m,c)(t)

for f in Hardy space H2(R) ⊂ L2(R).

I Grossmann interpreted Morlet’s continuous wavelettransform as a “coherent state” for the Lie group of affinemotions t 7→ mt + c, m > 0.

I Studied for quantum mechanics by Erik W. Aslaksen andJohn R. Klauder (1968/1969).

Discrete wavelet series

I For a scaling factor s > 1, Morlet sought to approximatedouble integral over (m, c) ∈ (0,∞)× R by series such as

f (t) ?=∑j,k

Wfj,k sj/2ψ(sj t − k)

for j , k ∈ Z, where

ψ(m,c)(t) = sj/2ψ(sj t − k)

is dilated by m = 1/sj and translated by c = k/sj .I How to determine wavelet coefficients Wfj,k numerically?I How large can s be? Is Shannon limit s = 2 possible?

Dyadic (s = 2) time-frequency tiling

I Corresponds to tiling time-frequency plane by area-preserving modifications of Heisenberg box of ψ(t).

I Narrower time intervals at high frequencies.

ω0/2ω0

0 t0 2t0 3t0 4t0 5t0

Calderón’s identity

I Yves Meyer recognized Grossmann-Morlet’s formula asAlberto Calderón’s reproducing identity (1964)

∫ ∞0

Qm(Q∗m(f ))dmm

valid for all f ∈ L2(R).I Here ψ(t) ∈ L2(R) and we assume that∫ ∞

0|ψ(mω)|2 dm

for almost all ω ∈ R.I Operator Qm : f 7→ ψm ∗ f is convolution withψm(t) = 1

mψ( tm ), and Q∗m is its adjoint.

Yves Meyer:

“I recognized Calderón’s reproducing identity and I could notbelieve that it had something to do with signal processing.

I took the first train to Marseilles where I met Ingrid Daubechies,Alex Grossmann and Jean Morlet. It was like a fairy tale.

This happened in 1984. I fell in love with signal processing. I feltI had found my homeland, something I always wanted to do.”

Marseille group

Ingrid Daubechies Alex Grossmann Jean Morlet

A collective enterprise

Yves Meyer:

“A last advice to young mathematicians is to simply forget thetorturing question of the relevance of what they are doing.

It is clear to me that the progress of mathematics is a collectiveenterprise.

All of us are needed.”

Yves Meyer (1939-, Abel Prize 2017)

References

I “Ondelettes et Opérateurs”, Yves Meyer, Herman, 1990.I “A Wavelet Tour of Signal Processing”, Stéphane Mallat,

Academic Press, 2009.I “Interview/Essay of/by Yves Meyer”, Ulf Persson,

Medlemsutskicket, Svenska Matematikersamfundet, 15maj 2011.

Outline

Fourier transform

Gabor atoms

Wavelet transform

Meyer’s first wavelet

42−2

Translations

ψ(t)ψ(t + 1) ψ(t − 1)

42−2

Dilations

ψ(t)21/2ψ(2t)

2−1/2ψ(t/2)

42−2

Orthonormal wavelet basis

I Let ψ(t) ∈ L2(R). Functions

ψj(t) = 2j/2ψ(2j t)

dilate ψ(t) by a factor 1/2j , and normalize.I Functions

ψj,k (t) = ψj(t − k)

translate ψj by k .I Call ψ(t) a wavelet if

ψj,k (t) = 2j/2ψ(2j t − k) for j , k ∈ Z

is an orthonormal basis for L2(R).

Discrete wavelet transform (DWT)

I Discrete wavelet transform

Wfj,k = 〈f , ψj,k 〉 =

∫ +∞

−∞f (t)ψj,k (t) dt for j , k ∈ Z

of f ∈ L2(R) is then given by sampling Wf (m, c).I Reconstruction formula

f (t) =∑j,k

Wfj,k ψj,k (t) =∑j,k

〈f , ψj,k 〉ψj,k (t)

Alexandre Grossmann (1930-) Jean Morlet (1931-2007)

Scale spaces, I

I Scale = 1/2j+1 subspace

Wj = span{ψj,k (t) | k ∈ Z} ⊂ L2(R) for j ∈ Z

I Orthogonal decomposition⊕j∈Z

Wj ⊂ L2(R) dense

I Scale ≥ 1/2j subspace

Vj =⊕i<j

Wi for j ∈ Z

I Nested subspaces

0 ⊂ · · · ⊂ Vj ⊂ Vj+1 ⊂ · · · ⊂ L2(R)

Scale spaces, II

I Orthogonal sumVj+1 = Vj ⊕Wj

I Scale ≥ 1/2j projection

projVj(f ) =

∑i<j

〈f , ψi,k 〉ψi,k

I Scale = 1/2j+1 details

projWj(f ) =

〈f , ψj,k 〉ψj,k

Haar basis

I Simplest wavelet, studied by Alfréd Haar (1909):

ψ(t) =

+1 for 0 ≤ t < 1/2−1 for 1/2 ≤ t < 10 otherwise.

