Multifractal processes: from turbulence to natural images and textures.

Multifractal processes: from turbulence to natural images and textures. Pierre CHAINAIS Université Clermont-Ferrand II LIMOS UMR 6158 France

Thanks Patrice ABRY Rolf RIEDI François SCHMITT Vladas PIPIRAS Laure COUTIN Emilie KOENIG Véronique DELOUILLE Jean-François HOCHEDEZ Samuel GISSOT

Planned trajectory 1 From turbulence to images 2 Multifractal processes 3 Texture synthesis 4 Quiet Sun

Motivations Signals, images and irregular objects turbulence : 1D signals, ideally ND, e.g. ε(x, t)), or vectorial, e.g. v(x, t)) Internet traffic, financial data, biological signals (heart rythm...), natural images : statistical models for vision, bayesian inference... texture : 2D, 3D, 2D+t, 3D+t... disordered media : clouds, porous media,...

Fully developed turbulent flows Re Re c Re = 9, 6 Re = 13, 1 Re = 26 Re = 2 Re = 1

From turbulence to images Multifractal processes Texture synthesis Quiet Sun Fully developed turbulent flows Observations : disorder Earth Simulator, Yokokawa et al. 23 P. Chainais - LIMOS Universite Blaise Pascal / Clermont II

Fully developed turbulent flows Observations : randomness 6 5 4 ε [U.A.] 3 2 dissipation 1.5.1.15.2.25.3.35.4 t [s] 1 9 8 7 v [m.s 1 ] 6 5 4 velocity 3 2 1.5.1.15.2.25.3.35.4 t [s]

Fully developed turbulent flows Re Re c : towards a statistical approach... experimental studies expensive and heavy, numerical simulations time consuming and expensive, (Earth Simulator = 512 proc., 1 4 Go!!), theoretical approaches difficult and limited... Pb = there are too many degrees of freedom ex : Re 1 9 = N d.d.l. 1 2...towards a statistical approach...

A. N. Kolmogorov (193 1987) Fully developed turbulent flows Observations Very : influential scale invariance 1941 theory of homogeneous, isotropic, incompressible turbulence based on Richardson's ideas Integral scale Inertial range Dissipative scale Energy input log E(k) Dissipationless turbulent cascade k -5/3 Viscous dissipation into heat log k -1-1 Energy is added to the fluid k 5/3 on spectrum the inertial scale and is dissipated as he dissipative scale. Energy transfer between eddies on intermediate scale

Fully developed turbulent flows Observations : intermittency 2.5 ζ(q) 2 1.5 1.5 K41 KO62 Proba. Density. Function 4 LARGE scales small scales 1 2 3 4 5 6 7 q 8 1 5 5 1 velocity increments ζ(q) q evolution of PDF 3 IE δv r q = IE v(x + r) v(x) q r ζ(q) IEε q r r τ(q)

1 9 8 7 6 5 4 3 2 1.5.1.15.2.25.3.35.4 t [s] Energy input Dissipationless turbulent cascade -1-1 4 Viscous dissipation into heat 8 1 5 5 1 Fully developed turbulent flows Main properties Observations exhibit a stochastic nature, Power law spectrum k 5/3 : scale invariance, Rare events are not so rare : departures from Gaussian distributions v [m.s 1 ] A. N. Kolmogorov (193 1987) Very influential 1941 theory of homogeneous, isotropic, incompressible turbulence based on Richardson's ideas log E(k) Integral Inertial Dissipative scale range scale k -5/3 log k Energy is added to the fluid on the inertial scale and is dissipated as heat on the dissipative scale. Energy transfer between eddies on intermediate scales is lossless. small scales Proba. Density. Function LARGE scales velocity increments Remark : Kolmogorov 1941 = Gaussian scale invariance [ fbm in 194] Kolmogorov 1962 = scale invariance + non Gaussian, Data analyses and descriptive models, but no constructive model...

Natural images Copyright Z. Chi

Why study the statistics of natural images? statistical inference for analyzing and understanding images using bayesian procedures : object tracking, pattern recognition, analytical performance analysis, denoising, compression... architecture of animal visual systems : visual processing strategies, biological structure of animal visual systems (e.g., retina).

