Year: 2019

Abstract:
For numerous domains, including for instance earth observation, medical imaging, astrophysics,..., available image and signal datasets often irregular spacetime sampling patterns and large missing data rates. These sampling properties is a critical issue to apply state-of-the-art learning-based (e.g., auto-encoders, CNNs,...) to fully benefit from the available large-scale observations and reach breakthroughs in the reconstruction and identification of processes of interest. In this paper, we address the end-to-end learning of representations of signals, images and image sequences from irregularly-sampled data, i.e. when the training data involved missing data. From an analogy to Bayesian formulation, we consider energy-based representations. Two energy forms are investigated: one derived from auto-encoders and one relating to Gibbs energies. The learning stage of these energy-based representations (or priors) involve a joint interpolation issue, which resorts to solving an energy minimization problem under observation constraints. Using a neural-network-based implementation of the considered energy forms, we can state an end-to-end learning scheme from irregularly-sampled data. We demonstrate the relevance of the proposed representations for different case-studies: namely, multivariate time series, 2 images and image sequences.

text: For numerous domains, including for instance earth observation, medical imaging, astrophysics,..., available image and signal datasets often irregular spacetime sampling patterns and large missing data rates. These sampling properties is a critical issue to apply state-of-the-art learning-based (e.g., auto-encoders, CNNs,...) to fully benefit from the available large-scale observations and reach breakthroughs in the reconstruction and identification of processes of interest. In this paper, we address the end-to-end learning of representations of signals, images and image sequences from irregularly-sampled data, i.e. when the training data involved missing data. From an analogy to Bayesian formulation, we consider energy-based representations. Two energy forms are investigated: one derived from auto-encoders and one relating to Gibbs energies. The learning stage of these energy-based representations (or priors) involve a joint interpolation issue, which resorts to solving an energy minimization problem under observation constraints. Using a neural-network-based implementation of the considered energy forms, we can state an end-to-end learning scheme from irregularly-sampled data. We demonstrate the relevance of the proposed representations for different case-studies: namely, multivariate time series, 2 images and image sequences.

heading: 1 INTRODUCTION
text: In numerous application domains, the available observation datasets do not involve gap-free and regularly-gridded signals or images. The irregular-sampling may result both from the characteristics of the sensors and sampling strategy, e.g. considered orbits and swaths in spacebone earth observation and astrophysics, sampling schemes in medical imaging, as well as environmental conditions which may affect the sensor, e.g. atmospheric conditions and clouds for earth observation.
A rich literature exists on interpolation for irregularly-sampled signals and images (also referred to as inpainting in image processing (4)). A classic framework states the interpolation issue as the miminisation of an energy, which may be interpreted in a Bayesian framework. A variety of energy forms, including Markovian priors (12), patch-based priors (20), gradient norms in variational and/or PDE-based formulations (4), Gaussian priors () as well as dynamical priors in fluid dynamics (3). The later relates to optimal interpolation and kriging (8), which is among the state-of-the-art and operational schemes in geoscience (10). Optimal schemes classically involve the inference of the considered covariance-based priors from irregularly-sampled data. This may however be at the expense of Gaussianity and linearity assumptions, which do not often apply for real signals and images. For the other types of energy forms, their parameterization are generally set a priori and not learnt from the data. Regarding more particularly data-driven and learning-based approaches, most previous works (2; 11; 20) have addressed the learning of interpolation schemes under the assumption that a representative gap-free dataset is available. This gap-free dataset may be the image itself (9; 20; 18). For numerous application domains, as mentionned above, this assumption cannot be fulfilled. Regarding recent advances in learning-based schemes, a variety of deep learning models, e.g. (7; 16; 24; 23), have been proposed. Most of these works focus on learning an interpolator. One may however expect to learn not only an interpolator but also some representation of considered data, which may be of interest for other applications. In this respect, RBM models (Restricted Boltzmann
Machines) (22; 6) are particularly appealing at the expense however of computationally-expensive MCMC schemes.
In this paper, we aim to learn representations of signals or images from irregularly-sampled observation datasets. Our contribution is three-fold:
• an end-to-end learning of energy-based representations from irregularly-sampled training data. Based on a neural-network architecture, it jointly embeds the considered energy form and an associated interpolation scheme.
• besides classic auto-encoder representations, we introduce NN-based Gibbs-Energy representations, which relate to Markovian priors embedded in CNNs.
• the demonstration of the relevance of the proposed end-to-end learning framework for different data types, namely time series, images and image sequences, with possibly very high missing data rates.
The remainder is organized as follows. Section 2 formally states the considered issue. We introduce the proposed end-to-end learning scheme in Section 3. We report numerical experiments in Section 4 and discuss our contribution with respect to related work in Section ??.

