AN IMPROVED VARIATIONAL MODE DECOMPOSITION METHOD FOR INTERNAL WAVES SEPARATION

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1 3rd Eropean Signal Processing Conference (EUSIPCO AN IMPROVED VARIATIONAL MODE DECOMPOSITION METHOD FOR INTERNAL WAVES SEPARATION Jeremy Schmitt, Ernesto Horne, Nelly Pstelni, Sylvain Jobad, Philippe Odier Laboratoire de Physiqe de l Ecole Normale Spériere de Lyon, UMR CNRS 567, F-6936 Lyon, France, firstname.lastname@ens-lyon.fr ABSTRACT This paper proposes to revisit the -D Variational Mode Decomposition (-D VMD in order to separate the incident and reflected waves in experimental images of internal waves velocity field. -D VMD aims at splitting an image into a seqence of oscillating components which are centered arond specific spatial freqencies. In this wor we develop a proximal algorithm with local convergence garantees, allowing more flexibility in order to deal with modes having different spectral properties and to add some optional constraints modeling prior informations. Or method is compared with the standard -D VMD and with a Hilbert based strategy sally employed for processing internal waves images. Index Terms Optimisation, mode decomposition, internal waves.. INTRODUCTION The aim of this stdy is to extract reflecting and incident waves from experimental images modeling the velocity field of oceanic internal wave reflection. Internal waves are waves propagating in the bl of a flid whose density varies with depth (ocean, atmosphere, interior of stars []. De to their intrinsic properties, these waves, as they reflect pon an inclined slope, can ndergo a strong focalization []. An experiment designed to stdy this phenomenon consists in observing the waves throgh a measrement of the flid velocity [3]. The main challenge in this stdy is to separate the incident and reflected waves, which have the same temporal freqency. The incident wave is localized in the spatial Forier spectrm, while the reflected wave is not. The classical procedre to perform this separation is based on a Hilbert transform [] bt it fails to achieve this separation properly by introdcing some bondary artefacts and ndesirable oscillations. This qestion can be formlated as an inverse problem that consists in extracting K oscillating components, denoted ( K with R N N, from the observed data z This wor is spported by the ANR-3-BS3- TRES grant and ANR--BS-6- ONLITUR R N N sch that z = K + ε, = where ε models an additive noise. The stdy of internal wave reflection corresponds to the specific case K =, where z models the velocity field, and each component is centered arond an nnown freqency ω = (ω,, ω, so that, for every location (l, l,..., N,..., N,,(l,l = a,(l,l cos ( v,,(l,l l + v,,(l,l l + ϕ, where a R N N models the amplitde changes and, for i,, the mean vale of v,i R N N is close to ω,i, and ϕ is a phase term. In the recent literatre of component extraction for image analysis, efficient methods are -D synchrosqeezing [5] or -D empirical mode decomposition [6]. The drawbac of the first one is to reqire a spervised domain selection in order to extract each component while the second one may sffer lac of robstness when noise is involved. Another strategy has been proposed in [7] based on monogenic analysis and in [8] for estimating jointly ( K and (ω K. However, all these for methods are not designed to extract components that can be spread ot on the freqency spectrm in one direction. In this wor, we propose to revisit the wor by Dragomirestiy and Zosso, initially formlated for analyzing -D signals [9] and extended to image analysis in [8], in order to incorporate constraints imposed by internal waves analysis. In section, we focs on the design of a new criterion adapted to the problem of extracting reflecting and incident waves from velocity field data. In Section 3, we derive an algorithm solving the associated minimization problem insring some local convergence garantees and in Section, experimental reslts demonstrate the efficiency of the proposed method compared to state-of-the-art strategies. A conclsion is drawn in Section 5. Notation A matrix in R N N is denoted in bold as and its components are ( (l,l L with L =,..., N,..., N. The Forier transform of the image is denoted û and the freqency indices are (ν, ν V = N /,..., N / N /,..., N / /5/$3. 5 IEEE 98

