The CIRFA-Cruise 2022 with RV Kronprins Haakon to the north-eastern coast of Greenland in the period April 22nd to May 9th 2022 was organised to perform measurements and make observations which allow for validation of sea ice remote sensing information and forecast products resulting from work in the Centre for Integrated Remote Sensing and Forecasting for Arctic Operations (CIRFA), a Centre for Research-based Innovation at UiT the Arctic University of Norway. This paper uses data collected during the cruise to investigate questions related to the interpretation and temporal consistency of polarimetric features computed from a series of quad-pol Radarsat-2 (RS-2) images, which was collected over a fast ice site in the Belgica Bank area in the western Fram Strait. The CIRFA-2022 Cruise team visited this fast ice site in the end of April 2022. The time series covers the transition from cold winter conditions in April to melting in mid June. This transition impacts radar backscat-tering, as can be clearly seen in the Pauli decomposition of quad-pol images.

This paper presents some results obtained during the 2022 CIRFA arctic cruise concerning the measurements of the radar response of different types of sea-ice using a high-resolution Ground-Based radar system operating at C band. This MIMO device was able to directly measure tomograms, or slices of reflectivity in the elevation-ground plane, allowing to quantitatively appreciate the penetration of radar waves into sea-ice types having complex geometrical features, and to assess the dominant contributions measured by radar devices at C band.

The synergistic use of Landsat-8 operational land imager (OLI) and Sentinel-2 multispectral instrument (MSI) data products provides an excellent opportunity to monitor the dynamics of aquatic ecosystems. However, the merging of data products from multisensors is often adversely affected by the difference in their spectral characteristics. In addition, the errors in the atmospheric correction (AC) methods further increase the inconsistencies in downstream products. This work proposes an improved spectral harmonization method for OLI and MSI-derived remote sensing reflectance (${R_{rs}}$) products, which significantly reduces uncertainties compared to those in the literature. We compared${R_{rs}}$retrieved via state-of-the-art AC processors, i.e., Acolite, C2RCC, and Polymer, against ship-based in situ${R_{rs}}$observations obtained from the Barents Sea waters, including a wide range of optical properties. Results suggest that the Acolite-derived${R_{rs}}$has a minimum bias for our study area with median absolute percentage difference (MAPD) varying from 9% to 25% in the blue–green bands. To spectrally merge OLI and MSI, we develop and apply a new machine learning-based bandpass adjustment (BA) model to near-simultaneous OLI and MSI images acquired in the years from 2018 to 2020. Compared to a conventional linear adjustment, we demonstrate that the spectral difference is significantly reduced from$\sim 6$% to 12% to$\sim 2$% to${< } {10\%}$in the common OLI-MSI bands using the proposed BA model. The findings of this study are useful for the combined use of OLI and MSI${R_{rs}}$products for water quality monitoring applications. The proposed method has the potential to be applied to other waters.

Present and future sensors are diversifying from traditional quad polarimetric mode of synthetic aperture radar acquisition. Thus, an approach that is interpretative in nature and applicable across polarimetric modes is required. In this context, the geodesic distance (GD)-based approach within the polarimetric synthetic aperture radar (PolSAR) literature is seen as an eigenvalue-decomposition free approach to interpret and analyze quad PolSAR data. This approach is highly adaptive toward applications due to its ability to compare the SAR observation with a known scatterer/model, or with another SAR observation in general providing a means for direct interpretation. In this work, we show that the GD (originally defined for the quad polarization mode) is generalizable across any arbitrary SAR polarimetric mode while retaining its simple form for ready computation. We show that the GD-based approach provides level ground for comparison of different polarimetric modes given a fixed application. We demonstrate it using change detection as the chosen application. In addition, we show how the behavior of the three roll-invariant GD-based parameters change under different polarimetric modes (e.g., quad, dual, and compact polarization modes). We also discuss how the GD-based approach can also be adapted to ground range detected (GRD) product data, which is presently available from Sentinel-1 and widely used in many applications. However, in this case, we show that only one of the three GD-derived parameters can be defined. We believe this work will make the GD-based approach important for present and future PolSAR applications cutting across sensors and its available polarimetric modes.

Monitoring sea ice in polar regions is critical for understanding global climate change and supporting marine navigation. Recently, researchers started to utilize machine/deep learning methodologies to automate the separation of sea ice and open water in synthetic aperture radar imagery. However, this requires a large amount of reliably labeled training data. We here propose an augmentation routine for Sentinel-1 data which incorporates physical principles of radar backscatter into the augmentation procedure. Firstly, we apply an incidence angle aware algorithm to segment Sentinel-1 images into separate clusters. We compute the corresponding slopes of backscatter intensity with the incidence angle for each cluster. Secondly, the slopes are used as prior information to project labeled pixels and segments to different incidence angles and thus further enrich the labeled data. We then apply a simplified U-Net for pixel-wise classification of Sentinel-1 images into sea ice or open water. The performance of our model is evaluated by visual inspection as well as comparison with an available product from the Chinese Academy of Science (CAS). The results indicate that the physics-based data augmentation improves the model performance compared to training with data without augmentation. The inferred ice edge is in line with the inference for other available data sets (CAS), but with a finer spatial resolution. Finally, we also found the inference of our model is highly correlated with the visual interpretation of overlapping optical observations. Overall, the proposed methodology provides an alternative for the automated separation of sea ice/open water at fine spatial resolution.

Sea ice mapping in polar region is crucial for understanding the global climate change, benefiting disaster control for local community as well as providing accurate navigation for mariners. Currently, meteorological ice services in many countries manually create ice-charts, which is time-consuming work from domain experts. Machine learning and deep learning methodology can be utilized to automate ice charting, however, large amounts of reliable training data is crucial for implementing the technology successfully, and training data itself is sparse and costly to obtain in the Arctic areas. To obtain more training data, we proposed a two-stage methodology which incorporate the physics into training dataset generation procedure. Firstly, a physics-based incidence angle aware algorithm was employed for generating better connected and fewer reference classes to be manually labelled for training. Secondly, we enrich the training data to a balanced training set by using the physical knowledge regarding incidence angle dependence of SAR intensity. This many-fold enriched dataset can then be used for pixel-wise sea ice versus water classification by using a reduced UNet architecture. The primary result shows the sea-ice and water edges can be reasonably delineated by utilizing the physical based training dataset generation together with the reduced UNet architecture.

