Chenqi Qian

Philipp Del Hougne
email: philipp.del-hougne@univ-rennes1.fr.

Noise-Adaptive Intelligent Programmable Meta-Imager

We present an intelligent programmable computational meta-imager that tailors its sequence of coherent scene illuminations not only to a specific information-extraction task (e.g., object recognition) but also adapts to different types and levels of noise. We systematically study how the learned illumination patterns depend on the noise, and we discover that trends in intensity and overlap of the learned illumination patterns can be understood intuitively. We conduct our analysis based on an analytical coupled-dipole forward model of a microwave dynamic metasurface antenna (DMA); we formulate a differentiable end-to-end information-flow pipeline comprising the programmable physical measurement process including noise as well as the subsequent digital processing layers. This pipeline allows us to jointly inverse-design the programmable physical weights (DMA configurations that determine the coherent scene illuminations) and the trainable digital weights. Our noise-adaptive intelligent meta-imager outperforms the conventional use of pseudo-random illumination patterns most clearly under conditions that make the extraction of sufficient task-relevant information challenging: latency constraints (limiting the number of allowed measurements) and strong noise. Programmable microwave meta-imagers in indoor surveillance and earth observation will be confronted with these conditions.

Introduction

Any measurement process is inevitably corrupted by noise of some type and level. We hypothesize that the optimal coherent illumination patterns to be used by an intelligent programmable metaimager to efficiently extract task-specific information from a scene will profoundly depend on the type and level of noise [START_REF] Saigre-Tardif | Intelligent meta-imagers: From compressed to learned sensing[END_REF]. We consider multi-shot programmable computational-imaging systems based on a single transmitter and a single detector; such systems are of particular relevance to the microwave domain where transceivers are costly and programmable metasurface apertures can synthesize (capture) coherent wavefronts from (with) a single radiofrequency chain. Envisioned deployment scenarios of programmable microwave meta-imagers include indoor surveillance and earth observation, where stringent latency constraints (limiting the number of allowed measurements) and significant amounts of noise combined with weak signals require that the utilized sequence of scene illuminations optimally extracts task-relevant information.

To systematically evaluate our hypothesis, we consider a prototypical object-recognition problem in which one microwave dynamic metasurface antenna (DMA) radiates a sequence of coherent wavefronts to the scene, and a second DMA coherently captures the reflected waves. For a given number of allowed measurements, we learn a task-specific and noise-specific sequence of optimized DMA configurations and benchmark its performance against conventional compressed sensing with random configurations. Our learned-sensing approach is advantageous whenever the amount of information that can be extracted from the scene is limited through latency constraints and/or noise such that it becomes crucial to maximize the amount of task-relevant information in the measured data. Moreover, we investigate how overlap and intensities of the learned illumination patterns depend on the noise type and level. We discover that these trends are intuitively understandable despite the black-box nature of the learned-sensing approach and the complexity of both metamaterial hardware and recognition task.

Compressive vs. Intelligent Meta-Imagers. Computational imagers multiplex scene information across a "coded aperture" onto a detector such that there is no one-to-one mapping between scene voxels and measured data. The coded aperture can be part of the transmit and/or receive hardware. Compressive imagers, a subset of computational imagers, take advantage of the inherent sparsity of most natural scenes to linearly embed the scene in a lower-dimensional space.

Random masks were shown to enable such isometric embeddings which are distance-preserving transformation; in other words, multiplexing the scene information across random masks compresses the scene information without distorting it. The ability to recover a scene from highly incomplete measurements sparked considerable interest, and much effort in the metamaterials community went into conceiving metamaterial hardware that can emulate random maskssee Sec. II in Ref. [START_REF] Saigre-Tardif | Intelligent meta-imagers: From compressed to learned sensing[END_REF]. Importantly for the current work, Ref. [START_REF] Sleasman | Dynamic metamaterial aperture for microwave imaging[END_REF] introduced programmable meta-imager hardware that can synthesize or capture different pseudo-random coherent wavefronts at a single frequency by randomly reconfiguring in situ the scattering properties of programmable meta-atoms in the transceiver hardware.

Two broad classes of programmable meta-imager hardware exist. A first class of programmable meta-imagers consists of structures that guide or trap waves; these structures are patterned with individually addressable meta-atoms that leak out (capture) wave energy toward (from) the scene via those meta-atoms that are configured to be resonant. The DMAs considered in this work are 2D parallel-plate waveguides with sub-wavelength 1-bit programmable meta-atoms patterned into one of the conducting surfaces [START_REF] Pulido-Mancera | Analytical Modeling of a Two-Dimensional Waveguide-Fed Metasurface[END_REF]. A second class of programmable meta-imagers consists of programmable metasurface reflect-or transmit-arrays, nowadays also coined reconfigurable intelligent surfaces, that sculpt waves from a feed antenna in order to flexibly illuminate a scene [START_REF] Li | Transmission-Type 2-Bit Programmable Metasurface for Single-Sensor and Single-Frequency Microwave Imaging[END_REF]. Programmable meta-imagers open the door for various kinds of optimizations of the used configurations to illuminate the scene, going beyond the use of random configurations from Refs. [START_REF] Sleasman | Dynamic metamaterial aperture for microwave imaging[END_REF][START_REF] Li | Transmission-Type 2-Bit Programmable Metasurface for Single-Sensor and Single-Frequency Microwave Imaging[END_REF]. For example, the configurations can be optimized to minimize the overlap of subsequent scene illuminations and thereby avoid the acquisition of redundant information [START_REF] Watts | Terahertz compressive imaging with metamaterial spatial light modulators[END_REF][START_REF] Del Hougne | Optimal Multiplexing of Spatially Encoded Information across Custom-Tailored Configurations of a Metasurface-Tunable Chaotic Cavity[END_REF]. It is also possible to use a specific, as opposed to arbitrary, orthogonal basis of scene illuminations in which a typical scene can be expressed as linear superposition of a small number of "principle" patterns [START_REF] Liang | Reconfigurable Array Design to Realize Principal Component Analysis (PCA)-Based Microwave Compressive Sensing Imaging System[END_REF][START_REF] Li | Machine-learning reprogrammable metasurface imager[END_REF].

The use of meta-imager configurations yielding pseudo-random, orthogonal or principlecomponent-based scene illuminations isometrically embeds all scene information in a lowerdimensional space and is hence by definition agnostic to the task and the noise. However, many deployment scenarios involve latency constraints and noise, both of which limit the amount of information that can be extracted from the scene. In such situations, it is desirable to instead use a purposefully non-isometric embedding that discriminates between task-relevant and taskirrelevant information as much as possible (see Supplementary Note 2 for an illustration). This learned-sensing approach uses the same programmable meta-imager hardware but judiciously tailors the utilized configurations such that the measurement process becomes simultaneously a task-specific "over-the-air" analog processing step. The selection of task-relevant information during the measurement, as opposed to an indiscriminate acquisition of all information, distinguishes intelligent meta-imagers from compressive meta-imagers [START_REF] Saigre-Tardif | Intelligent meta-imagers: From compressed to learned sensing[END_REF]. The concept of intelligent meta-imaging was introduced in Ref. [START_REF] Del Hougne | Learned Integrated Sensing Pipeline: Reconfigurable Metasurface Transceivers as Trainable Physical Layer in an Artificial Neural Network[END_REF] where an end-to-end pipeline including both the physical measurement process, parametrized by the meta-atom configurations, and the digital processing layer, parametrized by digital weights, was optimized with respect to the performance on a specific task. The resulting task-aware meta-imager configurations yielded substantial accuracy improvements under latency constraints, a result that was subsequently confirmed experimentally in Ref. [START_REF] Li | Intelligent Electromagnetic Sensing with Learnable Data Acquisition and Processing[END_REF]. However, while Refs. (9, 10) learned task-specific configurations for their programmable metaimagers, they were limited to scenarios without noise or with negligible noise, respectively. Yet, according to our aforementioned hypothesis, noise may profoundly impact the optimal metaimager configurations because, besides latency constraints, noise also limits the amount of information that can be extracted from the scene. As stated above, many envisioned microwave meta-imaging applications will inevitably be confronted with strong noise. Signal averaging to reduce the noise may not be optimal due to the inevitable latency penalty associated with repeated measurements. Moreover, in the remaining scenarios, one may desire to trade off the emitted signal power against performance in order to improve metrics like energy consumption, radiation exposure and spectrum allotment. Hence, understanding how the performance of intelligent metaimagers depends not only on latency constraints but also on the noise is important.

