This is the accepted version of the article: García-Sobrino, Joaquín; Laparra, Valero; Serra Sagristà, Joan; [et al.]. «Im- proved Statistically Based Retrievals via Spatial-Spectral Data Compression for IASI Data». IEEE transactions on geoscience and remote sensing, Vol. 57, Issue 8 (August 2019), p. 5651 - 5668. DOI 10.1109/TGRS.2019.2901396 This version is avaible at https://ddd.uab.cat/record/215775 under the terms of the license IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. XX, NO. Y, MONTH Z 2018 1 Improved Statistically Based Retrievals via Spatial-Spectral Data Compression for IASI data Joaquín García-Sobrino, Valero Laparra, Joan Serra-Sagristà, Senior Member, IEEE, Xavier Calbet, and Gustau Camps-Valls, Senior Member, IEEE Abstract—In this paper we analyze the effect of spatial and through a lossy compression stage. Lossy compression entails spectral compression on the performance of statistically based re- to lose some information in the reconstructed data, which may trieval. Although the quality of the information is not completely compromise the quality of the products in later processing preserved during the coding process, experiments reveal that a certain amount of compression may yield a positive impact on stages. Therefore, a careful evaluation of the impact of the the accuracy of retrievals. We unveil two strategies, both with compression process applied to the data is needed to determine interesting benefits: either to apply a very high compression, the quality of the information and of the derived products. which still maintains the same retrieval performance as that Although lossy compression implies going through a dis- obtained for uncompressed data; or to apply a moderate to tortion process, the quality of the recovered data can be high compression, which improves the performance. As a second contribution of this work, we focus on the origins of these still adequate for the intended specific use. If the amount benefits. On the one hand, we show that a certain amount of of the inherent data acquisition noise is large, the signal noise is removed during the compression stage, which benefits the removed during the coding process is mostly noise for certain retrievals performance. On the other hand, we analyze the effect compression ratios [2], [3]. Therefore, the amount of useful of compression on spectral/spatial regularization (smoothing). We signal lost is usually low and the quality of the recovered quantify the amount of information shared among the spatial neighbours for the different methods and compression ratios. products may be yet appropriate, or even better suited, to feed We also propose a simple strategy to specifically exploit spectral a subsequent information extraction stage. Of course, if the and spatial relations and find that, when these relations are compression ratio is very high, a lot of useful information is taken into account beforehand, the benefits of compression are removed and the quality of the reconstructed data is seriously reduced. These experiments suggest that compression can be compromised. understood as an indirect way to regularize the data and exploit spatial neighbours information, which improves the performance The impact of lossy compression on information extraction of pixel-wise statistics based retrieval algorithms. systems has been investigated in several application areas. Ryan and Arnold discussed the lossy compression of remotely Index Terms—Infrared Atmospheric Sounding Interferometer (IASI), Lossy Compression, Spectral Transforms, JPEG 2000, sensed images using vector quantization in the context of Statistically Based Retrieval, Kernel Methods, Regression. image classification in [4]. The impact of compression was analyzed on the maximum likelihood classification of the I. I NTRODUCTION recovered data concluding that the loss in the classification accuracy was not significant (less than 8%). Sánchez and Per- In recent decades, advances in remote sensing technol- ronnin evaluated two lossy compression techniques, one based ogy have made it possible to collect information from the on product quantizers and another based on dimensionality electromagnetic spectrum with unprecedented accuracy and reduction, to assess large-scale image classification in [5]. resolution [1]. Infrared sounder instruments usually produce Results reported that compression ratios between 64:1 and large volumes of data, which is costly to manage in an 128:1 produced little loss in the classification performance. operational context, i.e., for transmission, processing, and Similar results were obtained in [6], [7], [8], [9], [10], [11], storage. An effective strategy to alleviate the problems derived [12], where lossy compression did not markedly reduce clas- from the data size is to compress the data according to sification accuracy. the specifics needs of the final users. However, in order to Besides, several studies have analyzed the effect of near- achieve high compression ratios, it is typically needed to go lossless and lossy compression techniques based on JPEG Manuscript received December 17, 2019; 2000 standard within the framework of feature extraction, JGS and JSS are with the Group on Interactive Coding of Images, classification, and anomaly detection tasks. Pal et al. analyzed Universitat Autònoma de Barcelona, Spain. Web: http://www.gici.uab.cat. E- the performance of supervised, unsupervised, and hybrid clas- mail: joaquin.garcia.sobrino@deic.uab.cat, joan.serra@uab.cat. VL and GCV are with the Image Processing Laboratory (IPL), Universi- sification processes in [13]. Results reflected that classification tat de València, Spain. Web: http://isp.uv.es. E-mail: {valero.laparra, gus- accuracy was still reliable even at low bit-rates (high compres- tau.camps}@uv.es. sion ratio). A compression scheme based on Principal Compo- XC is with AEMET, C/ Leonardo Prietro Castro, 8, 28071 Madrid, Spain. E-mail: xcalbeta@aemet.es. nent Analysis (PCA) was proposed in [14] producing compet- This work was supported in part by the Spanish Ministry of Economy itive results in terms of information preservation in anomaly and Competitiveness and by the European Regional Development Fund under detection tasks. Supervised and unsupervised classification of Grants TIN2015-71126-R and RTI2018-095287-B-I00, by the Catalan Gov- ernment under Grant 2017SGR-463, and by the European Research Council reconstructed data was evaluated in [15]. The experimental under Consolidator Grant SEDAL ERC-2014-CoG 647423. results showed competitive classification performance after the 2 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. XX, NO. Y, MONTH Z 2018 compression stage. Analogous results were reported in [16], [31], [32] we observed that data that had gone through a near- where the proposed compression algorithm for hyperspectral lossless and lossy compression process enabled achieving, images did not significantly reduce the performance of hard for certain compression ratios, improved statistically based classification, linear spectral unmixing, and anomaly detection. retrieval performance compared to the retrieval on the original An interesting observation was reported in [17], where products. It was suggested that the spectral/spatial regulariza- the impact of lossy compression of hyperspectral data was tion and the noise filtering produced by the compression stage analyzed on multiclass classification, classification of mixed benefit the performance of the statistical retrieval algorithms. pixels via spectral unmixing, binary hard classification, and However, these counterintuitive observations were scarcely anomaly detection. Experimental results revealed that for studied. In this paper, we provide appropriate responses to multiclass classification and spectral unmixing, the Signal-to- the observations reported in [31], [32], investigate the actual Noise ratio (SNR) was a reasonable indicator of classification impact of compression in the retrievals, and supply results performance; for binary hard classification, the performance and discussion to understand why lossy compression may give was little dependent on the SNR; and for anomaly detec- rise to improved retrievals. This led us to lay out the main tion, the compression algorithms that produced the best rate- contributions of this paper: distortion performance were not the best choice. Actually, the 1) While experiments in [31], [32] were carried out in an effect of lossy compression on supervised classification and ideal scenario, where samples to define the training, vali- spectral unmixing using support vector machines was quan- dation and testing subsets were acquired under the same tified in [2]. The assessment reported that lossy compression conditions, here, we assume a more realistic scenario, can produce accurate results even at high compression ratios where samples used to define the training, the validation (above 16:1). and the testing subsets come all from disjoint orbits. The estimation of atmospheric parameters from remote Therefore, conclusions are not biased by an eventual risk sensing data is an inverse problem where we retrieve the of model overfitting. physical parameters given a set of observations. Inverting 2) Three different multi-component transforms have been the physical (radiative transfer) model through look-up-tables employed in the compression process, all bringing forth or optimal estimation models (OEM) [18] are standard ap- improved performance. proaches, but they lead to much higher computational costs 3) The causes of the benefits on the retrieval performance than those required for statistical