heading: 2 PROBLEM STATEMENT
text: In this section, we formally introduce the considered issue, namely the end-to-end learning of representations and interpolators from irregularly-sampled data. Within a classic Bayesian or energybased framework, interpolation issues may be stated as a minimization issue
X̂ = arg min X Uθ (X) subject to XΩ = YΩ (1)
where X is the considered signal, image or image series (referred to hereafter as the hidden state), Y the observation data, only available on a subdomain Ω of the entire domain D, and Uθ() the considered energy prior parameterized by θ. As briefly introduced above, a variety of energy priors have been proposed in the literature, e.g. (4; 20; 5).
We assume we are provided with a series of irregularly-sampled observations, that is to say a set {Y (i),Ω(i)}i∈{1,...,N}, such that Ω(i) ⊂ D and Y (i) is only defined on subdomain Ω(i). Assuming that all X(i) share some underlying energy representation Uθ(), we may define the following operator I
I ( Uθ, Y (i),Ω(i) )
= arg min X
Uθ (X) subject to XΩ(i) = Y (i) Ω(i) (2)
such that I(Y (i),Ω(i)) = X(i). Here, we aim to learn the parameters θ() of the energy Uθ() from the available observation dataset {Y (i),Ω(i)}i. Assuming operator I is known, this learning issue can be stated as the minimization of reconstruction error for the observed data
θ̂ = arg min θ ∑ i ∥∥∥Y (i) − I (Uθ, Y (i),Ω(i))∥∥∥2 Ω(i)
(3)
where ‖.‖2Ω refers to the L2 norm evaluated on subdomain. Learning energy Uθ() from observation dataset {Y (i),Ω(i)}i clearly involves a joint interpolation issue solved by operator I. Given this general formulation, the end-to-end learning issue comes to solve minimization (3) according to some given parameterization of energy Uθ(). In (3), interpolation operator I is clearly critical. In Section 3, we investigate a neural-network implementation of this general framework, which embeds a neural-network formulations both for energy Uθ() and interpolation operator I.

heading: 3 PROPOSED END-TO-END LEARNING FRAMEWORK
text: In this section, we detail the proposed neural-network-based implementation of the end-to-end formulation introduced in the previous section. We first present the considered paramaterizations for energy Uθ() in (3) (Section 3.1). We derive associated NN-based interpolation operators I (Section 3.2) and describe our overall NN architectures for the end-to-end learning of representations and interpolators from irregularly-sampled datasets (Section 3.3).

heading: 3.1 NN-BASED ENERGY FORMULATION
text: We first investigate NN-based energy representations based on auto-encoders (15). Let us denote by φE and φD the encoding and decoding operators of an auto-encoder (AE), which may comprise both dense auto-encoders (AEs), convolutional AEs as well as recurrent AEs when dealing with time-related processes. The key feature of AEs is that the encoding operator φE maps the state X into a low-dimensional space. Auto-encoders are naturally associated with the following energy
Uθ(X) = ‖X − φD (φE (X))‖2 (4) Minimizing (1) according to this energy amounts to retrieving the hidden state whose lowdimensional representation in the encoding space matches the observed data in the original decoded space. Here, parameters θ refer to the parameters of the encoder φE and decoder φD, respectively θE and θD.
The mapping to lower-dimensional space may be regarded as a potential loss in the representation potential of the representation. Gibbs models provide an appealing framework for an alternative energy-based representation, with no such dimensionality reduction constraint. Gibbs models introduced in statistical physics have also been widely explored in computer vision and pattern recognition (13) from the 80s. Gibbs models relate to the decomposition of Uθ as a sum of potentials Uθ(X) = ∑ c∈C Vc (Xc) where C is a set of cliques, i.e. a set of interacting sites (typically, local neighbors), and Vc the potential on clique c. In statistical physics, this formulation states the global energy of the system as the sum of local energies (the potential over locally-interacting sites). Here, we focus on the following parameterization of the potential function
Uθ(X) = ∑ s∈D ‖Xs − ψ (XNs) ‖2 (5)
with Ns the set of neighbors of site s for the entire domain D and ψ a potential function. Lowenergy state for this energy refers to state X which operator ψ provides a good prediction at any site s knowing the state in the neighborhood Ns of s. This type of Gibbs energy relates to Gaussian Markov random fields, where the conditional likelihood at one site given its neighborhood follows a Gaussian distribution. We implement this type of Gibbs energy using the following NN-based parameterization of operator ψ: ψ(X) = ψ2 (ψ1(X)) (6) It involves the composition of a space and/or time convolutional operator ψ1 and a coordinate-wise operator ψ2. The convolutional kernel for operator ψ1 is such that the coefficients for the center of convolutional window are set to zero. This property fulfills the constraint that X(s) is not involved in the computation of ψ (XNs) at site s. As an example, for a univariate image, ψ1 can be set as a convolutional layer with NF filters with kernels of size 3x3x1, such that for each kernel Kf Kf (1, 1, 0) = 0 (the same applies to biases). In such a case, operator ψ2 would be a convolution layer with one filter with a kernel of size 1x1xNF . Both ψ1 and ψ2 can also involve non-linear activations. Without loss of generality, given this parameterization for operator ψ, we may rewrite energy Uθ as Uθ(X) = ‖X − ψ (X) ‖2 where ψ (X)) at site s is given by ψ (XNs). Overall, we may use the following common formulation for the two types of energy-based representation Uθ(X) = ‖X − ψ (X) ‖2 (7) They differ in the parameterization chosen for operator ψ.