Properties of natural images Natural images have two main properties : non Gaussian 1 2 scale invariant S(k) 1/k 2 η 1 19 log 1 PDF 1 2 3 log 1 Spectrum(I) 1 18 1 17 1 16 1 15 1 14 1 13 slope ~ 1.8 4 8 6 4 2 2 4 6 8 Intensity in gradient image 1 12 1 1 1 1 2 1 3 log 1 k Field 87, Ruderman 94, Grenander & Srivastava 1, Mumford & Gidas 1...

8-1 -1 1 5 5 1 112 1 11 12 13 1 2 3 4 4 From turbulence to natural images The analogy between turbulence and natural images is fruitful. A. N. Kolmogorov (193 1987) Turbulence : ε r (x) = locally averaged dissipation Very influential 1941 theory of homogeneous, isotropic, incompressible turbulence based on Richardson's ideas Integral scale Energy input log E(k) Inertial Dissipative range scale Dissipationless Viscous dissipation turbulent cascade into heat k -5/3 Proba. Density. Function LARGE scales small scales log k Energy is added to the fluid on the inertial scale and is dissipated as heat on the dissipative scale. Energy transfer between eddies on intermediate scales is lossless. velocity increments Natural images : ε r (x) = locally averaged intensity/contrast. 1 2 1 19 log 1 Spectrum(I) 1 18 1 17 1 16 1 15 1 14 slope ~ 1.8 log 1 PDF 1 13 S(k) 1 log 1 k 8 6 4 2 2 4 6 8 Intensity in gradient image k 2 η [Ruderman 94, Turiel Mato Parga 98] non Gaussian

Intermittent models? 2 essential properties : non gaussian scale invariant Question How to build a signal / an image / an object with a non gaussian and scale invariant prescribed behaviour? with some desirable properties : homogeneity, isotropy, etc... Multiplicative cascades and multifractal processes

Trajectory 1 From turbulence to images 2 Multifractal processes 3 Texture synthesis 4 Quiet Sun

From self-similarity to multifractality Let T X (x, r) a quantity based on X (x) measured at scale r. e.g., increments X (x + r) X (x), local averages < X (x) > r, wavelet coeff... Strict scale invariance : = the distributions of T X (r) are all the same. Example : Normal laws with variance σ 2 (r) r 2H (fbm). Intermittency = less and less Gaussian at small scales = evolution of the distributions of T X (r), = evolution of the moments of T X (r), = scaling of IET X (r) q r ζ(q), q R.

From self-similarity to infinitely divisible multiscaling Self similarity IET X (r) q = c q r qh ex : f.b.m., Linear Fractional Stable Motion... Multifractal processes IET X (r) q = c q r ζ(q) ex : binomial cascade, random wavelet casc., MRW... Infinitely divisible multiscaling IET X (r) q = c q exp[ ζ(q) n(r)] multifractal si n(r) = log r, a priori n(r) log r.

Multifractal analysis to characterize intermittency Proba. Density. Function 4 LARGE scales small scales 8 1 5 5 1 velocity increments exponents ζ(q) = characteristic parameters. The more different from qh the ζ(q), the more intermittent and multifractal the signal. Estimation : log IET X (r) q = ζ(q) log r + cste Remark : multifractal formalism = ζ(q) D(h)

Compound Poisson cascades = multiplicative cascades Big whirls have little whirls, Which feed on their velocity ; And little whirls have lesser whirls, And so on to viscosity. Poem by L.F. Richardson (1922) 1.5 W W 1 1.5 2 W W 1 W 1 W 11 1 1 1 3 2 1 phenomenological picture, binomial multiplicative cascade. 1

.3.25.2.15.1.5 1 2 3 4 5 6 Binomial cascades dyadic tree irregular signal 1.5 W W 1 1.5 2 W W 1 W 1 W 11 1 1 3 1 = 2 1 1 Turbulence de Jet, R λ 58 Dissipation temps(s)

Towards infinitely divisible cascades Binomial Compound Poisson Infinitely divisible r=1 ( (k+1/2)2 j, 2 j ) r 1 (t i, r i ) 1 C r (t) 2 1 2 2... Q r (t) = j t Λ j (t) { Qr (t k k(t)), r = 2 j only. Mandelbrot 74 Q r (t) t (t i,r i ) C r (t) t i : uniform stationary r i : 1/r 2 scaling Barral & Mandelbrot 2 W i r Q r (t) exp M(C r (t)) continuous mult. cascades Schmitt & Marsan 1 Muzy & Bacry 2 Chainais & Riedi & Abry 3 t

