Online learning of the transfer matrix of dynamic scattering media wavefront shaping meets multidimensional time series Lorenzo Valzania1and Sylvain Gigan1

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Online learning of the transfer matrix of dynamic scattering media: wavefront shaping

meets multidimensional time series

Lorenzo Valzania1, ∗and Sylvain Gigan1, ∗

1Laboratoire Kastler Brossel, ´

Ecole Normale Sup´erieure - Paris Sciences et Lettres (PSL) Research University,

Sorbonne Universit´e, Centre National de la Recherche Scientiﬁque (CNRS) UMR 8552,

Coll`ege de France, 24 rue Lhomond, 75005 Paris, France

(Dated: October 11, 2022)

Thanks to the latest advancements in wavefront shaping, optical methods have proven crucial to

achieve imaging and control light in multiply scattering media, like biological tissues. However,

the stability times of living biological specimens often prevent such methods from gaining insights

into relevant functioning mechanisms in cellular and organ systems. Here we present a recursive

and online optimization routine, borrowed from time series analysis, to optimally track the transfer

matrix of dynamic scattering media over arbitrarily long timescales. While preserving the advantages

of both optimization-based routines and transfer-matrix measurements, it operates in a memory-

eﬃcient manner. Because it can be readily implemented in existing wavefront shaping setups,

featuring amplitude and/or phase modulation and phase-resolved or intensity-only acquisition, it

paves the way for eﬃcient optical investigations of living biological specimens.

I. INTRODUCTION

Optical methods are an irreplaceable tool to investi-

gate biological media. They deliver images at numer-

ous contrast mechanisms [1], and can activate injected

biomolecules [2] and ﬂuorescent markers [3]. However,

precisely delivering light in space and time through bio-

logical tissues is not straightforward, as photons get mul-

tiply scattered by heterogeneities of tissues, limiting their

penetration depth [4].

Another current challenge lies in tracking the scat-

tering behaviour of living specimens, with decorrelation

times up to only a few ms [5]. This proves crucial to

understand the functioning mechanisms of cells and or-

ganisms, which requires their observation at extremely

diﬀerent timescales, from nanoseconds (at a molecular

level) to minutes (for organ systems) [6]. The need for

fast data acquisitions results, in turn, in measurements

with inherently low signal-to-noise ratios, and requires

solving long and multidimensional time series [7], whose

prohibitive size can make their evaluation problematic.

Wavefront shaping techniques have established them-

selves as the tools of choice to guide light in scattering

media [8]. The transmission of arbitrary ﬁelds [9], point-

spread-function (PSF) engineering [10], imaging [11], as

well as tuning energy transmission through scattering

media [12], become all accessible if the transfer matrix of

the medium is measured [8, 13]. However, conventional

methods to retrieve the transfer matrix yield sub-optimal

solutions in noisy environments [8]. Those optimization

routines which can compensate for noise in the trans-

fer matrix [14], however, require storing in memory the

whole history of past measurements, making them un-

suited with long streams of data.

∗Correspondence to: lorenzo.valzania1@gmail.com,

sylvain.gigan@lkb.ens.fr

Iterative, optimization-based, sequential algorithms to

focus through scattering media yield an increase in the

focus intensity already at their early iterations, which

makes them the preferred option on dynamic media. Im-

portantly, they are cast as recursive procedures, i.e.,

computing the new estimate of the solution only requires

the previous estimate and the new data point. Unfortu-

nately, their stochastic nature makes optimization over a

set of output modes less reliable and the transmission of

arbitrary ﬁelds prohibitive. Moreover, these procedures

rely on maximizing a given metric, limiting light control

to one predeﬁned task. Various implementations derived

from genetic algorithms [15, 16] have shown better re-

silience to noise than sequential algorithms, however at

the cost of a higher computational complexity and careful

choice of several adjustable parameters.

In signal processing, communications and ﬁnance,

where most datasets are multidimensional time series,

the recursive least-squares (RLS) algorithm has played a

central role for system identiﬁcation and prediction [17–

19]. It allows optimal learning of linear predictors in an

online manner—predictors are updated every time a new

piece of data is sequentially made available, however past

data do not need to be stored in memory. Consequently,

its computational complexity is independent of the length

of the time series, so iterations can be run over and over,

ideally at the same rate as data acquisition (real-time

operation).

Here, we demonstrate that the RLS algorithm repre-

sents a valuable tool to optimally estimate the transfer

matrix of dynamic scattering media online and recur-

sively. The least-squares optimization ensures resilience

to noise. The algorithm is provided with a tunable mem-

ory, such that the dynamics of the scattering medium is

accounted for. By doing so only the most reliable data

points, i.e., those acquired within the stability time of

the medium, are used during the optimization. We jus-

tify how the RLS model can ﬁt a wide variety of dy-

namic mechanisms happening in scattering media. Its

arXiv:2210.04033v1 [physics.optics] 8 Oct 2022

performance is showcased with both simulated and exper-

imental results, tracking the transmission matrix and the

time-gated reﬂection matrix at realistic noise levels and

stability times. We further show how light optimization

can be achieved with binary amplitude or phase mod-

ulation and with phase-resolved or intensity-only mea-

surements. Based on its computational complexity, we

discuss its feasibility for light control in living biologi-

cal specimens at large ﬁelds of view. Its simple imple-

mentation and the low number of adjustable parameters

(whose choice is motivated in the next sections) make our

proposed method readily applicable in existing wavefront

shaping setups.