I Discontinuous. Localized in time, but not in frequency.

ψ(t)+1

Alfréd Haar (1885-1933)

Strömberg basis

I Contrary to Balian-Low prediction, orthonormal waveletbases with more regular ψ(t) can be found.

I 1981: Jan-Olov Strömberg found a continuous,piecewise-linear function ψ(t) with rapid decay such that

ψj,k (t) = 2j/2ψ(2j t − k) for j , k ∈ Z

is an orthonormal wavelet basis for H1(R).

Jan-Olov Strömberg (1947?-)

Meyer wavelet

I 1985: Yves Meyer found a C∞ smooth wavelet ψ(t) withrapid decay such that ψj,k (t) form an orthonormal basis forL2(R).

ψ(t)1

42−2

Yves Meyer

Meyer basis

I Meyer wavelets provide a universal unconditional basis foralmost all classical Banach spaces (Lp for 1 < p <∞,Hardy, BMO, Hölder, Sobolev and Besov spaces).

I Extended to functions on Rn for n ≥ 2 by Pierre-GillesLemarié* and Yves Meyer (December 1985).

Pierre Gilles Lemarié-Rieusset*

Multifractal analysis

I Hölder regularity of e.g. Brownian motion can be detectedby decay rate of wavelet coefficients Wfj,k .

I Stéphane Jaffard* used wavelet analysis to determineHölder exponents, varying with position, of multifractalfunctions.

I Used by Marie Farge as a tool for studying turbulence.I Energy is transferred from large to small scales,

suggesting time-scale analysis is appropriate.

Stéphane Jaffard* (1962-) Marie Farge (1953-)

Wavelets and operators

I Fourier analysis is adapted to time-invariant operators,which diagonalize functions g(t) = eiωt .

I Wavelet analysis is adapted to operators that are (nearly)diagonalized by a wavelet basis ψj,k (t).

I Generate algebra of “Calderón-Zygmund operators”,including pseudo-differential operators and singularintegral operators.

I Need criteria for L2-continuity, to replace Fourier transform.I Theorem T (1) of Guy David* and Jean-Lin Journé*

provides one such tool.

Guy David*(1957-) Jean-Lin Journé* (1957-2016)

Outline

Fourier transform

Gabor atoms

Wavelet transform

Wavelets and multiresolution approximations

I 1985: Coifman and Meyer axiomatized approximation ofL2(R) by nested subspaces of functions at varyingresolutions = inverse scales.

I Aim: Fill gap between few wavelets then at hand, and ageneral formalism satisfied by all orthonormal waveletbases.

Multi-resolution

approximations

Orthonormalwaveletbases

Ronald Raphael Coifman (1941-)

Multi-resolution approximation

I LetV0 ⊂ L2(R)

be a space of “scale ≥ 20 = 1 approximations”.I Define space Vj of “scale ≥ 1/2j approximations” by

f (t) ∈ V0 ⇐⇒ f (2j t) ∈ Vj

I Assume V0 ⊂ V1, so that

· · · ⊂ Vj ⊂ Vj+1 ⊂ . . .

I Assume⋂

j Vj = 0 and⋃

j Vj is dense in L2(R).

Scaling function

I A function φ(t), such that integer translates

φn(t) = φ(t − n) for n ∈ Z

form an orthonormal basis for V0, is called a scalingfunction for the multiresolution approximation.

I The functions

φj,n(t) = 2j/2φ(2j t − n) for n ∈ Z

then form an orthonormal basis for Vj .I Fourier transform φ(ω) satisfies∑

|φ(ω + 2πk)|2 = 1 .

Haar approximations, I

I Haar’s wavelet basis fits into this framework.I V0 ⊂ L2(R) is space of functions that are constant on each

interval [n,n + 1), for n ∈ Z.I Scaling function

φ(t) =

{+1 for 0 ≤ t < 10 otherwise.

I Vj is space of functions that are constant on each interval[n/2j , (n + 1)/2j).

Haar approximations, II

I Note that V0 ⊂ V1.I Translates ψn(t) = ψ(t − n) of Haar wavelet

ψ(t) =

+1 for 0 ≤ t < 1/2−1 for 1/2 ≤ t < 10 otherwise

form orthonormal basis for W0 = V1 − V0.