Noise in Task-Specific Sensing. A related strand of research in the optical domain already routinely includes noise in similar end-to-end task-specific optimizations of physical hardware and digital processing layers [START_REF] Chakrabarti | Learning Sensor Multiplexing Design through Back-propagation[END_REF][START_REF] Horstmeyer | Convolutional neural networks that teach microscopes how to image[END_REF][START_REF] Sitzmann | End-to-end optimization of optics and image processing for achromatic extended depth of field and super-resolution imaging[END_REF][START_REF] Hershko | Multicolor localization microscopy and point-spread-function engineering by deep learning[END_REF][START_REF] Lin | End-to-end nanophotonic inverse design for imaging and polarimetry[END_REF][START_REF] Tseng | Neural nano-optics for high-quality thin lens imaging[END_REF][START_REF] Arya | End-to-End Optimization of Metasurfaces for Imaging with Compressed Sensing[END_REF]. However, typically only a single low noise level is (sometimes somewhat arbitrarily) chosen and optimized for in these works, such that there is no systematic investigation of the effect of noise, especially not of strong noise, on the optimized illumination patterns. Moreover, these optical learned-sensing schemes differ conceptually from the present work in that they operate with a single-shot detector-array measurement. By construction, these schemes hence do not face a latency-induced limitation on the amount of information that can be extracted from the scene, unlike the typical multi-shot single-detector programmable meta-imagers considered in the present work that must limit their number of measurements to avoid prohibitive latencies. This difference originates from the fact that the optical regime is confronted with different hardware constraints: while it is costly to measure phase, detector arrays like CCD cameras are available at low or moderate cost. Only a single illumination pattern is hence optimized in typical optical learned sensing, precluding the investigation of intriguing aspects such as the overlap of subsequent learned illumination patterns in multi-shot schemes that are discussed in the present work. Moreover, this single illumination pattern is often implemented with a static optimized optical element (e.g., a metasurface) as opposed to an in-situ reconfigurable measurement setup as considered in this work. Departing from the single-shot approach, Refs. [START_REF] Muthumbi | Learned sensing: jointly optimized microscope hardware for accurate image classification[END_REF][START_REF] Nehme | Learning Optimal Wavefront Shaping for Multi-Channel Imaging[END_REF] from the optical domain optimized exactly two illumination patterns for a specific task and one chosen noise level; again, no systematic analysis of the role of noise on the learned illumination patterns was presented and strong-noise regimes were not considered.

The complexity of typical information-extraction problems implies that there is, in general, no guarantee to converge toward the globally optimal set of scene illuminations. However, heuristic evidence suggests that different realizations converge to distinct local optima of roughly equal quality [START_REF] Del Hougne | Learned Integrated Sensing Pipeline: Reconfigurable Metasurface Transceivers as Trainable Physical Layer in an Artificial Neural Network[END_REF]. Nonetheless, in a few special cases, provably optimal "maximum information states" can be identified analytically. If the task is to distinguish between exactly two specific scene configurations without any realization-to-realization variation using a single-shot coherent illumination, and assuming that the difference Δ𝑇 between the multi-channel input-output relation of the system can be measured without any noise for these two configurations, then it follows from basic linear algebra that the first eigenvector of (Δ𝑇) † (Δ𝑇) is the optimal illumination pattern [START_REF] Bouchet | Optimal Control of Coherent Light Scattering for Binary Decision Problems[END_REF].

A similar technique can be applied to the precise estimation of a small perturbation of a single parameter in the vicinity of a known value [START_REF] Bouchet | Maximum information states for coherent scattering measurements[END_REF]. However, we are generally interested in sensing non-perturbative changes with a certain degree of realization-dependence and more than two possible outcomes, and without reliance on noise-free characterization measurements. Therefore, the identification of task-specific illumination patterns generally involves an optimization problem.

Our Contributions. In this paper, we systematically explore how the combination of latency constraints and noise impacts intelligent multi-shot programmable meta-imagers. Considering microwave DMA hardware for a prototypical object-recognition problem, we benchmark the performance of a noise-adaptive intelligent programmable meta-imager against the conventional use of random configurations. We consider the entire range of conceivable noise levels for two distinct noise types: signal-independent additive Gaussian noise and signal-dependent additive Gaussian noise. We analyze how the average overlap and the average intensity of the learned scene illuminations vary with the number of allowed measurements, the noise level, and the noise type.

Thereby, we discover intuitively understandable trends. This paper is organized as follows. First, we describe the considered system model. Second, we formulate the end-to-end information flow through our wave-based information-extraction problem. Third, we discuss the end-to-end task-specific optimization. Fourth, we present our analysis of performances and illumination-pattern sequences for the case of signal-independent additive noise. Fifth, we perform the same analysis for the case of signal-dependent additive noise.

Sixth, we study the performance outside the trained noise regime. Finally, we conclude with a discussion and summary.

Results

System Model. We consider the microwave computational programmable meta-imager system introduced in Ref. [START_REF] Del Hougne | Learned Integrated Sensing Pipeline: Reconfigurable Metasurface Transceivers as Trainable Physical Layer in an Artificial Neural Network[END_REF]. As illustrated in Figure 1, a scene in free space 1 is exposed to a sequence of M coherent continuous-wave illuminations at 10 GHz generated by a transmitting (TX) DMA, and the reflected waves are coherently captured by a second receiving (RX) DMA. The DMAs are parallel-plate waveguide structures connected to a single-mode coaxial cable and patterned with programmable meta-atoms on one of their surfaces [START_REF] Pulido-Mancera | Analytical Modeling of a Two-Dimensional Waveguide-Fed Metasurface[END_REF][START_REF] Del Hougne | Learned Integrated Sensing Pipeline: Reconfigurable Metasurface Transceivers as Trainable Physical Layer in an Artificial Neural Network[END_REF]. Each meta-atom is individually 1-bit programmable and approximated as a discrete dipole; in its ON (OFF) state, a programmable metaatom is (is not) resonant and hence has a finite (zero) dipole moment such that it couples (does not couple) waveguide modes to modes propagating in free space. The dipole moment of a resonant meta-atom depends on the incident field and hence on (i) its location on the waveguide surface, and (ii) the configuration of the remaining meta-atoms due to multiple scattering. The 𝑖th measurement yields a single complex-valued scalar 𝑚 𝑖 that is corrupted by additive noise 𝑛 𝑖 . 2 Under the first Born approximation,

𝑚 𝑖 = ∫ 𝐸 𝑖 TX (𝐫)𝐸 𝑖 RX (𝐫)𝜎(𝐫)d𝐫 scene = ∫ ℐ 𝑖 (𝐫)𝜎(𝐫)d𝐫 scene , (1)
where 𝐫 denotes a coordinate in 3D space, 𝐸 𝑖 TX (𝐫) and 𝐸 𝑖 RX (𝐫) are the electric field patterns of the TX DMA and the RX DMA in the plane of the scene, respectively, and 𝜎(𝐫) is the scene reflectivity [START_REF] Lipworth | Comprehensive simulation platform for a metamaterial imaging system[END_REF]. Moreover, we define the 𝑖th coherent illumination pattern ℐ 𝑖 (𝐫) = 𝐸 𝑖 TX (𝐫)𝐸 𝑖 RX (𝐫). I and Q components of the M-element complex-valued vector of measured data, [𝑚 1 + 𝑛 1 , 𝑚 2 +

1 Computational meta-imagers are to date mainly studied for operation in free space where the Green's function between metasurface aperture and scene is known analytically. Interestingly, first investigations of computational meta-imaging in complex scattering environments indicate that the reverberation in such conditions can strongly improve the achievable precision because it introduces a generalized interferometric sensitivity [START_REF] Del Hougne | Deeply Subwavelength Localization with Reverberation-Coded Aperture[END_REF]. 2 In principle, the noise can also be non-additive (e.g., in Poisson-distributed processes). In the most general formulation, the measured complex-valued scalar is 𝑓(𝑚 𝑖 ), where 𝑓(•) denotes the arbitrarily complex noise function acting on 𝑚 𝑖 . In the present work, we describe the entire physical measurement process with an analytical coupled-dipole forward model that approximates the subwavelength meta-atoms as dipoles. This involves determining the dipole moments of each meta-atom for each configuration (3), as well as the use of Eq. ( 1) to propagate the field from the TX DMA to the scene, and back from the scene to the RX DMA [START_REF] Lipworth | Comprehensive simulation platform for a metamaterial imaging system[END_REF]. We assume that the amount of wave energy injected into the TX DMA is always the same throughout this work. For the sake of brevity, we do not repeat the mathematical details of our analytical forward model here; instead, we refer the reader to Ref. [START_REF] Del Hougne | Learned Integrated Sensing Pipeline: Reconfigurable Metasurface Transceivers as Trainable Physical Layer in an Artificial Neural Network[END_REF] where they are provided and explained in depth. Note that our results presented below rely on having a differentiable forward model, but there is no requirement for this model to be analytical; instead, one could, for instance, also learn a forward model [START_REF] Li | Intelligent Electromagnetic Sensing with Learnable Data Acquisition and Processing[END_REF]. For future experimental implementations, a physics-based learned forward model of the physical measurement process could be an enticing option to exploit partial physics knowledge in combination with fine tuning learned from calibration data of the experimental system [START_REF] Shlezinger | Model-Based Deep Learning[END_REF].

Information Flow. To implement a task-specific end-to-end optimization of both physical and digital layers, a prerequisite is the formulation of the pipeline through which information flows in the considered problem. The essential elements of this pipeline are the same for all wave-based information-extraction problems, including imaging, sensing, localization, and object recognition.

The only significant difference lies in the task-specific cost function that is to be optimized for good performance. Two broad classes of tasks can be distinguished: classification-type tasks and regression-type tasks. Classification-type tasks include, for instance, object recognition on which we focus for concreteness in the following. For classification tasks, the cost function determines the precision with which a given scene is correctly determined to belong to one out of P pre-defined classes. Regression-type tasks include, for instance, the estimation of a continuous parameter (e.g., the location of an object) as well as image reconstruction. For regression tasks, the cost function quantifies the difference between the true and estimated variable(s), such as an object location or the pixel values in an image.

The starting point of the information flow is the sought-after latent information of interest 𝜒.

The latter exists in some abstract latent information space. In our prototypical example, we seek to recognize metallic digits in the scene. The latent information of interest is hence the fact that the digit belongs to one of the 𝑃 = 10 possible classes. As illustrated in Figure 2, the latent information of interest could hence, for instance, be "3". Some scene function 𝒮 maps the latent information space to the experimental reality by encoding 𝜒 in the scene reflectivity 𝜎(𝐫). This scene function 𝒮 usually describes some natural process, in our case handwriting, that can have a certain realization-to-realization variation (i.e., different people write digits to some extent differently) and is, in general, not parametrized by any controllable parameters that could be optimized. The scene reflectivity 𝜎(𝐫) is hence the physically encoded information of interest, 𝒮(𝜒).

