Koopman Ensembles for Probabilistic Time Series Forecasting

In the context of an increasing popularity of data-driven models to represent dynamical systems, many machine learning-based implementations of the Koopman operator have recently been proposed. However, the vast majority of those works are limited to deterministic predictions, while the knowledge of uncertainty is critical in fields like meteorology and climatology. In this work, we investigate the training of ensembles of models to produce stochastic outputs. We show through experiments on real remote sensing image time series that ensembles of independently trained models are highly overconfident and that using a training criterion that explicitly encourages the members to produce predictions with high inter-model variances greatly improves the uncertainty quantification of the ensembles.

MultiHU-TD: Multifeature Hyperspectral Unmixing Based on Tensor Decomposition

Hyperspectral unmixing allows representing mixed pixels as a set of pure materials weighted by their abundances. Spectral features alone are often insufficient, so it is common to rely on other features of the scene. Matrix models become insufficient when the hyperspectral image (HSI) is represented as a high-order tensor with additional features in a multimodal, multifeature framework. Tensor models such as canonical polyadic decomposition allow for this kind of unmixing but lack a general framework and interpretability of the results. In this article, we propose an interpretable methodological framework for low-rank multifeature hyperspectral unmixing based on tensor decomposition (MultiHU-TD) that incorporates the abundance sum-to-one constraint in the alternating optimization alternating direction method of multipliers (ADMM) algorithm and provide in-depth mathematical, physical, and graphical interpretation and connections with the extended linear mixing model. As additional features, we propose to incorporate mathematical morphology and reframe a previous work on neighborhood patches within MultiHU-TD. Experiments on real HSIs showcase the interpretability of the model and the analysis of the results. Python and MATLAB implementations are made available on GitHub.

Neural Koopman prior for data assimilation

With the increasing availability of large scale datasets, computational power and tools like automatic differentiation and expressive neural network architectures, sequential data are now often treated in a data-driven way, with a dynamical model trained from the observation data. While neural networks are often seen as uninterpretable black-box architectures, they can still benefit from physical priors on the data and from mathematical knowledge. In this paper, we use a neural network architecture which leverages the long-known Koopman operator theory to embed dynamical systems in latent spaces where their dynamics can be described linearly, enabling a number of appealing features. We introduce methods that enable to train such a model for long-term continuous reconstruction, even in difficult contexts where the data comes in irregularly-sampled time series. The potential for self-supervised learning is also demonstrated, as we show the promising use of trained dynamical models as priors for variational data assimilation techniques, with applications to e.g. time series interpolation and forecasting.

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.

Learning Sentinel-2 reflectance dynamics for data-driven assimilation and forecasting

Over the last few years, massive amounts of satellite multispectral and hyperspectral images covering the Earth's surface have been made publicly available for scientific purpose, for example through the European Copernicus project. Simultaneously, the development of self-supervised learning (SSL) methods has sparked great interest in the remote sensing community, enabling to learn latent representations from unlabeled data to help treating downstream tasks for which there is few annotated examples, such as interpolation, forecasting or unmixing. Following this line, we train a deep learning model inspired from the Koopman operator theory to model long-term reflectance dynamics in an unsupervised way. We show that this trained model, being differentiable, can be used as a prior for data assimilation in a straightforward way. Our datasets, which are composed of Sentinel-2 multispectral image time series, are publicly released with several levels of treatment.

The ImSPOC snapshot imaging spectrometer: image formation model and device characterization

In recent years, the demand for capturing spectral information with finer detail has increased, requiring hyperspectral imaging devices capable of acquiring the required information with increased temporal, spatial and spectral resolution. In this work, we present the image acquisition model of the Image SPectrometer On Chip (ImSPOC), a novel compact snapshot image spectrometer based on the interferometry of Fabry-Perot. Additionally, we propose the interferometer response characterization algorithm (IRCA), a robust three-step procedure to characterize the ImSPOC device that estimates the optical parameters of the composing interferometers' transfer function. The proposed algorithm processes the image output from a set of monochromatic light sources, refining the results through nonlinear regression after an ad-hoc initialization. Experimental analysis confirms the performances of the proposed approach for the characterization of four different ImSPOC prototypes. The source code associated to this paper is available at https://github.com/danaroth83/irca.