approaches [19]. In the last due to a compression stage are thoroughly investigated decade, statistical model inversion based on machine learning and discussed, and reasons of the statistically based has provided excellent performance in accuracy and efficiency retrieval improvement are provided. terms. Statistically based atmospheric parameter retrieval from IASI and AIRS data was first conducted using vanilla neural In this paper, a wide spread lossy compression technique networks trained with backpropagation [20], [21], [22]. In a and two different statistical retrieval algorithms are evaluated set of previous comparisons [19], [23], [24], [25], [26] we on Infrared Atmospheric Sounding Interferometer (IASI) L1C used other forms of nonlinear and nonparametric machine data [33]. For lossy compression, JPEG 2000 standard [34] learning regression like more advanced neural nets, kernel is paired along with three different spectral transforms: the ridge regression and Gaussian processes. Other strategies to Discrete Wavelet Transform (DWT) [35], the Pairwise Or- plain regression consider introducing a dimensionality reduc- thogonal Transform (POT) [36]), and the Multilevel Cluster- tion before the regression. Smart approaches consider spatially ing Karhunen-Loève Transform (Multilevel Clustering KLT) and noise-aware transforms like in [25], [27], or more ad- [37]. For atmospheric variables prediction, a linear and a vanced nonlinear dimensionality reduction approaches based nonlinear statistically based retrieval algorithm, i.e., the Linear on kernel machines [28]. Regression (LR) and the Kernel Ridge Regression (KRR) The functional sliced inverse regression (FSIR) in [29] is [38], respectively, are employed in the retrieval of physical a dimensionality reduction method that generalizes PCA by information (temperature and dew point temperature profiles) designing a spectral information driven by the parameters to from the reconstructed data. be retrieved. FSIR is actually related to linear discriminant To investigate the origin of the improved retrieval perfor- analysis (LDA) and partial least squares (PLS) approaches, mance produced by lossy compression, two approaches are being a kind of supervised dimensionality reduction approach, followed. On the one hand, we analyze the noise level remain- and provides noise-aware feature components (hence being ing in the recovered spectra. On the other hand, we compare related to minimum noise fraction -MNF- methods). Never- the performance of the retrieval algorithms when a compressed theless, the standard PCA approach is still the most widely (and reconstructed) data is employed, with a simple method adopted and studied for dimensionality reduction in atmo- which specifically takes into account information from the spheric parameter retrieval. In [30] the impact of spectral spatial and spectral neighbours. Experiments reveal that the compression on statistically based retrievals was analyzed, improvement in retrieval accuracy is motivated by: 1) lossy investigating the use of Principal Components Analysis (PCA) compression performing some noise filtering, which typically based methods for compressing high-resolution infrared mea- improves regression and function approximation results; and surements before performing linear regression. It was shown 2) compression being an indirect way to exploit spatial fea- that a compression ratio of 15:1 can be performed with low ture relations, which, generally, helps in pixel-wise retrieval degradation of the temperature and water vapour retrievals. In algorithms. GARCIA-SOBRINO: IMPROVED STATISTICALLY BASED RETRIEVALS VIA SPATIAL-SPECTRAL DATA COMPRESSION FOR IASI DATA 3 The remainder of the paper is organized as follows. Sec- (usually the peak absolute error) must be preserved in the tion II introduces the proposed sequential approach detailing recovered data. the compression scheme and the statistically based retrieval Here we will focus on lossy compression because it is methods used in the experiments. Section III reports the data a widely accepted strategy when large volumes of remote collection used in the experiments. Section IV reports and ana- sensing data need be compressed, as witnessed in the today- lyzes the experimental results. Section V provides an extensive in-use pipeline for IASI data [45], [46] (recall that we will use discussion. Finally, Section VI draws some conclusions. IASI data for the experiments). 1) Lossy Compression: Lossy compression approach allows II. M ETHODOLOGY to achieve high compression ratios at the expense of recovering This section describes the methods and techniques used in a reconstructed data not identical to the original ones. It is ex- the experiments. For the compression stage, the JPEG 2000 pected that the reconstructed data preserve enough information standard is paired along with three spectral transforms to to be used for the intended specific purpose, while alleviating exploit the high spectral redundancy present in IASI L1C data. problems derived from the transmission, handling, and storage For atmospheric prediction, two different statistically based of large volumes of information. retrieval algorithms, which have provided competitive perfor- The international standard JPEG 2000 [34] is used to carry mance in atmospheric parameter retrieval [39], are employed: a out the lossy compression stage. JPEG 2000 was published by standard least squares LR and a kernel ridge regression (KRR) the Joint Photographic Experts Group (JPEG) in year 2000 as method. the successor of classical JPEG coding standard, and provides Figure 1 illustrates the proposed sequential scheme. In a a wide diversity of features and functionalities in a single first stage, the data is lossy compressed. At the receiver compressed code-stream: among others, lossless and lossy side, data is decompressed to produce the reconstructed data. compression, progressive lossy-to-lossless coding, robustness Finally, the statistical parameter retrieval is carried out on the to the presence of errors, region-of-interest coding, and pro- reconstructed data. gressive transmission. JPEG 2000 is employed in a wide range of applications, e.g., remote sensing, medical imagery, mobile Encoded applications, digital library, digital photography, etc. Input Lossy Lossy compression compression data (spectral (spectral transform transform ++ Decompression Decompression To achieve competitive compression performance in data data JPEG JPEG 2000) 2000) formed by thousands of spectral components, like IASI L1C Reconstructed data products, it is of paramount importance to exploit the re- dundancy present in the spectral dimension. Three spectral transforms are paired along with JPEG 2000: the Discrete Statistical Statistical Prediction - Temperature Wavelet Transform (DWT), the Pairwise Orthogonal Trans- parameter parameter retrieval retrieval (LR - Dew point temperature form (POT), and the Multilevel Clustering Karhunen-Loève (LR and and KRR) KRR) Transform (Multilevel Clustering KLT). Fig. 1. Proposed sequential chain. DWT decomposes the processed signal into different sub- bands, which allows to decorrelate the data to be encoded. When the transform is applied, the signal is decomposed into two set of coefficients: the low and high frequency A. Data Compression subbands. The low frequency subbands (L) have a coarse Compression of remote sensing products is an efficient frequency resolution, while the high frequency subbands (H) strategy to reduce the large size of the collected data. In recent represent fine details of the data. The high frequency subbands years, various compression strategies have been adopted in contain coefficients that can be discarded or quantized to several standards and have been implemented both on board achieve efficient lossy compression. Often, competitive data the satellites and on-the-ground processing stations [40], [41], quality can be achieved from a small amount of transform [42], [43], [44]. coefficients, which makes DWT a suitable transform for lossy Three compression approaches can be considered, namely, compression [47]. DWT can be applied on the transform lossless, lossy, and near-lossless. Lossless compression is the coefficients (low frequency subbands) in successive levels mandatory approach when the original signal must be fully producing further decorrelation, although the optimal number preserved, however, the achievable compression ratios are of transform levels shall depend on the particularities of the rather low (below 5:1). Lossy compression can be a desirable data. DWT can be used too to transform two dimensional approach in scenarios where high compression ratios are signals by applying a one-dimensional transform in the vertical required and losing some information in the signal might be direction and a one-dimensional transform in the horizontal admissible. Near-lossless compression is also a lossy compres- direction, which usually improves the coding results. sion approach, where some information is removed from the Both POT and Multilevel Clustering KLT are affordable original data, but the compression ratios achieved are closer approximations of the computationally demanding Karhunen- to lossless compression than to typical lossy compression. Loève Transform (KLT) [48], which is optimal decorrelating Near-lossless compression is an adequate approach when Gaussian sources. A critical consideration when applying a higher compression ratios compared to lossless compression spectral transform on data volumes with large spectral dimen- are demanded and, at the same time, a specific fidelity criterion sion