Pansharpening of images acquired with color filter arrays

International audienceIn remote sensing, a common scenario involves the simultaneous acquisition of a panchromatic (PAN), a broad-band high spatial resolution image, and a multispectral (MS) image, which is composed of several spectral bands but at lower spatial resolution. The two sensors mounted on the same platform can be found in several very high spatial resolution optical remote sensing satellites for Earth observation (e.g., Quickbird, WorldView and SPOT) In this work we investigate an alternative acquisition strategy, which combines the information from both images into a single band image with the same number of pixels of the PAN. This operation allows to significantly reduce the burden of data downlink by achieving a fixed compression ratio of 1/(1 + b/ρ 2) compared to the conventional acquisition modes. Here, b and ρ denote the amount of distinct bands in the MS image and the scale ratio between the PAN and MS, respectively (e.g.: b = ρ = 4 as in many commercial high spatial resolution satellites). Many strategies can be conceived to generate such a compressed image from a given set of PAN and MS sources. A simple option, which will be presented here, is based on an application of the color filter array (CFA) theory. Specifically, the value of each pixel in the spatial support of the synthetic image is taken from the corresponding sample either in the PAN or in a given band of the MS upsampled to the size of the PAN. The choice is deterministic and done according to a custom mask. There are several works in the literature proposing various methods to construct masks which are able to preserve as much spectral content as possible for conventional RGB images. However, those results are not directly applicable to the case at hand, since it deals with i) images with different spatial resolution, ii) potentially more than three spectral bands and, iii) in general, different radiometric dynamics across bands. A tentative approach to address these issues is presented in this work. The compressed image resulting from the proposed acquisition strategy will be processed to generate an image featuring both the spatial resolution of the PAN and the spectral bands of the MS. This final product allows a direct comparison with the result of any standard pansharpening algorithm; the latter refers to a specific instance of data fusion (i.e., fusion of a PAN and MS image), which differs from our scenario since both sources are separately taken as input. In our setting, the fusion step performed at the ground segment will jointly involve a fusion and reconstruction problem (also known as demosaicing in the CFA literature). We propose to address this problem with a variational approach. We present in this work preliminary results related to the proposed scheme on real remote sensed images, tested over two different datasets acquired by the Quickbird and Geoeye-1 platforms, which show superior performances compared to applying a basic radiometric compression algorithm to both sources before performing a pansharpening protocol. The validation of the final products in both scenarios allows to visually and numerically appreciate the tradeoff between the compression of the source data and the quality loss suffered

Joint demosaicing and fusion of multiresolution coded acquisitions: A unified image formation and reconstruction method

Novel optical imaging devices allow for hybrid acquisition modalities such as compressed acquisitions with locally different spatial and spectral resolutions captured by a single focal plane array. In this work, we propose to model the capturing system of a multiresolution coded acquisition (MRCA) in a unified framework, which natively includes conventional systems such as those based on spectral/color filter arrays, compressed coded apertures, and multiresolution sensing. We also propose a model-based image reconstruction algorithm performing a joint demosaicing and fusion (JoDeFu) of any acquisition modeled in the MRCA framework. The JoDeFu reconstruction algorithm solves an inverse problem with a proximal splitting technique and is able to reconstruct an uncompressed image datacube at the highest available spatial and spectral resolution. An implementation of the code is available at https://github.com/danaroth83/jodefu.Comment: 15 pages, 7 figures; regular pape

Image fusion and reconstruction of compressed data: A joint approach

International audienceIn the context of data fusion, pansharpening refers to the combination of a panchromatic (PAN) and a multispectral (MS) image, aimed at generating an image that features both the high spatial resolution of the former and high spectral diversity of the latter. In this work we present a model to jointly solve the problem of data fusion and reconstruction of a compressed image; the latter is envisioned to be generated solely with optical on-board instruments, and stored in place of the original sources. The burden of data downlink is hence significantly reduced at the expense of a more laborious analysis done at the ground segment to estimate the missing information. The reconstruction algorithm estimates the target sharpened image directly instead of decompressing the original sources beforehand; a viable and practical novel solution is also introduced to show the effectiveness of the approach