In this study, we compare the retrieval results for the dielectric properties of verified oil slick, acquired using airborne multifrequency synthetic aperture radar. A polarimetric two-scale model was used to invert the radar imagery by first employing solely the co-polarization channels, and then by employing the co-polarization channels in conjunction with the cross-polarization channels, and thereby employing the full suite of polarization information available. The goal is to show that the inversion results obtained from both methods are consistent. Given that the ocean surface is a highly nondepolarizing surface scatter, the signal return within the cross-polarization channels is usually negligible and of no practical use when trying to invert the returned backscatter into useful quantities such as the dielectric constant. In this article, we employ F-SAR data, which was acquired in X-, S-, and L-bands and has an extremely low noise floor, implying that the cross-polarization ratio can be employed. A signal-to-noise analysis showed that only the L-band acquisitions were suitable for analysis in this article. The retrieval results are comparable for the two methods in the case of low dielectric values.

The vast amount of spectral information provided by hyperspectral images can be useful for different applications. However, the presence of redundant bands will negatively affect application performance. Therefore, it is crucial to select a relevant subset that preserves the information of the original set. In this paper, we present an automatic and accurate band selection method based on Graph Laplacians. Unlike existing band selection methods, this method exploits two similarity measures simultaneously. Furthermore, it is performed on a superpixel level, so it allows us to preserve not only global but contemporaneously local particularities of original data. Experiments show the importance of measuring the relevance of the bands at local and global scales and the ability of the method to minimize intercorrelation among selected bands, hence improving the selection of the most informative spectral channels.

DLRs F-SAR instrument acquired a timeseries of simultaneous quad-polarimetric radar imagery of verified oil slick in S- and L-band during the NORSE2019 oil-on-water experiment conducted in the North Sea. Deriving maps of internal zones of consistently thick oil, from a freely floating oil slick, with no fixed reference point, by exploiting both the temporal and multifrequency aspect of the damping ratio of the acquired data, is the aim of this paper. Internal zones that are stable over time are indicated by the stability level which requires a threshold to be applied to the feature under consideration. We verify our findings by comparing results to aerial photography, taken at the same time as one of the SAR acquisitions, and demonstrate that the method can be applied to freely floating oil slick.

The additive system noise in synthetic aperture radar (SAR) imagery is a challenging problem for the operational use of SAR data for sea ice classification. This noise degrades the performance of the sea ice classification models. The most common way of dealing with this is to remove mean noise profiles from the backscatter intensities as a preprocessing step. In this study we investigate how including the nominal noise profiles as a feature directly into the model affects the classification. Our noise-aware approach can be used in conjunction with any other deep learning model for sea ice classification. Hence our findings pave the way for getting refined and smoother sea ice maps for ice charting. For experimentally evaluating our proposed approach, we train our noise-aware deep model using carefully labeled data consisting of both sea ice data and noise profile. We present validation results considering separate validation data. Our empirical study confirms the superior performance of the CNN model driven by noise-aware characteristics.

This work introduces a novel method that combines machine learning (ML) techniques with dual-polarimetric (dual-pol) synthetic aperture radar (SAR) observations for estimating quad-polarimetric (quad-pol) parameters, which are presumed to contain geophysical sea ice information. In the training phase, the output parameters are generated from quad-pol observations obtained by Radarsat-2 (RS2), and the corresponding input data consist of features obtained from overlapping dual-pol Sentinel-1 (S1) data. Then, two, well-recognized ML methods are studied to learn the functional relationship between the output and input data. These ML approaches are the Gaussian process regression (GPR) and neural network (NN) for regression models. The goal is to use the aforementioned ML techniques to generate Arctic sea ice information from freely available dual-pol observations acquired by S1, which can, in general, only be generated from quad-pol data. Eight overlapping RS2 and S1 scenes were used to train and test the GPR and NN models. Statistical regression performance measures were computed to evaluate the strength of the ML regression methods. Then, two scenes were selected for further evaluation, where overlapping optical images were available as well. This allowed the visual interpretation of the maps estimated by the ML models. Finally, one of the methods was tested on an entire S1 scene to perform prediction on areas outside of the RS2 and S1 overlap. Our results indicate that the studied ML techniques can be utilized to increase the information retrieval capacity of the wide swath dual-pol S1 imagery while embedding physical properties in the methodology.

Over recent years, rapid environmental changes in the Arctic and subarctic regions have caused significant alterations in the ecosystem structure and seasonality, including the primary productivity of the Barents Sea. This work aims at improving methodology for studying these features, by estimating chlorophyll-a (chl-a) concentrations in the transitional Barents Sea by remotely sensing its optical properties, in order to better understand the large-scale algal bloom dynamics in the region. The in-situ measurements of chl-a are collected from the year 2016 to 2018 over a wide area of the Barents Sea to cover the spatial and temporal variations in chl-a concentration. Optical images of the Barents Sea are captured by the Multi-Spectral Imager Instrument on Sentinel-2. Using these remotely sensed optical images and the in-situ measurements, we propose a match-up dataset creation method based on the distribution of the remotely sensed reflectance spectra. Different Machine Learning (ML) techniques are assessed to estimate concentration of chl-a using the match-up dataset. Most of these techniques have not been investigated before in the subarctic region such as the Barents Sea. The Ocean Color Net (OCN) regression model proposed in this study has outperformed other ML-based techniques including Support Vector Regression, Gaussian Process Regression, and the globally trained Case-2 Regional/Coast Colour (C2RCC) processing chain model C2RCC-Nets, as well as empirical methods based on spectral band ratios. A wide range of experiments has demonstrated the effectiveness of the proposed OCN for ocean color remote sensing in the subarctic region. The performance of the OCN is also presented spatially by computing chl-a maps in the Barents Sea.