is the computational complexity [49]. Some spectral trans- 4 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. XX, NO. Y, MONTH Z 2018 forms like the KLT may prove unusable for large dimensions least squares solution, which depends on the inversion of such as in IASI data. POT and Multilevel Clustering KLT (X> X). In order to ensure inversion we use the classical rely on the implementation of a divide-and-conquer strategy Tikhonov regularized solution: to the KLT. While a classic KLT decorrelates all components with each other regardless of how much energy they share, WL = (X> X + λI)−1 X> Y. (2) a divide-and-conquer strategy implements a mechanism to decorrelate several spectral components with high shared en- Kernel methods are based on defining a mapping function ergy and ignore the other components. The resulting transform φ(x) for the input samples to a Hilbert space, H, of very high is the composition of smaller KLT transforms, where each (possibly infinite) dimensionality DH . KRR can be defined composition is processed as a classic KLT. If a multilevel mode following the same procedure as in the LR case as a linear least is used, most of the data energy flows across the composition squares regression in the Hilbert space. If we map the samples of transforms up to the last level, because most of the signal using the mapping function we have Φ = φ(X) ∈ RN ×DH , energy is grouped in the first few decorrelated components, and Φ∗ = φ(X∗ ) ∈ RM ×DH . Therefore the prediction model which are further decorrelated in upper levels. POT uses a is given by: two-component KLT transform for every pair of consecutive components and works in a multilevel mode. In the case of Ŷ∗ = Φ∗ W. (3) Multilevel Clustering KLT, each composition of KLT can be formed by a number of spectral components between two and As in the LR case, we can apply the Tikhonov regularized all the spectral components. Figure 2 illustrates the structure solution to find the weights: of a classical KLT, of a POT and of a Multilevel Clustering KLT. W = (Φ> Φ + λI)−1 Φ> Y. (4) Note that this problem is not solvable as the inverse runs B. Statistically Based Parameter Retrieval on matrix ΦΦ> , which is of size DH × DH , and Φ is in In the last decades, statistically based retrieval has proved principle unknown. However, if we use a λ that ensures that to be very useful to solve inverse problems using remote the matrix can be inverted, the solution is equivalent to: sensing data. Different algorithms have been considered, but mainly restricted to neural networks [20], [21], [22], [25] W = Φ> (ΦΦ> + λI)−1 Y. (5) and kernel machines and Gaussian processes [19], [23], [24], [26]. In particular, kernel methods are well suited due to If we summarize the right part of the equation by α = its ability to deal with high dimensional data [50]. Kernel (ΦΦ> + λI)−1 Y, the predictions for the text input samples methods generalize linear algorithms while still relying on is: linear algebra. This is why we focus on two approaches for (first guess) parameter estimation: least squares (regularized) Y∗ = Φ∗ Φ> α. (6) linear regression and its generalization using kernels. We will use classical linear regression (LR) and kernel methods in this Note that even though the mapping Φ is unknown, one work to implement the step of retrieving physical parameters can replace this inner product matrix with a similarity matrix from the data. We will use the Kernel Ridge Regression (KRR) between samples, which is known as the kernel matrix K. In algorithm [38], [51], which has shown very good performance this case we can replace the Gram matrix as K = ΦΦ> , and in prediction when used on IASI L1 data on different problems equivalently K∗ = Φ∗ Φ> . Therefore the predictions can be [39], [52], [53]. KRR was the method employed in [31], [32] computed by: to analyze the effect of compression on physical parameters retrieval at different compression ratios. The KRR has the Ŷ∗ = K∗ α. (7) advantage of generalizing least squares linear regression to the nonlinear regression case. In KRR we only need a kernel function, k(xi , xj ), accord- 1) Statistically Based Retrieval Methods: Here we define ing to the Mercer’s theorem [54]. Kernel methods literature is the notation and review the formulation of KRR from the LR full of examples of proper kernel matrices. Here we will use solution. We will denote xi ∈ RDx as the inputs and yi ∈ RDy the most standard one, the Gaussian Function (Radial Basis as the desired outputs. In matrix notation, we will denote the Function, RBF) kernel k(xi ,xj ) = exp(-kxi − xj k2 /(2σ 2 )), training input samples as X = [x1 . . . xN ] ∈ RN ×Dx , the which has only one free parameter, σ. Therefore, in LR one training desired outputs as Y = [y1 . . . yN ] ∈ RN ×Dy , the parameter is tuned: the regularization parameter λ. In KRR, test input samples as X∗ = [x∗1 . . . x∗N ] ∈ RM ×Dx , and the two free parameters are tuned: the regularization parameter λ test desired outputs Y∗ = [y1∗ . . . yN∗ ] ∈ RM ×Dy . Using the and the kernel parameter σ. matrix notation, the application of the linear model is: In both cases we used a cross-validation strategy for their Ŷ∗ = X∗ WL , (1) optimization using one half of the data for training and the other half for validating. A MATLAB implementation of KRR where we have discarded the bias term and WL ∈ RDx ×Dy and other regression algorithms can be found at http://isp.uv. are the regression weights. We fit the weights using classical es/soft_regression.html. GARCIA-SOBRINO: IMPROVED STATISTICALLY BASED RETRIEVALS VIA SPATIAL-SPECTRAL DATA COMPRESSION FOR IASI DATA 5 C3,1 C2,1 C2,2 C2,1 C1,1 C1,2 C1,3 C1,4 C1,1 C1,2 a) Classical KLT b) POT c) Multilevel Clustering KLT Fig. 2. Structure of a classical KLT, POT, and Multilevel Clustering KLT. This example decorrelates eight spectral components. Each arrow denotes a component and each coloured rectangle represents the computation of a KLT. In the case of POT, three levels of transform are applied, with a different number of clusters per level, and all clusters transforming two components. In the case of Multilevel Clustering KLT, two levels of multilevel clustering are applied and two clusters are defined in the first level. III. DATA AND E XPERIMENTAL S ETTING testing. The training set is composed by seven orbits and This section introduces the data collection used in the the atmospheric parameter retrieval test is carried out on six experiments and the parameter configuration employed in the different orbits. All data have been produced by the IASI- compression and in the statistically based regression stages. B instrument, implemented on the MetOp-B satellite. Table I First, a description of the main characteristics of the IASI reports the characteristics of the 13 IASI orbits used in the instrument is given, along with the specific products used in experiments. In order to isolate the results for cloud free and the experiments. cloudy areas we use the fractional cloud cover data provided in the L2 level [62]. A. Data Collection Usually, the Band 3 of the IASI spectrum and some channels IASI is the main payload instrument carried on the MetOp from the Band 1 and from the Band 2 are not used for satellite series [55]. Data provided by the instrument represent temperature and dew point temperature retrieval because they a significant improvement in the retrieval accuracy and vertical are influenced by solar radiation and trace gases such as CO, resolution of atmospheric parameters (temperature and water CH4 , etc. Following the feature selection in [39] we performed vapour concentration mostly) with respect to previous lower feature selection removing the most noisy bands and keeping spectral resolution instruments, such as HIRS, SEVIRI, etc. 4699. This has in turn improved the output from numerical weather prediction models and atmospheric chemistry studies. B. Setting and Parameter Configuration for Lossy Compres- IASI produces rich spectral information to derive temper- sion ature and moisture profiles. The spectra is also used for the determination of trace gases such as CO, CO2 , CH4 , SO2 , Lossy compression is carried out through JPEG 2000 stan- N2 O, HNO3 , NH3 , OCS, and CF4 [56], [57], [58], [59], [60], dard. It is meaningful to consider that, in hyperspectral data [61], land and sea surface temperatures, and cloud properties. with such a large number of spectral components as IASI The instrument collects data with high spectral, spatial, and products, it is of utmost importance to exploit the high spectral temporal resolution producing large volumes of information redundancy inherent to the data. As mentioned, to achieve (about 16 Gigabytes per day generated by each of the IASI-A improved coding performance, JPEG 2000 is paired along with and IASI-B instruments). IASI covers the spectral range be- three spectral transforms: for DWT, 10 levels of 9/7 DWT tween 645 and 2760 cm−1 yielding 8461 spectral components. are applied in the spectral dimension; POT is run using the Such an amount of information is costly to manage, hence the default parameter settings; and for Multilevel Clustering KLT, need to search for efficient strategies to reduce the large size 100 clusters are defined in the first level. To account for the of the data for improved processing, transmission, and storage. spatial redundancy, a 9/7 DWT with five levels is applied An effective strategy to alleviate the large volume of in- in the spatial dimension. Nine target bit-rates are analyzed, formation produced by the IASI instrument is to compress distributed from 2 to 0.0025 bit per pixel per component the data according