The scene is now probed with M coherent illumination patterns {ℐ 𝑖 (𝐶 𝑖 )} which are generated with M configurations {𝐶 𝑖 } of the DMAs, as detailed in the previous section. 3 Given the singledetector nature of the considered measurement process, each measurement yields a single complexvalued scalar measurement that depends on both the scene and the utilized configuration of the TX DMA and the RX DMA: 𝑚 𝑖 (𝒮(𝜒), 𝐶 𝑖 ). In addition, the measurement is corrupted by additive noise 𝑛 𝑖 , as mentioned previously. The measurement process is hence parametrized by trainable physical weights, the M DMA configurations {𝐶 𝑖 }, which are highlighted in green font in Figure 2 and can be optimized. Overall, we summarize the mapping of the scene 𝒮(𝜒) to the measured data vector through the measurement function ℳ.

Finally, a digital-processing function 𝒟 acts on the measurement data to obtain an estimate 𝜒 ̃ of the information of interest. This digital processing step is parametrized by a set of trainable digital parameters {𝜙 𝑖 }, for instance, the weights of a fully connected artificial neural network (ANN) in our present work, which are highlighted in pink font in Figure 2 and can be optimized. One complication of typical programmable meta-imager hardware is the 1-bit or few-bit (as opposed to continuous) programmability of the meta-atoms that may appear incompatible with error backpropagation. To overcome this challenge, we follow Refs. [START_REF] Del Hougne | Learned Integrated Sensing Pipeline: Reconfigurable Metasurface Transceivers as Trainable Physical Layer in an Artificial Neural Network[END_REF][START_REF] Chakrabarti | Learning Sensor Multiplexing Design through Back-propagation[END_REF] and use a "temperature parameter" that gradually drives the values of the trainable physical parameters from a continuous distribution to the desired discrete distribution. Algorithmic details are available in Ref. [START_REF] Del Hougne | Learned Integrated Sensing Pipeline: Reconfigurable Metasurface Transceivers as Trainable Physical Layer in an Artificial Neural Network[END_REF]. We use the MNIST data set with 70,000 examples of handwritten digits for training (51,000), validation (9,000) and testing (10,000).

Signal-Independent Additive Noise. We begin by considering the most common model for noise in the kind of microwave measurements we are concerned with: signal-independent additive Gaussian noise. Specifically, we consider independent and identically distributed zeromean Gaussian noise with standard deviation 𝜌 on the I and Q components of the complex-valued measured data. In other words, Re(𝑛 𝑖 ) and Im(𝑛 𝑖 ) are drawn from a zero-mean Gaussian distribution with standard deviation 𝜌. However, more meaningful than the absolute noise level is the relative noise level compared to the signal level. The SNR is defined as the ratio of the signal variance to the noise variance. Yet, we cannot simply use the SNR here because the standard deviation of the measured signal is strongly dependent on the choice of DMA configurations (random vs. non-random). To understand the latter, it is easy to think of extreme cases in which the standard deviation of the measured signal is minimal, e.g., if non-random DMA configurations differing only in the configuration of a single meta-atom are used.

Recall that such differences are not related to the power injected into the TX DMA which is kept fixed throughout this work. To define a universal reference signal level, we hence introduce the auxiliary quantity

𝜅 = SD ( Re(𝑚 𝑖 )+Im(𝑚 𝑖 ) 2 ) random DMA configurations
, that is, the standard deviation of the averaged I and Q components of the signals measured with many random DMA configurations. We can now define 𝜌′ = 𝜌/𝜅 as meaningful dimensionless quantification of the noise level. Moreover, we note that considering non-zero-mean Gaussian distributions would yield the same results in our system model due to (i) the above-mentioned normalization of the measured data before it is fed into the digital layers and

(ii) the fact that we assume lossless analog-to-digital conversion [START_REF] Viswanathan | Optimization of a Sparse Aperture Configuration for Millimeter-Wave Computational Imaging[END_REF], which is effectively available, for instance, with commercially available vector network analyzers in the microwave domain.

We expect the use of end-to-end optimized task-specific DMA configurations to outperform the conventional use of random configurations in conditions under which only a limited amount of information can be extracted from the scene: when the number of allowed measurements M is limited due to latency constraints and/or when the noise is strong. Therefore, we explore the performance of these two approaches (learned illuminations vs. random illuminations) as a function Let us begin by examining the extreme cases. For a low noise level (𝜌′ = 0.1) and many measurements (𝑀 > 10), both approaches achieve close to unity accuracy because sufficient taskrelevant information is included in so many measurements even if they are not task-specific. For a high noise level and few measurements, the results of both techniques approach the random-guess baseline of 10 % accuracy (the 𝑃 = 10 classes are equiprobable). Using few and very noisy measurements does not allow the system to extract sufficient task-relevant information to achieve good performance, although the use of learned patterns remains superior to the use of random patterns, as expected. The most interesting and application-relevant regime is between these two extreme cases. We note that the accuracy with the learned illuminations is always superior to the random-illumination benchmark. For 𝜌′ = 10, the learned patterns outperform the random patterns most clearly for the largest considered value of 𝑀 = 150 (46 % vs. 18 %). For 𝜌′ = 1, the largest absolute performance gain from using learned patterns is seen at 𝑀 = 8 (72 % vs. 40 %). For 𝜌′ = 0.1, the largest absolute performance gain is at 𝑀 = 2 (70 % vs. 50 %) and 𝑀 = 3 (83 % vs. 63 %). The performance gap peak hence occurs at lower values of 𝑀 for lower values of 𝜌′. Naturally, the advantage of learned patterns over random patterns occurs in a regime in which the total amount of information that can be extracted from the scene is limited -but not too limited such that even the learned patterns struggle to capture sufficient task-relevant information.

We now aim to gain insights into how the end-to-end task-specific optimization achieves these remarkable performance improvements over the conventional random patterns. Given the complexity of the DMA hardware as well as the recognition task, we do not hope to comprehend "microscopically" every meta-atom's specific learned configuration; instead, we now analyze "macroscopically" each sequence of 𝑀 illumination patterns, {ℐ 𝑖 }, in terms of the illumination patterns' mutual overlaps and their intensities. Specifically, we evaluate the mean illumination pattern overlap within a given sequence,

𝒪 = ⟨ | | ∫ (ℐ 𝑖 (𝐫)) † ℐ 𝑗 (𝐫)d𝐫 scene √ ∫ |ℐ 𝑖 (𝐫)| 2 d𝐫 scene ∫ |ℐ 𝑗 (𝐫)| 2 d𝐫 scene | | ⟩ 𝑖≠𝑗 , ( 2
)
where † denotes the complex conjugate and the angle brackets denote averaging over all possible choices of two distinct patterns within a given sequence; note also that |ℐ 𝑖 (𝐫)| 2 = (ℐ 𝑖 (𝐫)) † ℐ 𝑖 (𝐫).

Furthermore, we define for a given sequence the corresponding mean illumination pattern intensity,

𝐼 = ⟨∫ |ℐ 𝑖 (𝐫)| 2 d𝐫 scene ⟩ 𝑖 , ( 3
)
where the angle brackets denote averaging over all 𝑀 patterns of the given sequence. Note that even though the input power injected into the TX DMA is fixed throughout this work, 𝐼 can significantly vary depending on (i) how many meta-atoms are configured to be resonant ("ON"), and (ii) whether they create constructive or destructive interferences in the illumination patterns.

In the most extreme case with all TX DMA and/or all RX DMA elements configured to be nonresonant, 𝐼 = 0.

In the second row of Figure 3, we plot the average over the ten realizations of the mean illumination pattern overlap 𝒪. If random illumination patterns are used, the average overlap is by construction independent of M and 𝜌′ (since neither of these influences the choice of illumination patterns) at a value of 29 %. For the low-noise regime (𝜌′ = 0.1), the average overlap of the learned patterns is identical to that of the random patterns and also independent of M. This is consistent with the findings in Ref. [START_REF] Del Hougne | Learned Integrated Sensing Pipeline: Reconfigurable Metasurface Transceivers as Trainable Physical Layer in an Artificial Neural Network[END_REF] for noiseless operation. Remarkably, concerns about the acquisition of redundant information in subsequent measurements with partially overlapping illumination patterns are apparently traded off against more important other benefits identified by the intelligent meta-imager in its learned illuminations. We speculate that the system's reason for preferring a finite overlap relates to the fact that the scene contains both relevant and irrelevant details with respect to our specific task. If irrelevant scene details were illuminated with different patterns, varying contributions to the measured signal originating from the irrelevant details would act as an effective source of noise. It is hence a plausible strategy to always illuminate the irrelevant details in the same way, so that signal variations solely originate from the relevant scene details. This would then indeed imply that optimized task-specific scene illuminations have a non-zero overlap even in the low-noise or noise-free regime.

For the moderate-noise and high-noise regimes, we observe that the average overlap increases with decreasing M, all the way up to 84 % for 𝑀 = 2 and 𝜌′ = 10. Hence, the intelligent metaimager deliberately increases the overlap of its illumination patterns as the amount of information that it can extract from the scene becomes more and more limited. This choice can be understood intuitively because subsequent measurements with high overlap are essentially a kind of signal averaging, a well-known human strategy to limit the corruption of measured data through strong noise. However, while signal averaging would correspond to 100 % overlap, the intelligent metaimager gradually adapts the level of overlap to the noise level and the number of allowed measurements, hence finding an optimized trade-off between some sort of signal averaging and the acquisition of non-redundant information, going well beyond mere signal averaging.

