Mathematics of Digitalization: Case Study

Massimo Fornasier Chair of Applied Numerical Analysis Email: [email protected] ——————– First lecture

Munich, Oct. 17 2017 – Fallstudien der Modellbildung 1 INTRODUCTION

Main topics: 1. After a short introduction on the Hilbert spaces, we describe the concepts of Fourier series, Fourier transforms and their discrete implementation, in particular the algorithm of the Fast Fourier Transform (FFT). 2. To connect the continuous Fourier transforms and the discrete one, we need to address the Shannon sampling theory (analog-to-digital conversion). For that it will be fundamental to understand the Poisson summation formula which will allow us to estimate also the errors in the analog-to-digital conversion, the so-called aliasing errors. 3. We shall introduce systems for robust analog-to-digital

INTRODUCTION 2 conversion called frames in Hilbert spaces. We address special cases of frames, in particular Gabor frames in their continuous and discrete versions and the implementation of their so-called canonical duals. Applications to audio signals and in image compression will be showed.

4. We address in more detail the study case of the fresco restoration and how to use the tools we considered so far to solve it.

INTRODUCTION 3 THE IMPORTANCE OF HARMONIC ANALYSIS

We can synthesize a variety of complicated functions by • means of pure sinusoids in the same way we can produce a chord of C major pushing the keyboard (C, E, G) of a piano.

It was Joseph Fourier (http://www-gap.dcs.st- • and.ac.uk/ history/Mathematicians/Fourier.html) who ∼ first developed the modern methods of trigonometric series and integrals in the study of the heat conduction in the solids (Analytical Theory of Heat, 1815).

THE IMPORTANCE OF HARMONIC ANALYSIS 4 SOME OF THE MILESTONES

The most cited paper in mathematics • J. W. Cooley and J. W. Tukey, An algorithm for the machine computation of complex Fourier series, Math. Comp., 19 (1965), 297-301.

Almost 3/4 of the Nobel prizes in Physics are given on • work using methods and tools from Fourier analysis. F. N. Magill, ed., The Nobel Prize Winners — Physics, Vol 1-3, Salem Press. Englewood Cliffs, NJ. 1989.

Francis Crick, James Watson and Maurice Wilkins got the • Nobel prize for the medicine in 1962 for the discovery of the molecular structure of the DNA. This was the first example of the use of Fourier methods on data coming from X ray diffraction.

SOME OF THE MILESTONES 5 Herbert Hauptmann (a mathematician) and Jerome • Karle shared in 1985 the Nobel prize for chemistry after having shown how to use systematically Fourier analysis in order to determine the structure of large molecules from X ray diffraction data. W. A. Hendrickson, the 1985 Nobel Prize in Chemistery, Science 231 (1986), 362-364.

The starting of the time-frequency analysis and the • wavelets (1960-1990).

A large number of technological products (CD-DVD, high • definition TV, digital phones, medical imaging instruments ...) are based on discrete Fourier analysis.

SOME OF THE MILESTONES 6 HILBERT SPACES

Let be a vector space. A scalar product u,v is a map H h i from to C such that: H×H (i) au + bv,z = a u,z + b v,z for all u,v,z and a,b C. h i h i h i ∈H ∈ (ii) u,v = v,u for all u,v . h i h i ∈H (iii) u,u R, u,u 0 for all u and u,u =0 if u =0. h i∈ h i≥ ∈H h i 6 6

HILBERT SPACES 7 We recall that a scalar product verify the Cauchy-Schwarz inequality (which will be proved in the exercises):

u,v u,u 1/2 v,v 1/2 u,v . (1) |h i|≤h i h i ∀ ∈H and that u = u,u 1/2 define a norm for . k k h i H

A Hilbert spaces is a vector space endowed with a scalar H product u,v and is complete with respect to the norm h i u,u 1/2. h i

HILBERT SPACES 8 Example 0.1 Some relevant examples:

Let Ω Rn. The vector space of functions • ⊂ L2(Ω) = f :Ω C f 2dx < is a Hilbert spaces with { → | Ω | | ∞} the scalar product: R

u,v = u(x)v(x)dx. h i ZΩ Let Ω Zn. The vector space • d ⊂ 2 C 2 ℓ (Ωd)= f :Ωd k Ωd f(k) < is a Hilbert { → | ∈ | | ∞} spaces with the scalar product: P u,v = u(k)v(k). h i k Ωd X∈ In particular if Ω = d< then ℓ2(Ω )= Cd. | d| ∞ d

HILBERT SPACES 9 Definition 0.2 Let E be a topological vector space. We say

that E′ is the topological dual space of E, i.e., E′={ϕ from E to C ϕ is continuous and linear}. Given two Hilbert spaces | , , the set of linear operators T : , i.e., H K H→K T (au + bv)= aT (v)+ bT (v), which are also continuous is a vector space which, endowed with the norm Tu K k k is a complete normed space, T supu ,u=0 u H k| |kH→K ≡ ∈H 6 k k i.e., a Banach space, and we indicate it with ( , ). In L H K particular, if = the we set ( ) ( , ) and if = C H K L H ≡L H H K then we have ( , )= ′. L H K H