heading: 3.2 NN-BASED INTERPOLATOR
text: Besides the NN-based energy formulation, the general formulation stated in (3) involves the definition of interpolation operator I, which refers to minimization (1). We here derive NN-based interpolation architectures from the considered NN-based energy parameterization.
Given parameterization (7), a simple fixed-point algorithm may be considered to solve for (3). This algorithm at the basis of DINEOF algorithm and XXX for matrix completion under subspace constraints (2; 14) involves the following iterative update
X (k+1) p = ψ ( X(k) ) X(k+1) (Ω) = Y (Ω)
X(k+1) ( Ω ) = X (k+1) p ( Ω ) (8)
Interestingly, the algorithm is parameter-free and can be readily implemented in a NN architecture given the number of iterations to be considered.
Given some initialisation, one may typically consider an iterative gradient-based descent which applies at each iteration k
X (k+1) p = X(k) − λJUθ
( X(k) ) X(k+1) (Ω) = Y (Ω)
X(k+1) ( Ω ) = X (k+1) p ( Ω ) (9)
with JUθ the gradient of energy Uθ w.r.t. state X , λ the gradient step and Ω the missing data area. Automatic differentiation tool embedded in neural network frameworks may provide the numerical computation for gradient JUθ given the NN-based parameterization for energy Uθ. This proved numerically too expensive and was not further investigated in our experiments. Given the considered form for energy Uθ, its gradient w.r.t. X decomposes as a product
JUθ (X) = Jψ (X) (X − ψ (X)) (10)
and X − ψ (X) may be regarded as a suboptimal gradient descent. Hence, rather than considering the true Jacobian Jψ for operator ψ, we may consider an approximation through a trainable CNN G() such that the gradient descent becomes
X (k+1) p = X(k) −G ( X(k), ψ ( X(k) )) X(k+1) (Ω) = Y (Ω)
X(k+1) ( Ω ) = X (k+1) p ( Ω ) (11)
where G(X(k), ψ(X(k))) = G̃(X(k) − ψ(X(k))) and G̃ is a CNN to be learnt jointly to ψ during the learning stage. Interestingly, this gradient descent embeds the fixed-point algorithm when G̃ is the identity.
Let us denote respectively by IFP and IG the fixed-point and gradient-based NN-based interpolators, which implement NI iterations of the proposed interpolation updates. Below, INN will denote both IFP and IG. Whereas IFP is parameter-free, IG involves the parameterization of operator G. We typically consider a CNN with ReLu activations with increasing numbers of filter through layers up to the final layer which applies a linear convolutional with a number of filters given by the dimension of the state.