2 3rd Eropean Signal Processing Conference (EUSIPCO. CRITERION The soltion proposed in [8] aims to estimate jointly ( K and (ω K by solving z K min (,ω K + α K D ( = =,(l,l e j(ω,l +ω, l (l,l L, (,where the D analytic signal is defined in the Forier domain, for every freqency (ν, ν V, as û,(ν,ν = ( + sign(ω,ν + ω, ν û,(ν,ν. The D analytic signal is chosen to set to zero one half-plane of the freqency domain relatively to the freqency vector ω. D models the gradient operator and the coefficient α > denotes a reglarization parameter allowing to adjst the selectivity of the filter. The limitations related to this criterion are essentially twofold. First, incident and reflected wave have different spectral behaviors. In particlar, the spectrm of the reflected wave is very compact horizontally bt not vertically. We introdce parameters α,i, depending of the mode and the coordinate i,, in order to separately adjst the horizontal and vertical spectral compacity of each mode. Second, we wold lie to add some prior information lined to the physics of internal waves. In particlar, the reflected wave only propagates along the reflection slope and it is nll far away from the slope. This prior information can be introdced by means of a penalty term f (. For instance it can be chosen as an indicator fnction i C ( whose vale is if C = R N ( (l, l S (l,l = and + otherwise. For sch a choice of the penalty f, we impose the component to be zero in the set of indices S. Sch a choice of C denotes a non-empty, closed and convex sbset of R N N. Bt more generally, we assme that f is at least convex, lower semi-continos and proper from R N N to ], + ]. According to previos remars, the criterion we derive is: min f ( + λ z,ω + D ( α,,(l,l e jω,l (l,l L K D ( + α,, ( =,(l,l e jω,l (l,l L where D and D denote respectively the horizontal and vertical gradient operator. The parameters α, and α, allowing to adjst the selectivity are chosen positive. The parameter λ allows s to adjst the fidelity to data. Similarly to (, this criterion is also non-convex. In what follows, we propose an algorithmic soltion to estimate a local minimizer. 3. ALGORITHM A poplar approach for solving ( consists in alternating the minimization over ( K and (ω K. For every iterations m and each mode,..., K, we denote φ,m ( = f ( + λ + i + α, D ( (l,l e jω[m] [m] i z + α, D ( (l,l e jω[m] ψ,m (ω = α, D (,[m+],(l,l + α, D (,[m+],(l,l l, (l,l L, l (l,l L e jωl (l,l L e jωl (l,l L (3, (. Based on this fnctions, an alternating procedre is smmarized in Algorithm. Algorithm Gass-Seidel algorithm For m =,,... For =,..., K [m+] Argmin φ,m ( R N N ω [m+] Argmin ψ,m (ω ω R This algorithmic strategy is sed in [8] when α = α, = α, and f =. For this configration, sch a minimization strategy provides satisfactory reslts however it involves parameters tricy to adjst. Moreover, it is well nown that sch an alternating minimization procedre reqires restrictive conditions to garantee convergence to a local minimizer as discssed in [, ]. A simple soltion to overcome these difficlties is to replace each minimisation step by a proximity operator step as sggested in [] and described in Algorithm. Algorithm has attractive convergence properties. The convergence to a global minimm is ensred in the case where the criterion is coercive and semi-algebraic []. Algorithm Proximal alternating algorithm Set µ > and ρ > For m =,,... For =,..., K [m+] ω [m+] = arg min φ,m ( + ρ [m] R N N = arg min ψ,m (ω + µ ω ω[m] ω R In order to solve each step efficiently, we rewrite them in the Forier domain. We denote F the Forier transform and F the inverse Forier transform. De to the Parseval/Plancherel theorem, both sbproblems can be written in 98