Next, we consider in the third row of Figure 3 the average over the ten realizations of the mean illumination pattern intensity 𝐼. Recall that the wave energy injected into the TX DMA is always the same throughout this work. For random DMA configurations, the average value of 𝐼 is again by construction independent of M and 𝜌′, and we normalize it to unity for convenience. For learned DMA configurations, even in the low-noise regime (𝜌′ = 0.1), as M is decreased from 150, the average intensity increases to almost twice that obtained with random DMA configurations for moderately low values of M. Surprisingly, for very small values of M the average intensity drops again, presumably because the limited available control over the illumination pattern is allocated to optimizing the pattern as opposed to the signal strength in this regime. For the moderate-noise regime (𝜌′ = 1), the average intensity for learned DMA configurations increases from twice to four times that obtained with random DMA configurations as M is reduced from 150 to 1. In the highnoise regime (𝜌′ = 10), the averaged intensity with learned DMA configurations is four times that obtained with random DMA configurations for all considered values of M. Again, this choice of the intelligent meta-imager to deliberately increase the intensity is intuitively understandable: as the amount of information that can be extracted from the scene gets more limited due to more noise and/or lower M, boosting the signal strength helps to overcome some noise corruption and extract more information per measurement. Human operators routinely choose to focus illumination patterns on regions of interest (ROI) in order to probe them with improved SNR. For instance, in Refs. [START_REF] Li | Intelligent metasurface imager and recognizer[END_REF][START_REF] Ruan | Microwave Speech Recognizer Empowered by a Programmable Metasurface[END_REF], in a first step a programmable meta-imager illuminates a scene with random patterns in order to identify the ROI; in a second step, waves are then focused on this ROI. In Refs. [START_REF] Li | Intelligent metasurface imager and recognizer[END_REF][START_REF] Ruan | Microwave Speech Recognizer Empowered by a Programmable Metasurface[END_REF], the choice to focus in the second step was explicitly imposed by a human operator whereas the intensity increases we observe in Figure 3 occur spontaneously without any human instructions.

How does the intelligent meta-imager increase 𝐼 by a factor of four on average if the wave energy injected into the TX DMA is always the same? An important aspect are certainly constructive interferences that can be tailored to occur in the scene. However, a second aspect relates to the number of radiating meta-atoms. Clearly, with more radiating elements, on average more energy leaks out of the DMA and toward the scene. While the average ON ratio of the metaatoms is obviously 50% for random illuminations, it gradually increases up to 68 % as M gets smaller and the noise stronger -see the fourth row in Figure 3. Nonetheless, given that the ON ratio only increases from 50 % to 68 %, the four-fold increase in 𝐼 must mainly be attributed to tailored constructive interferences.

We have hence discovered two "macroscopic" trends in the learned illumination patterns that can be understood intuitively: as the amount of information that can be extracted gets limited by latency and noise, the intelligent meta-imager tends to increase overlap and intensity of its illumination patterns. While we can intuitively comprehend these choices, these results arise spontaneously in the optimization process without any explicit constraints from us regarding these mechanisms. This observation is reminiscent of other optimization problems in wave engineering where intuitively understandable designs emerged spontaneously; for instance, the inverse design of structures aimed at backscatter-protected transport yielded topological insulators [START_REF] Christiansen | Topological Insulators by Topology Optimization[END_REF].

Moreover, the insights into the learned illumination patterns give us confidence that average global ("macroscopic") features can be understood intuitively, in contrast to "microscopic" details or realization-specific global features. With respect to the latter, it is interesting to note that while the average accuracy fluctuates little across the ten realizations, the fluctuations in overlap are notable and the fluctuations in intensity are very large. This is also evident upon visual inspection of the provided example illumination-pattern magnitudes in Figure 3. Overall, these findings hint at analogies with statistical physics and thermodynamics, which are indeed known to be intimately connected to machine-learning concepts [START_REF] Carleo | Machine learning and the physical sciences[END_REF].

In Figure 4, we plot the same quantities as in Figure 3 but for a fixed 𝑀 = 3 and sweeping the noise level 𝜌′ from 10 -2 to 10 2 . It is apparent that for noise levels below 10 -1 , the results become independent of 𝜌′ because the noise becomes negligible. For noise levels above 10, we see that the previously discussed trends in terms of overlap and intensity gradually disappear again. This can be attributed to excessively strong noise such that the system cannot successfully operate anymore. Signal-Dependent Additive Noise. Having studied signal-independent additive noise in the previous section, we now consider a type of noise that is additive and signal-dependent. The purpose of this section is to demonstrate the generality of the noise-adaptiveness of our intelligent programmable meta-imager, and to investigate whether the above-discussed trends for signalindependent noise also hold for other noise types. The signal-dependent noise model we consider draws Re(𝑛 𝑖 ) and Im(𝑛 𝑖 ) from zero-mean Gaussian distributions with standard deviations 𝛽|Re(𝑚 𝑖 )| and 𝛽|Im(𝑚 𝑖 )|, respectively. The noise distribution is hence parametrized by the dimensionless parameter 𝛽. In this section, we fix 𝑀 = 3 and sweep 𝛽 from 10 -2 to 10 2 . We do not explicitly claim that this noise model is relevant to a specific microwave experiment, but we do note that this type of noise model may arise, for instance, in certain natural processes [START_REF] Harris | Signal-dependent noise determines motor planning[END_REF].

The trends observed with this signal-dependent noise type in the right column of Figure 4 are qualitatively clearly different from those previously found for signal-independent noise in the left column of the same figure. Our intelligent meta-imager hence adapts not only to the noise strength but also to the noise type. With the utilized signal-dependent noise model and for 𝑀 = 3, the absolute performance gain with learned illuminations is roughly constant for all considered values of 𝛽. For 𝛽 → 0, this noise model approaches the noiseless regime and hence yields results comparable to the signal-independent noise model from the previous section with 𝜌′ → 0. The average overlap slightly decreases as 𝛽 is increased. More notable is that the average intensity approaches zero as 𝛽 is increased. This trend is again intuitively understandable. For high values of 𝛽, strong signals are severely punished with very strong noise such that it appears advantageous to limit the signal intensity. This is achieved through a notable reduction of the ON ratio to below 25 %, but certainly also through tailored destructive interferences in the scene: For 𝛽 = 10 2 , a quarter of the meta-atoms leaks energy to the scene but the average intensity in the scene is close to zero. In contrast, for 𝛽 = 10 -2 , there is no strong-signal penalty and a mildly enhanced intensity with respect to the use of random DMA configurations is seen in Figure 4.

Performance Outside the Trained Noise Regime. Finally, we explore how the recognition performance changes if the intelligent meta-imager is forced to operate outside the trained noise regime. The results displayed in Figure 5 reveal that in the case of signal-independent additive Gaussian noise, operating at a noise level that is stronger than in the trained noise regime yields deteriorated performance, as expected. Operating at a lower noise level than during training does not significantly impact the performance. The use of learned as opposed to random illuminations is always advantageous. In the very-strong-noise regime (𝜌′ > 10), the average accuracy is close to the random-guess baseline.

A qualitatively different behavior is seen in the case of signal-dependent additive Gaussian noise. Here, for low noise levels (𝛽 < 1), we also observe that testing at a noise level lower (higher)

than the training noise level does not (does) deteriorate the average accuracy. However, for very strong noise levels (𝛽 > 10), the accuracy is above the random-guess baseline, in contrast to the signal-independent additive noise. Moreover, we observe that in the strong-noise regime the accuracy rapidly falls off as the noise level is detuned in either direction, and detuning toward lower noise levels leads to a faster deterioration of performance. These observations can be understood from the fact that the system can extract some information from the noise because the noise is signal-dependent and hence bears some correlation with the signal. For instance, a high measured value is unlikely to arise from a low signal in the considered signal-dependent noise model. Yet, if the system has carefully adjusted to extract some information from the noise, it is more vulnerable to noise detuning.

Discussion

So far, we tested the performance of a programmable meta-imager that generates a sequence of scene illuminations that are specific to a chosen task and a chosen type and level of noise. The transition toward a system that self-adaptively detects the type and level of noise and updates accordingly its utilized sequence of DMA configurations without additional human input is straightforward. First, before runtime, one establishes a codebook of sequences of DMA configurations that are optimized for the various types and levels of noise that one expects to possibly arise during runtime. For instance, in the common case of signal-independent additive complex Gaussian noise, this will involve repeating the end-to-end task-specific optimization for the range of different noise levels that may be encountered. During runtime, the current type and level of noise is simply determined from repeated measurements of the same scene with the same illumination pattern, to then choose accordingly a suitable sequence of DMA configurations from the codebook. This sensing of the noise level should be repeated in regular intervals that correspond to the frequency with which the noise can change in the working environment. Given achievable refresh rates of the DMAs of at least a few tens of kHz, it is possible to perform the noise sensing while a typical scene is effectively static. Therefore, our intelligent programmable meta-imager is not only noise-specific but also straightforwardly noise-adaptive.

Throughout this work, we have focused on different types of detector noise that arise during the detection of a signal. In principle, noise can also arise prior to the signal detection, namely during the signal generation or the signal propagation. Signal-generation noise could arise at the TX DMA and would be multiplexed across the scene illumination patterns, an effect known as noise folding [START_REF] Arias-Castro | Noise Folding in Compressed Sensing[END_REF]. Signal-propagation noise would occur if a given instance of the scene reflectivity was not perfectly static but to some extent fluctuating, a phenomenon that may arise, for instance, for operation in dynamically evolving indoor environments as opposed to free space [START_REF] Del Hougne | Robust position sensing with wave fingerprints in dynamic complex propagation environments[END_REF] or if the location of the digit in the scene plane dynamically fluctuates. Both signal-generation noise and signal-propagation noise cannot trivially be mapped into detector noise, but by accounting for them in the information-flow pipeline, our end-to-end optimization technique would adapt to those types of noise as well. A preliminary investigation of the impact of fluctuations of the digit's location in the scene plane is presented in Supplementary Note 1 and suggests that the ability to select predominantly task-relevant information remains a strong advantage under such conditions.