heading: 3.3 END-TO-END ARCHITECTURE AND IMPLEMENTATION DETAILS
text: Given the parameterizations for energy Uθ and the associated NN-based interpolators presented previously, we design an end-to-end learning for energy representation Uθ and associated interpolator
INN , which uses as inputs an observed sample Y (i) and the associated missing-data-free domain Ω(i). Using a normalization preprocessing step, we initially fill missing data with zeros to provide an initial interpolated state to the architecture. We provide a sketch of the architecture in Fig.1.
Regarding implementation details, beyond the design of the architectures, which may be applicationdependent for operators ψ and G̃ (see Section 4), we consider an implementation under keras using tensorflow as backend. Regarding the training strategy, we use adam optimizer. We iteratively increase the number of blocksNI (number of gradient steps) to avoid the training to diverge. Similarly, we decrease the learning rate accross iterations, typically from 1e-3 to 1e-6. In our experiments, we typically consider from 5 to 15 blocks. All the experiments were run under workstations with a single GPU (Nvidia GTX 1080 and GTX 1080 Ti).

heading: 4 EXPERIMENTS
text: In this section, we report numerical experiments on different datasets to evaluate and demonstrate the proposed scheme. We consider three different case-studies: an image dataset, namely MNIST; a multivariate time-series through an application to Lorenz-63 dynamics (17) and an image sequence dataset through an application to ocean remote sensing data with real missing data patterns. In all experiments, we refer to the AE-based framework, respectively as FP(d)-ConvAE and G(d)-ConvAE using the fixed-point or gradient-based interpolator where the value of d refers to the number of interpolation steps. Similarly, we refer to the Gibbs-based frameworks respectively as FP(d)-GENN and G(d)-GENN.

heading: 4.1 MNIST DATASETS
text: We evaluate the proposed framework on MNIST datasets for which we simulate missing data patterns. The dataset comprises 60000 28x28 grayscale images. For this dataset, we only evaluate the AE-based setting. We consider the following convolutional AE architecture with a 20-dimensional encoding space:
• Encoder operator φE : Conv2D(20)+ ReLU + AvPooling + Conv2D(40) + ReLU + AveragePooling + Dense(80) + ReLU + Dense(20);
• Decoder operator φE : Conv2DTranspose(40) + ResNet(2), ResNet: Conv2D(40)+ReLU+Conv2D(20)
We generate random missing data patterns composed of NS squares of size WSxWS , the center of the square is randomly sampled uniformly over the image grid. As illustrated in Fig.3, we consider four missing data patterns: NS = 20 and WS = 5, NS = 30 and WS = 5, NS = 3 and WS = 9, NS = 6 and WS = 9. As performance measure, we evaluate an interpolation score (I-score), a global reconstruction score (R-score) for the interpolated images and an auto-encoding (AE-score) score of the trained auto-encoder applied to gap-free data, in terms of explained variance. We also evaluate a classification score (C-score), in terms of mean accurcay, using the 20-dimensional encoding space as feature space for classification with a 3-layer MLP. We report all performance measures for both the test dataset in Tab.1 for MNIST dataset. For benchmarking purposes, we also report the performance of DINEOF framework, which uses a 20-dimensional PCA trained on the gap-free dataset, the auto-encoder architecture trained on gap-free dataset as well as the considered convolutional auto-encoder trained using an initial zero-filling for missing data areas and a training loss computed only of observed data areas. The later can be regarded as a FP(1)-ConvAE architecture using a single block in Fig.1. Overall, these results illustrate that representations trained from gap-free data may not apply when considering significant missing data rates as illustrated by relatively poor performance of PCA-based and AE schemes, when trained from gap-free data. Similarly, training an AE representations using as input a zero-filling strategy lowers the auto-encoding power when applied to gap-free data. Overall, the proposed scheme guarantees a good representation in terms of AE score with an additional gain in terms of interpolation performance, typically between ≈ 15% and 30% depending of the missing data patterns, the gain being greater when considering larger missing data areas.