φ(t)+1

ψ(t)+1

Wavelets and conjugate mirror filters

I 1986: Stéphane Mallat and Yves Meyer connectedmultiresolution approximations to conjugate mirror filters.

I Parallel between multiresolution approximations andLaplacian pyramid scheme for numerical imageprocessing, introduced by Peter J. Burt and Edward H.Adelson (1983).

I Wavelet coefficients can be calculated by a generalizationof quadrature mirror filters, invented by D. Esteban and C.Galand for compression and transmission algorithms fordigital telephones (1977).

Stéphane Mallat (1962-)

Orthogonal projections

0 ⊂ . . . ⊂ Vj ⊂ Vj+1 ⊂ . . . ⊂ L2(R)

0 . . .oo Vjoo

��

Vj+1aoo

d��

. . .oo L2(R)oo

. . . Wj−1 Wj . . .

Conjugatemirrorfilters

Multi-resolution

approximationsks Orthonormal

waveletbases

Haar approximations and details

I Filter input (x , y) ∈ R2.I Approximation and details:

y and d =12

x − 12

��

I Recovery: x = a + d and y = a− d .

Repeat

I Signal (fk )k for 1 ≤ k ≤ NI Details (dk )k for 1 ≤ k ≤ N − 1 and final approximation

aN−1.

��

�� ''

ww��

�� ''

ww��

��++

ss��

Photo by Radka Tezaur

Vertical Haar approximation and detail

Vertical and horizontal Haar approximation and detail

Twofold vertical and horizontal Haar approximation and detail

Threefold vertical and horizontal Haar approximation and detail

Threefold Haar approximation (rescaled)

Scaling equation

I Since V0 ⊂ V1, we can write

φ(t) =∑

hn ·√

2φ(2t − n)

for a sequence (hn)n ∈ `2(Z)

I Fourier transform

h(ω) =∑

hne−inω

satisfiesφ(ω) =

Quadrature condition

Proposition (Mallat, Meyer)

Fourier transform h(ω) =∑

n hne−inω satisfies

|h(ω)|2 + |h(ω + π)|2 = 2 (1)

for all ω ∈ R, andh(0) =

√2 . (2)

Proof.

(1) Use φ(ω) = 1√2h(ω/2)φ(ω/2) and

∑k |φ(ω + 2πk)|2 = 1.

(2) Completeness of multiresolution approximation implies|φ(0)| = 1 6= 0.

Approximation as convolution

I Orthogonal projection V1 → V0 takes a signal

f (t) =∑

fn ·√

2φ(2t − n)

m am · φ(t −m), where

am = 〈f (t), φ(t −m)〉 =∑

fnhn−2m .

I Mallat recognized (fn)n 7→ (am)m as approximation (lowpass) half of a conjugate mirror filter.

I (am)m is given by convolving (fn)n with (h−n)n and onlykeeping even terms (downsampling).

Wavelet construction

Theorem (Mallat, Meyer (1986))

Let (gn)n ∈ `2(Z) and ψ(t) ∈ L2(R) have Fourier transforms

g(ω) =∑

gne−iωn = e−iωh∗(ω + π)

ψ(ω) =1√2

Then the translates

ψn(t) = ψ(t − n) for n ∈ Z

form an orthonormal basis for W0 ⊂ V1, and the ψj,k (t) form anorthonormal wavelet basis for L2(R).

Detail as convolution

I Orthogonal projection V1 →W0 takes a signal

f (t) =∑

fn ·√

2φ(2t − n)

m dm · ψ(t −m), where

dm = 〈f (t), ψ(t −m)〉 =∑

fngn−2m .

I Hence (fn)n 7→ (dm)m is detail (high pass) half of aconjugate mirror filter, given by convolving with (g−n)n anddownsampling.

I Detail filter is conjugate mirror of approximation filter:

gn = (−1)n−1h1−n .

Cascade

Fast wavelet transformation (FWT)

I Start with a signal in VH , at high resolution 2H

f (t) =∑

fn · φH,n(t)

I Convolve (fn)n with (gn)n to get details in WH−1∑m

dm · ψH,m(t)

I Convolve (fn)n with (hn)n to get approximation at next lowerresolution, and repeat. Store details at each level.

I Finish with details in WH−1, . . . ,WL, and approximation inVL, at a low resolution 2L.

Performance

I Many coefficients Wfj,k = 〈f (t), ψj,k (t)〉 may be neglible,e.g., if f is regular near k at scale 1/2j , where the neededregularity depends on ψ(t).