This work evaluates three machine learning methods with respect to their ability of learning the functional relationship between dual-polarimetric (dual-pol) input data and quad-polarimetric (quad-pol) output parameters. We chose to study and compare the learning strength of a Neural Network (NN) approach, two kernel-methods, the Support Vector Machine (SVM) and the Gaussian Process Regression (GPR). Overlapping quad-pol Radarsat-2 (RS2) and dual-pol ScanSAR Sentinel-l (S1) sea ice Synthetic Aperture Radar (SAR) scenes, with 20 minutes time difference, were used for establishing the relationship between the dual-pol input data and corresponding quad-pol output parameters. We then used the learned relationship to predict quad-pol parameters for the overlapping S1 dual-pol scene, and show the results of the three machine learning methods, visually, by showing images of the predicted polarimetric features, and quantitively, by computing statistical performance measures. The results indicate that all three methods have strong learning capacity, however, the computed statistical measures and the visual comparisons suggest the best performance for the GPR model.

We propose the use of a polarimetric two-scale surface scattering model to retrieve the dielectric parameters of oil slick from the polarimetric synthetic aperture radar. The ocean surface is modeled as an ensemble of randomly orientated, slightly roughened, tilted facets, for which the small perturbation model is assumed to be valid under the condition of no tilt. The orientation of the random facets causes a variation in the in-plane and out-of-plane tilt angles. As the original model utilizes both the co-polarization and cross-polarization channels to determine both the dielectric and roughness characteristics simultaneously from a series of look-up tables, the model is adapted from its original form in order to estimate the roughness characteristics of the scattering surface first, before the dielectric properties are inferred. The performance of the altered scattering model is then evaluated by applying it to multiple sets of quad-polarimetric data containing verified oil slicks, acquired from oil-on-water clean-up exercises in the North Sea. Histograms of retrieved values for the modulus of the dielectric constant indicate that the model is able to invert for values similar to the actual value of 2.3, the dielectric constant of pure crude oil at the lower limit, with successively higher values being found up to values of approximately 20 at the edges of the slicks. An error evaluation is also presented and demonstrates that sources of error are related to the alteration of the model to suit co-polarimetric data and the variance of the speckle that is related to the size of the averaging window. While the results are interesting, the approach is limited to the use of only the ratio of the co-polarimetric channels, from which two unknowns are estimated.

Copernicus is the European programme for monitoring the Earth.It consists of a set of systems that collect data from satellites and in-situ sensors, process this data and provide users with reliable and up-to-date information on a range of environmental and security issues.The data and information processed and disseminated puts Copernicus at the forefront of the big data paradigm, giving rise to all relevant challenges, the so-called 5 Vs: volume, velocity, variety, veracity and value.In this short paper, we discuss the challenges of extracting information and knowledge from huge archives of Copernicus data.We propose to achieve this by scale-out distributed deep learning techniques that run on very big clusters offering virtual machines and GPUs.We also discuss the challenges of achieving scalability in the management of the extreme volumes of information and knowledge extracted from Copernicus data.The envisioned scientific and technical work will be carried out in the context of the H2020 project ExtremeEarth which starts in January 2019.

In this paper, we evaluate the possibility of using a machine learning Gaussian Process Regression (GPR) approach to monitor Chlorophyll-a content in Arctic waters by using the Sentinel 3 Ocean and Land Color Instrument. We develop the GPR model on a synthetic dataset, which represents both open ocean and coastal Arctic waters. This allows the model to be exposed to and trained on data from both kinds of aquatic environments. The chosen GPR model has previously been trained and tested in a different aquatic environment, representing a variety of complexity conditions, where it was demonstrated to have strong generalization capabilities. Our results suggest that this model can also be used for diverse Arctic water conditions as well.

In this paper, we introduce a new approach for investigation of polarimetric Synthetic Aperture Radar (PolSAR) images for oil slick analysis. Our method aims at enhancing discrimination of oil types by exploring the polarimetric features that can be produced by processing PolSAR scenes without dimensionality reduction. Taking advantage of a mixture description of the interactions among classes within the dataset and a characterization of their intra- and inter-class variability, our algorithm is able to quantify the areal coverage of different elements. These estimates can be used to hence improve classification. Experimental results on a PolSAR dataset acquired by unmanned aerial vehicle (UAV) on oil slicks in open water show the capacity of our method.

This letter presents a new general framework for solving polarimetric target decompositions that extends them to use more statistical information and include radar texture models. Polarimetric target decomposition methods generally have more physical parameters than equations and are, thus, underdetermined and have no unique solution. The common approach to solve them is to make certain assumptions, thus fixing some parameters, allowing the other parameters to be solved freely. This letter explains how to obtain additional equations from several statistical moments to find unique solutions and to address the issue of textured product models. The current work extends our previous conference works [1]-[3]. Preliminary results are demonstrated for a well-known real polarimetric synthetic aperture radar scene for the three-component Freeman-Durden decomposition.

Calving of icebergs at the tidewater glacier fronts is a component of the mass loss in Polar regions. Studying the regional distribution of icebergs, their volume, motion, and interaction with the environment is of interest. Here, we present the results from a fieldwork campaign conducted in Kongsfjorden, Svalbard in April 2016, where both satellite and ground-based remote sensing instruments were used to observe dynamics of sea ice, icebergs, and growlers. We used a ground-based radar system, imaging the study area every second minute during five days. During the same observation period, we collected four RADARSAT-2 (RS-2) quad-pol images, that are used for automatic detection of icebergs. In addition, the fieldwork team collected GPS positions of some drifting and grounded icebergs in the fjord to be used as ground-truth data. The comparison and combination of satellite, ground-based radar, and in-situ data contribute to cross-validate the results.

This paper addresses the problem of up-scaling full polarimetric (quad-pol) parameters from small quad-pol synthetic aperture radar (SAR) scenes to large dual-pol scenes, using a sophisticated Machine Learning (ML) method, namely the Gaussian Process Regression (GPR). The approach is to let the GPR model learn the relationships between the dual-pol input data and the quad-pol parameters on a quad-pol scene, and then extrapolate the relationships to the whole dual-pol scene. We demonstrate the procedure on two pairs of quadpol Radarsat-2 (RS2) and dual-pol ScanSAR Sentinel-1 (S1) scenes, acquired less than 20 minutes apart. The results are visualised as pixel-wise parametric maps, supported by three quantitative regression performance measures. In addition, we show certainty level maps for the estimated parameters. Our results indicate the potential of using the ML GPR model to upscale quad-pol scenes to large dual-pol images.