Compound Poisson Cascades (CPC) Basic version CPC are easy to use : (x Q l (x) = i,r i ) C l (x) W i [ ] IE (x i,r i ) C l (x) W i Taking the logarithm : log Q l (x) = (x i,r i ) C l (x) log W i + K = superposition of transparent cylinders in 3D! multiplicative model for intensity I(x) Q l (x) additive model for contrast φ(x) log I(x) log Q l (x)

Compound Poisson Cascades (CPC) Basic version CPC are easy to use : Q l (x) = IE Taking the logarithm : log Q l (x) = (x i,r i ) WI1 D(x i,r i ) i [ (xi,ri) WI1 D(x i,r i ) i (x i,r i ) C l (x) ] log W i I1 D(xi,r i ) + K = superposition of transparent cylinders in 3D! multiplicative model for intensity I(x) Q l (x) additive model for contrast φ(x) log I(x) log Q l (x)

Compound Poisson Cascades (CPC) Generalized version Adding some geometrical kernel f(x) : (x i,r i ) Wf((x x i)/r i ) i Q l (x) = [ ] IE (xi,ri) Wf((x x i)/r i ) Taking the logarithm : log Q l (x) = ( ) x xi log W i f + K (x i,r i ) = superposition of transparent objects in 3D! f(x) permits to add some anisotropy and regularity. ( visual effects in 2D) r i i

Scaling laws for infinitely divisible cascades scale invariance 9 multifractal scaling log 1 Spectrum(Q l ) 8 7 6 slope~ 1.84 ~ (2+tau(2)) log 1 E ε r q 2 1 slope τ(q) q=5 5 1 2 log 1 (k) q=1 2 1 log r 1 Spectrum of Q l (x) Box averages

Examples 1D 8 7 6 5 Q r (t) 4 3 2 1 1 2 3 4 5 6 7 8 t

Examples 2D

Examples 3D density isosurface

Porous media? Experimental observations. Figure 9: Part of a typical planar-section through the Savonnier oolithic sandstone. The side-length of the image is 1 cm. The pore-space is shown in black and rock matrix in white.

A good model for a turbulent signal inf. div. random walk V H = B H ( t Q) and velocity v SIMULATION (IDC) experiment simulation slope 5/3 6 5 experiment theory simulation V sim t EXPERIMENT (MODANE) log (PSD) 5/3 ζ(q)/ζ(1) 4 3 2 V exp 1 t log (frequency) 1 2 3 4 5 6 q V H versus Modane Spectrum Exponents ζ(q) = simulated signal obeying the She-Lévêque model [ Chainais Riedi Abry, 23 ]

A good model for natural images Infinitely divisible cascades appear as a good model for various disordered physical systems (e.g., turbulence, natural images) They feature altogether many non trivial good properties : physical interpretation (superposition of objects in 3D), non Gaussian, scale invariance, higher order statistics, meet all usual properties of natural images, connect consistently to other existing models. [ Chainais 27 ]

Trajectory 1 From turbulence to images 2 Multifractal processes 3 Texture synthesis 4 Quiet Sun

Examples of textures

Some interesting possibilities Controlling inhomogeneities

Some interesting possibilities Controlling local anisotropy

Some interesting possibilities Controlling local anisotropy (thanks to Emilie Koenig)

Multifractal stochastic animations : 2D+t

Texture synthesis on the sphere...on a variety? [ See Emilie Koenig s poster ]

Modeling images of the quiet Sun in the extreme UV Pierre Chainais, Emilie Koenig, Université Clermont-Ferrand II LIMOS UMR 6158 - France V. Delouille, J.-F. Hochedez, Royal Observatory of Belgium Bruxelles, Belgique

A 1 billion Dollar question! SOlar Heliospheric Orbiter, a space-based observatory, at the Lagrange point between Earth and Sun (1.5 million kilometers sunward of the Earth). EIT : telescope onboard SoHO mission (ESA/NASA), Ultra Violet at λ = { 17.1 ; 19.5 ; 28.4 ; 3.4 } nm. What would a high resolution observation mission observe? Should one finance such a mission? (...1 billion Dollars...)

