dataesr/scientific-paragraphs-categorization huggingface dataset:
After training, the learned reconstruction function f θ = G θ • A † can be directly deployed either on the training samples of observations or on new previously unseen observations to recover their respective ground-truth signals. Algorithm 1 provides the pseudo-code of the Equivariant Imaging (EI) where L is the mean squared error (MSE). The signal set consists of different scaling of a triangular signal. On the left, the dataset does not enjoy any invariance, and hence it is not possible to learn the data distribution in the nullspace of A. In this case, the network can inpaint the signal in an arbitrary way (in green), while achieving zero data consistency loss. On the right, the dataset is shift invariant. The range of A is shifted via the transformations Tg, and the network inpaints the signal correctly.
Sparse-view CT The imaging physics model of X-ray computed tomography (CT) is the discrete radon transform. The physics model A is the radon transformation where 50 views (angles) are uniformly subsampled to generate the sparse-view sinograms (observations) y. The Filtered back projection (FBP) function, i. e. iradon, is used to approximate A †. In this task, we exploit the invariance of the CT images to rotations 2  , and G is the group of rotations by 1 degree (|G|=360). We use the CT100 dataset  [33] , a public real CT clinic dataset which comprises 100 real in-vivo CT images collected from the cancer imaging archive  3     patients. The CT images are resized to 128 × 128 pixels and we then apply the radon function on them to generate the 50-views sinograms. We used the first 90 sinograms for training while the remaining 10 sinograms for testing.
A qualitative comparison is presented in Figure  4. The sparse-view FBP contains the line artifacts. Both the FBP-ConvNet and our methods significantly reduce these artifacts, giving visually indistinguishable results. Figure  5  shows the value of PSNR of reconstruction on the training measurements and test measurements and we have the following observations: (i) We would naturally expect the network trained with ground truth data to perform the best. However, we note that the equivariant test error is almost as good despite having no access to ground truth images and only learning on the sparse sinogram data. Furthermore the EI solution is about 7 dB better than the FBP, clearly demonstrating the correct learning of the null space component of the image. (ii) We note that there is a significant gap between training and test error for the FBPConvNet, suggesting that the network may be overfitting.
Image inpainting As a proof-of-concept of the generality of the method, we also applied our method on an image inpainting task with a fixed set of deleted pixels. This is relevant for example to the problem of reconstructing images from cameras with hot or dead pixels. In the image inpainting task, the corrupted measurement is given by y = b x where b is a binary mask, is the Hadamard product, and the associated operator A = diag(b) and A = A †. Here, we consider pixelwise inpainting where we randomly drop 30% of pixel measurements. We train our model by applying random shift transformations. We evaluate the reconstruction performance of our approach and other learning methods using the Ur-ban100  [35]  natural image dataset. For each image, we cropped a 512x512 pixel area at the center and then resized it to 256x256 for the ground truth image.
A. Training Details We first provide the details of the network architectures and hyperparameters of Figs. 4 5 6 and Table  1  of the main paper. We implemented the algorithms and operators (e. g. radon and iradon) in Python with PyTorch 1. 6 and trained the models on NVIDIA 1080ti and 2080ti GPUs. Figure  7  illustrates the architecture of the residual U-Net used  [28]  in our paper. For the sparse-view CT task, we used the Adam optimizer with a batch size of 2 and an initial learning rate of 0. 0005. The weight decay is 10 -8. The distribution strength β is 10 -8 for EI adv. We trained the networks for 5000 epochs, keeping the learning rate constant for the first 2000 epochs and then shrinking it by a factor of 0. 1 every 1000 epochs. More reconstruction examples are presented in Figure  9.
g. those with skipconnections). Moreover, this approach is constrained to a single image and cannot incorporate additional training data. In contrast, we show that our method can learn beyond the range space without heavily relying on the inductive bias of an specific autoencoder architecture. Moreover, we show that EI outperforms the best DIP architecture as it leverages the full compressed training dataset. We compare our method with the DIP on the 50-views CT image reconstruction task. For our method, we use the same residual U-Net as in the other experiments. We build the DIP using two architectures: the same residual U-Net used in EI (which we denote DIP-1) and the best autoencoder network suggested in  [31]  (which we denote DIP-2). Following  [31] , we input iid Gaussian noise to both DIP-1 (1 channel) and DIP-2.