In this paper, the retrieval of sea-ice surface heights from the interferometric TanDEM-X data is investigated. The data were acquired over fast and drifting ice in Fram Strait located between Greenland and Svalbard. Additional measurements of the sea-ice surface topography were carried out using a stereo camera and a laser altimeter. The comparison of the surface elevation retrieved from TanDEM-X imagery with the results of the stereo camera measurements revealed that sea-ice ridges greater than 0.5 m can be estimated with a root-mean-square error of 0.3 m or less with the error decreasing as a function of ridge height. Although the helicopter-borne laser data are only available as 1-D profiles with a much higher across-track spatial resolution than the TanDEM-X data, they proved to be useful for the validation. The need for multilook averaging to reduce the phase noise is identified as the main challenge in achieving the spatial resolution necessary for retrieving sea-ice surface topography using synthetic aperture radar interferometry.

In this paper we investigate interferometric pairs of SAR images acquired by Tandem-X with the monostatic pursuit and the alternative bistatic modes for sea ice segmentation. The individual SAR images are modelled as non-Gaussian, and from the modelled data different features are extracted, stacked together and clustered. The interferometric coherence is regarded as an additional feature and utilized for clustering. In addition to complementing the information extracted from the individual images, the interferometric coherence is found to be capable of discriminating between open water and sea ice, as well as between different ice types.

This paper presents new experimental results of 3-D imaging using tomographic techniques over a snow covered sea ice medium, sensed with an X-band radar system. The available data are from a ground-based synthetic aperture radar data collection campaign carried out over Kattfjord, Tromsø, Norway. Direct imaging of the vertical structures of the radar reflectivity of the snow and sea ice layers is achieved by focusing the signal from a 2-D synthetic array in the 3-D space. The effect of a change in propagation velocity of the wave inside the considered medium is investigated in the focusing process, and the tomograms are effectively corrected for this effect. The distribution of the scattering contributions in the vertical direction reveals a strong response from the sea ice cover. Tomograms at two different polarizations are investigated and compared. The results and the interpretations are also supported by the simulated data from the same system.

This letter discusses cross-correlation features derived from near-coincident RADARSAT-2 quad-polarimetric and RISAT-1 hybrid-polarity (HP) measurements collected during the NOrwegian Radar oil Spill Experiment in 2015 (NORSE2015). We show that the imaginary part of the cross-correlation between RH and RV is an HP parallel to the real part of the cross-correlation between HH and VV earlier proposed for oil spill characterization. We compared the RADARSAT-2 and RISAT-1 scenes, separated in time by less than an hour, and the results show a clear difference between the slicks across these acquisitions. The development of the oil spills was closely monitored during NORSE2015. Due to the evolving nature of the oil spills and the weathering processes acting upon the spills, our results also indicate an importance of a high synthetic aperture radar sampling rate during an actual oil spill event.

In this paper, we propose a new test statistic for unsupervised change detection in polarimetric radar images. We work with multilook complex covariance matrix data, whose underlying model is assumed to be the scaled complex Wishart distribution. We use the complex-kind Hotelling-Lawley trace (HLT) statistic for measuring the similarity of two covariance matrices. The distribution of the HLT statistic is approximated by a Fisher-Snedecor distribution, which is used to define the significance level of a false alarm rate regulated change detector. Experiments on simulated and real PolSAR data sets demonstrate that the proposed change detection method gives detection rates and error rates that are comparable with the generalized likelihood ratio test.

In this paper, we present an unsupervised change detection method for polarimetric synthetic aperture radar (Pol-SAR) images based on the relaxed Wishart distribution. Most polarimetric change detectors assume the Gaussian-based complex Wishart model for multilook covariance matrices, which is only satisfied for homogeneous areas with fully developed speckle and no texture. Liu et al. recently proposed a new change detection algorithm under the multilook product model (MPM) to describe the heterogeneous clutters. The improvement has come at the expense of higher computational cost since the similarity measure is based on more advanced models accounting for texture, and they contain some mathematical special functions that is difficult to evaluate such similarity measures. In this paper, we will demonstrate the ability of the relaxed Wishart distribution for textured change detection analysis. Change results on simulated and real data demonstrate the effectiveness of the algorithm.

In this paper, we further analyse the problem that polarimetric target decomposition methods in general have more physical parameters than equations, making the decomposition under-determined and hence have no unique solution. The common approach to get around this problem is to make certain assumptions, thus fixing one or more parameters, allowing the other free parameters to be solved from the set of expressions. We recently showed how to obtain additional information from fourth-order statistics to find a unique solution to model-based polarimetric decompositions ([1]). We previously showed a fourth-order unique solution that was valid only for Gaussian data, and indicated that non-Gaussian data led to an over-estimation in many of the parameters. This work describes our new method to obtain a generic textured data solution through an optimisation approach and presents preliminary results for a sea ice specific model.

In this paper we present experimental results relative to the vertical structure of snow covered lake ice and sea ice, sensed with X-band radar system operated in a tomographic configuration. The available data are from a Ground-Based SAR campaign carried out over Prestvannet frozen lake and Kattfjord, both in Tromso, Norway. Direct imaging of the vertical structures of the snow and ice layers is achieved by focusing the signal from a 2D synthetic array in the 3D space. By making use of a priori information about the depth of snow and ice, the refractive index of snow and sea ice/lake ice is estimated from a single polarization tomographic measurement, and the results are in good agreement with previous experimental results. We have also shown that air bubbles in low salinity ice are the main contributors for the backscatter signal from ice.