Learned sensing can achieve a remarkable reduction of the number of required measurements in order to achieve a given performance goal as long as only a (relatively small) fraction of all scene information is relevant to successfully complete the task. Besides the improved latency, this reduction of the number of required measurements also favorably impacts many other metrics like energy consumption, radiation exposure, as well as memory and computing requirements during runtime. Indeed, given that less data is acquired, storing and digitally processing the data is less costly. Moreover, our digital processing layer requires only a single forward pass during runtime to estimate the sought-after parameterin contrast, for instance, to popular iterative image reconstruction algorithms. But, of course, all these benefits during runtime of a task-specific endto-end optimized system do not come for free but originate from the availability of a priori knowledge and a one-off computationally intensive optimization of the DMA configurations before runtime.

To summarize, we have presented a noise-adaptive and task-specific intelligent programmable meta-imager, considering a prototypical object-recognition task with microwave DMA hardware.

The amount of information that can be extracted from the scene is limited both through latency constraints and noise in such multi-shot single-detector schemes which are typical for microwavedomain meta-imaging applications. Remarkable performance improvements of our intelligent meta-imager over conventional compressive meta-imagers arise under these conditions. We have demonstrated these performance gains for a signal-independent and a signal-dependent additive noise type. Moreover, we have discovered that "macroscopic" features of the learned illumination patterns, namely their overlaps and intensities, are on average intuitively understandable. We faithfully expect that our results can be transposed to information-extraction problems based on other wave phenomena (e.g., optics, acoustics, elastics, quantum mechanics) and/or with other types of in-situ programmable measurement hardware.

Figure 1 :

 1 Figure 1: Schematic Overview. An information of interest 𝜒, "3", is physically encoded in a scene reflectivity of a corresponding handwritten metallic digit. The scene is probed with M measurements. For each measurement, a TX DMA radiates a coherent field toward the scene, and the reflection is coherently captured by a RX DMA, yielding a single complex-valued scalar 𝑚 𝑖 that is corrupted by noise 𝑛 𝑖 . The coherently radiated and captured fields of the TX and RX DMAs in the 𝑖th measurement are determined by the configuration of the DMAs' meta-atoms, 𝐶 1 TX and 𝐶 1 RX , respectively. The wave energy injected into the TX DMA is always the same. The measured data from the M measurements is injected into a fully-connected digital-processing neural network in order to output an estimate 𝜒 ̃ of the sought-after information of interest. The overall procedure is hence parametrized by physical weights (the meta-atom configurations) and digital weights (the digital ANN weights). The inset shows the considered DMA hardware: a 2D parallel-plate waveguide with 16 1-bit programmable meta-atoms patterned into the surface facing the scene. Depending on the bias voltage applied across the PIN diode, a given meta-atom is resonant (blue) or not (red), as shown for an example configuration on the left. The corresponding dipole moments of the meta-atoms are shown on the right as phasors.

Figure 2 :

 2 Figure 2: Information-Flow Pipeline for Task-Specific and Noise-Adaptive Coherent Wave-Based Information Extraction.A batch {𝜒} of instances of the latent information of interest 𝜒, an example being "3" for our handwritten digit recognition task, is mapped via the scene function 𝒮 to the real world; in our case, the scene reflectivity physically encodes the information of interest. Next, a measurement function ℳ, parametrized by a set of physical parameters {𝐶 𝑖 } (in our case the sequence of DMA configurations; green), maps the scene reflectivity to a complex-valued vector of measured noise-corrupted data. Finally, a digitalprocessing function 𝒟, parametrized by a set of digital parameters {𝜙 𝑖 } (in our case the ANN weights; pink), attempts to extract an estimate 𝜒 ̃ from the measured data. The cost function evaluates how close the

  of M and 𝜌′ in the following. To begin, we consider the achieved accuracy as a function of M for the three representative noise levels of 𝜌′ = 0.1, 1 and 10. The corresponding results are plotted in the top row of Figure3and show the average and standard deviation over 10 separate optimizations for each combination of M and 𝜌′. To avoid that the results are specific to a DMA layout, we choose different random locations of the 16 meta-atoms on the TX and RX DMA apertures for each of the ten optimizations.

Figure 3 :

 3 Figure 3: Intelligent Meta-Imaging Adaptation to Signal-Independent Additive Gaussian Noise. For three representative noise levels (𝜌′ = 0.1, 1, 10), the dependence of achieved accuracy (top row), average illumination pattern overlap (second row), average illumination pattern intensity (third row), and average ON ratio of the meta-atoms (fourth row) is plotted as a function of the number of allowed measurements M. We contrast our task-specific end-to-end optimized approach ("Learned Illuminations", red) with the conventional compressive-sensing approach of using random DMA configurations ("Random Illuminations",

Figure 4 :

 4 Figure 4: Intelligent Meta-Imaging Adaptation to Signal-Independent vs. Signal-Dependent Additive Gaussian Noise. For 𝑀 = 3, the dependence of achieved accuracy (top row), average illumination pattern overlap (second row), average illumination pattern intensity (third row), and average ON ratio of the metaatoms (fourth row) is plotted as a function of the noise strength for the two noise types. Each data point represents the average over 20 realizations, each with randomly chosen meta-atom locations. The error bars indicate the standard deviation across these 20 realizations.

Figure 5 :

 5 Figure 5: Performance of Noise-Adaptive Intelligent Meta-Imaging Outside the Trained Noise Regime. The intelligent meta-imager is adapted to the trained noise regime (horizontal axis) and then tested across a wide range of noise levels beyond the one from the training. For 𝑀 = 3, we show the average accuracy over 20 realizations for random (top row) vs. learned (bottom row) illuminations and the cases of signal-independent (left column) vs. signal-dependent (right column) noise. The white dashed lines indicate the training noise regime.

For the sake of compactness, we denote with 𝐶 𝑖 = {𝐶 𝑖 TX , 𝐶 𝑖 RX } jointly the 𝑖th configuration of the TX DMA, 𝐶 𝑖 TX , and the 𝑖th configuration of the RX DMA, 𝐶 𝑖 RX .


Chloé Saigre-Tardif

Philipp Del Hougne

Self-Adaptive RISs Beyond Free Space: Convergence of Localization, Sensing and Communication under Rich-Scattering Conditions

We discuss the need for a confluence of localization, sensing and communications if RISs are to be deployed in a self-adaptive manner in the dynamically evolving richscattering settings that are typical for important 6G deployment scenarios such as factories. We establish that in such problems the rich-scattering wireless channels are subject to a highly nonlinear deterministic double-parametrization through both the RIS and uncontrolled moving objects. Therefore, acquiring full context-awareness through localization and sensing is a prerequisite for self-adaptive RIS-empowered communications. Yet, the byproducts of this daunting communications overhead can feed many appliances that require context awareness, such that overhead concerns may vanish. We illustrate the essential steps for operating a self-adaptive RIS under rich scattering based on a prototypical case study. We discover that self-adaptive RISs outperform context-ignorant RISs only below a certain noise threshold that depends, among other factors, on how strongly uncontrolled perturbers impact the wireless channel. We also discuss ensuing future research directions that will determine the conditions under which self-adaptive RISs may serve as technological enabler of 6G networks.

I. INTRODUCTION

While fifth generation (5G) wireless communication networks are still being rolled out, researchers already actively explore how the anticipated needs for sixth generation (6G) wireless communication networks can be met. New applications like virtual reality and the wireless networked control of robotic systems (smart factory, automated driving, etc.) are expected to require up to 1000-fold increases of peak data rates and at least 10-fold improvements of energy and spectral efficiencies [START_REF] You | Towards 6G wireless communication networks: Vision, enabling technologies, and new paradigm shifts[END_REF]. One promising technological enabler for 6G may be reconfigurable intelligent surfaces (RISs) which endow wireless environments with programmability [START_REF] Renzo | Smart radio environments empowered by reconfigurable ai meta-surfaces: An idea whose time has come[END_REF]. Thereby, the physical layer system design is no longer restricted to the radiated waveforms but additionally includes the wireless channels, enabling a clear paradigm shift "beyond Shannon" who's famous law assumes uncontrolled channels.

But the full potential of this "smart radio environment" concept can only be reaped in practice if the RIS can selfadapt its configuration to a dynamically evolving deployment setting. A self-adaptive RIS (SA-RIS) is equipped with sensing units and a microprocessor such that during runtime it can autonomously adapt its configuration based on the sensor data, Chloé Saigre-Tardif and Philipp del Hougne are with Univ Rennes, CNRS, IETR -UMR 6164, F-35000, Rennes, France.