Leveraging Neural Koopman Operators to Learn Continuous Representations of Dynamical Systems from Scarce Data

Over the last few years, several works have proposed deep learning architectures to learn dynamical systems from observation data with no or little knowledge of the underlying physics. A line of work relies on learning representations where the dynamics of the underlying phenomenon can be described by a linear operator, based on the Koopman operator theory. However, despite being able to provide reliable long-term predictions for some dynamical systems in ideal situations, the methods proposed so far have limitations, such as requiring to discretize intrinsically continuous dynamical systems, leading to data loss, especially when handling incomplete or sparsely sampled data. Here, we propose a new deep Koopman framework that represents dynamics in an intrinsically continuous way, leading to better performance on limited training data, as exemplified on several datasets arising from dynamical systems.

Joint Demosaicing and Fusion of Multiresolution Compressed Acquisitions: Image Formation and Reconstruction Methods

Novel optical imaging devices allow for hybrid acquisition modalities such as compressed acquisitions with locally different spatial and spectral resolutions captured by the same focal plane array. In this work, we propose to model a multiresolution compressed acquisition (MRCA) in a generic framework, which natively includes acquisitions by conventional systems such as those based on spectral/color filter arrays, compressed coded apertures, and multiresolution sensing. We 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.

An Introduction to Deep Morphological Networks

The recent impressive results of deep learning-based methods on computer vision applications brought fresh air to the research and industrial community. This success is mainly due to the process that allows those methods to learn data-driven features, generally based upon linear operations. However, in some scenarios, such operations do not have a good performance because of their inherited process that blurs edges, losing notions of corners, borders, and geometry of objects. Overcoming this, non-linear operations, such as morphological ones, may preserve such properties of the objects, being preferable and even state-of-the-art in some applications. Encouraged by this, in this work, we propose a novel network, called Deep Morphological Network (DeepMorphNet), capable of doing non-linear morphological operations while performing the feature learning process by optimizing the structuring elements. The DeepMorphNets can be trained and optimized end-to-end using traditional existing techniques commonly employed in the training of deep learning approaches. A systematic evaluation of the proposed algorithm is conducted using two synthetic and two traditional image classification datasets. Results show that the proposed DeepMorphNets is a promising technique that can learn distinct features when compared to the ones learned by current deep learning methods.

Dynamic Multi-Scale Segmentation of Remote Sensing Images based on Convolutional Networks

Semantic segmentation requires methods capable of learning high-level features while dealing with large volume of data. Towards such goal, Convolutional Networks can learn specific and adaptable features based on the data. However, these networks are not capable of processing a whole remote sensing image, given its huge size. To overcome such limitation, the image is processed using fixed size patches. The definition of the input patch size is usually performed empirically (evaluating several sizes) or imposed (by network constraint). Both strategies suffer from drawbacks and could not lead to the best patch size. To alleviate this problem, several works exploited multi-scale information by combining networks or layers. This process increases the number of parameters resulting in a more difficult model to train. In this work, we propose a novel technique to perform semantic segmentation of remote sensing images that exploits a multi-scale paradigm without increasing the number of parameters while defining, in training time, the best patch size. The main idea is to train a dilated network with distinct patch sizes, allowing it to capture multi-scale characteristics from heterogeneous contexts. While processing these varying patches, the network provides a score for each patch size, helping in the definition of the best size for the current scenario. A systematic evaluation of the proposed algorithm is conducted using four high-resolution remote sensing datasets with very distinct properties. Our results show that the proposed algorithm provides improvements in pixelwise classification accuracy when compared to state-of-the-art methods.