to the specific requirements of the end- (bpppc), corresponding to compression ratios from 8:1 to user applications. As outputs of the models, we use the 6,400:1. Table II summarizes the main characteristics of the physical variables (temperature and dew point temperature) three compression settings studied in the experiments. given by the analyses of the model of the European Centre In the experiments, Kakadu software [63] has been used for Medium-Range Weather Forecasts (ECMWF). The model for JPEG 2000, Pairwise Orthogonal Transform software [64] provides estimations for 137 different pressure levels between has been run for POT, and Spectral Transform software [65] [10−2 · · · 103 ] hPa in the atmosphere and spatial resolution of has been employed for Multilevel Clustering KLT. The JPEG 0.5 degrees. We co-registered the predictions supplied with the 2000 standard provides a multicomponent extension in its Part scenes acquired by IASI instrument with the analyses of the 2 [66], which has been used to apply the DWT. ECMWF. It is worth mentioning that, in our lossy compression The experiments have been conducted in a realistic context, approach, we apply a decoding stage prior to the statistically where data from disjoint orbits are used for training and based retrieval, so that retrieval works with the reconstructed 6 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. XX, NO. Y, MONTH Z 2018 TABLE I IASI L1C PRODUCTS USED IN THE EXPERIMENTS . I DENTIFIERS AND SIZES ARE PROVIDED . M IS THE NUMBER OF SPECTRAL COMPONENTS , N S IS THE NUMBER OF SCAN LINES , N-FOR S IS THE NUMBER OF ELEMENTARY FIELDS OF REGARD (FOR) PER LINE , AND N-IFOV S IS THE NUMBER OF INSTANTANEOUS FIELDS OF VIEW (IFOV) PER FOR. Size Orbit identifiers (M × Ns × N-FORs × N-IFOVs) Training set IASI_xxx_1C_M01_20131017012656Z_20131017030856Z_N_O_20131017021701Z (8461 × 765 × 30 × 4) IASI_xxx_1C_M01_20131017045352Z_20131017063552Z_N_O_20131017054413Z (8461 × 765 × 30 × 4) IASI_xxx_1C_M01_20131017081752Z_20131017095656Z_N_O_20131017090901Z (8461 × 743 × 30 × 4) IASI_xxx_1C_M01_20131017113856Z_20131017131752Z_N_O_20131017123024Z (8461 × 742 × 30 × 4) IASI_xxx_1C_M01_20131017145656Z_20131017163856Z_N_O_20131017155030Z (8461 × 765 × 30 × 4) IASI_xxx_1C_M01_20131017181752Z_20131017195952Z_N_O_20131017190946Z (8461 × 765 × 30 × 4) IASI_xxx_1C_M01_20131017214152Z_20131017232352Z_N_O_20131017223236Z (8461 × 765 × 30 × 4) Test set IASI_xxx_1C_M01_20131017030856Z_20131017045352Z_N_O_20131017035958Z (8461 × 787 × 30 × 4) IASI_xxx_1C_M01_20131017063552Z_20131017081752Z_N_O_20131017072637Z (8461 × 765 × 30 × 4) IASI_xxx_1C_M01_20131017095656Z_20131017113856Z_N_O_20131017104905Z (8461 × 765 × 30 × 4) IASI_xxx_1C_M01_20131017131752Z_20131017145656Z_N_O_20131017141023Z (8461 × 743 × 30 × 4) IASI_xxx_1C_M01_20131017163856Z_20131017181752Z_N_O_20131017173059Z (8461 × 742 × 30 × 4) IASI_xxx_1C_M01_20131017195952Z_20131017214152Z_N_O_20131017205049Z (8461 × 765 × 30 × 4) TABLE II C ONFIGURATION OF THE COMPRESSION SETTING USED IN THE EXPERIMENTS . Compression Target bit-rates Spectral transform Spatial transform Compression ratios technique (bpppc) DWT 10 levels Default parameter POT 2, 0.5, 0.25, 0.1, 8, 32, 64, 160, 320, settings 5 levels of DWT JPEG 2000 0.05, 0.03, 0.01, 533, 1,600, 3,200, 100 clusters in the 0.005, and 0.0025 and 6,400 Multilevel first level and Clustering KLT multilevel mode data and not in the compressed domain. These reconstructed We trained one model for each coding setting and each data have exactly the same spectral and spatial dimensions compression ratio. Note that the input data to the retrieval as the original data, i.e., no spectral or spatial dimensionality process is the reconstructed spectra, i.e., compressed and reduction occur. It is the quality of the observations that is decompressed data. The models were trained using a simple being modified, not the size of the scenes. cross-validation scheme with half of the data used for training and the other half used for validating the parameters. C. Settings of Statistically Based Regression IV. E XPERIMENTAL R ESULTS The retrieval experiments have been carried out in a realistic This section presents the experimental results. We report scenario, i.e., data from disjoint IASI orbits have been used results for atmospheric parameter retrieval over the recon- for training and testing. The statistical retrieval algorithms are structed spectra and investigate the origin of the improvement applied on IASI data hyperpixels at different pressure levels in the retrievals produced using the reconstructed spectra. considering all the spectral components at a particular spatial Results are evaluated in terms of noise removal and shared position in the regression. spatio/spectral information to assess the impact of compression In the training stage 8,000 samples are randomly selected on the retrievals. This study is conducted through several from each of the training orbits (seven products), which experiments: produces a training set of 56,000 samples. To allow a fair 1) We evaluate the performance of the retrievals produced comparison, the positions of the samples used for training using reconstructed spectra. Different compression set- are kept constant for all the experiments, i.e., all three multi- tings and retrieval algorithms are used. The experiment component transforms and all compression ratios. Six different is carried out in a realistic scenario where data used for orbits are used for testing. The prediction is carried out on the testing comes from orbits different from those used for whole orbit. To assess the retrieval performance, the RMSE training. between the ground-truth values of temperature/dew point 2) We analyze the amount of noise left in the data after temperature and the predictions produced by the regression being processed with each compression setting and at models is computed in each pressure level. different compression ratios. GARCIA-SOBRINO: IMPROVED STATISTICALLY BASED RETRIEVALS VIA SPATIAL-SPECTRAL DATA COMPRESSION FOR IASI DATA 7 3) We analyze the spectral and spatial information in the the compression ratio is very high, the retrieval performance compression settings. First we quantify the amount of in- begins to deteriorate because too much distortion is introduced formation shared among the spatial neighbours. We then in the reconstructed data. This behaviour occurs for both LR propose an scheme which specifically enforces different and KRR, although the retrieval improvements are larger for levels of spatial and spectral relations in a simple way, LR (in the sense that higher compression ratios can be applied and compare the performance of the retrieval algorithms before the retrieval performance starts to deteriorate). These when using data processed following this scheme and results are consistent with the observations reported in [32]. when using reconstructed data from the compression However, note that here the situation is more realistic than algorithms. in [32], since we use different orbits for training, validation, Results are reported for one IASI L1C orbit and one and testing. By doing so we ensure that no information physical variable (dew point temperature) in most of the about the true values due to the blurring introduced by the experiments due to restrictions in page length. Conclusions compression settings can be used in the predictions. Results from the results for the remainder orbits and for temperature are also consistent for the new compression settings evaluated. are similar. We report all these results in the supplementary An interesting analysis was introduced in [27] about the material at http://isp.uv.es/spatio_spectral_compression.html. correlation between the dimensions of the predictions, where an index (iD ) of non-diagonality of the covariance matrix of A. Retrieval Assessment the predictions for different pressure levels, ΣŶ , was proposed: This section reports the results of estimating atmo- Bŷd spheric physical variables through LR and KRR using iD = (8) reconstructed IASI L1C spectra. Results are presented Sŷ for the orbit IASI_xxx_1C_M01_20131017030856Z_2013- where Sŷ = ||Σŷ ||, Σŷ is the covariance matrix of the 1017045352Z_N_O_20131017035958Z (see table I) and dew predictions, and Bŷd is the biggest value in the diagonal of point temperature prediction. Results for all the other orbits ΣŶ (see [27] for details). Figure 4 shows the value of the and variables are very similar and reported as supplementary iD index for the different compression configurations and the material. different regression algorithms taken into consideration in this A first assessment of the reconstructed radiances is reported paper. We provide the index without the normalization by in the Appendix at http://gici.uab.cat/pub/spatio_spectral_ the number of output dimensions. We can see that given a compression/appendix.html, where the normalized radiance particular regression method, the differences in the correlation residuals statistics and the spectral signature of the recon- between the outputs for different compression ratios (and structed radiances are analyzed. Next, Figure 3 shows the also for different compression settings) are negligible for retrieval performance of dew point temperature prediction the compression ranges with low prediction errors, i.e., until for the lossy compression settings proposed. Plots show the compression ratios about 300:1. average of the RMSE prediction over the different pressure Several conclusions can be drawn: levels between 1100 and 100 hPa. We analyze four different 1) Statistical atmospheric parameter retrieval benefits