heading: 4.2 MULTIVARIATE TIME SERIES
text: We present an application to the Lorenz-63 dynamics (17), which involve a 3-dimensional state governed by the following ordinary differential equation: dXt,1 dt = σ (Xt,2 −Xt,2) dXt,2 dt = ρXt,1 −Xt,2 −Xt,1Xt,3
dXt,3 dt = Xt,1Xt,2 − βXt,3
(12)
Under parameterization σ = 10, ρ = 28 and β = 8/3 considered here, Lorenz-63 dynamics are chaotic dynamics, which make then challening in our context. They can be regarded as a reducedorder model of turbulence dynamics. We simulate Lorenz-63 time series of 200 time steps using a Runge-Kutta-4 ODE solver with an integration step of 0.01 from an initial condition in the attractor. For a given experiment, we first subsample the simulated series to a given time step dt and then generate using a uniform random samplong a missing data mask accounting for 75% of the data. Overall, training and test time series are formed by subsequences of 200 time steps. We report experiments with the GE-NN setting. The AE-based framework showed lower performance and is not included here. The considered GE-NN architecture is as follows: a 1D convolution layer with 120 filters with a kernel width of 3, zero-weight-constraints for the center of the convolution kernel and a Relu activation, a 1D convolution layer with 6 filters a kernel width of 1 and a Relu activation, a residual network with 4 residual units using 6 filters with a kernel width of 1 and a linear activation. The last layer is a convolutional layer with 3 filters, a kernel width of 1 and a linear activation.
For benchmarking purposes, we report the interpolation performance issued from an ensemble Kalman smoother (EnKS) (10) knowing the true model, regarded as a lower-bound of the interpolation performance. The parameter setting of the EnKS is as follows: 200 members, noise-free dynamical model and spherical observation covariannce to 0.1 · I . We also compare the proposed approach to DINEOF (2; 21). Here, the learning of the PCA decomposition used in the DINEOF scheme relies on gap-free data. Fig.2 illustrates this comparison for one sequence of 200 time steps with dt = 0.02. In this example, one can hardly distinguish the interpolated sequence using the proposed approach (FP(15)-GE-NN). By contrast, DINEOF scheme cannot retrieve some of the largest deviations. We report in Appendix Tab.3 the performance of the different interpolation schemes. The proposed approach clearly outperforms DINEOF by about one order of magnitude for the experiments with a time-step dt = 0.02. The interpolation error for observed states (first line in Tab.3) also stresses the improved prior issued from the proposed Gibbs-like energy setting. For chaotic dynamics, global PCA representation seems poorly adapted where local representations as embedded by the considered Gibbs energy setting appear more appealing.

heading: 4.3 IMAGE SEQUENCE DATASET WITH VERY LARGE MISSING DATA RATES
text: The third case-study addresses satellite-derived Sea Surface Temperature (SST) image time series. Due to their sensitivity to the cloud cover, such SST datasets issued from infrared sensors may involve large missing data rates (typically, between 70% and 90%, Fig.?? for an illustration). For evaluation purposes, we build a groundtruthed dataset from high-resolution numerical simulations, namely NATL60 data (1), using real cloud masks from METOP AVHRR sensor (19). We sample 128x512 images over five consecutive days from June to August at a 0.05◦resolution in an open ocean region in the North-East Atlantic from (40.58◦N,46.3◦W) to (53.04◦N, 16.18◦W). Overall, we randomly sample 400 128x512x11 image series as training data and 150 as test dataset.
For this case-study, we consider the following four architectures for the AEs and the GE-NNs:
• ConvAE1: the first convolutional auto-encoder involves the following encoder architecture: five consecutive blocks with a Conv2D layer, a ReLu layer and a 2x2 average pooling layer, the first one with 20 filters the following four ones with 40 filters, and a final linear convolutional layer with 20 filters. The output of the encoder is 4x16x20. The decoder involves a Conv2DTranspose layer with ReLu activation for an initial 16x16 upsampling stage a Conv2DTranspose layer with ReLu activation for an additional 2x2 upsampling stage, a Conv2D layer with 16 filters and a last Conv2D layer with 5 filters. All Conv2D layers use 3x3 kernels. Overall, this model involves ≈ 400,000 parameters.
• ConvAE2: we consider a more complex auto-encoder with an architecture similar to ConvAE1 where the number of filters is doubled (e.g., The output of the encoder is a 4x16x40 tensor). Overall, this model involves ≈ 900,000 parameters.
• GE-NN1,2: we consider two GE-NN architectures. They share the same global architecture with an initial 4x4 average pooling, a Conv2D layer with ReLu activation with a zeroweight constraint on the center of the convolution window, a 1x1 Conv2D layer with N filters, a ResNet with a bilinear residual unit, composed of an initial mapping to an initial 32x128x(5*N) space with a Conv2D+ReLu layer, a linear 1x1 Conv2D+ReLu layer with N filters and a final 4x4 Conv2DTranspose layer with a linear activation for an upsampling to the input shape. GE-NN1 and GE-NN2 differ in the convolutional parameters of the first Conv2D layers and in the number of residual units. GE-NN1 involves 5x5 kernels, N = 20 and 3 residual units for a total of ≈ 30,000 parameters. For GE-NN2, we consider 11x11 kernels, N = 100 and 10 residual units for a total of ≈ 570,000 parameters.
These different parameterizations were selected so that ConvAE1 and GE-NN2 involve a modeling complexity in the same range. We may point out that the considered GE-NN architecture are not applied to the finest resolution but to downscaled grids by a factor of 4. The application of GENNs to the finest resolution showed poor performance. This is regarded as an illustration of the requirement for considering a scale-selection problem when applying a given prior. The upscaling involves the combination of a Conv2DTranspose layer with 11 filters, a Conv2D layer with a ReLu activation with 22 filters and a linear Conv2D layer with 11 filters.
Similarly to MNIST dataset, we report the performance of the different models in terms of interpolation score (I-score), reconstruction score (R-score) and auto-encoding score (AE-score) both for the training and test dataset. We compare the performance of the four models using the fixed-point and gradient-based interpolation. Overall, we can draw conlusions similar to MNIST case-study. Representations trained from gap-free data lead to poor performance and the proposed scheme reaches the best performance (gain over 50% in terms of explained variance for the interpolation and reconstruction score). Here, models trained with a zero-filling strategy show good interpolation and reconstruction performance, but very poor AE score, stressing that cannot apply beyond the considered interpolation task. When comparing GE-NN and AE settings, GE-NNs show slightly better performance with a much lower complexity (e.g., 30,000 parameters for GE-NN1 vs. 400,000 parameters for ConvAE1). Regarding the comparison between the fixed-point and gradient-based interpolation strategies, the later reaches slightly better interpolation and reconstruction score. We may point out the significant gain w.r.t. OI, which is the current operational tool for ocean remote sensing data. We illustrate these results in Appendix (Fig.6), which further stresses the gain w.r.t. OI for the reconstruction of finer-scale structures.