In this paper, we compare satellite-borne Cand X-band synthetic aperture radar (SAR) data for marine oil spill observation. During large-scale oil-on-water exercises in the North Sea, quad-polarization Radarsat-2 (C-band) and dual-polarization TerraSAR-X (X-band) data were acquired with temporal distances of less than 24 min. The objective is to characterize and quantify differences in the Radarsat-2 and TerraSAR-X measurements. Three scene pairs are compared in terms of data quality and signal characteristics, including statistical properties and selected multipolarization (HH, VV) parameters. The signal characteristics are also compared among low-backscatter features of various origin within the individual pairs. No viable argument for selecting one sensor above the other is identified in the data quality study. In the statistical analysis, investigation of logcumulants indicates a larger deviation from Gaussian statistics in the TerraSAR-X data compared with Radarsat-2 measurements. Log-cumulant diagrams are also shown to be a useful tool for discrimination between oil spills and a simulated biogenic slick in both sensors. Multipolarization features show enhanced slick-sea contrasts and a better discrimination between mineral oil spills and other low-backscatter features in Radarsat-2 compared with TerraSAR-X. The presence of a non-Bragg scattering component in the data is revealed for both sensors. The relative contribution of non-Bragg scattering to the total backscatter is found to be higher in the TerraSAR-X data than in the Radarsat-2 data. In general, the non-Bragg component is found to account for a larger part of the backscatter in slick-covered areas compared with clean sea.

In recent years, we have presented many algorithms for polarimetric SAR image segmentation that show the continually improving developments in the field. However, there are two distinct and divergent approaches - one using highly flexible textured models for the covariance matrix statistics (such as the Wishart, K-Wishart, and U-distribution), and the other using simple features extracted from such data (the Extended Polarimetric Feature Space method). In this study we will present a summary and comparison of both approaches and discuss the pros and cons for each with respect to image segmentation applications.

In this paper we present experimental results relative to the vertical structure of snow covered sea ice, sensed with X-band microwaves in a tomographic configuration. The available data are from a Ground-Based SAR campaign carried out by a team from the IETR, University of Rennes 1 in March 2013, over Kattfjord, Tromso, Norway, in collaboration with members of the EO lab at University of Tromso. Direct imaging of the vertical structures of the snow and sea ice layers is achieved by focusing the signal from a 2D synthetic array in the 3D space. The effect of propagation velocity in the focusing process is investigated and the tomograms are effectively corrected for this effect. The vertical structure of the medium reveals a strong response from sea ice. The effect of Brewster angle on the appearance of the tomograms is also investigated. Tomograms at two different polarizations are investigated and compared with dual-pol TerraSAR-X measurements.

This paper presents a processing chain for the change detection of Arctic glaciers from multitemporal multipolarization synthetic aperture radar (SAR) images. We produce terrain-corrected multilook complex covariance data by including the effects of topography on both geolocation and SAR radiometry as well as azimuth slope variations on polarization signature. An unsupervised contextual non-Gaussian clustering algorithm is employed for the segmentation of each terrain-corrected polarimetric SAR image and subsequently labeled with the aid of ground-truth data into glacier facies. We demonstrate the consistency of the segmentation algorithm by characterizing the expected random error level for different SAR acquisition conditions. This allows us to determine whether an observed variation is statistically significant and therefore can be used for the postclassification change detection of Arctic glaciers. Subsequently, the average classified images of succeeding years are compared, and changes are identified as the detected differences in the location of boundaries between glacier facies. In the current analysis, a series of dual-polarization C-band ENVISAT ASAR images over the Kongsvegen glacier, Svalbard, is used for demonstration.

A statistical model for multilook polarimetric radar data is presented where the polarimetric channels are associated with individual texture variables having potentially different statistical properties. The feasibility of producing closed-form probability density functions under certain restrictions is outlined. Mellin kind statistics is derived under various assumptions on the texture variables, and the potential for model fit assessment and hypothesis testing in the Mellin domain is demonstrated. Application to real data proves the usefulness of the analytic approach.

In this paper, we study surface slick characterization in polarimetric C-band synthetic aperture radar (SAR) data. The objective is to identify the most powerful multipolarization SAR descriptors for mineral oil spill versus biogenic slick discrimination. A systematic comparison of eight well-known multipolarization features is provided. The analysis is performed on data that we collected during a large-scale oil spill exercise at the Frigg field situated northwest of Stavanger, in June 2011. Controlled oil spills and simulated look-alikes were simultaneously captured within fine quad-polarization Radarsat-2 acquisitions during this experiment. Multipolarization features derived from only the copolarized complex scattering coefficients are explored. We find that the two most powerful multipolarization features extracted from this data set are the geometric intensity, measuring the combined intensity based on the determinant of the coherency matrix, and the real part of the copolarization cross product, which is related to the scattering behavior of the target. We show that these two features can distinguish between the simulated biogenic slicks and mineral oil types such as Balder and Oseberg blend, and that the discriminative power seems to be persistent with time.

In this paper we propose a new test statistic for unsupervised change detection in polarimetric synthetic aperture radar (Pol-SAR) data. We work with multilook complex (MLC) covariance matrix data, whose underlying model is assumed to be the scaled complex Wishart distribution. We use the complex kind Hotelling-Lawley (HL) trace statistic for measuring the similarity of two covariance matrices. The sampling distribution of the HL trace is approximated by a Fisher-Snedecor distribution, which is used to define the significance level of a constant false alarm rate change detector. The performance of the proposed method is tested on simulated and real PolSAR data sets and compared to the likelihood ratio test statistic.

This work extends upon our simple feature-based multi-channel SAR segmentation method to incorporate highly desirable statistical properties into a computationally simple approach. The desirable properties include Markov random field contextual smoothing and goodness-of-fit testing to automatically obtain the significant number of classes. To achieve this we need to find an explicit class model to fit these non-Gaussian, non-symmetric or skewed feature space clusters. We take the skewed scale mixture of Gaussian scheme to model our classes and approximate it by a number of constrained Gaussians, thereby retaining much of the speed and simplicity of the original feature space method. The algorithm will be demonstrated on a real data and compared to an automatic Gaussian model.