Corresponding Author: Philipp del Hougne (e-mail: philipp.delhougne@univ-rennes1.fr).

without human intervention [START_REF] Ma | Smart metasurface with self-adaptively reprogrammable functions[END_REF]. First concepts for SA-RISs are being studied for deployment scenarios in free space [START_REF] Alexandropoulos | Hybrid Reconfigurable Intelligent Metasurfaces: Enabling Simultaneous Tunable Reflections and Sensing for 6G Wireless Communications[END_REF], that is, within an essentially empty and thus very simple radio environment. Corresponding experimental efforts are conducted in anechoic (echo-free) rooms [START_REF] Ma | Smart metasurface with self-adaptively reprogrammable functions[END_REF], [START_REF] Ma | Smart sensing metasurface with self-defined functions in dual polarizations[END_REF]. In such free-space scenarios, the RIS-parametrized channel is a threeelement linear cascade: i) propagation from the base station (BS) to the RIS, ii) modulation by the RIS, and iii) propagation from the RIS to the user equipment (UE). Therefore, a freespace self-adaptive RIS must sound two channels (BS-RIS and RIS-UE) in order to update its configuration, which is recognized as implying substantial overhead that may even outweigh the benefits of RIS-parametrized channels.

However, many indoor deployment scenarios of 6G are not compatible with this free-space hypothesis. 6G is expected to consist of all-spectra integrated networks, ranging from microwave systems that offer wide coverage at low cost, via millimeter wave systems, all the way to even terahertz or optical systems with unprecedented bandwidth [START_REF] You | Towards 6G wireless communication networks: Vision, enabling technologies, and new paradigm shifts[END_REF]. At microwave frequencies, even an office room can give rise to substantial rich scattering that invalidates the free-space assumption, as seen in experiments at 2.45 GHz [START_REF] Del Hougne | Optimally diverse communication channels in disordered environments with tuned randomness[END_REF]. Moreover, 6G is expected to fully realize the Industry 4.0 revolution that already started with 5G [START_REF] You | Towards 6G wireless communication networks: Vision, enabling technologies, and new paradigm shifts[END_REF]; the envisioned factory settings are full of metallic elements that certainly strongly scatter microwaves and millimeter waves. Therefore, the analysis of SA-RISs must be fundamentally rethought beyond the free-space hypothesis, and surprisingly optimistic conclusions about the associated overhead may follow.

A pivotal point is the qualitatively very different parametrization of rich-scattering wireless channels through RISs. As noted above, free-space channels are (usually) linearly parametrized by a RIS. In contrast, the RISparametrization of rich-scattering channels is highly nonlinear because any given ray can encounter multiple RIS elements as it ricochets on its way from BS to UE. A rich-scattering self-adaptive RIS must develop a detailed deterministic understanding of this nonlinear parametrization. Merely sounding BS-RIS and RIS-UE channels would not be of much help. Instead, a rich-scattering SA-RIS must essentially gain complete context awareness (localize transceivers and scatterers, sense shapes and postures, etc.) and combine these insights with its understanding of the nonlinearly parametrized channel in order to self-adapt. The complexity of these localization and sensing tasks may appear daunting, especially with regard to overhead concerns.

Fortunately, the results from localization and sensing tasks that a rich-scattering SA-RIS must perform as auxiliary step to serve its primary communication purpose are actually themselves highly sought after. Gaining context awareness is essential for emerging services in areas including smart health care, touchless human-machine interaction, autonomous mobility, and security. The potential of RISs for localization [START_REF] Wymeersch | Radio localization and mapping with reconfigurable intelligent surfaces: Challenges, opportunities, and research directions[END_REF]- [START_REF] Del Hougne | Deeply Subwavelength Localization with Reverberation-Coded Aperture[END_REF] and sensing [START_REF] Saigre-Tardif | Intelligent meta-imagers: From compressed to learned sensing[END_REF] has not gone unnoticed, but these explorations are so far decoupled from communications-related RIS deployments. Yet, outside the specific RIS context, there is already a general tendency toward Integrated Sensing and Communication (ISAC) [START_REF] Ma | Joint Radar-Communication Strategies for Autonomous Vehicles: Combining Two Key Automotive Technologies[END_REF], [START_REF] Lima | Convergent Communication, Sensing and Localization in 6G Systems: An Overview of Technologies, Opportunities and Challenges[END_REF] that is fueled by striking similarities between sensing and communication systems, and motivated by limited spectral resources and power consumption constraints. Of course, this ISAC trend is expected to extend to RIS-empowered systems. But given the above analysis, context-aware services can be fed with localization and sensing information as a byproduct of SA-RIS-empowered communication systems under richscattering conditions, implying virtually zero overhead. In SA-RIS-empowered rich-scattering settings, sensing is thus a prerequiste for communication, in contrast to typical ISAC concepts that seek to unify two systems serving different purposes, one for sensing and one for communication.

In this paper, we introduce the concept of SA-RISs beyond the oftentimes questionable free-space assumption. We begin by reviewing the usually overlooked nonlinearity of the parametrization of rich-scattering wireless channels. Then, with the help of an illustrative case study, we introduce the essential ingredients of operating a SA-RIS under rich-scattering conditions. We explain how its overhead automatically provides full context awareness. We conclude by looking forward to key research challenges and open questions that will determine the feasibility of deploying SA-RISs in 6G networks under realistic conditions.

II. PARAMETRIZED RICH-SCATTERING CHANNELS

The emerging RIS paradigm introduces a deterministic parametrization of wireless channels that is fundamentally incompatible with conventional statistical models of fading channels. Current attempts at marrying together these two worlds take the free-space linear RIS-parametrization from Fig. 1a as starting point to imagine a linear cascade of fading, interaction with the RIS, and further fading. Despite the appealing simplicity of this approach, it is unsuitable for analysing RIS-parametrized rich-scattering environments because it violates fundamental (wave) physics. The most important shortcoming is that linear cascaded models fail to capture mesoscopic correlations, originating from ricocheting rays, that yield a highly nonlinear RIS-parametrization of the wireless channel. As sketched in Fig. 1b, rays connecting a transmitter to a receiver via the ith RIS element are typically also bouncing off other RIS elements, such that the wireless channel cannot be formulated as a linear function of individual RIS element configurations.

It is important to not confuse the uncontested linearity of the wave equation (i.e., the superposition principle) with the nonlinear parametrization of the wireless channel. The nonlinearity we discuss refers to how the channel (rather than the output) depends on the boundary conditions (rather than the input). The degree of nonlinearity of a wireless channel's RIS-parametrization depends on the amount of reverberation in the environment. This is intuitively obvious: if an environment presents less attenuation, the lifetime of rays is longer such that they can encounter more RIS elements during their lifetime. In the limiting case of free space without any reverberation, the linear RIS-parametrization is recovered. In enclosed metallic environments such as inside vessels or factories, the attenuation is weaker than in partially open and lossy environments like office rooms.

From the electromagnetic perspective, RISs can be understood as arrays of purposefully controlled perturbers of the wireless channel. However, fading rich-scattering environments are additionally parametrized by uncontrolled perturbers whose locations and shapes dynamically evolve. An example would be the body of a human who moves inside an office room and assumes various postures (upright, sitting, etc.). The wireless channel is therefore subject to a deterministic non-linear double-parametrization through both the RIS configuration and a collection of parameters that describes locations and shapes of the uncontrolled perturbers. In order to determine a suitable RIS configuration that supports a desired communications objective, the SA-RIS must i) deterministically understand this complex parametrization, and ii) estimate the current perturber parameters. The latter implies the acquisition of complete context awareness.

III. CASE STUDY A. Scenario

To demonstrate the essential aspects of operating a selfadaptive RIS under rich-scattering conditions, we present a case study of the prototypical example sketched in Fig. 2: an enclosed indoor environment in which an object moves and changes shape (representing, for instance, a human). The uncontrolled perturbations are thus parametrized through:

1) Object location along its allowed trajectory. 2) Object shape among its three allowed shapes.

In this example, the first item is a continuous parameter whereas the second one is a discrete parameter.

The rich scattering is evident upon inspection of the field map in Fig. 2. The field strength does not depend on the distance from the BS but fluctuates on a halfwavelength scale. The isotropic field maxima, especially in the vicinity of the UE, evidence that rays from all possible directions are incident -in sharp contrast to what would happen in free space. In our case study, the RIS' desired communications functionality is binary amplitudeshift-keying (BASK) backscatter communication. Therein, digital information is encoded into the waves not at the BS but by the RIS-equipped propagation environment. Specifically, the RIS is used as massive backscatter array to maximize ('1') or minimize ('0') the received signal strength indicator (RSSI) at the UE [START_REF] Zhao | Metasurface-assisted massive backscatter wireless communication with commodity Wi-Fi signals[END_REF]. Relative to the average RSSI, maximization yields an improvement of around 10 dB whereas minimization achieves an RSSI close to zero -see Fig. 3a. But the necessary RIS configurations for this purpose strongly depend on the perturber parameters. Indeed, the two optimal configurations for the perturber parameters in Fig. 3a are impossible to distinguish in terms of the corresponding RSSI for another combination of perturber parameters, as seen in Fig. 3b. The strong dependence of the wireless channel on both the RIS configuration and the perturber parameters is thus apparent in Fig. 3. The SA-RIS must hence perform both localization and sensing tasks to estimate the perturber parameters as prerequisite in order to subsequently optimize its configuration for the desired communications functionality.

B. Channel Model

We use the only (to the best of our knowledge) to date available physics-compliant channel model for RISparametrized rich scattering: PhysFad [START_REF] Faqiri | PhysFad: Physics-Based End-to-End Channel Modeling of RIS-Parametrized Environments with Adjustable Fading[END_REF] is derived from first principles based on a coupled-dipole formalism and describes all wireless entities as dipoles or collections of dipoles with suitable properties (resonance frequency, absorption, etc.). As seen in Fig. 2, continuous surfaces of scattering objects are discretized and represented as dipole fences. As detailed in Ref. [START_REF] Faqiri | PhysFad: Physics-Based End-to-End Channel Modeling of RIS-Parametrized Environments with Adjustable Fading[END_REF], the configuration of a RIS element is controlled via the resonance frequency of the corresponding dipole. The underlying Lorentzian resonator model automatically accounts for frequency selectivity and the intertwinement between phase and amplitude response. For simplicity, and in line with most existing experimental prototypes (e.g., Refs. [START_REF] Del Hougne | Optimally diverse communication channels in disordered environments with tuned randomness[END_REF], [START_REF] Del Hougne | Deeply Subwavelength Localization with Reverberation-Coded Aperture[END_REF]), we limit ourselves to 1-bit programmable RIS elements.