3 3rd Eropean Signal Processing Conference (EUSIPCO the Forier domain: û [m+] = arg min û û [m] û + f (F û ρ + λ û + û [m] i ẑ i ( + α, j(ν ω [m], ( + sign(ν û (ν,ν ( +α, j(ν ω [m], (+sign(ν û (ν,ν ω [m+] = arg min ω R µ ω[m] ω + α, ( j(ν ω ( + sign(ν û [m+],(ν,ν ( + α, j(ν ω ( + sign(ν û [m+],(ν,ν (ν,ν V (ν,ν V, (ν,ν V (ν,ν V The estimation step of ω [m+] has an analytic expression that is, for every i,, ω [m] ω [m+],i + 8µα,i,i = + 8µα,i (ν,ν Ω [m] where Ω [m] Ω [m] (û[m+] (ν,ν Ω ν [m] i,(ν,ν (û[m+],(ν,ν, (5 denotes the half-plane in the Forier domain: = ν = (ν, ν V ν ω [m], + ν ω [m],. On the other hand, the estimation of [m+] is obtained throgh forward-bacward iterations generating a seqence (v [p] p bild as v [p+] ( [p] = prox γρf v + γb γf AF v [p], (6 where the proximity operator step allows s to deal with the non-smooth fnction f. It is defined as ( x R N N prox f x = arg min y f (y + x y, which is redced to the projection onto a closed convex set when f = ι C. In (6, A models a linear operator defined as A A: û (ν,ν û (ν,ν if (ν, ν Ω [m], otherwise, with A (ν,ν = +ρλ+ρ(α, (ν ω, +α, (ν ω, and b R N N is sch that b (l,l = [m],(l + ρλ(z,l (l,l [m] i,(l,l i. Algorithm 3 Prox-D-VMD algorithm Choose the parameters λ >, ρ >, µ > Set the parameters α, >, α, > Initialize ( [] K, ( ω [] K For every m =,,... For every,..., K Set v [] = [m] Set γ ] ; / A [ For p =,,... Update v [p] Set [m+] as detailled in (6 = lim p + v [p] Compte ω [m+], and ω [m+], sing (5. EXPERIMENTAL RESULTS Experiments are performed on a critical reflection velocity field image z of size N = 3 and N = 38 illstrated on Fig.(a. Or goal is to extract the incident wave and the reflected wave (ths, K = from z. Or method is compared with a method based on Hilbert-transform, which is sally sed to process the internal waves velocity field [] and with the -D VMD proposed in [8] where the data fidelity constraint is enforced strictly throgh an agmented Lagrangian. An illstration of each strategy is illstrated in Fig.. One can observe that the Hilbert based strategy leads to mode mixing, while the standard -D VMD provides ndesirables oscillations in the top of the -D field. The proposed method removes all these artefacts. In order to evalate the impact of the proposed constraints, we perform three different experiments. In the first experiment, there is no optional constraint (f = f = and no directionality (α,i α. In the second experiment, we add the directionality constraint for the second component by setting α, = α/, since the spectrm of the reflected wave is expected to be more spread ot horizontally than vertically. In the third experiment, we combine the directionality with the zero constraint by setting S to be the top first 7 rows of (α, = α/ and f = i C. The reslts are illstrated in Figre, which shows mean vertical profiles of the incident wave (solid line and the reflected wave (dashed line. Mean profiles are obtained by averaging the absolte vale on each row of the images. In each plot, we compare the proposed soltion with the ones obtained with the Hilbert filtering and -D VMD. The left colmn presents the reslts of the proposed method where α is fixed and λ varies. On the right colmn, λ is fixed and α varies. In the first experiment, we can observe that for larger λ, the soltion is closer to the standard -D VMD that imposes the exact eqality between the data and the extracted components. Here, varying λ allows s some flexibility, for instance when noise is involved. A similar behavior is observed regarding the parameter α. One can observe that the basic Hilbert filtering strongly attenates the incident wave The iterations are smmarized in Algorithm 3. 98

4 3rd Eropean Signal Processing Conference (EUSIPCO Data z Incident wave Reflected wave Hilbert (a (b (c -D VMD (d (e Proposed method (f (g Fig.. Reslts on internal waves velocity field with critical wave reflection. st row: reslts obtained throgh a temporal and spatial Hilbert filtering. nd row: -D VMD. 3rd row: proposed method with α, = α/ and zero vale constraint on the 7 first rows of mode (f = i C. on the bondaries and leads to ndesirable oscillations on the reflected wave. In the second experiment, the directionality constraint allows s to redce the oscillations of the first IMF on the first 3 pixels near the slope. Moreover, adding this flexibility in the directionality allows s to be less sensitive to the choices of λ and α, which is why the profiles for different vales of parameters are sperposed. In