I These can then be ignored, allowing for compression.I Remaining coefficients can be analyzed to detect transient

features.I FWT requires O(N) multiplications for input of size N.I Proportionality constant given by length (= taps) of filter

(hn)n, or by length of support of ψ(t).

Fast Fourier transform (FFT)

Algorithmic structure of orthonormal wavelet basis makes theFWT about as fast as the Gauss/Cooley-Tukey FFT.

C.F. Gauss (1777-1855) J. Cooley (1926-2016) J. Tukey (1915-2000)

Vanishing moments

Definition

ψ(t) has m vanishing moments if∫ ∞−∞

t rψ(t) dt = 0

for 0 ≤ r < m.

I Haar wavelet has only one vanishing moment.I Meyer wavelet has infinitely many vanishing moments.I If f (t) is Cm−1 smooth near t0, then

Wfj,k = 〈f (t), ψj,k (t)〉

is small for 2j t0 near k .

Regularity

If ψ(t) has m vanishing moments, h(ω) =∑

n hne−inω has azero of order m at ω = π, and we can factor

h(ω) =√

2(1 + e−iω

)m· R(e−iω) .

Theorem (Tchamitchian* (1987))

Let B = supω |R(e−iω)|. Functions φ(t) and ψ(t) are uniformlyCα Hölder/Lipschitz regular for

α < m − log2(B)− 1 .

Philippe Tchamitchian* (1957-)

Compactly supported wavelets

I Consider wavelets ψ(t) with m ≥ 1 vanishing moments.I Filter (hn)n of minimal length corresponds to polynomial

R(z) of minimal degree in z = e−iω.I For (hn)n real

|R(z)|2 = P(y)

is polynomial in y = sin2(ω/2).I Quadrature condition (1) equivalent to

(1− y)mP(y) + ymP(1− y) = 1 (3)

with P(y) ≥ 0 for y ∈ [0,1].

Daubechies wavelets

Theorem (Daubechies (1988))

Real conjugate mirror filters (hn)n, such that ψ(t) has mvanishing moments, have length ≥ 2m. Equality is achieved for

P(y) =m−1∑k=0

(m + k − 1

Associated Daubechies wavelets have compact support, ofminimal length for given number m of vanishing moments.

Daubechies scaling function φ(t) and wavelet ψ(t), for m = 2

Ingrid Daubechies (1954-)

Wavelet bases are multiresolution approximations

1988: Pierre Gilles Lemarié* proved that all orthonormalwavelet bases (for wavelets with sufficiently fast decay) arisefrom multiresolution approximations.

Conjugatemirrorfilters

Multi-resolution

approximationsks ks +3 Orthonormal

waveletbases

Which conjugate mirror filters come frommultiresolution approximations?

Theorem (Mallat, Meyer (1986))

If h(ω) =∑

n hne−iωn with |h(ω)|2 + |h(ω + π)|2 = 2 andh(0) =

√2 satisfies

I h(ω) is C1 near 0I h(ω) is bounded away from 0 on [−π/2, π/2]

φ(ω) =∞∏

h(ω/2j)√2

is the Fourier transform of a scaling function φ(t).

Stable QMFs are FWTs

1989: Albert Cohen* found a necessary and sufficient conditionfor a conjugate mirror filter (hn)n to come from a multiresolutionanalysis, hence also from an an orthonormal wavelet basis.

Stableconjugate

mirrorfilters

ks +3Multi-

resolutionapproximations

ks +3 Orthonormalwaveletbases

Cohen’s condition identifies numerically stable filters among allpossible conjugate mirror filters. Previously, tuning coefficientsof a quadrature mirror filter was done empirically.

Albert Cohen* (1965-)

Later developments

I 1992: Cohen*-Daubechies-Feauveau biorthogonalwavelets (used in JPEG2000)

I 1994: Donoho-Johnstone wavelet shrinkage (noisereduction)

I 1997: F.G. Meyer-Coifman: Brushlets (texture analysis)I 2000: Candés-Donoho: Curvelets (curve singularities)I 2003: Donoho-Elad sparse representation via`1-minimization

I 2006: Donoho, Candès-Romberg-Tao compressivesensing (superresolution)

Yves Meyer (ca. 2013)

References

I “Multiresolution Approximations and Wavelet OrthonormalBases of L2(R)”, Stéphane Mallat, Transactions of theA.M.S., 1989.

I “A Wavelet Tour of Signal Processing”, Stéphane Mallat,Academic Press, 2009.

I “Image Compression”, Patrick J. Van Fleet,www.whydomath.org, 2011.

Yves Meyer: restoring the role of mathematics in signal ... · mathematics in signal and image...

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