Model-based demosaicking for acquisitions by a RGBW color filter array

Microsatellites and drones are often equipped with digital cameras whose sensing system is based on color filter arrays (CFAs), which define a pattern of color filter overlaid over the focal plane. Recent commercial cameras have started implementing RGBW patterns, which include some filters with a wideband spectral response together with the more classical RGB ones. This allows for additional light energy to be captured by the relevant pixels and increases the overall SNR of the acquisition. Demosaicking defines reconstructing a multi-spectral image from the raw image and recovering the full color components for all pixels. However, this operation is often tailored for the most widespread patterns, such as the Bayer pattern. Consequently, less common patterns that are still employed in commercial cameras are often neglected. In this work, we present a generalized framework to represent the image formation model of such cameras. This model is then exploited by our proposed demosaicking algorithm to reconstruct the datacube of interest with a Bayesian approach, using a total variation regularizer as prior. Some preliminary experimental results are also presented, which apply to the reconstruction of acquisitions of various RGBW cameras

Interferometer response characterization algorithm for multi-aperture Fabry-Perot imaging spectrometers

In recent years, the demand for hyperspectral imaging devices has grown significantly, driven by their ability of capturing high-resolution spectral information. Among the several possible optical designs for acquiring hyperspectral images, there is a growing interest in interferometric spectral imaging systems based on division of aperture. These systems have the advantage of capturing snapshot acquisitions while maintaining a compact design. However, they require a careful calibration to operate properly. In this work, we present the interferometer response characterization algorithm (IRCA), a robust three-step procedure designed to characterize the transmittance response of multi-aperture imaging spectrometers based on the interferometry of Fabry-Perot. Additionally, we propose a formulation of the image formation model for such devices suitable to estimate the parameters of interest by considering the model under various regimes of finesse. The proposed algorithm processes the image output obtained from a set of monochromatic light sources and refines the results using nonlinear regression after an ad-hoc initialization. Through experimental analysis conducted on four different prototypes from the Image SPectrometer On Chip (ImSPOC) family, we validate the performance of our approach for characterization. The associated source code for this paper is available at https://github.com/danaroth83/irca.Comment: 20 pages, 11 figures. (Revised structure, added experiments

Resolution Enhancement of Hyperspectral Data Exploiting Real Multi-Platform Data

Multi-platform data introduce new possibilities in the context of data fusion, as they allow to exploit several remotely sensed images acquired by different combinations of sensors. This scenario is particularly interesting for the sharpening of hyperspectral (HS) images, due to the limited availability of high-resolution (HR) sensors mounted onboard of the same platform as that of the HS device. However, the differences in the acquisition geometry and the nonsimultaneity of this kind of observations introduce further difficulties whose effects have to be taken into account in the design of data fusion algorithms. In this study, we present the most widespread HS image sharpening techniques and assess their performances by testing them over real acquisitions taken by the Earth Observing-1 (EO-1) and the WorldView-3 (WV3) satellites. We also highlight the difficulties arising from the use of multi-platform data and, at the same time, the benefits achievable through this approach

Coastal Sea Level Monitoring in the Mediterranean and Black Seas

Spanning over a century, a traditional way to monitor sea level variability by tide gauges is – in combination with modern observational techniques like satellite altimetry – an inevitable ingredient in sea level studies over the climate scales and in coastal seas. The development of the instrumentation, remote data acquisition, processing and archiving in last decades allowed for extending the applications towards a variety of users and coastal hazard managers. The Mediterranean and Black50 seas are an example for such a transition – while having a long tradition for sea level observations with several records spanning over a century, the number of modern tide gauge stations are growing rapidly, with data available both in real-time and as a research product at different time resolutions. As no comprehensive survey of the tide gauge networks has been carried out recently in these basins, the aim of this paper is to map the existing coastal sea level monitoring infrastructures and the respective data availability. The survey encompasses description of major monitoring networks in the Mediterranean and Black55 seas and their characteristics, including the type of sea level sensors, measuring resolutions, data availability and existence of ancillary measurements, altogether collecting information about 236 presently operational tide gauge stations. The availability of the Mediterranean and Black seas sea level data in the global and European sea level repositories has been also screened and classified following their sampling interval and level of quality-check, pointing to the necessity of harmonization of the data available with different metadata and series at different repositories. Finally, an assessment of the networks’ capabilities60 for their usage in different sea level applications has been done, with recommendations that might mitigate the bottlenecks and assure further development of the networks in a coordinated way, being that more necessary in the era of the human-induced climate changes and the sea level ris

Techniques de traitement du signal basées modèles pour systèmes d'imagerie optique non conventionnels