This paper proposes a novel unsupervised, non-Gaussian, and contextual segmentation method that combines an advanced statistical distribution with spatial contextual information for multilook polarimetric synthetic aperture radar (PolSAR) data. This extends on previous studies that have shown the added value of both non-Gaussian modeling and contextual smoothing individually or for intensity channels only. The method is based on a Markov random field (MRF) model that integrates aK-Wishart distribution for the PolSAR data statistics conditioned to each image cluster and a Potts model for the spatial context. Specifically, the proposed algorithm is constructed based upon the stochastic expectation maximization (SEM) algorithm. A new formulation of SEM is developed to jointly perform clustering of the data and parameter estimation of theK-Wishart distribution and the MRF model. Experiments on simulated and real PolSAR data demonstrate the added value of using an appropriate statistical representation, in combination with contextual smoothing.

This paper investigates the impact of terrain correction on change detection results. We firstly assess the effects of topography on radar brightness and show how we can produce the radiometrically terrain corrected multilook complex (MLC) covariance data. Next, changes on radar cross section as a function of polarization states due to azimuth slope variations will be studied. Finally experimental results on glacier dataset are shown by focusing on change detection results before and after terrain corrections.

The previously proposed multi-texture model for multi-looked PolSAR data statistics [1] is hereby implemented into an advanced statistical clustering algorithm and tested on several real PolSAR images. The multi-texture model is based on the product model for SAR statistics, yet allows the possibility of different texture parameters for the co-polarized (co-pol) and cross-polarized (cross-pol) channels. The implementation automatically determines the most appropriate texture model between the proposed “dual-texture” model and the traditional “scalar-texture” model. The clustering algorithm is implemented as a multi-texture version of [2]. It incorporates the flexible U-distribution, contextual smoothing with Markov random fields, and determines the number of classes with goodness-of-fit tests. The real SAR examples indicate that multi-texture is not generally required and we discuss the possible mis-interpretation of multi-texture in alternative window-based estimation methods, due to mixing of different polarimetric classes.

The Shannon entropy has previously been used in the analysis of polarimetric synthetic aperture radar (PolSAR) data, assuming a complex Gaussian distribution for the scattering vector. According to the maximum entropy characterization theorem, the deviation from Gaussian statistics can be quantified as a reduction of the Shannon entropy. This paper derives the Shannon entropy within the framework of the non- Gaussian, multivariate product model and discusses information theoretical interpretations. An experiment demonstrates the information contents associated with the non-Gaussian entropy component.

In this study a decision fusion strategy is proposed for a multisensor data fusion process using optical and polarimetric multifrequency synthetic aperture radar data for forest classification. Instead of utilizing one classifier for all available features, grouped features are classified by using individual classifiers. A qualified majority voting (QMV) consensual rule, derived from the confusion matrix is utilized in the subsequent fusion process to combine decisions. With this approach, more consistent results are obtained than using all features as a stacked vector and maximum likelihood classification (MLC).

Single-polarization (VV or HH) C-band synthetic aperture radar (SAR) sensors have conventionally been utilized in remote sensing of marine oil pollution. This paper examines the capability of combining the complex VV and HH channels in C-band and X-band SAR for oil spill characterization and for discrimination between mineral oil spills and biogenic slicks. The two frequency bands are evaluated from a theoretical point of view and subsequently experimentally compared using a truly unique data set, consisting of quasi-simultaneous Radarsat-2 and TerraSAR-X data acquired during the June 2011 oil-on-water exercise in the North Sea. Multi-polarization features for the two frequencies are compared based on classification results. A potential for discriminating biogenic films from mineral oil slicks is found. The analysis shows that some slicks have internal zones that correlate well with expected thickness variations.

A clustering method that combines an advanced statistical distribution with spatial contextual information is proposed for multilook polarimetric synthetic aperture radar (PolSAR) data. It is based on a Markov random field (MRF) model that integrates a K-Wishart distribution for the PolSAR data statistics conditioned to each image cluster and a Potts model for the spatial context. Specifically, the proposed algorithm is constructed based upon the expectation maximization (EM) algorithm. A new formulation of EM is developed to jointly address parameter estimation in the K-Wishart distribution and the spatial context model, and also minimization of the energy function. Experiments are presented with simulated and real quad-pol L-band data.

A statistical model for multilook polarimetric radar data is presented where the polarimetric channels are associated with individual texture variables having potentially different statistical properties. The feasibility of producing closed form probability density functions under certain restrictions is out lined. Mellin kind statistics are derived under various assumptions on the texture variables, and the potential for model fit assessment and hypothesis testing in the Mellin domain is demonstrated. Application to real data proves the usefulness of the analytic approach.

The advent of polarimetric synthetic aperture radar has spurred a growing interest in statistical models for complex-valued covariance matrices, which is the common representation of multilook polarimetric radar images. In this paper, we respond to an emergent need by proposing statistical tests for the simple and composite goodness-of-fit (GoF) problem for a class of compound matrix distributions. The tests are based on Mellin-kind matrix cumulants. These are derived from a novel characteristic function for positive definite Hermitian random matrices, defined in terms of a matrix-variate Mellin transform instead of the conventional Fouriér transform, and belong to a new framework for statistical analysis of multilook polarimetric radar data recently introduced by the authors. The cumulant-based tests are easy to compute, and the asymptotic sampling distribution of the test statistic is chi-square distributed in the simple hypothesis case. Under the composite hypothesis, the sampling distribution is obtained by Monte Carlo simulations. We evaluate the power of the proposed GoF tests with simulated data. We also use them to assess the fit of several matrix distributions to real data acquired by Radarsat-2 in fine-quad polarization mode.

In this paper, we propose to use a matrix-variate Mellin transform in the statistical analysis of multilook polarimetric radar data. The domain of the transform integral is the cone of complex positive definite matrices, which allows for transformation of the distributions used to model the polarimetric covariance and coherency matrix. Based on the matrix-variate Mellin transform, an alternative characteristic function is defined, from which we can retrieve a new kind of matrix log-moments and log-cumulants. It is demonstrated that the matrix log-cumulants are of great value to analysis of polarimetric radar data, and that they can be used to derive estimators for the distribution parameters with low bias and variance.