In a given realization of the RIS-parametrized richscattering environment, the wireless channel depends in a nontrivial manner on the locations and properties of all dipoles needed to describe the considered scenario. The self-consistent PhysFad formalism involves a matrix inversion to correctly capture all mesoscopic correlations. Mathematical details are explained in Ref. [START_REF] Faqiri | PhysFad: Physics-Based End-to-End Channel Modeling of RIS-Parametrized Environments with Adjustable Fading[END_REF] and not repeated here for the sake of brevity. Overall, PhysFad outputs the wireless channel H between the designated transmitting and receiving dipoles for the specified double-parametrization (RIS and perturbers) of the considered wireless environment. Throught this paper, we consider single-tone operation and are interested in the RSSI at the UE, i.e., |H + n| assuming unity transmit power, where H is the BS-UE channel and n denotes noise.

C. Operation Principle

The operation principle of a rich-scattering SA-RIS can be broken down into two steps: localization and sensing to gain complete context awareness, followed by the identification of suitable RIS configurations for the desired communications functionality. Both steps require an offline calibration to adjust to the specific wireless environment under consideration as well as an online operation during runtime that is repeated once per coherence time.

Localization and Sensing. We tackle localization and sensing in rich-scattering conditions under a unified conceptual and technical framework as inverse problem: which perturber parameters explain the available field measurements? This approach strongly differs from its free-space counterpart where RIS-empowered localization [START_REF] Wymeersch | Radio localization and mapping with reconfigurable intelligent surfaces: Challenges, opportunities, and research directions[END_REF] and sensing [START_REF] Saigre-Tardif | Intelligent meta-imagers: From compressed to learned sensing[END_REF] rely on raytracing techniques (angle of arrival, etc.) and the first Born approximation (absence of multiple scattering), respectively. These underlying assumptions are incompatible with richscattering conditions where waves are strongly ricocheting: rays from all possible angles are incident and any coherent wavefront gets completely scrambled. Nonetheless, the soughtafter location and sensing information is encoded in the scrambled waves -hence the inverse-problem perspective. The key insight is that the field measured at an arbitrary location in a rich-scattering environment is the superposition of all reflections, and reflections off the objects of interest depend on their location, shape and orientation.

A series of field measurements can hence act as a wave fingerprint that enables the unambiguous identification of the perturber parameters. This model-agnostic approach equally applies to bi-static (moving object) and mono-static (moving UE) scenarios. Thanks to the configurational diversity offered by the RIS, this series of measurements can be performed at a single frequency and with a single or a few node(s). To that end, we must learn to recognize the characteristic fluctuations of the wireless channel(s) over a fixed series of random RIS configurations that are specific to each value of the sensing parameter. To avoid that the SA-RIS requires cooperation with the UE, we endow it with its own auxiliary receivers. Given the statistical uniformity of random wave fields, these auxiliary receivers (AuxRXs) could be placed anywhere. Therefore, our scheme can be implemented i) by replacing a few RIS elements with sensing units (similar to Ref. [START_REF] Ma | Smart sensing metasurface with self-defined functions in dual polarizations[END_REF]), ii) by using the hybrid meta-atoms from Ref. [START_REF] Alexandropoulos | Hybrid Reconfigurable Intelligent Metasurfaces: Enabling Simultaneous Tunable Reflections and Sensing for 6G Wireless Communications[END_REF] that simultaneously capture and reflect waves, or iii) by placing sensors at arbitrary locations away from the RIS. For concreteness, we consider the first option in the following.

The required number of the auxiliary measurements for precise localization and sensing depends on how much information about the perturber parameter of interest can be extracted per auxiliary measurement:

1) Amount of reverberation. The longer the waves reverberate before being attenuated, the more sensitive the field becomes to a given perturber parameter [START_REF] Del Hougne | Deeply Subwavelength Localization with Reverberation-Coded Aperture[END_REF]. 2) Perturbation size. The larger the perturbation is, the more strongly it affects the field. 3) Signal-to-noise ratio (SNR). The higher the SNR is, the more information can be extracted per measurement. To solve said inverse problems, fully connected artificial neural networks (ANNs) can be trained as surrogate inverse models. We use the auxiliary field measurements as input to the ANN, and the output is either the prediction of a continuous variable (perturber location) or a classification (perturber shape). Fully connected ANN architectures are deliberately chosen because due to the rich scattering the sought-after information is presumable not encoded in local features but long-range correlations [START_REF] Del Hougne | Deeply Subwavelength Localization with Reverberation-Coded Aperture[END_REF], [START_REF] Del Hougne | Robust position sensing with wave fingerprints in dynamic complex propagation environments[END_REF]. Because we train one ANN per perturber parameter, to robustly estimate the assigned parameter, each ANN should be insensitive to fluctuations of the other perturber parameter(s) [START_REF] Del Hougne | Robust position sensing with wave fingerprints in dynamic complex propagation environments[END_REF]. A training data set with sufficiently representative parameter combinations is required.

Communication. The second step confronts us with an inverse-design problem: which RIS configurations best satisfy the needs of the desired communications functionality (here BASK backscatter communication)? The answer strongly depends on both the current perturber parameters and the specific wireless environment, as seen in Fig. 3. Therefore, both the acquisition of full context awareness (via online localization and sensing as discussed above) and an offline calibration to the specific environment are unavoidable prerequisites for SA-RIS-assisted communications in dynamic rich-scattering conditions. Two options exist for the offline calibration:

1) Open loop. During the offline calibration, a surrogate forward model is trained. This model takes RIS configuration and perturber parameters as inputs to predict the wireless channel in the specific setting.

During runtime, the SA-RIS makes use of this model, in combination with the sensed perturber parameters, to optimize its configuration once per coherence time. 2) Closed loop. During the offline calibration, optimal RIS configurations are learned for each possible combination of perturber parameters. During runtime, based on the sensed pertuber parameters, the SA-RIS chooses the most appropriate configuration out of the pre-stored code book.

Successfully learning a surrogate forward model can be extremely challenging for the highly nonlinear parametrization associated with strong reverberation. Moreover, the online optimization once per coherence time can be costly in terms of latency and computational burden. Especially when the perturber parameter space is of manageable size, the closedloop approach is hence very attractive: the nonlinear channel parametrization is not learned explicitly and looking up a suitable code-book entry during runtime is very fast. However, during the calibration, the closed-loop approach requires control over the RIS to measure the channels corresponding to specific RIS configurations in order to optimize the RIS configurations. In contrast, the open-loop approach offers the possibility of updating the calibration through observations with arbitrary RIS configurations during runtime.

The closed-loop approach must inevitably discretize continuous perturber parameters (in our example the object location) because the code book can only have a finite number of entries. For each combination of discrete perturber parameter values (shape and location), optimized RIS configurations for the desired communications functionality are identified during calibration. These optimizations are difficult due to the nonlinear parametrization and the 1-bit programmability constraint. An example of a simple iterative approach to such an optimization is shown in Fig. 4a. First, the BS-UE channel is measured for 100 random RIS configurations. Second, the best configuration is taken as starting point for an element-by-element iterative optimization. The latter must loop multiple times over each RIS element before convergence -a clear signature of the nonlinear channel parametrization: the optimal configuration of the ith RIS element depends on the configuration of the other RIS elements. During runtime, the object will continuously vary its position and shape; the localization and sensing step will predict a shape class and a continuous position value. The communication step will then look up the code-book entry that most closely corresponds to this set of perturber parameters. The expected performance loss due to the limited code-book resolution with respect to the object location is plotted in Fig. 4b. A sub-wavelength code-book resolution is clearly needed to avoid strong performance penalties. Thus, the closed-loop approach is suitable when the perturber parameter space is of a manageable size. Note that interpolating between code-book entries is not easily realizable due to the 1-bit programmability of the RIS elements.

Simultaneous Localization, Sensing and Communication. So far, we have discussed a SA-RIS system in which communications are interrupted once per coherence time in order to perform localization and sensing with a dedicated series of RIS configurations, to subsequently update the RIS configurations used for communications. If the coherence time is long and if the RIS sequence for localization and sensing is short, then these interruptions are certainly tolerable. Nonetheless, the time-sequential use of the same hardware for sensing and communications raises the question of how to balance the two: more time spent on sensing improves the sensing precision and thereby ultimately the communication quality, but reduces the time available to communicate. To avoid such trade-offs, it is appealing to perform sensing solely based on the signals and RIS configurations used for communications. Without any signals or RIS configurations exclusively dedicated to sensing, interruptions of the communications to perform localization and sensing are avoided.

The associated challenge lies in solving the inverse problem that underlies the localization and sensing. The previously discussed surrogate inverse model maps auxiliary channel measurements obtained for a fixed series of RIS configurations to the sought-after perturber parameters. Given the strong channel dependence on the RIS configuration (see Fig. 3), transposing this technique to auxiliary channel measurements obtained for an arbitrary series of RIS configurations is very challenging. An open-loop approach is to train ANNs that take both the auxiliary measurements and the corresponding RIS configurations as input. A closed-loop approach suitable for manageable perturber parameter spaces is to train separate ANNs for each configuration from the communications code book. Even if the latter involves training a few hundred ANNs, it may still be an easier task than learning to be able to deal with any out of all possible 2 25 configurations in the open-loop approach.