from scenarios, i.e., land and cloud free, land and cloudy, ocean spectral/spatial lossy compression. It is clear from Fig. 3 and cloud free, and ocean and cloudy conditions. Cloud free that using reconstructed data after a compression process conditions is equivalent to cloud fraction equals to 0%. Other- allows to achieve improved statistically based retrieval wise, it is considered cloudy conditions. Ocean is equivalent performance. While moderate and medium compression to land fraction equals to 0%. Otherwise, it is considered ratios enable to achieve at least the same performance land. Table III reports the number of spectra analyzed in than the original spectra, when the distortion level intro- each scenario. The compression settings POT + JPEG 2000 duced in the data is high, the accuracy in the retrievals and Multilevel Clustering KLT + JPEG 2000 achieve lower decreases. In all the scenarios analyzed, a compression maximum compression ratios compared to DWT + JPEG 2000 ratio of, approximately, 100:1 allows to achieve the same because side-information needs to be transmitted in addition or improved retrieval results compared to retrieval results to the compressed data. obtained from uncompressed spectra (original data). TABLE III 2) KRR produces more accurate predictions than LR. For N UMBER OF SPECTRA IN EACH OF THE SCENARIOS ANALYZED . T HE all plots, KRR consistently yields better retrieval perfor- PERCENTAGE OF THE ORBIT IN EACH SCENARIO IS REPORTED IN BRACKETS . mance than LR. These results are constant for all the data (original and reconstructed spectra) and scenarios Land - Cloud Land - Ocean - Ocean - analyzed. The difference is specially significant for free Cloudy Cloud free Cloudy cloudy conditions. 10,218 28,613 6,658 48,951 3) Lossy compression enables more significant retrieval (10.8%) (30.3%) (7.1%) (51.8%) improvements for LR compared to KRR. The retrieval improvement is more evident for LR than for KRR when One can see that while low compression ratios do not reduce a compression process has taken place, which suggests the accuracy in the retrievals, the prediction results are im- that the spectral/spatial regularization produced by the proved as the compression ratio increases. Nonetheless, when compression stage particularly benefits LR. 8 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. XX, NO. Y, MONTH Z 2018 Land - Cloud free Land - Cloudy Ocean - Cloud free Ocean - Cloudy 4.8 4.6 4.4 Mean RMSE 4.2 JPEG 2000 4 3.8 DWT + 3.6 3.4 3.2 3 4.8 4.6 4.4 Mean RMSE 4.2 4 JPEG 2000 3.8 POT + 3.6 3.4 3.2 3 4.8 4.6 4.4 Mean RMSE 4.2 4 MLC KLT + JPEG 2000 3.8 3.6 3.4 3.2 3 101 102 103 101 102 103 101 102 103 101 102 103 Compression ratio Compression ratio Compression ratio Compression ratio Fig. 3. Dew point temperature (in kelvin) retrieval performance for different lossy compression settings using LR (solid red lines) and KRR (solid blue lines). In all the plots, the vertical axis represents the averaged RMSE over the different pressure levels between 1100 and 100 hPa and the horizontal axis represents the compression ratio. Ranges are the same in all the plots to ease the comparison. Each row shows the results for a particular compression setting and each column shows the results for a particular scenario. Results using uncompressed spectra (original data) for LR (dashed red lines) and KRR (dashed blue lines) are plotted for comparison purposes. DWT + JPEG 2000 POT + JPEG 2000 MLC KLT + JPEG 2000 0.12 0.12 0.12 0.1 0.1 0.1 0.08 0.08 0.08 iD iD iD 0.06 0.06 0.06 0.04 0.04 0.04 101 102 103 101 102 103 101 102 103 Compression ratio Compression ratio Compression ratio Fig. 4. Index iD for different lossy compression settings using LR (solid red lines) and KRR (solid blue lines). In all the plots, the vertical axis represents the iD index and the horizontal axis represents the compression ratio. Ranges are the same in all the plots to ease the comparison. Results using uncompressed spectra (original data) for LR (dashed red lines) and KRR (dashed blue lines) are plotted for comparison purposes. 4) LR achieves competitive performance when the spectra level Clustering KLT + JPEG 2000 leads to the best have been compressed at higher compression ratios retrieval improvements for land and cloudy, ocean and compared to KRR. Fig. 3 illustrates that while KRR cloud free, and ocean and cloudy conditions. For land is able to achieve improved retrieval results when data and cloud free conditions, both DWT + JPEG 2000 compressed at a compression ratio of, approximately, and POT + JPEG 2000 slightly overcome Multilevel 124:1 are used, LR still produces competitive perfor- Clustering KLT + JPEG 2000. If the interest is to mance when spectra compressed at a compression ratio optimize the retrieval performance in all the scenarios, of, approximately, 203:1 is employed. Multilevel Clustering KLT + JPEG 2000 is the best 5) Multilevel Clustering KLT + JPEG 2000 leads to the compression setting. best prediction results. The compression setting Multi- 6) POT + JPEG 2000 allows to achieve higher com- GARCIA-SOBRINO: IMPROVED STATISTICALLY BASED RETRIEVALS VIA SPATIAL-SPECTRAL DATA COMPRESSION FOR IASI DATA 9 pression ratios leading at least to the same retrieval a compact representation of the data by eliminating redundant performance than the original spectra. If the interest information, preserving as much as possible the characteristics is to achieve very high compression ratios, POT + of the observation needed by the intended data user. JPEG 2000 would be the best compression setting. A Figure 7 reports an estimate of the noise level present in the compression ratio of 421:1 allows to achieve at least reconstructed spectra for the compression settings analyzed. the same prediction results compared to the original In the experiments it is assumed that the noise model in the data for all the scenarios analyzed. Higher compression IASI data comes from an additive white Gaussian distribution, ratios decrease the retrieval performance because useful and also independent and identically distributed. This is a information is missing in the recovered spectra. realistic assumption for an interferometer and is the model 7) The effect of lossy compression is alike for all the adopted in [46], [67], [68], [69]. To compute the noise standard compression settings and scenarios analyzed. One can deviation, the Anisotropic Nonparametric Image Restoration see in Fig. 3 that, depending on the compression set- toolbox [70] was used. This method provides a robust estimate ting applied and the scenario analyzed, the results can of the standard deviation based on the well-known median of slightly vary, but the impact of lossy compression on the absolute deviation (MAD) [71]. This technique is used retrieval performance is similar for all cases. along with an orthogonal wavelet transform in such a way 8) The effect of the compression on the predictions corre- that the median estimator is applied on the high frequency lation is negligible for useful compression ratios. If we subbands (fine details) of the transform domain to reduce the take in consideration the range of compression ratios impact of the features present in the signal. In the experiments, for low prediction error (i.e. [0-300], see Fig. 3), the a Daubechies wavelet transform was used. Specific and more difference in correlation between the predicted outputs detailed description of the adopted strategy can be found when using and when not using compression is very in [72], [73]. low (Fig. 4). The same effect can be observed if we We can see how part of the improvement in the retrievals compare this difference with the difference inferred by is due to the filtering of noise performed by the compression using different prediction methods. process. It is interesting to compare Fig. 7 with Fig. 3. It is Figure 5 shows the bias and the RMSE results for the range clear from Fig. 7 that, as the compression ratio increases, the of pressure levels corresponding to the troposphere (between noise level in the reconstructed data decreases. While most 1100 and 100 hPa). For each compression setting and each of the noise is removed at moderate compression ratios, the scenario, results for the compression ratio that produces the noise level keeps unchanged for extremely high compression best prediction performance are reported. ratios because almost all the noise has already been removed Reconstructed spectra produce better (or at least the same) from the data. When data is compressed in moderate to high results compared to the original data for all pressure levels and compression ratios (i.e., [10:1-300:1]), most of the content all compression settings. Generally, improvements are more removed by the compression process is noise, keeping the evident in the mid-low troposphere. As illustrated in Fig. 3, retrieval performance competitive or even getting better results the improved prediction is more significant for LR. Results than using uncompressed data. However, if the compression ra- are usually better over ocean areas than over land, and better tio is extremely high (i.e., [300:1-2,000:1]), useful information on cloud free situations rather than in cloudy scenarios. is also eliminated, degrading the accuracy of the retrievals. We The Mean Absolute Error (MAE) between the dew point can suppose that noise filtering is one of the reasons why re- temperature values provided by the ECMWF analyses (and constructed compressed spectra from high compression ratios taken here as the ground truth), and the predicted values pro- can result in better performance compared to uncompressed duced by LR and KRR