There is an increasing demand for images with higher spectral and spatial resolution for applications in several domains such as health, environment, quality checking and natural disasters monitoring. Hyperspectral imagery provides the necessary spectral diversity to recover the composition of materials on site for applications such as the detection of fires, anomalies, chemical agents, targets and changes in the scene.The requirement for cheaper and more compact devices (e.g. to be embarked on low cost satellites and airborne platform) which are capable of capturing this information has led to the development of nonconventional innovative design concepts to overcome the technological limitations of traditional cameras. Data acquired by such novel imaging devices following the computational imaging paradigm are typically not readily exploitable for the final application. A computational phase is hence needed for extracting useful information from the raw acquisitions.This thesis addresses this issue by setting up an inversion problem. The general approach is to characterize the data fidelity term with a physical model, describing the underlying optical transformations performed by the device. The challenge is then shifted on the regularization step to properly characterizes the features of the quantities of interest and improve the accuracy of the estimation, which can be tackled with variational techniques.The analysis is applied to two novel concepts for nonconventional optical devices. The first one is a novel compressed acquisition imaging system based on color filter arrays, which embeds information from sensors with different spatial and spectral characteristics into a single mosaiced product. As opposed to existing compressed sensing based devices, the goal is not to recover the original uncompressed multiresolution sources, but instead to directly recover a synthetic fused image with both high spatial and spectral resolution.The proposed solution relies on the total variation regularization and is the subject of a detailed analysis, comparing its compressive power with straightforward software alternatives, evaluating its performances as the amount of channels changes, and validating its efficiency in comparison to state of the art methods when applied to classical fusion or mosaicing algorithms separately.The second class of devices is based on the ImSPOC patent, a design concept for a low finesse snapshot imaging spectrometer based on the interferometry of Fabry-Pérot. Its ideal behaviour follows the principle of the Fourier Transform Spectroscopy, as its acquisition can be interpreted as a sampled version of an interferogram, arranged across different sub-images distributed on the same focal plane.After defining a physical model based on optical geometry, its validity is evaluated over real acquisitions by setting up a Bayesian inference problem to determine its parameters, with approaches based on maximum likelihood estimators, regular-grid searches and nonlinear regression.A variety of preliminary tests are then carried out on the inversion method, with approaches based on singular value decomposition and sparse-inducing regularizers, accompanied by a analysis of their robustness to model mismatches.Il existe une demande croissante d'images avec une résolution spectrale et spatiale plus élevée pour des applications dans plusieurs domaines tels que la santé, l'environnement, le contrôle qualité et la surveillance des catastrophes naturelles. L'imagerie hyperspectrale fournit la diversité spectrale nécessaire pour récupérer la composition des matériaux sur site pour des applications telles que la détection d'incendies, d'anomalies, d'agents chimiques, de cibles et de changements de scène.L'exigence de dispositifs moins chers et plus compacts (par exemple, pour être embarqués sur des satellites à faible coût et une plateforme aéroportée) capables de capturer ces informations a conduit au développement de concepts de conception innovants non conventionnels pour surmonter les limitations technologiques des caméras traditionnelles.Les données acquises à partir de ces nouveaux dispositifs d'imagerie suivant le paradigme d'imagerie informatique ne sont généralement pas facilement exploitables pour l'application finale.Une phase de calcul est nécessaire pour extraire des informations utiles des acquisitions brutes.Cette thèse aborde cette question en mettant en place un problème d'inversion. L'approche générale consiste à caractériser le terme de fidélité des données avec un modèle physique, décrivant les transformations optiques sous-jacentes effectuées par le dispositif. Le défi est ensuite déplacé vers l'étape de régularisation pour bien caractériser les caractéristiques des quantités d'intérêt et améliorer la précision de l'estimation, ce qui peut être abordé avec des techniques variationnelles.L'analyse est appliquée à deux nouveaux concepts de dispositifs optiques non conventionnels.Le premier est un nouveau système d'imagerie d'acquisition compressé basé sur des matrices de filtres de couleur, qui intègre des informations provenant de capteurs avec différentes caractéristiques spatiales et spectrales dans un seul produit mosaïqué. Contrairement aux dispositifs existants basés sur la détection compressée, l'objectif n'est pas de récupérer les sources multirésolutions non compressées d'origine, mais plutôt de récupérer directement une image fusionnée synthétique avec une résolution spatiale et spectrale élevée.La solution proposée repose sur la régularisation de la variation totale et fait l'objet d'une analyse détaillée, comparant sa puissance de compression avec des alternatives logicielles simples, évaluant ses performances au fur et à mesure que le nombre de canaux change, et validant son efficacité par rapport aux méthodes de l'état de l'art lorsque appliqué séparément aux algorithmes classiques de fusion ou de mosaïquage.La deuxième classe d'appareils considérée dans ce travail est basée sur le brevet ImSPOC, un concept de conception pour un spectromètre imageur instantané de faible finesse basé sur l'interférométrie de Fabry-Pérot. Son comportement idéal suit le principe de la spectroscopie à transformée de Fourier, car son acquisition peut être interprétée comme une version échantillonnée d'un interférogramme, disposée sur différentes sous-images réparties sur le même plan focal.Après avoir défini un modèle physique basé sur la géométrie optique, sa validité est évaluée sur des acquisitions réelles en mettant en place un problème d'inférence bayésienne pour déterminer ses paramètres, avec des approches basées sur des estimateurs du maximum de vraisemblance, des recherches en grille régulière et une régression non linéaire.Divers tests préliminaires sont ensuite menés sur la méthode d'inversion, avec des approches basées sur la décomposition en valeurs singulières et les régularisations creuses, accompagnées d'une analyse de leur robustesse aux mésappariements de modèles