HILBERT SPACES 10 Theorem 0.3 (Representation theorem of Riesz-Frechet) Let H be a Hilbert space. For every ϕ ′ there exists a unique ∈H f such that ∈H ϕ,v ϕ(v)= v,f v . h i≡ h i ∀ ∈H Moreover we have

f = ϕ ′ sup ϕ,v . k k k kH ≡ v =1 |h i| k k

HILBERT SPACES 11 Proposition 0.4 (Bessel inequality) Let uα α A be an { } ∈ orthonormal system in , i.e., uα,uβ = δα,β where δ , is the H h i · · Kronecker symbol. Then for x : ∈H x,u 2 x 2. |h αi| ≤k k α A X∈ In particular {α x,u =0} is countable. |h αi 6

HILBERT SPACES 12 Theorem 0.5 (Fourier) Let uα α A be a countable { } ∈ orthonormal system. Then the following are equivalent:

(i) x = α A x,uα uα for all x . ∈ h i ∈H P 2 2 (ii) (Parseval idendity) x = α A x,uα for all x . k k ∈ |h i| ∈H (iii) (Completeness) If x andP if x,u =0 for all α, then ∈H h αi x =0.

2 The sequence x,uα is such that α A x,uα < , then h i ∈ |h i| ∞ the series α A x,uα uα is convergent in and converges to ∈ h i P H x. A orthonormal set which verifies one of the previous P conditions is called an orthonormal basis for the Hilbert space . H

HILBERT SPACES 13 Theorem 0.6 Every Hilbert space has an orthonormal basis.

Definition 0.7 A Hilbert space is separable if an only if it has H a countable orthonormal basis.

HILBERT SPACES 14 FOURIERSERIESANDTRANSFORM

A trigonometric series (or Fourier series) in complex form of period τ > 0 is a function series of the type: 1 T (x)= c e2πinx/τ (2) √τ n n Z X∈ Set

a0 =2c0, an = cn + c n, bn = i(cn c n) (3) − − − we have, by the Euler formulas (eiw = cos(w)+ i sin(w)), that

1 a0 ∞ T (x)= + (ak cos(2πkx/τ)+ bk sin(2πkx/τ)) (4) √τ 2 ! kX=1

FOURIERSERIESANDTRANSFORM 15 1 2πinx/τ We observe now that e n Z e { √τ } ∈ τ τ cos(2πkx/τ) k N sin(2πkx/τ) k N,k=0 are an { 2 } ∈ { 2 } ∈ 6 orthonormal system in the Hilbert space = L2(0,τ). For p S p H example

1 1 1 τ e2πinx/τ , e2πimx/τ = e2πinx/τ e2πimx/τ dx h√τ √τ i τ 0 Z τ 1 2πi(n m)x/τ = e − dx τ Z0 1 2πi(n m)x = e − dx = δm,n. Z0

FOURIERSERIESANDTRANSFORM 16 We wonder whether they consitute systems of orthonormal bases. We consider the function πx x2 f(x)= . 2 − 4 Note that it is symmetric and therefore the coefficients b (f)= f, τ sin(2πkx/τ) vanish for all k. By computing k h 2 i (Exercise! Integrate twice by parts) instead the coefficients p a (f)= f, τ cos(2πkx/τ) we obtain the trigonometric series k h 2 i associated to f given by p

π2 ∞ cos(kx) T (x)= . (5) f 6 − k2 kX=1

FOURIERSERIESANDTRANSFORM 17 Theorem 0.8 (Pointwise convergence) Let f : R C periodic → with period τ > 0 and locally integrable. Assume that in + x R the limits f(x ) e f(x−) exist and are finite, and that 0 ∈ 0 0 the left and right finite difference

+ f(x + h) f(x ) f(x h)+ f(x−) 0 − 0 , 0 − 0 , h h

are bounded for any h> 0 small. Then the series Tf of f

converges in x0 to the mean of the values of the right and

left limits of f in x0:

+ 1 f(x ) f(x−) T (x )= f,e2πinξ/τ e2πinx/τ = 0 − 0 . f 0 τ h i 2 n Z X∈

In particular if f ′(x0) exists then Tf (x0)= f(x0).

FOURIERSERIESANDTRANSFORM 18 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6

2 2 πx x π cos (kx) Figure 1: f(x)= = ∞ 2 . 2 − 4 6 − k=1 k P

FOURIERSERIESANDTRANSFORM 19 The set of step functions S are dense in L2(0,τ); • We have • 2πna 1 a2 τ π2 ∞ cos( ) χ , e2πinx/τ 2 = + τ |h [0,a] √τ i| τ π2 6 − n2 n Z n=1 ! X∈ X From the previous theorem applied on f • 1 a = χ 2 = χ , e2πinx/τ 2. k [0,a]k2 |h [0,a] √τ i| n Z X∈

1 2πinx/τ By density of S we conclude that e n Z is an • { √τ } ∈ orthonormal basis.