when uncompressed and reconstructed data. It is also clear that increasing the compression ratio spectra are employed in the retrievals is illustrated in Fig. 6. beyond a certain point has no effect on the elimination of Maps plot the MAE for all the analyzed orbits. Results are noise and therefore the only content removed from now on reported only for the original data and Multilevel Clustering is relevant information. Therefore, the retrievals performance KLT + JPEG 2000, which is the compression technique that decreases beyond this point. Note that the compression settings produces the most competitive performance, due to restrictions POT + JPEG 2000 and Multilevel Clustering KLT + JPEG in page length. As expected, KRR produces smaller errors than 2000 achieve lower maximum compression ratios compared LR (more bluish areas), and maps show that the MAE is very to DWT + JPEG 2000 because side-information needs to be similar when the original data and the reconstructed spectra are transmitted in addition to the compressed data. used. Errors occur in the same geographic areas and the error C. Exploiting Spectral/Spatial Relations through Compression pattern is similar for both the original and the reconstructed data. It is acknowledged that parameter retrieval and model inversion applications largely improve when exploiting spatial information [74], [75], [76]. Compression is an indirect way B. Noise Reduction (denoising) through Compression to stress relevant feature relations and to enforce smoothness This section is specifically dedicated to analyze the effect in the reconstructed data, which can be exploited by pixelwise of the compression process on the noise present in the data. retrieval methods. Lossy compression is known to be an indirect way of per- In this section we analyze the impact that the analyzed forming signal filtration and denoising. Compression aims for compression approaches have on sharing spectral and spatial 10 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. XX, NO. Y, MONTH Z 2018 Land - Cloud free Land - Cloudy Ocean - Cloud free Ocean - Cloudy 102 LR (CR = 1623:1) LR (CR = 64:1) LR (CR = 64:1) LR (CR = 1623:1) KRR (CR = 161:1) KRR (CR = 32:1) KRR (CR = 64:1) KRR (CR = 32:1) p[hPa] JPEG 2000 DWT + 103 102 LR (CR = 421:1) LR (CR = 62:1) LR (CR = 278:1) LR (CR = 278:1) KRR (CR = 148:1) KRR (CR = 148:1) KRR (CR = 62:1) KRR (CR = 62:1) p[hPa] JPEG 2000 POT + 103 102 LR (CR = 203:1) LR (CR = 124:1) LR (CR = 203:1) LR (CR = 124:1) KRR (CR = 124:1) KRR (CR = 124:1) KRR (CR = 124:1) KRR (CR = 124:1) p[hPa] MLC KLT + JPEG 2000 103 -2 0 2 4 6 8 -2 0 2 4 6 8 -2 0 2 4 6 8 -2 0 2 4 6 8 RMSE (K) RMSE (K) RMSE (K) RMSE (K) Fig. 5. Dew point temperature (in kelvin) RMSE profiles and bias. The compression ratio (CR) with the best average RMSE is reported for LR (thick, solid and red lines) and KRR (thick, solid and green lines). Results using the original data are shown as well for comparison purposes when LR (thick, dashed and red lines) and KRR (thick, dashed and green lines) are used for the predictions. The bias of the reconstructed data are plotted with thin and dash-dot lines for LR (red) and KRR (green). information about the neighboring pixels and the effect it we used 94, 440 samples for each component and compute has in the retrieval performance. We first asses how much the average among all the components. Particular values are information is shared between spatial neighbours. Then we different for the three compression settings analyzed, however, propose a simple method which specifically takes into account the trend is the same for all of them. For low to high information from the spatial and spectral neighbours, and we compression ratios (i.e. [30:1-500:1]) each coefficient contains compare these results with the results obtained when we do the more information about its neighbours than the original scenes. retrieval over the recovered coefficients after compression. We When the compression ratio is extremely high (i.e., higher than also investigate how the benefits of compression (sec. IV-A) 500:1), the shared information decreases dramatically. This are affected when we have already exploited specifically the behavior is also easy to see in the detailed squares below the spatial/spectral information in the data. graphs. In these squares we show the spatial pattern of the mutual information. It is easy to see that closer coefficients are 1) Assessing the amount of Spectral/Spatial Information more related (as expected), and that only the global intensity shared through Compression: Here we use information theory of the relations changes (i.e., the pattern of the relations is [77] measures in order to quantify the amount of information similar for different compression ratios). that spatial neighbours share when use different compression approaches and different compression ratios. In particular we 2) A Simple Way to Exploit Spectral/Spatial Relations: compute the mutual information between one pixel and its In the following we investigate the impact that specifically neighbours. The same procedure has been used to measure exploiting spectral and spatial information has in the retrievals. the amount of information shared between spatial, spectral and Elaborated models that take into account the dependence of orientation neighbours in the wavelet domain [78] and in the the the spatial smoothness with the spectral channel [80], normalized domain [79]. or avoid the neighbour pixels corrupted by clouds could be In Fig. 8 we show the results when the spatial neighbours used. However, for simplicity we present results for the less are taken into account. We computed the mutual information complex way of computing the average between neighboring between each pixel and its neighbours in a neighbouring of pixels. We choose here to simply convolve the data with two 11 × 11. For the experiment we used the data contained in the Gaussian filters, one over the spectral domain and one with whole orbit, i.e., for the computation of the mutual information spatial neighboring pixels. We define the σse and the σsa GARCIA-SOBRINO: IMPROVED STATISTICALLY BASED RETRIEVALS VIA SPATIAL-SPECTRAL DATA COMPRESSION FOR IASI DATA 11 LR, Original Data, Uncompressed LR, MLC KLT + JPEG 2000, CR = 160:1 KRR, Original Data, KRR, MLC KLT + JPEG 2000, Uncompressed CR = 160:1 0 5 10 15 Fig. 6. MAE for dew point temperature over the whole profile. Plots report MAE for uncompressed and reconstructed spectra and LR and KRR algorithms. Larger errors are represented by red color and smaller errors by blue color. CR reports the compression ratio achieved in the compression stage. DWT + JPEG 2000 POT + JPEG 2000 MLC KLT + JPEG 2000 Noise standard deviation Noise standard deviation Noise standard deviation 10 10 10 8 8 8 6 6 6 4 4 4 2 2 2 0 0 0 101 102 103 101 102 103 101 102 103 Compression ratio Compression ratio Compression ratio Fig. 7. For each compression setting, the Noise Standard Deviation is plotted (solid blue lines). In all the plots, the vertical axis represents the noise level (noise standard deviation of IASI raw data) in the reconstructed spectra and the horizontal axis represents the compression ratio. Noise estimation for uncompressed data (dashed green lines) is reported as well for comparison purposes. parameters for the standard deviation of the Gaussian filter in Apodization introduces spatial correlation, σap . So the total the spectral and the spatial dimension, respectively. σse and spatial correlation would be σtot = σap + σsa . Or equivalently σsa are defined in the image domain. In particular σse refers we cannot start our experiment from zero correlation since to distance in the spectral dimension, σse = 1 is equal to one the apodization process has already introduced some spatial channel distance, which corresponds to ∼ 0.5cm−1 . σsa is correlation. defined regarding the distances between pixels, for instance Table IV reports the retrieval performance for dew point σsa = 1 is equal to the distance between one pixel and the temperature for different combinations of the parameters val- neighbouring one, which corresponds to ∼ 0.5◦ . Combinations ues. While exploiting both spectral and spatial features benefits of five different σse and seven different σsa are evaluated. The the retrieval performance for LR, exploiting spectral features minimum values, σse = 0.01 and σsa = 0.01, correspond to does not improve the results for KRR. In the case of LR, the a identity function, i.e., no spectral/spatial features are taken performance is slightly improved when σse is large and σsa into account. is small. However, for KRR the retrieval results are generally Note that the IASI data we are using is already apodised. degraded when σse is increased. 