Model Based Signal Processing Techniques for Nonconventional Optical Imaging Systems

Il existe une demande croissante d'images avec une résolution spectrale et spatiale plus élevée pour des applications dans plusieurs domaines tels que la santé, l'environnement, le contrôle qualité et la surveillance des catastrophes naturelles. L'imagerie hyperspectrale fournit la diversité spectrale nécessaire pour récupérer la composition des matériaux sur site pour des applications telles que la détection d'incendies, d'anomalies, d'agents chimiques, de cibles et de changements de scène.L'exigence de dispositifs moins chers et plus compacts (par exemple, pour être embarqués sur des satellites à faible coût et une plateforme aéroportée) capables de capturer ces informations a conduit au développement de concepts de conception innovants non conventionnels pour surmonter les limitations technologiques des caméras traditionnelles.Les données acquises à partir de ces nouveaux dispositifs d'imagerie suivant le paradigme d'imagerie informatique ne sont généralement pas facilement exploitables pour l'application finale.Une phase de calcul est nécessaire pour extraire des informations utiles des acquisitions brutes.Cette thèse aborde cette question en mettant en place un problème d'inversion. L'approche générale consiste à caractériser le terme de fidélité des données avec un modèle physique, décrivant les transformations optiques sous-jacentes effectuées par le dispositif. Le défi est ensuite déplacé vers l'étape de régularisation pour bien caractériser les caractéristiques des quantités d'intérêt et améliorer la précision de l'estimation, ce qui peut être abordé avec des techniques variationnelles.L'analyse est appliquée à deux nouveaux concepts de dispositifs optiques non conventionnels.Le premier est un nouveau système d'imagerie d'acquisition compressé basé sur des matrices de filtres de couleur, qui intègre des informations provenant de capteurs avec différentes caractéristiques spatiales et spectrales dans un seul produit mosaïqué. Contrairement aux dispositifs existants basés sur la détection compressée, l'objectif n'est pas de récupérer les sources multirésolutions non compressées d'origine, mais plutôt de récupérer directement une image fusionnée synthétique avec une résolution spatiale et spectrale élevée.La solution proposée repose sur la régularisation de la variation totale et fait l'objet d'une analyse détaillée, comparant sa puissance de compression avec des alternatives logicielles simples, évaluant ses performances au fur et à mesure que le nombre de canaux change, et validant son efficacité par rapport aux méthodes de l'état de l'art lorsque appliqué séparément aux algorithmes classiques de fusion ou de mosaïquage.La deuxième classe d'appareils considérée dans ce travail est basée sur le brevet ImSPOC, un concept de conception pour un spectromètre imageur instantané de faible finesse basé sur l'interférométrie de Fabry-Pérot. Son comportement idéal suit le principe de la spectroscopie à transformée de Fourier, car son acquisition peut être interprétée comme une version échantillonnée d'un interférogramme, disposée sur différentes sous-images réparties sur le même plan focal.Après avoir défini un modèle physique basé sur la géométrie optique, sa validité est évaluée sur des acquisitions réelles en mettant en place un problème d'inférence bayésienne pour déterminer ses paramètres, avec des approches basées sur des estimateurs du maximum de vraisemblance, des recherches en grille régulière et une régression non linéaire.Divers tests préliminaires