heading: 5 CONCLUSION
text: In this paper, we have addressed the learning of energy-based representations of signals and images from observation datasets involving missing data (with possibly very large missing data rates). Using the proposed architectures, we can jointly learn relevant representations of signals and images while jointly providing the associated interpolation schemes. Our experiments stress that learning representations from gap-free data may lead to representations poorly adapted to the analysis of data with large missing data areas. We have also introduced a Gibbs priors embedded in a neural network architecture. Relying on local characteristics rather than global ones as in AE schemes, these priors involve a much lower complexity. Our experiments support their relevance for addressing inverse problems in signal and image analysis. Future work may further explore multi-scale extensions of the proposed schemes along with couplings between global and local energy representations
and hybrid minimization schemes combining both gradient-based and fixed-point strategies in the considered end-to-end formulation.

heading: A APPENDIX
text: A.1 SUPPLEMENTARY FOR MNIST DATASET
We illustrate below both the considered masking patterns as well as reconstruction examples for the proposed framework applied to MNIST dataset.
A.2 SUPPLEMENTARY FOR LORENZ-63 DYNAMICS
We report below a Table which details the interpolation performance of the proposed GE-NN representation applied to Lorenz-63 time series in comparison with a PCA-based scheme and a lowerbound provided by the interpolation assuming the ODE model is known.
A.3 SUPPLEMENTARY FOR SST DATASET
We report below reconstruction examples for the application of the proposed GE-NN approach to SST time series with real missing data masks, which involve very large missing data rates (typically above 80%). The consistency between the interpolation results and the reconstruction of the gap-free image from the learnt energy-based representation further stresses the ability of the proposed approach to extract a generic representation from irregularly-sampled data. These reulsts
also emphasize a much greater ability of the proposed learning-based scheme to reconstruct finescale structures, which can hardly be reconstructed by an OI scheme with a Gaussian space-time covariance model. We may recall that the later is the stae-of-the-art approach for the processing of satellite-derived earth observation data (8).

Review summaries:
The paper proposes a framework to learn representations of signals when they the signals are irregularly sampled.  They propose to do this by using some modified iteration steps from DINEOF algorithm. In addition to this, they propose a new energy function which is inspired by the idea of Gibbs distribution. The parametrize the energy function by some convolutional filters with a constraint. As claimed by the paper in the introduction, they only use the under sampled signals to learn as opposed to using the fully sampled ground truth signals in the previous deep learning basic approaches.