This paper presents an automatic image segmentation method for polarimetric synthetic aperture radar data. It utilizes the full polarimetric information and incorporates texture by modeling with a non-Gaussian distribution for the complex scattering coefficients. The modeling is based upon the well-known product model, with a Gamma-distributed texture parameter leading to the K-Wishart model for the covariance matrix. The automatic clustering is achieved through a finite mixture model estimated with a modified expectation maximization algorithm. We include an additional goodness-of-fit test stage that allows for splitting and merging of clusters. This not only improves the model fit of the clusters, but also dynamically selects the appropriate number of clusters. The resulting image segmentation depicts the statistically significant clusters within the image. A key feature is that the degree of sub-sampling of the input image will affect the detail level of the clustering, revealing only the major classes or a variable level of detail. Real-world examples are shown to demonstrate the technique.

In this paper, we consider statistical analysis of PolSAR data in the framework of the multivariate product model. The complex scattering vector is here considered as a double stochastic circular Gaussian variable, in which the variance is linearly scaled by a common stochastic scaling factor z. The scaling factor is associated with texture. We discuss various parametric probability density functions for z, and indicate how model parameters can be estimated from data by a simple moment based method. Experimental analysis shows that for some surface covers, certain texture distributions fit better than others. Then, polarimetric covariance matrix data analysis is addressed in the framework of product models, and we propose a processing scheme which perform image segmentation using a stochastic EM approach.

This paper addresses estimation of the equivalent number of looks (ENL), an important parameter in statistical modeling of multilook synthetic aperture radar (SAR) images. Two new ENL estimators are discovered by looking at certain moments of the multilook polarimetric covariance matrix, which is commonly used to represent multilook polarimetric SAR (PolSAR) data, and assuming that the covariance matrix is complex Wishart distributed. First, a second-order trace moment provides a polarimetric extension of the ENL definition and also a matrix-variate version of the conventional ENL estimator. The second estimator is obtained from the log-determinant matrix moment and is also shown to be the maximum likelihood estimator under the Wishart model. It proves to have much lower variance than any other known ENL estimator, whether applied to single-polarization or PolSAR data. Moreover, this estimator is less affected by texture and thus provides more accurate results than other estimators should the assumption of Gaussian statistics for the complex scattering coefficients be violated. These are the first known estimators to use the full covariance matrix as input, rather than individual intensity channels, and therefore to utilize all the statistical information available. We finally demonstrate how an ENL estimate can be computed automatically from the empirical density of small sample estimates calculated over a whole scene. We show that this method is more robust than procedures where the estimate is calculated in a manually selected region of interest.

We present two new estimators for the equivalent number number of looks (ENL) designed for polarimetric synthetic aperture radar data. These are the first known estimators to utilise the full multilook polarimetric covariance matrix, and both are derived from moments of the Wishart distribution. The first estimator is obtained from the second-order trace moment, which provides a multivariate generalisation of the conventional ENL definition. The second estimator is found from the log-determinant moment, and is also shown to be the maximum likelihood estimator. The latter estimator proves to have superior statistical properties to any other known ENL estimator. These properties are decisive for the good performance obtained with a proposed procedure for unsupervised estimation. We demonstrate on airborne AIRSAR data how a robust ENL estimate can be extracted from the distribution of small sample estimates collected over the whole scene.

We apply a polarimetric classification scheme to Envisat Alternating Polarisation mode ASAR images and compare to ground truth data. By analysing images with a range of acquisition conditions and comparing the classification accuracy with the ground truth data, we investigate the influence of the acquisition conditions for glacier facies discrimination. We find that the main influence is the image swath angle, which affects the image pixel geometry. There was no obvious preference for the different polarisation channels, although the dual polarisation classifications were consistently better than single polarisation classifications.

In this paper we present the normal variance-mean mixture model as a framework for analyzing SAR data. The complex envelope of the echo signal is considered as a double stochastic circular Gaussian variable, in which both the variance and the mean are linearly scaled by a stochastic scaling factor Z. We then derive the generalized K amplitude model, and indicate how its parameters can be estimated from data. Some preliminary results show that this model represents the amplitude of SAR data well.

In this paper, we present a generalized Wishart classifier derived from a non-Gaussian model for polarimetric synthetic aperture radar (PolSAR) data. Our starting point is to demonstrate that the scale mixture of Gaussian (SMoG) distribution model is suitable for modeling PolSAR data. We show that the distribution of the sample covariance matrix for the SMoG model is given as a generalization of the Wishart distribution and present this expression in integral form. We then derive the closed-form solution for one particular SMoG distribution, which is known as the multivariateK-distribution. Based on this new distribution for the sample covariance matrix, termed as theK-Wishart distribution, we propose a Bayesian classification scheme, which can be used in both supervised and unsupervised modes. To demonstrate the effect of including non-Gaussianity, we present a detailed comparison with the standard Wishart classifier using airborne EMISAR data.

In this paper, we develop a Wiener-type speckle filter that operates in the stationary wavelet domain. We denote it as thestationarywavelet-domainWiener(SWW) speckle filter. We assume that both the speckle-free image and the speckle contribution have spatial correlations and utilize well-established models for the power density spectrum of the radar cross section to estimate the autospectra that define the filter. It turns out that the filter is independent of the wavelet-domain scale level, i.e., the filter is the same at all scale levels. The SWW filter works on nonoverlapping blocks in the wavelet domain, which are obtained by a quadtree algorithm. Due to the dyadic support of the wavelet coefficients, a natural smoothing is carried out on the boundaries between neighboring blocks, and no visual boundary effects can be observed. The SWW filter is unbiased and shows good performance in despeckling synthetic aperture radar (SAR) images. It smooths homogeneous areas while preserving textured areas and point scatterers. In contrast to most other speckle filters, the SWW filter requires the SAR data to be given in single-look complex form.

In this paper we present a generalised Wishart classifier derived from a non-Gaussian model for polarimetric synthetic aperture radar (POLSAR) data. Our starting point is to demonstrate that the scale mixture of Gaussian (SMoG) distribution model is suitable for modelling POLSAR data. We show that the distribution of the sample covariance matrix for the SMoG model is given as a generalisation of the Wishart distribution, and present this expression in integral form. We then derive the closed form solution for one particular SMoG distribution, known as the multivariate K-distribution. Based on this new distribution, termed the K-Wishart distribution, we propose a Bayesian classification scheme, which can be used in both supervised and unsupervised mode. Modelling and classification is tested on airborne EMISAR data.