The AlpArray Gravity Research Group (AAGRG), as part of the European AlpArray program, focuses on the compilation of a homogeneous surface-based gravity data set across the Alpine area. In 2017 10 European countries in the Alpine realm agreed to contribute with gravity data for a new compilation of the Alpine gravity field in an area spanning from 2 to 23 • E and from 41 to 51 • N. This compilation relies on existing national gravity databases and, for the Ligurian and the Adriatic seas, on shipborne data of the Service Hydrographique et Océanographique de la Marine and of the Bureau Gravimétrique International. Furthermore,.
for the Ivrea zone in the Western Alps, recently acquired data were added to the database. This first pan-Alpine gravity data map is homogeneous regarding input data sets, applied methods and all corrections, as well as reference frames. Here, the AAGRG presents the data set of the recalculated gravity fields on a 4 km × 4 km grid for public release and a 2 km × 2 km grid for special request. The final products also include calculated values for mass and bathymetry corrections of the measured gravity at each grid point, as well as height. This allows users to use later customized densities for their own calculations of mass corrections. Correction densities used are 2670 kg m -3 for landmasses, 1030 kg m -3 for water masses above the ellipsoid and -1640 kg m -3 for those below the ellipsoid and 1000 kg m -3 for lake water masses.
The correction radius was set to the Hayford zone O 2 (167 km). The new Bouguer anomaly is station completed (CBA) and compiled according to the most modern criteria and reference frames (both positioning and gravity), including atmospheric corrections. Special emphasis was put on the gravity effect of the numerous lakes in the study area, which can have an effect of up to 5 mGal for gravity stations located at shorelines with steep slopes, e. g. , for the rather deep reservoirs in the Alps. The results of an error statistic based on cross validations and/or "interpolation residuals" are provided for the entire database. As an example, the interpolation residuals of the Austrian data set range between about -8 and +8 mGal and the cross-validation residuals between -14 and +10 mGal; standard deviations are well below 1 mGal.
The accuracy of the newly compiled gravity database is close to ±5 mGal for most areas. A first interpretation of the new map shows that the resolution of the gravity anomalies is suited for applications ranging from intra-crustal-to crustal-scale modeling to interdisciplinary studies on the regional and continental scales, as well as applications as joint inversion with other data sets. The data are published with the DOI https://doi. org/10. 5880/fidgeo. 2020. 045 (Zahorec et al. , 2021)  via GFZ Data Services.
The AlpArray Gravity Research Group With respect to the national expertise and databases available in the Alpine countries, the formation of an international research group (AlpArray Gravity Research Group; AAGRG) was decided within the framework of activities in the European AlpArray program and established at an EGU splinter meeting in 2017. In the subsequent workshops in Bratislava (Slovakia) in 2018, and two further technical meetings of the group (again in Bratislava in 2018 and in Sopron, Hungary, in 2019), the organizational, scientific, and numerical requirements for the compilation of the new pan-Alpine digital gravity database were established, which consists of Bouguer and free-air anomalies  (BA and FA)  and values of mass correction. Although most of the national group members were extensively involved in the processing of data, we would like to remind with gratitude that by far the most intensive part of the processing was done by the group members from Bratislava and Banská Bystrica (Slovakia).
Publication layout We document in detail our procedures, from raw data to final high-resolution gravity maps. The referencing and quality assessment of various gravity databases and digital Earth surface models are discussed in Sect. 2. The equations and their implementations to obtain various gravity anomaly products, as well as the reprocessing of original raw data and of the related corrections, are described in Sect. 3. Section 4 presents the new, homogenized Bouguer gravity map for the Alps. In Sect. 4. 3 we describe the attached Bouguer map, together with an accompanying description and interpretation of the gravity anomalies in the Alps and their surroundings. Notes on the uncertainty of the compilation are given in Sect. 5. We conclude on the listing and availability of the new gravity data (Sect.