sont ensuite menés sur la méthode d'inversion, avec des approches basées sur la décomposition en valeurs singulières et les régularisations creuses, accompagnées d'une analyse de leur robustesse aux mésappariements de modèles.There is an increasing demand for images with higher spectral and spatial resolution for applications in several domains such as health, environment, quality checking and natural disasters monitoring. Hyperspectral imagery provides the necessary spectral diversity to recover the composition of materials on site for applications such as the detection of fires, anomalies, chemical agents, targets and changes in the scene.The requirement for cheaper and more compact devices (e.g. to be embarked on low cost satellites and airborne platform) which are capable of capturing this information has led to the development of nonconventional innovative design concepts to overcome the technological limitations of traditional cameras. Data acquired by such novel imaging devices following the computational imaging paradigm are typically not readily exploitable for the final application. A computational phase is hence needed for extracting useful information from the raw acquisitions.This thesis addresses this issue by setting up an inversion problem. The general approach is to characterize the data fidelity term with a physical model, describing the underlying optical transformations performed by the device. The challenge is then shifted on the regularization step to properly characterizes the features of the quantities of interest and improve the accuracy of the estimation, which can be tackled with variational techniques.The analysis is applied to two novel concepts for nonconventional optical devices. The first one is a novel compressed acquisition imaging system based on color filter arrays, which embeds information from sensors with different spatial and spectral characteristics into a single mosaiced product. As opposed to existing compressed sensing based devices, the goal is not to recover the original uncompressed multiresolution sources, but instead to directly recover a synthetic fused image with both high spatial and spectral resolution.The proposed solution relies on the total variation regularization and is the subject of a detailed analysis, comparing its compressive power with straightforward software alternatives, evaluating its performances as the amount of channels changes, and validating its efficiency in comparison to state of the art methods when applied to classical fusion or mosaicing algorithms separately.The second class of devices is based on the ImSPOC patent, a design concept for a low finesse snapshot imaging spectrometer based on the interferometry of Fabry-Pérot. Its ideal behaviour follows the principle of the Fourier Transform Spectroscopy, as its acquisition can be interpreted as a sampled version of an interferogram, arranged across different sub-images distributed on the same focal plane.After defining a physical model based on optical geometry, its validity is evaluated over real acquisitions by setting up a Bayesian inference problem to determine its parameters, with approaches based on maximum likelihood estimators, regular-grid searches and nonlinear regression.A variety of preliminary tests are then carried out on the inversion method, with approaches based on singular value decomposition and sparse-inducing regularizers, accompanied by a analysis of their robustness to model mismatches

Techniques de traitement du signal basées modèles pour systèmes d'imagerie optique non conventionnels