D. Results

Having established the operation principle of a SA-RIS under rich-scattering conditions, we present indicative results for our prototypical scenario. Given the manageable perturber parameter space in our case study, we opt for the closedloop approaches discussed above. In particular, thanks to the eight auxiliary receivers, sensing and localization are possible based on a single RIS configuration at the targeted 20 dB level of SNR in our problem. Hence, with each transmitted communications symbol, the SA-RIS can update its knowledge about the environmental context (object position and shape) based on the eight auxiliary field measurements and the set of sensing and localization ANNs corresponding to the RIS configuration used to transmit the communication symbol. Thereby, without any signals or RIS configurations dedicated to sensing, the SA-RIS acquires full context-awareness and can adapt its configuration for the transmission of the next communication symbol to the environment's evolution. In other words, localization and sensing do not interrupt communications.

In Fig. 5a, the accuracy with which the SA-RIS recognizes the object's shape is plotted. For SNR values above the targeted 20 dB, the accuracy is close to unity; at low SNRs, it approaches the random-guess baseline. Similarly, in Fig. 5b, the average error of the estimated object location is plotted in units of wavelength. Above the targeted SNR of 20 dB, the average error is on the order of a tenth of the wavelength. Note that this is well below the half-wavelength resolution baseline which is frequently assumed to be a fundamental bound -however, there is no half-wavelength resolution limit because reverberation endows us with interferometric sensitivity and deeply subwavelength resolution under richscattering conditions [START_REF] Del Hougne | Deeply Subwavelength Localization with Reverberation-Coded Aperture[END_REF]. The first result is therefore that highfidelity sensing and localization, equivalent to full contextawareness, are possible without dedicated RIS configurations, thus enabling uninterrupted communications. This real-time context awareness is a prerequisite for robust communications in dynamic environments but can simultaneously feed services in health care, human-machine interaction and many other appliances that require context-awareness.

Next, let us turn our attention to the RSSI at the UE which is minimized (maximized) to transfer the symbol '0' ('1') in BASK. The two green curves in Fig. 5c evidence that the two RSSI levels are easily distinguishable above the targeted SNR value of 20 dB. Correspondingly, the bit error rate (BER) plotted in Fig. 5d is very low in this regime. As the SNR is reduced toward 0 dB, the lowest (highest) achievable RSSI increases (decreases), making the two symbols less distinguishable and hence causing the BER to increase. Eventually, below 0 dB, the curves begin to overlap and steeply rise, being dominated by the additive Gaussian noise, and the BER reaches its upper bound of 0.5.

The reasons for the reduced symbol distinguishability at lower SNR values is twofold: on the one hand, the noise increasingly disturbs the desired destructive (constructive) interferences for '0' ('1'); on the other hand, the sensing and localization information based on which these interferences are designed is itself less accurate. To evidence the interplay of localization, sensing and communications more clearly, we plot in Fig. 5c and Fig. 5d additionally two benchmarks. The curves in purple correspond to a scenario with perfect context awareness. This benchmark only slightly outperforms the SA-RIS above an SNR of 20 dB because its location and sensing information is only marginally more accurate than our estimates. Below roughly 10 dB in SNR, the difference between the two scenarios grows, quantifying the impact of the imperfectly estimated localization and sensing information on the communications.

Another important benchmark is that of a context-ignorant RIS (blue). Such a RIS uses two fixed configurations for '0' and '1' that are optimized to yield the best performance on average over many perturber parameter combinations. This strategy makes use of the subset of rays that bounce off RIS elements but not the perturbing object. Clearly, a contextignorant RIS is expected to perform worse because it fails to control the rays that interact with the uncontrolled perturbers. Indeed, its BER is twice that of the SA-RIS for SNRs above 20 dB. Interestingly, however, below SNRs of 10 dB the context-ignorant RIS performs better because the estimated localization and sensing information of the SA-RIS becomes too imprecise. The benefits of SA-RISs hence unfold only above a threshold-SNR level. This behavior is expected to be general, though the specific threshold-SNR level depends on many parameters: if there were more perturbing objects, fewer rays could avoid all of them, hence the context-ignorant RIS would perform worse.

IV. CHALLENGES, OPEN QUESTIONS,

AND RESEARCH DIRECTIONS The above discussions and results provide an initial understanding of the difficulties and potential benefits of operating SA-RISs under the rich-scattering conditions encountered in many realistic 6G deployment scenarios. Important open research questions and challenges remain in order to determine whether RIS-ISAC can help to reach the ambitious goals of 6G:

Scalability: The scalability of the discussed approach to yet more complex scenarios with many more independent perturbers should be investigated. Fundamental limits should be identified on the degree of complexity of channel parametrizations that is learnable with acceptable effort. Naturally, once an individual perturber's impact on the wireless channel is below the noise level, it cannot be sensed efficiently anymore. Moreover, the degree of variability of the perturber parameters may determine the success of the scheme. While the possible combinations of perturber parameters may be clearly defined in a factory setting, an occupied rail car's variations may be too strong. This analysis will hence clarify in which settings a SA-RIS can be deployed and which settings' dynamics are too complex to be suited for a SA-RIS deployment.

Motion Forecasting: The discussed scheme reacts to motion by sensing in order to adapt to it. It is enticing to transition to an approach that anticipates motion, and hence the evolution of the wireless channels. Given the natural origin of motion (e.g., humans), the use of recurrent neural networks for time-series forecasting is a promising tool to this end. Again, beyond its benefits for communications, these forecasting "skills" of a SA-RIS will simultaneously serve other context-aware appliances, too.

Impact of Reverberation Strength: The reverberation strength in a rich-scattering environment is a double-sided sword for the discussed RIS-ISAC technique. On the one hand, the longer the waves reverberate before being attenuated, the more nonlinear the channel parametrization becomes, making it harder to learn, be it explicitly or implicitly. On the other hand, the longer the waves reverberate, the more sensitive they become to details of the perturbers such that they encode the sought-after location and sensing information more efficiently in the multiplexed field measurements at the auxiliary receivers [START_REF] Del Hougne | Deeply Subwavelength Localization with Reverberation-Coded Aperture[END_REF]. Therefore, it is of great importance to understand these trade-offs for 6G RIS-ISAC under richscattering conditions.

Channel Modelling and Experimentation: The importance of physics-based channel models of realistic wireless environments and corresponding real-life experimentation is currently underestimated. To reap the full potential of RISs, it must be ensured that developed signal-processing algorithms are compatible with both physics and the targeted deployment scenarios. Accounting for multiple scattering in a rigorous and deterministic manner calls for the development of a new generation of channel models, as well as experimentations outside echo-free test environments.

V. CONCLUSION

If RISs are to unfold their full potential as technological enabler of 6G, they must self-adaptively operate in highly complex settings that can differ remarkably from free space in which RISs are studied to date. In this paper, we have explained why the strongly nonlinear parametrization of richscattering wireless channels both through RIS elements and uncontrolled perturbing objects makes full context awareness a prerequisite for SA-RIS-assisted communications. We presented a case study illustrating the resulting interplay of localization and sensing with communications. Our analysis suggests that their convergence in an ISAC fashion is pivotal for the deployment of SA-RISs in commonly discussed 6G settings (e.g., factories) and not merely motivated by limited resources. Looking forward, we discussed key research directions that will determine the feasibility of rich-scattering ISAC and hence SA-RISs in commonly envisioned 6G scenarios.

Fig. 1 .

 1 Fig. 1. (a) A quasi free-space wireless channel is linearly parametrized by a RIS. (b) A fading rich-scattering wireless channel is nonlinearly doubleparametrized by a RIS and moving perturbers.

Fig. 2 .

 2 Fig.2. Considered rich-scattering scenario in PhysFad[START_REF] Faqiri | PhysFad: Physics-Based End-to-End Channel Modeling of RIS-Parametrized Environments with Adjustable Fading[END_REF], comprising a many-wavelengths-large irregularly-shaped reverberant room (see wavelength λ scalebar), a base station (BS), a user equipment (UE), a conformal fourpart distributed RIS with eight auxiliary receivers (AuxRXs), and an object moving along the indicated trajectory. In addition, the object can assume three different shapes: cross (shown), circle, or square. The RIS consists of 25 1-bit programmable macro-pixels, each grouping together four adjacent elements (encircled elements are ON resonance, the others OFF resonance). The sketch is overlayed on the corresponding field magnitude map.

Fig. 3 .

 3 Fig. 3. RIS-empowered BASK in double-parametrized wireless channel H. For two different choices of perturber parameters (see bottom left insets), (a) and (b) show the wireless channel for 100 random RIS configurations (blue), as well as for configurations optimized to minimize ('0') or maximize ('1') the RSSI. The optimized configurations are displayed as inset. The wireless channels for the optimized configurations from (a) in the setting of (b) are also shown. A dot's distance from the origin is the RSSI.

Fig. 4 .

 4 Fig. 4. Closed-loop self-adaptive rich-scattering RIS scheme for the backscatter BASK functionality. (a) Example iterative optimization of the RIS configuration for the parameter settings from Fig. 3b. (b) Average performance dependence on the code-book resolution.

Fig. 5 .

 5 Fig. 5. ISAC with self-adaptive rich-scattering RIS performing BASK backscatter communication. (a) Object shape recognition accuracy. (b) Average object localization error. (c) Measured magnitude (signal plus noise) for the two possible BASK symbols ('0' and '1') for the discussed selfadaptive RIS (green) and two benchmarks: a RIS operated with perfect context awareness (purple) and a RIS operated without any context awareness (blue). (d) Corresponding BER curves for the three scenarios from (c).