Problems with positioning, heights and gravity data One of the key problems in the unification of gravimetric databases is the homogenization of position, height and gravimetric coordinate systems used in each database. Through its historical development, each country has used and sometimes still uses local systems and their realization (frame), which are often based on the established principles of reference systems using older ellipsoids or older geodetic reference networks and projections. These systems and their realizations thus contain several differences which are responsible for large inhomogeneities, shifts and errors in position, height and gravity. These errors are most evident in the mutual comparison of data from individual countries. To avoid these problems in the position of gravimetric points, all position data were transformed from local systems to the European Terrestrial Reference System 1989 (ETRS89), which is accurate, homogeneous and recommended for all European countries  (Altamimi, 2018).
A similar situation is in the height systems in that countries use different types of physical heights, they are linked to different tide gauges, and each country has a different practical implementation of the relevant height system. The solution is again the transformation to a uniform platform in the form of ellipsoidal heights in the ETRS89 system based on the ellipsoid GRS80  (Moritz, 2000). The situation is similar in gravimetric reference systems, in which especially the gravimetric databases that have been created for decades often use old gravimetric systems linked to the Potsdam system. An important step was therefore to convert these data into gravimetric systems which are connected to absolute gravimetric points and measurements, such as IGSN71  (Morelli et al. , 1972)  or modern national systems connected with the recent absolute measurements, which are verified by international comparisons of absolute gravimeters  (Francis et al.
, 2015). For these transformations, national transformation services were used (operated by national mapping services, e. g. , SAPOS, SKPOS) or transformations implemented into standard GIS tools or our own software implementations based on national standards, information and experience of individual responsible institutions. The transformation from physical heights in national vertical systems to ellipsoidal heights in the ETRS89 system, ellipsoid GRS80, was realized using available local geoid and quasi-geoid models available through transformation services or implemented in current geodetic processing programs (e. g. , Trimble Business Center, Leica Infinity). If a local geoid or quasi-geoid model was not available for some areas, then the global geopotential model EIGEN-6C4  (Förste et al.
3. 1). In addition, emphasis was put on the calculation of the gravimetric effects of the Alpine lakes on the basis of bathymetric data of the region (Sect. 3. 2). To complete the AAGRG database an old CBA map from the former Socialist Federal Republic (SFR) of Yugoslavia  (Bilibajkič et al. , 1979)  (Sect. 3. 3) was digitized. Further improvements of the new CBA map are the refined calculations of an atmospheric correction and the future containment of distant terrain and bathymetry effects (Sect. 3. 4). The basic formula for the CBA calculation was adopted from  Meurers et al. (2001) : BA(λ, ϕh E ) =g(λ, ϕh E ) -γ (ϕh E ) -δg M (λ, ϕh E ) + δg B (λ, ϕ, H ) + δg A (λ, ϕ, H ), (1) γ (ϕh E ) = γ 0 (ϕ) + ∂γ ∂h E | 0 h E + 1 2 ∂ 2 γ ∂h 2 E | 0 h 2 E , (2) where γ 0 (ϕ) results from the well-known Somigliana formula  (Somigliana, 1929)  for the normal gravity acceleration of a rotational ellipsoid at its surface  (Heiskanen and Moritz, 1967) : γ 0 (ϕ) = a γ E cos 2 ϕ + c γ P sin 2 ϕ a 2 cos 2 ϕ + c 2 sin 2 ϕ , (3) and higher vertical derivatives of γ (ϕ, h E ) are given by ∂γ ∂h E 0 = - 2γ 0 a 1 + f -2 f sin 2 ϕ + 3 2 f 2 -2f 2 sin 2 ϕ + 1 2 f 2 sin 4 ϕ -2ω 2 , (4) ∂ 2 γ ∂h 2 E 0 = 6γ 0 a 2 1 -f sin 2 ϕ 2.