There is an increasing demand for images with higher spectral and spatial resolution for applications in several domains such as health, environment, quality checking and natural disasters monitoring. Hyperspectral imagery provides the necessary spectral diversity to recover the composition of materials on site for applications such as the detection of fires, anomalies, chemical agents, targets and changes in the scene.The requirement for cheaper and more compact devices (e.g. to be embarked on low cost satellites and airborne platform) which are capable of capturing this information has led to the development of nonconventional innovative design concepts to overcome the technological limitations of traditional cameras. Data acquired by such novel imaging devices following the computational imaging paradigm are typically not readily exploitable for the final application. A computational phase is hence needed for extracting useful information from the raw acquisitions.This thesis addresses this issue by setting up an inversion problem. The general approach is to characterize the data fidelity term with a physical model, describing the underlying optical transformations performed by the device. The challenge is then shifted on the regularization step to properly characterizes the features of the quantities of interest and improve the accuracy of the estimation, which can be tackled with variational techniques.The analysis is applied to two novel concepts for nonconventional optical devices. The first one is a novel compressed acquisition imaging system based on color filter arrays, which embeds information from sensors with different spatial and spectral characteristics into a single mosaiced product. As opposed to existing compressed sensing based devices, the goal is not to recover the original uncompressed multiresolution sources, but instead to directly recover a synthetic fused image with both high spatial and spectral resolution.The proposed solution relies on the total variation regularization and is the subject of a detailed analysis, comparing its compressive power with straightforward software alternatives, evaluating its performances as the amount of channels changes, and validating its efficiency in comparison to state of the art methods when applied to classical fusion or mosaicing algorithms separately.The second class of devices is based on the ImSPOC patent, a design concept for a low finesse snapshot imaging spectrometer based on the interferometry of Fabry-Pérot. Its ideal behaviour follows the principle of the Fourier Transform Spectroscopy, as its acquisition can be interpreted as a sampled version of an interferogram, arranged across different sub-images distributed on the same focal plane.After defining a physical model based on optical geometry, its validity is evaluated over real acquisitions by setting up a Bayesian inference problem to determine its parameters, with approaches based on maximum likelihood estimators, regular-grid searches and nonlinear regression.A variety of preliminary tests are then carried out on the inversion method, with approaches based on singular value decomposition and sparse-inducing regularizers, accompanied by a analysis of their robustness to model mismatches.Il existe une demande croissante d'images avec une résolution spectrale et spatiale plus élevée pour des applications dans plusieurs domaines tels que la santé, l'environnement, le contrôle qualité et la surveillance des catastrophes naturelles. L'imagerie hyperspectrale fournit la diversité spectrale nécessaire pour récupérer la composition des matériaux sur site pour des applications telles que la détection d'incendies, d'anomalies, d'agents chimiques, de cibles et de changements de scène.L'exigence de dispositifs moins chers et plus compacts (par exemple, pour être embarqués sur des satellites à faible coût et une plateforme aéroportée) capables de capturer ces informations a conduit au développement de concepts de conception innovants non conventionnels pour surmonter les limitations technologiques des caméras traditionnelles.Les données acquises à partir de ces nouveaux dispositifs d'imagerie suivant le paradigme d'imagerie informatique ne sont généralement pas facilement exploitables pour l'application finale.Une phase de calcul est nécessaire pour extraire des informations utiles des acquisitions brutes.Cette thèse aborde cette question en mettant en place un problème d'inversion. L'approche générale consiste à caractériser le terme de fidélité des données avec un modèle physique, décrivant les transformations optiques sous-jacentes effectuées par le dispositif. Le défi est ensuite déplacé vers l'étape de régularisation pour bien caractériser les caractéristiques des quantités d'intérêt et améliorer la précision de l'estimation, ce qui peut être abordé avec des techniques variationnelles.L'analyse est appliquée à deux nouveaux concepts de dispositifs optiques non conventionnels.Le premier est un nouveau système d'imagerie d'acquisition compressé basé sur des matrices de filtres de couleur, qui intègre des informations provenant de capteurs avec différentes caractéristiques spatiales et spectrales dans un seul produit mosaïqué. Contrairement aux dispositifs existants basés sur la détection compressée, l'objectif n'est pas de récupérer les sources multirésolutions non compressées d'origine, mais plutôt de récupérer directement une image fusionnée synthétique avec une résolution spatiale et spectrale élevée.La solution proposée repose sur la régularisation de la variation totale et fait l'objet d'une analyse détaillée, comparant sa puissance de compression avec des alternatives logicielles simples, évaluant ses performances au fur et à mesure que le nombre de canaux change, et validant son efficacité par rapport aux méthodes de l'état de l'art lorsque appliqué séparément aux algorithmes classiques de fusion ou de mosaïquage.La deuxième classe d'appareils considérée dans ce travail est basée sur le brevet ImSPOC, un concept de conception pour un spectromètre imageur instantané de faible finesse basé sur l'interférométrie de Fabry-Pérot. Son comportement idéal suit le principe de la spectroscopie à transformée de Fourier, car son acquisition peut être interprétée comme une version échantillonnée d'un interférogramme, disposée sur différentes sous-images réparties sur le même plan focal.Après avoir défini un modèle physique basé sur la géométrie optique, sa validité est évaluée sur des acquisitions réelles en mettant en place un problème d'inférence bayésienne pour déterminer ses paramètres, avec des approches basées sur des estimateurs du maximum de vraisemblance, des recherches en grille régulière et une régression non linéaire.Divers tests préliminaires sont ensuite menés sur la méthode d'inversion, avec des approches basées sur la décomposition en valeurs singulières et les régularisations creuses, accompagnées d'une analyse de leur robustesse aux mésappariements de modèles