12 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. XX, NO. Y, MONTH Z 2018 DWT + JPEG 2000 POT + JPEG 2000 MLC KLT + JPEG 2000 2.5 2.5 2.4 2.3 2 2 2.2 Bits/dim Bits/dim Bits/dim 1.5 2.1 1.5 1 2 0.5 1 2 3 1 1 2 3 1.9 1 2 3 10 10 10 10 10 10 10 10 10 Compression ratio Compression ratio Compression ratio Fig. 8. Spatial information shared for each compression approach and different compression ratios. Blue solid curve shows the average amount of mutual information (in number of bits) shared between one coefficient and its spatial neighbours. Green dashed line represents the same quantity for the uncompressed image. Below each plot, the detailed shared information between the central pixel and each neighbour is reported. In each square, the amount of black corresponds to the mutual information between this spatial position and the central pixel. Each square corresponds to a different compression ratio in the plot above, and are sorted left to right as the dots in the blue solid curve in the plot. TABLE IV RMSE OF DEW POINT TEMPERATURE PREDICTIONS ( IN KELVIN ) FOR DIFFERENT COMBINATIONS OF σsa AND σse . T HE IMPROVEMENT OVER THE ORIGINAL DATA IS REPORTED IN PERCENTAGE IN BRACKETS . B EST ( GREEN ) AND WORST ( RED ) PERFORMANCE ARE REPORTED . LR KRR σse (spectral) σse (spectral) 0.01 0.25 1 5 10 0.01 0.25 1 5 10 4.515 4.540 4.370 4.362 3.345 3.376 3.389 3.379 0.01 4.515 0.01 3.345 (0%) (-0.6%) (3.2%) (3.4%) (0%) (-0.9%) (-1.3%) (-1.0%) σsa (spatial) σsa (spatial) 4.343 4.343 4.400 4.254 4.233 3.248 3.248 3.276 3.305 3.266 0.25 0.25 (3.8%) (3.8%) (2.6%) (5.8%) (6.2%) (2.9%) (2.9%) (2.1%) (1.2%) (2.4%) 4.061 4.061 4.097 4.036 3.996 3.082 3.082 3.109 3.113 3.109 1 1 (10.1%) (10.1%) (9.3%) (10.6%) (11.5%) (7.9%) (7.9%) (7.1%) (6.9%) (7.1%) 3.834 3.835 3.880 3.809 3.771 2.994 2.995 3.017 3.028 3.021 5 5 (15.1%) (15.1%) (14.1%) (15.6%) (16.5%) (10.5%) (10.5%) (9.8%) (9.5%) (9.7%) 3.742 3.740 3.795 3.724 3.711 3.007 3.007 3.025 3.035 3.031 10 10 (17.1%) (17.2%) (15.9%) (17.5%) (17.8%) (10.1%) (10.1%) (9.6%) (9.3%) (9.4%) 3.684 3.683 3.737 3.697 3.681 3.069 3.070 3.108 3.114 3.095 20 20 (18.4%) (18.4%) (17.2%) (18.1%) (18.4%) (8.3%) (8.2%) (7.1%) (6.9%) (7.5%) 3.669 3.668 3.736 3.711 3.694 3.138 3.138 3.139 3.160 3.152 30 30 (18.7%) (18.8%) (17.3%) (17.8%) (18.2%) (6.2%) (6.2%) (6.2%) (5.5%) (5.8%) It is clear that spatial features are more important than In order to analyze the performance of compression paired spectral features, as shown in [76] for hyperspectral data. It along with the exploitation of spectral/spatial feature relations, is interesting to note that, while the best prediction results we analyze several configurations. On the one hand we present for LR are yielded for large σsa , the best choice for KRR results for the already explored strategies: retrieval over the is significantly smaller σsa . Improvements for LR are more original data, retrieval over the compressed and recovered significant than for KRR. Note that even for this extremely data, and retrievals over the data processed using the method simple averaging procedure, results confirm the suggestion presented in the previous section. On the other hand we follow reported in [32]: exploitation of spectral and spatial regular- two extra strategies: 1) first, the Gaussian filter is applied to the ization (smoothing) improves the retrieval results. original data and then, the filtered data are compressed; 2) first, the original data are compressed and then, the reconstructed 3) Combining Compression Settings with the Specific Ex- spectra are filtered. Figure 9 illustrates the sequential chain of ploitation of Spectral/Spatial Relations: In this section we the last two strategies. show how the benefits due to compression are drastically re- duced when we have already taken into account spatial/spectral Figure 10 and Fig. 11 illustrate the retrieval performance information when pre-processing the data. We will show of dew point temperature for LR and KRR, respectively. Plots that compression is an effective, consistent, and cheap way show the average of the RMSE prediction over the different to exploit important spectral and spatial data relations and pressure levels. A configuration that produces competitive removing noise in the reconstructed spectra. However, it is performance for both LR and KRR is used in the filtering done in an indirect way and the benefits are limited. stage, i.e., σse = 0.25 and σsa = 10. Results are reported for GARCIA-SOBRINO: IMPROVED STATISTICALLY BASED RETRIEVALS VIA SPATIAL-SPECTRAL DATA COMPRESSION FOR IASI DATA 13 Filtered count local relations between neighboring pixels, which Gaussian Gaussian filter filter data Lossy Lossy compression compression Input (σ benefits the regression algorithms. The improvement is (σscsc == 0.25, 0.25, (spectral (spectral transform transform ++ data σσstst== 10) JPEG 10) JPEG 2000) 2000) more significant for cloudy conditions than for cloud Reconstructed data Encoded data free conditions. Decompression Decompression 3) LR benefits more significantly from spectral/spatial reg- ularization than KRR. According to the results reported Statistical Statistical Prediction - Temperature in the supplementary material at http://isp.uv.es/spatio_ parameter parameter retrieval retrieval (LR - Dew point temperature spectral_compression.html, it is clear from the individual (LR and and KRR) KRR) plots that, when spectral/spatial regularization is added, a) Spectral/spatial regularization + compression. First, we filter the data to larger improvements in the retrieval performance are take into account the spatial and spectral relations. Then, the filtered data are compressed and decompressed, finally statistical parameter retrieval is achieved for LR compared to KRR. Gains produced carried out over the reconstructed spectra. for LR are, approximately, twice the gains produced for KRR. Encoded 4) Filtering + compression yields better results than com- Input Lossy Lossy compression compression data data (spectral (spectral transform transform ++ Decompression Decompression pression + filtering. When the exploitation of features JPEG JPEG 2000) 2000) relations is paired along with compression, it is more Reconstructed Filtered data Gaussian Gaussian filter filter data efficient to perform first the spectral/spatial regulariza- (σ (σscsc == 0.25, 0.25, σσstst== 10) 10) tion and then compress the filtered data. When the compression stage is first carried out and the recon- Statistical Statistical Prediction - Temperature parameter parameter retrieval retrieval structed spectra are filtered, the prediction performance (LR - Dew point temperature (LR and and KRR) KRR) is degraded as the compression ratio increases. The loss of performance is clear even at low compression ratios. b) Compression + spectral/spatial regularization. First, a compression 5) Results are consistent for all the approaches and sce- and decompression stage is performed. Then, feature relations are taken narios analyzed. Conclusions are similar for all the into account in the reconstructed spectra using a Gaussian filter, finally plots, which suggests that the observed behaviour is statistical parameter retrieval is carried out over the filtered data. consistent for all the approaches, compression settings, and scenarios analyzed. Fig. 9. Adopted sequential chain when a spectral/spatial regularization stage and a coding process are carried out. Figure 12 illustrates the bias and the RMSE results for the range of pressure levels between 1100 and 100 hPa. The figure illustrates that the benefits of compression are compensated by pre-processing the data to take into account neighbouring four different scenarios: land and cloud free, land and cloudy, spatial and spectral features. This supports the idea that one ocean and cloud free, and ocean and cloudy. The compression important advantage of using recovered compressed data for settings POT + JPEG 2000 and Multilevel Clustering KLT retrieval is that the process exploits information about the + JPEG 2000 achieve lower maximum compression ratios neighbours in the data. compared to DWT + JPEG 2000 because side-information Both approaches achieve the similar performance for all needs to be transmitted in addition to the compressed data. pressure levels and scenarios, which indicates that the benefits Several conclusions can be extracted from this experiment: obtained from compression were mostly due to the indirect 1) Pre-process the data to take into account the spa- exploitaiton of spatial and spectral neighbouring pixels infor- tial and the spectral information about the neighbours mation. compensates the effect of using recovered data from compression. While using recovered data (maroon curve) V. D ISCUSSION improves the results obtained for the original data (red Lossy compression techniques aim at reducing the size of curve), this improvement vanishes when spectral/spatial the transmitted/stored information keeping the most informa- information is exploited, i.e., using a compression tive part of the signal. As a consequence, the data infor- method over data that have been already pre-processed mation becomes more compact, which helps the subsequent to take into account spectral/spatial information (blue information extraction steps. The multi-component transforms line) does not improve the performance (orange line). employed in the compression stage in this paper are based This observation indicates that when spectral/spatial either in the Karhunen-Loeve transform (KLT) or the wavelet regularization is carried out before compression, the transform. Both transformations have the effect of translating spectral/spatial transform applied in the compression the data to a domain where the statistical relations (or correla- stage is not able to exploit extra features. tions) are reduced [35], [81], [82]. When compressing in these 2) Exploiting feature relations achieves improved retrieval domains, the less relevant features are reduced, thus stressing performance. As we have already seen in section IV-C2, the important ones. When transforming back to the original exploiting extra features about the neighbouring pixels domain, the reconstructed data is a version of the original in the original spectra clearly benefits the prediction data where the less relevant features have also been reduced. results. The consistency and homogeneity of the data From the signal processing point of view, it can be seen as a is improved by pre-processing the data to take into ac- filtering process. 