Mass correction One of the main problems in the homogenization of data and recompilation of gravity fields was the use of different procedures for the calculation of mass correction (MC) and bathymetry correction (BC) by national operators/authorities. This meant that a complete recalculation had to be carried out for the new compilation based on the available point data and the best digital elevation models (DEMs) available. The proper choice of DEMs is discussed in Appendix C. An important first step before starting the recompilation was to test and select the available software to calculate the mass corrections. We compared two custom software packages developed by team members: Toposk software  (Zahorec et al. , 2017a)  and TriTop  (Holzrichter et al. , 2019). Considering the results of this comparison (refer to Appendix D), we decided to use Toposk based on ellipsoidal heights h E of the calculation points and ellipsoidal digital elevation models (using in the majority of cases local geoids for the transformation).
On the other hand, the bathymetric and simplified atmospheric corrections were calculated for physical heights H of the calculation points. Bathymetric corrections were also calculated by means of the Toposk software but in a slightly adjusted mode (see below and Fig. 3 ). If the normal field in Eq. (  1 ) is defined at the height above the surface ellipsoid, it is necessary to define the effects of terrain and bathymetry masses above the ellipsoid (not above the geoid). Therefore, the concept requires the use of ellipsoidal heights of the observation points, and at the same time it is necessary to transform the topography and bathymetry grids from physical to ellipsoidal heights. In the AlpArray area, the situation is more or less simple, the ellipsoid is below the geoid throughout the region (approx.
In    contrast to the digitization for the AAGRG described here, the Slovenian and Croatian database contains new data. The reprocessing included the identification and correction of individual steps in the frame of CBA calculations to ensure a processing status which complies with that of the recalculated anomaly of the new AlpArray map. Specifically, normal gravity was corrected for the difference between the IGF 1967 and the Somigliana/GRS80 equations. Then the simple free-air correction was replaced by a more accurate approach, and the sphericity of the Earth was taken into account. However, this was neglected in cases when simple planar Bouguer corrections in the original data were used. For the last two corrections, the approximate heights at the digitization points generated from the model MERIT (multi-error-removed improved-terrain) were used.
A short remark on future treatment of true atmosphere and distant relief effects As a challenge for the further development of the AlpArray CBA map, we also estimated the global effects of the true atmosphere and distant relief. Atmospheric correction is usually calculated based on a simple approximation according to  Wenzel (1985). By the term true atmosphere, we mean the model of the atmosphere derived from the effect of a spherical shell with radially dependent density using the US standard atmosphere 1976  (Karcol, 2011)  with an irregularly shaped bottom surface formed by the Earth's surface, calculated globally  (Mikuška et al. , 2008). The difference between atmospheric correction calculated by both approaches for the AlpArray region (calculated for selected database points) is shown in Fig.
(  7 ) to (  14 ), we have to separate liquid from solid parts, which leads to the following equation: BA ρ (x i , y j ) = BA int (x i , y j ) + 1 - ρ ρ 0 δg M (x i , y j ) 1 - ρ -ρ oc ρ 0 -ρ oc δg B (x i , y j ), ( 15 ) where ρ oc is the density of ocean water (1030 kg m -3 ). Equation (  15 ) neglects the small density difference between lake and ocean water. However, this leads to only small errors on the order of a few percent of the lake correction for reasonable crustal densities. To conclude, in addition to the methodological procedures just described, we will now describe another problem related to the gridding of our database. In the case of the AAGRG compilation, interpolation of original and gridded data has been done by an iterative procedure: a. Data providers, who were not allowed to release original information, created gridded data relying initially on their own scattered data and keeping only the nodes inside their own territory on a grid the AAGRG defined in common for the whole area.
7 km. The DTM2006 model was selected due to its close relationship with the creation of the model EGM2008. This model was originally compiled in a grid of 30 arcsec × 30 arcsec. For the purposes of our calculations, the model was transformed and resampled into a format corresponding to the calculation of the standard mass/bathymetric correction using Toposk. We calculated the gravity values g GGM using the software GrafLab  (Bucha and Janák, 2013)  using the maximum degree of spherical harmonic coefficients for a specific GGM. Calculations were performed in GRS80 ellipsoidal coordinates. Figure  12  shows a comparison of a BA map derived from terrestrial data with the map derived from the EIGEN-6C4 model (calculation points were made on a 2 km × 2 km grid) in the area covered by terrestrial data.