II. METHODS

The method bears similarities with conventional rou-

tines for the measurement of the transfer matrix, and

its working principle is graphically summarized in Fig.

1(a). However, here we allow the transfer matrix Xt∈

CM×Nof the scattering medium to be dynamic, where

we have denoted the number of output and input de-

grees of freedom with Mand N, respectively. At ev-

ery time step t, while probing the medium with the in-

put at∈CNand collecting the corresponding output

yt=Xtat∈CM, we aim to solve the optimization prob-

lem ˆ

Xt= arg minXtLt(Xt), with

Lt(Xt)≡

τ=1 λt−τ||yτ−Xtaτ||2+δλt||Xt||2

F,(1)

and where || · || and || · ||Fdenote the L2-norm of a vector

and the Frobenius norm of a matrix, respectively. Al-

though for sake of generality the inputs and the outputs

are assumed to be complex, we will also report an im-

plementation where they are real, meaning that only the

amplitude of the input beam is modulated and the in-

tensity of the output ﬁelds is measured. Equation 1 is

a linear least-squares loss function, featuring Tikhonov

regularization via the regularization constant δ. Note,

however, that each data-ﬁdelity term ||yτ−Xtaτ||2is

exponentially weighted in time, such that the old pieces

of data (corresponding to τt) are less relevant than

the most recent ones in the current estimation of the

transfer matrix at time t. In other words, the forgetting

factor λ≤1 endows the algorithm with a memory, which

allows it to cope with dynamic transfer matrices—at ev-

ery time step t, the optimization problem is solved anew,

using the whole history of past data, where more con-

tribution is given to newest data. Evidently, in the case

of a static scattering medium, all measurements can be

equally trusted, thus Eq. (1) reduces to a typical regu-

larized linear least-squares problem upon setting λ= 1.

Once λand δare ﬁxed, the least-squares problem has

a unique solution, provided the inputs are linearly in-

dependent, which is the case in conventional transfer-

matrix measurements, where the inputs are drawn from

the Hadamard basis of order N.

The choice of exponential weights for Eq. (1) is moti-

vated by the physics of our problem. We aim to follow

the evolution of the transfer matrix of dynamic scatter-

ing media, subjected to uncorrelated variations, whereby

the total transferred power fraction is constant in time.

These conditions apply in a wide variety of dynamic

mechanisms in scattering media investigated with visible

and near-infrared light, e.g. whenever their inner scatter-

ers move due to functional changes [5, 20], or even when

the sample drifts away from its initial position, suggest-

ing that our method can also be used as an online cali-

bration tool of imaging systems. In all these situations,

the transfer matrix can indeed be described by the time

series [21],

Xt=σX

pσ2

X+σ2

(Xt−1+Pt),(2)

where we assume that both the transfer matrix and the

perturbation matrix Ptare random variables indepen-

dently drawn from complex Gaussian distributions with

zero mean and constant variance σ2

Xand σ2

P, respectively

[22]. Equation 2 denotes an autoregressive model of or-

der 1, AR(1), whose autocovariance is proportional to

(σX/pσ2

X+σ2

P)t, justifying our exponentially weighted

model of Eq. (1). When focusing through dynamic scat-

tering media following Eq. (2), the stability time of the

enhancement is proportional to σ−2

P[21]. This means

that the optimal weight λshould follow the same de-

pendence, thus in principle requiring the knowledge of

the rate of change of the scattering medium. A strat-

egy for automatically tuning the forgetting factor will be

discussed in section IV.

Crucially, minimizing the loss function of Eq. (1) does

not require storing the whole history of past data. This

becomes apparent if we recall that the linear least-squares

estimate of Xt,ˆ

Xt, satisﬁes the normal equations,

Ctˆ

t=Kt,(3)

with the covariance matrix of inputs and the cross-

covariance matrix at time trespectively deﬁned as,

Ct≡

τ=1 λt−τaτaH

τ+δλtIN∈CN×N(4a)

Kt≡

τ=1

λt−τaτyH

τ∈CN×M,(4b)

with INdenoting the identity matrix of order Nand the

superscript Hstanding for Hermitian transposition. The

quantities calculated in Eqs. (4) can be both estimated

recursively, as follows:

Ct=λCt−1+ataH

t(5a)

Kt=λKt−1+atyH

t.(5b)

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Onlinelearningofthetransfermatrixofdynamicscatteringmedia:wavefrontshapingmeetsmultidimensionaltimeseriesLorenzoValzania1,andSylvainGigan1,1LaboratoireKastlerBrossel,EcoleNormaleSuperieure-ParisSciencesetLettres(PSL)ResearchUniversity,SorbonneUniversite,CentreNationaldelaRechercheScientique(CN...

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Online learning of the transfer matrix of dynamic scattering media wavefront shaping meets multidimensional time series Lorenzo Valzania1and Sylvain Gigan1

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