14 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. XX, NO. Y, MONTH Z 2018 Original data Compression Original data + Gaussian filter Gaussian filter + Compression Compression + Gaussian filter Land - Cloud free Land - Cloudy Ocean - Cloud free Ocean - Cloudy 4.8 4.6 4.4 Mean RMSE 4.2 JPEG 2000 4 DWT + 3.8 3.6 3.4 3.2 4.8 4.6 4.4 Mean RMSE 4.2 JPEG 2000 4 POT + 3.8 3.6 3.4 3.2 4.8 4.6 4.4 Mean RMSE 4.2 4 MLC KLT + JPEG 2000 3.8 3.6 3.4 3.2 101 102 103 101 102 103 101 102 103 101 102 103 Compression ratio Compression ratio Compression ratio Compression ratio Fig. 10. Dew point temperature (in kelvin) retrieval performance for different lossy compression settings using LR. In all the plots, the vertical axis represents the averaged RMSE over the different pressure levels and the horizontal axis represents the compression ratio. Ranges are the same in all the plots to ease the comparison. Each row shows the results for a particular compression setting and each column shows the results for a particular scenario. Each plot compares five different approaches, i.e., original data, compression of the original data, original data + Gaussian filter, Gaussian filter + compression, and compression + Gaussian filter. As an example, when using KLT on images, the transforma- ratios were very high, the retrieval performance was notably tion becomes very similar to the Fourier transformation [83], decreased, as expected. i.e., the signal is decomposed in different frequencies. The Then we analyzed the first effect: how certain compression high frequencies of the images have low magnitude in this ratios help to remove useless information while keeping the domain, which in general translates into a poor signal to noise relevant one. We analyzed the compression approaches as ratio –this serves to explain why most coding techniques aim at if they were denoising methods. When a signal is lossy reducing the energy of these frequencies–. As a consequence, compressed, the reconstructed (decompressed) signal has lost the reconstructed image has less energy in the high frequen- information. When this removed information is mostly noise cies, which stresses the contribution of the low frequencies. and only a small amount of useful information is removed, We investigated how much each of these aspects affects compression has a positive impact on the regression results. the performance of the statistically-based retrieval methods We showed that for certain compression ratios and compres- for sounder data. In particular, we analyzed the retrieval of sion settings, most of the removed information was noise. physical variables (temperature and dew point temperature) Figure 7 reported the noise level present in the reconstructed from IASI L1C data. spectra for different compression ratios and different compres- We first verified that applying a compression process to sion settings. Plots illustrated that low compression ratios of remote sensing scenes can improve the performance of the approximately 8:1 removed small amounts of noise, specially statistically based retrieval methods. We analyzed different in the case of POT + JPEG 2000 and Multilevel Clustering compression settings combined with different retrieval meth- KLT + JPEG 2000. Moderate and high compression ratios ods in a realistic scenario. This was illustrated in Figure 3 and were able to filter most of the noise present in the data. Figure 5. While low compression ratios kept the results almost However, when extremely high compression ratios (i.e., above unchanged, moderate and high compression ratios generally 500:1) were used in the compression stage, useful information enabled improved retrievals. However, when the compression was also removed and the quality of the reconstructed data was GARCIA-SOBRINO: IMPROVED STATISTICALLY BASED RETRIEVALS VIA SPATIAL-SPECTRAL DATA COMPRESSION FOR IASI DATA 15 Original data Compression Original data + Gaussian filter Gaussian filter + Compression Compression + Gaussian filter Land - Cloud free Land - Cloudy Ocean - Cloud free Ocean - Cloudy 3.6 3.5 3.4 Mean RMSE 3.3 JPEG 2000 3.2 DWT + 3.1 3 2.9 2.8 3.6 3.5 3.4 Mean RMSE 3.3 JPEG 2000 3.2 POT + 3.1 3 2.9 2.8 3.6 3.5 3.4 Mean RMSE 3.3 3.2 MLC KLT + JPEG 2000 3.1 3 2.9 2.8 101 102 103 101 102 103 101 102 103 101 102 103 Compression ratio Compression ratio Compression ratio Compression ratio Fig. 11. Dew point temperature (in kelvin) retrieval performance for different lossy compression settings using KRR. In all the plots, the vertical axis represents the averaged RMSE over the different pressure levels and the horizontal axis represents the compression ratio. Ranges are the same in all the plots to ease the comparison. Each row shows the results for a particular compression setting and each column shows the results for a particular scenario. Each plot compares five different approaches, i.e., original data, compression of the original data, original data + Gaussian filter, Gaussian filter + compression, and compression + Gaussian filter. Land - Cloud free Land - Cloudy Ocean - Cloud free Ocean - Cloudy 102 LR LR LR LR KRR KRR KRR KRR Original data + Gaussian filter p[hPa] 103 2 10 LR (CR = 271:1) LR (CR = 203:1) LR (CR = 203:1) LR (CR = 271:1) KRR (CR = 271:1) KRR (CR = 271:1) KRR (CR = 203:1) KRR (CR = 271:1) Gaussian filter + p[hPa] (MLC KLT + JPEG 2000) 10 3 -2 0 2 4 6 8 -2 0 2 4 6 8 -2 0 2 4 6 8 -2 0 2 4 6 8 RMSE (K) RMSE (K) RMSE (K) RMSE (K) Fig. 12. Dew point temperature (in kelvin) RMSE profiles and bias. Two approaches are compared: original data + Gaussian filter, and Gaussian filter + Multilevel Clustering KLT + JPEG 2000. LR (thick, solid and red lines) and KRR (thick, solid and green lines). The bias of the each approach is plotted with thin and dash-dot lines for LR (red) and KRR (green). For the compression setting, only the compression ratio (CR) with the best average RMSE is reported. Results using the original data are also displayed for comparison purposes, LR (thick, dashed and red lines) and KRR (thick, dashed and green lines). 16 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. XX, NO. Y, MONTH Z 2018 degraded. This explains why reconstructed spectra produce spatial feature relations in an indirect way, which benefits the accurate retrieval performance when the original data have retrieval methods (on average and using ECMWF analysis as been compressed at moderate and high compression ratios ground truth). We observed that moderate-to-high compression (between 32:1 and 420:1, approximately, depending on the ratios produced improved results in predicting temperature and compression setting). When extremely high compression ratios dew point temperature in the different scenarios analyzed. As were used, the removed signal was informative and therefore expected, when the compression ratio is extremely high, the retrieval performance decreased. benefits disappear because large amounts of useful information The last part of the paper was focused on the second are removed from the data. important effect of the lossy compression techniques. We posit Results reported that exploiting spatial relations between that the compaction of the information introduced by the neighboring pixels is more productive than exploiting spectral compression has the effect of exploiting spatial and spectral relations. While spectral regularization kept the results almost information about the neighbours in the reconstructed signal. unchanged, spatial regularization improved the predictions by First, we specifically computed the amount of information almost 20%. Spatial regularization is, hence, a key element about the spatial neighbours is introduced when compressing to be exploited in pixelwise retrieval algorithms, where the the data. After that, we showed that exploiting certain spec- spatial component is missing. tral/spatial regularization on hyperspectral data had a positive Experiments revealed that when the data is pre-processed effect in the retrievals (Table IV). In particular, the spatial to take into account the spectral and spatial feature relations relations were more important than the spectral relations. before compression, the retrieval performance was improved Then we evaluated the impact of compression in the retrievals compared to the results yielded by the original data even at when the data was already pre-processed to take into account high compression ratios. The compression setting Multilevel spectral/spatial feature relations. Results in Fig. 10 and Fig. 11 Clustering KLT + JPEG 2000 was able to significantly improve showed that when the data was pre-processed the effect the atmospheric predictions at compression ratios higher than of compression in retrieval performance are compensated. 200:1. In other words, an important advantage of using recovered The proposed methodology can be applied to other retrieval compressed data for retrieval is that the process exploits methods and benefits the development of current and up- information about the data neighbours. coming infrared sounding instruments. While current retrieval methods would benefit from efficiently compressed spectra, VI. C ONCLUSIONS savings in data transmission and storage would involve oper- ational improvements. This paper studied the impact of lossy compression on This study may have a deep impact on both currently statistically based regression algorithms for retrieving atmo- flying infrared sounding instruments (e.g., IASI and CrIS ) spheric profiles of temperature and dew point temperature or upcoming (e.g., MTG-IRS). using infrared sounder data (IASI L1C). 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