The maximum differences between grids are at the level of tens of milligals (RMS error is about 4 mGal) but without any systematic error. It follows that the GGM-derived map can be used to fill in gaps (marginal parts) in the terrestrial data. GGM data points located in gaps of the original gravity points were separated by the shortest distance criteria of 15 km using a standard database search query in QGIS. A data (Fig. 19 ) but at the same time not filling gaps that are too small between them, which could lead to local artificial anomalies.
Uncertainties of data and map The newly compiled gravity database of the Alps and their surroundings is based on decades of data collection and processing experience of the AAGRG members. The national gravity data, which were recompiled here under new, modern geophysical and geodetic aspects (Sects. 2 and 3), were collected with rather different instruments at different times over the last 70 years and processed with extremely different processing methods. At the end of the data processing, we therefore asked ourselves for what purposes it can be used and how accurate the new map actually is. The first question can be answered relatively easily: with mediumto large-scale modeling of the Alpine lithosphere and/or the Alpine Earth crust, as realized in the AlpArray initiative, there should be no problems with the final accuracy of the database: these errors are small compared to the uncertainties that result from modeling and simulation.
The second question about accuracy (uncertainty), which is caused by using extremely different data sets, is much more difficult to an-swer because in practice for all participating countries there are no exploitable metadata available for the national gravity databases. As desirable as it would have been for the submitted pan-Alpine gravity maps to present "uncertainty maps" at the same scale, this project is hindered due to the complexity of the task and the lack of information on errors and accuracies in the field campaigns and data processing of the individual countries. However, in order to obtain an estimate of the uncertainty, we have tried in the following section to list various aspects of error analysis by way of examples. It must be reserved for a later publication to present a numericalstatistical analysis of the map (e.
Furthermore, different numerical approaches that can be used for the data processing provide different results. In Appendix D we reported test investigations which led to the selection of the software for the calculation of the MC (Appendix D, Fig. D1 ). A comparison of the standard deviations (1. 95 mGal for the software TriTop and 0. 39 mGal for Toposk) also gives an indication of the achieved accuracy of the database -even if this can only be a partial aspect. Two other sources of error deserve a closer look. In Sect. 5. 2 we will discuss errors that occur when calculating the mass correction with different correction densities. Notes on the accuracy of the anomalies due to a 2D (on the map projection plane) and a 3D interpolation needed have already been given in Sect. 4. 1. Based on national investigations in the area of Austria, indications of the achieved numerical accuracy of the Bouguer anomalies are then given in Sect.
5. 3. Finally, in Sect. 5. 4 the results of an error statistic based on cross validations (CVs) are given for the entire database. However, it should not be forgotten that CV is a purely statistical measure and in minor amounts considers point data quality, which indicates that we cannot directly represent the quality of the newly compiled gravity fields from the CV. CV works well with dense station coverage; only then can we exclude large local anomalies, for example, due to geological causes. The less dense the coverage is, the less we can exclude the presence of local anomalies. Note that these local anomalies can easily be produced by selecting improper MC density, for example, in a station setting covering a valley and adjacent mountain flanks where densities differ from the assumed MC density remarkably.
Cross-validation error for the entire database As mentioned in the previous subsection both interpolation residuals and cross-validation methods provide some picture of data quality. At the same time, these methods can be used as a criterion for excluding gross errors from individual databases. Both methods give qualitatively similar results (see Fig. 15 ), with cross validation giving quantitatively more significant residuals. Since in the case of crossvalidation residuals (unlike interpolation residuals) it is possible to exchange data between grid providers in order to comply with the conditions of confidentiality of the original data, we show in Fig. 16  a complete map of cross-validation residuals for the whole area. While the standard deviation of these residuals is well below 1 mGal (comparable to Table 2), the extreme values reach tens of milligals (about 120 points exceed 10 mGal, 9 points exceed 20 mGal).