A solar cycle seen by E.I.T. (Extreme UV Imaging Telescope) at λ = 19.5 nm 1996-25

The quiet Sun

The quiet Sun Definition : no active region, looks like a turbulent background...

Properties of the quiet Sun Spectrum Histogram 1 14 6 x 15 1 13 5 1 12 log 1 S(k) 1 11 1 1 1 9 1 8 slope~ 2.9 HISTO (Intensity) 4 3 2 1 7 1 1 6 log 1 k 1 1 1 1 2 1 3 5 1 15 2 25 3 35 4 Intensity [a.u.] 1/ k η with η 2.9 I(x, y) ( lognormal) scale invariance : stochastic sub-pixel extrapolation?

Identification of a stochastic model : multifractal analysis log S(q, j) ζ(q) j ζ(q) = qh + τ(q) 7 6 q=3 2.5 2 5 1.5 4 q=2 1 log 2 S q (j) 3 2 q=1 ζ(q).5.5 1 1 1.5 1 q=.5 2 2 1 2 3 4 5 6 j 2.5 1 1 2 3 4 5 q

Identification of a stochastic model : multifractal analysis log S(q, j) ζ(q) j ζ(q) = qh + τ(q) 7 6 q=3 2.5 2 5 1.5 4 q=2 1 log 2 S q (j) 3 2 q=1 ζ(q).5.5 1 1 1.5 1 q=.5 2 2 1 2 3 4 5 6 j 2.5 1 1 2 3 4 5 q Quiet Sun images are multifractal : Frac. int. CPC = possible multifractal model

Fractionnally integrated compound Poisson cascades Construction 1 Identify the Hurst parameter H = ζ(1), and let τ(q) = ζ(q) qh, 2 Identify a CPC with the same τ(q), (1 + T )q e.g., τ(q) = 1 (for q q (1 + qt ) +) with T =.85, 3 Numerical simulation of the CPC, 4 Filtering 1/ k H of CPC, Fourier spectrum 1/ k 2+2H+τ(2) and ζ(q) = qh + τ(q).

Visual comparison Quiet Sun frac. int. CPC

Loss of information due to poor resolution

Towards a better virtual resolution...

Interpolation : no supplemental information

Stochastic scale invariant extrapolation

Conclusion & Perspectives Now we can consider the scale invariant subpixel extrapolation of images of the quiet Sun, the study of the evolution of the local contrast as resolution gets finer, the optimization and validation of processing techniques (e.g., Velociraptor), the calibration of future spatial instruments...

Conclusion Infinitely divisible cascades appear as a good model for various disordered physical systems (e.g., turbulence, natural images) Other possible applications : image processing based on the CPC model, texture synthesis on a curved variety, generalized scale invariance (anisotropy...), turbulence (still...) : vectorial fields... porous media (e.g., bones), http ://www.isima.fr/ pchainai

Results of multifractal analysis ζ(q) τ(q) = ζ(q) qζ(1) 2.5.5 2 1.5 1.5 ζ(q).5.5 τ(q) 1 1.5 1 2 1.5 2.5 2 2.5 1 1 2 3 4 5 q 3 1 1 2 3 4 5 q

Results of multifractal analysis ζ(q) τ(q) = ζ(q) qζ(1) 2.5.5 2 1.5 1.5.5 1 ζ(q).5 τ(q) 1.5 1 2 1.5 2 2.5 1 1 2 3 4 5 q Quiet Sun 97 qh 2.5 3 Quiet Sun 97 monofractal 1 1 2 3 4 5 q Quiet Sun images are multifractal

Results of multifractal analysis ζ(q) τ(q) = ζ(q) qζ(1) 2.5.5 2 1.5 1.5.5 1 ζ(q).5 τ(q) 1.5 1 1.5 2 Quiet Sun 97 Exp.(Frac. Gauss) Frac. Int. CPC α stable 2 2.5 Quiet Sun 97 Exp.(Frac. Gauss) Frac. Int. CPC α stable 2.5 1 1 2 3 4 5 q 3 1 1 2 3 4 5 q A possible multifractal stochastic : FI-CPC