FOURIERSERIESANDTRANSFORM 20 For functions f L2(0,τ) we define the Fourier transform ∈ : L2(0,τ) ℓ2(Z) as F →

1 2πinx/τ 1 2πinx/τ f(n)= f, e = f(x)e− dx. (6) F h √τ i √τ Z(0,τ)

1

0.8

0.6

0.4

0.2

0.5 1 1.5 2 2.5 3

Figure 2: Fourier series associtated to f(x)= x [x] . − −

FOURIERSERIESANDTRANSFORM 21 What happens if we make τ ? →∞ τ/2 1 2πinx/τ 1 2πinx/τ 1 − 2πinξ/τ 2πinx/τ f, e e = f(ξ)e− dξ e . h √τ i√τ τ n Z n Z τ/2 ! X∈ X∈ Z The last sums is in fact a Riemann sum. This makes us supposing that if f is an integrable function on R then we could write something like

f(x)= lim f(x)χ[τ/2, τ/2] = τ − →∞

1 2πinx/τ 1 2πinx/τ 2πiwξ 2πiξw = lim f, e e = f(ξ)e− dξ e dw. τ h √τ i√τ R R →∞ n Z   X∈ Z Z When and in which sense this writing makes sense is though still to be proved.

FOURIERSERIESANDTRANSFORM 22 Theorem 0.9 (Fourier transform and Plancherel identity) We define for f L1(R) the Fourier transform, given by ∈ 2πiwx f(w)= f(x)e− dx. (7) F R Z Moreover, if also f L1(R) then the Fourier transform is F ∈ invertible and

f(x)= f(w)e2πiwxdw. (8) R F Z If f L1 L2 then : L1 L2 L2 is a continuous linear ∈ F → operator which can be extended uniquely on the entire L2. T T In particular, is an isometric isomorphism, i.e., the F Plancherel identity holds:

f = f . (9) k k2 kF k2 Moreover, if f,g L2 then f,g = f, g . ∈ h i hF F i

FOURIERSERIESANDTRANSFORM 23 Remark 0.10 This definition is used in the literature of time-frequency analysis. The other definition

1 iωx fˆ(ω)= f(x)e− dx 2π R Z is also used, for instance in the literature related to wavelets. In these lectures we use both in a consistent way with the corresponding literature. Anyway observe that f(ω)=2πfˆ(2πω) and, up to rescaling, one can transform all F the result with one definition to results with another definition.

FOURIERSERIESANDTRANSFORM 24 What is the meaning of the Fourier transform? The Fourier transform represents the frequency content of a function/signal. It gives us which are the important oscillating building blocks of a signal and their dinstinctive frequency of oscillation. 8 1 0.5 6 0 4 -0.5 -1 2 -1.5 0 0 50 100 150 200 250 0 50 100 150 200 250

Figure 3: Sinosoidal signal affected by noise: there are two fundamental frequency components

FOURIERSERIESANDTRANSFORM 25 The fortune of Fourier analysis stays essentially in the fact that it is able to describe one of the most frequent behavior in the nature: the wave phanomena, most of which are governed by ovelapping laws of the type

αt 2πis0t e− e , t> 0 yα,s0 (t)=  0, t< 0. 

yα,s0 (t) is called Lorenzian function.

FOURIERSERIESANDTRANSFORM 26 When, for example, some molecules are excited by electomagnetic radiation, damped oscillations are induced

and described by the law yα,s0 (t).

Every building block of the molecule has its own unique special oscillations. The Lorenzians are called in this case the molecular spectrum.

From here comes the idea which “costed” the Nobel prize to Richard Ernst for chemistry (1991) for the development of a powerful instrument to determine the structure of large complex organic molecules.

FOURIERSERIESANDTRANSFORM 27 We want now to discuss how in practice we can use the tools of the Fourier analysis.

The execises we showed on simple functions show us that it is not so trivial to compute Fourier series or transforms.

1 For relatively simple functions such as f(x)= a2+x2 the computation of the Fourier transform is far from being trivial. π w a In this case we have f(w)= e−| | . F a

FOURIERSERIESANDTRANSFORM 28 A holomorphic complex function f(z) (i.e., differentiable) on a domain Ω except for a point a, called singularity, can be expanded in a power series, called the Laurent series, n f(z)= n Z cn(z a) . The specific coefficient c 1 is called ∈ − − residual and we write Res(f,a)= c 1. One can show that if f P − is holomorphic on a domain Ω which contains the halfplane Im(z) 0 except for a finite number of non real singularities { ≥ } and if the limit lim z ,Im(z) 0 f(z)=0 then we have for every | |→∞ ≥ α> 0,

r iαx iα lim f(x)e dx =2πi Res(f( )e ·,a), Im(a) > 0 . r →∞ r { · } Z− X Analogous result holds for α< 0 and for Im(z) < 0 , but { } changing the sign of the integral.

FOURIERSERIESANDTRANSFORM 29