the third experiment, we illstrate that the zero-constraint on the pixels of the second component which are far from the slope enables to cancel the ndesired oscillations of the reflected wave and the bondary effects. In or experiments, we observe that iterates are enogh to insre convergence. Withot the zero constraint, the convergence of the algorithm is fast (abot 3 seconds while with a zero constraint, the convergence is close to mintes (since the comptation of the modes involves forward-bacward iterations. In or experiments, the [] are set to zero, while the ω [] are initialized on the nit circle. however, we observe that the initialization of the and ω have no incidence on the final estimate. 5. CONCLUSION This paper presents an efficient method for the separation of incident and reflected wave in internal waves experimental images. This method revisits the -D VMD proposed by Dragomiretsiy and Zosso by adding stronger convergence garantees, giving more degrees of freedom for the setting of parameters and the possibility to add some prior nowledge throgh additional constraints. The proposed method has been tested on internal wave experimental images with critical reflection, and otperforms the classic Hilbert filtering since it strongly redces the bondary effects and the ndesired oscillations in the estimated reflected wave. REFERENCES [] B. R. Stherland, Internal gravity waves, Cambridge University Press,. [] OM Phillips, The dynamics of the pper ocean, Cambridge University Press, 966. [3] G. Bordes, A. Venaille, S. Jobad, P. Odier, and T. Daxois, Experimental observation of a strong mean flow indced by internal gravity waves., Physics of Flids, vol., no. 8, pp. Article No. 866, Agst. [] M.J. Mercier, N.B. Garnier, and T. Daxois, Reflection and diffraction of internal waves analyzed with the hilbert transform, Phys. Flids, vol., no. 8, pp. 866, 8. [5] M. Clasel, T. Oberlin, and V. Perrier, The monogenic synchrosqeezed wavelet transform: A tool for the decomposition/demodlation of AM-FM images, Appl. Comp. Harm. Anal.,, to appear. [6] J. Schmitt, N. Pstelni, P. Borgnat, P. Flandrin, and L. Condat, A -D Prony-Hang transform: A new tool for -D spectral analysis, IEEE Trans. Image Process., vol., pp , Oct.. [7] M. Unser, D. Sage, and D. Van De Ville, Mltiresoltion monogenic signal analysis sing the Riesz-Laplace wavelet transform, IEEE Trans. Image Process., vol. 8, no., pp. 8, November 9. [8] K. Dragomiretsiy and D. Zosso, Two-Dimensional Variational Mode Decomposition, in Energy Minimization Methods in Compter Vision and Pattern Recognition, Hong Kong, China, Janary [9] K. Dragomiretsiy and D. Zosso, Variational Mode Decomposition, IEEE Trans. Signal Process., vol. 6, no. 3, pp. 53 5, Febrary. [] T. F. Chan and C. K. Wong, Convergence of the alternating minimization algorithm for blind deconvoltion, Linear Algebra Appl, vol. 36, pp ,. [] J. Bolte, P. L. Combettes, and J.-C Pesqet, Alternating proximal algorithm for blind image recovery, in Proc. Int. Conf. Image Process., Hong Kong, China, September, pp , hal

5 3rd Eropean Signal Processing Conference (EUSIPCO α =, λ =,, 3 α =, 5,, λ =.8.6. Data Hilbert filtering D VMD Proposed method α =, λ = Proposed method α =, λ = Proposed method α =, λ = Data Hilbert filtering D VMD Proposed method α =, λ = Proposed method α = 5, λ = Proposed method α =, λ = Experiment Experiment Experiment Fig.. Mean profile over the vertical axis of the estimates for different vales of parameters α and λ. Blac solid line is the mean profile of data z. Color solid lines represent the extracted incident wave profiles with the different methods, while color dashed lines represent the extracted reflected waves profiles: brown for Hilbert filtering, red for -D VMD (α =, pin/ble/cyan for the proposed method. In the first colmn, λ = and α =, 5,. In the second colmn, α = and λ =,, 3. First row shows the profiles estimated by or method in the nidirectional and nconstrained case. Second row presents the profiles estimated with the proposed method in the directional case (α, = α/. Third row shows the profiles estimated with the proposed method in the directional and constrained case (f = i C. 98