We propose a new kernel-based data transformation technique. It is founded on the principle of maximum entropy (MaxEnt) preservation, hence named kernel MaxEnt. The key measure is Renyi's entropy estimated via Parzen windowing. We show that kernel MaxEnt is based on eigenvectors, and is in that sense similar to kernel PCA, but may produce strikingly different transformed data sets. An enhanced spectral clustering algorithm is proposed, by replacing kernel PCA by kernel MaxEnt as an intermediate step. This has a major impact on performance.

In this letter, we discuss the multivariate Laplace probability model in the context of a normal variance mixture model. We briefly review the derivation of the probability density function (pdf) and discuss a few important properties. We then present two methods for estimating its parameters from data and include an example of usage, where we apply the model to represent the statistics of the discrete Fourier transform coefficients of a speech signal. Since the pdf is given in closed form, and the model parameters can be easily obtained, this distribution may be useful for representing multivariate, sparsely distributed data, with mutually dependent components.

In this paper, a new statistical model for representing the amplitude statistics of ultrasonic images is presented. The model is called the Rician inverse Gaussian (RiIG) distribution, due to the fact that it is constructed as a mixture of the Rice distribution and the Inverse Gaussian distribution. The probability density function (pdf) of the RiIG model is given in closed form as a function of three parameters. Some theoretical background on this new model is discussed, and an iterative algorithm for estimating its parameters from data is given. Then, the appropriateness of the RiIG distribution as a model for the amplitude statistics of medical ultrasound images is experimentally studied. It is shown that the new distribution can fit to the various shapes of local histograms of linearly scaled ultrasound data better than existing models. A log-likelihood cross-validation comparison of the predictive performance of the RiIG, the K, and the generalized Nakagami models turns out in favor of the new model. Furthermore, a maximum a posteriori (MAP) filter is developed based on the RiIG distribution. Experimental studies show that the RiIG MAP filter has excellent filtering performance in the sense that it smooths homogeneous regions, and at the same time preserves details.

We develop a novel classifier in a kernel feature space defined by the eigenspectrum of the Laplacian data matrix. The classification cost function is derived from a distance measure between probability densities. The Laplacian data matrix is obtained based on a training set, while test data is mapped to the kernel space using the Nystrom routine. In that space, the test data is classified based on the angle between the test point and the training data class means. We illustrate the performance of the new classifier on synthetic and real data.

In this paper, we introduce a new statistical distribution for modeling non-Rayleigh amplitude statistics, which we have called the Rician inverse Gaussian (RiIG) distribution. It is a mixture of the Rice distribution and the inverse Gaussian distribution. The probability density function (pdf) is given in closed form as a function of three parameters. This makes the pdf very flexible in the sense that it may be fitted to a variety of shapes, ranging from the Rayleigh-shaped pdf to a noncentral chi2-shaped pdf. The theoretical basis of the new model is quite thoroughly discussed, and we also give two iterative algorithms for estimating its parameters from data. Finally, we include some modeling examples, where we have tested the ability of the distribution to represent locale amplitude histograms of linear medical ultrasound data and single-look synthetic aperture radar data. We compare the goodness of fit of the RiIG model with that of the K model, and, in most cases, the new model turns out as a better statistical model for the data. We also include a series of log-likelihood tests to evaluate the predictive performance of the proposed model.

We discuss a new information-theoretic framework for spectral clustering that is founded on the recently introduced information cut. A novel spectral clustering algorithm is proposed, where the clustering solution is given as a linearly weighted combination of certain top eigenvectors of the data affinity matrix. The information cut provides us with a theoretically well-defined graph-spectral cost function, and also establishes a close link between spectral clustering, and non-parametric density estimation. As a result, a natural criterion for creating the data affinity matrix is provided. We present preliminary clustering results to illustrate some of the properties of our algorithm, and we also make comparative remarks.

A new distance measure between probability density functions (pdfs) is introduced, which we refer to as the Laplacian pdf dis- tance. The Laplacian pdf distance exhibits a remarkable connec- tion to Mercer kernel based learning theory via the Parzen window technique for density estimation. In a kernel feature space defined by the eigenspectrum of the Laplacian data matrix, this pdf dis- tance is shown to measure the cosine of the angle between cluster mean vectors. The Laplacian data matrix, and hence its eigenspec- trum, can be obtained automatically based on the data at hand, by optimal Parzen window selection. We show that the Laplacian pdf distance has an interesting interpretation as a risk function connected to the probability of error. 1 Introduction In recent years, spectral clustering methods, i.e. data partitioning based on the eigenspectrum of kernel matrices, have received a lot of attention [1, 2]. Some unresolved questions associated with these methods are for example that it is not always clear which cost function that is being optimized and that is not clear how to construct a proper kernel matrix. In this paper, we introduce a well-defined cost function for spectral clustering. This cost function is derived from a new information theoretic distance measure between cluster pdfs, named the Laplacian pdf distance. The information theoretic/spectral duality is established via the Parzen window methodology for density estimation. The resulting spectral clustering cost function measures the cosine of the angle between cluster mean vectors in a Mercer kernel feature space, where the feature space is determined by the eigenspectrum of the Laplacian matrix. A principled approach to spectral clustering would be to optimize this cost function in the feature space by assigning cluster memberships. Because of space limitations, we leave it to a future paper to present an actual clustering algorithm optimizing this cost function, and focus in this paper on the theoretical properties of the new measure. Corresponding author. Phone: (+47) 776 46493. Email: [email protected] An important by-product of the theory presented is that a method for learning the Mercer kernel matrix via optimal Parzen windowing is provided. This means that the Laplacian matrix, its eigenspectrum and hence the feature space mapping can be determined automatically. We illustrate this property by an example. We also show that the Laplacian pdf distance has an interesting relationship to the probability of error. In section 2, we briefly review kernel feature space theory. In section 3, we utilize the Parzen window technique for function approximation, in order to introduce the new Laplacian pdf distance and discuss some properties in sections 4 and 5. Section 6 concludes the paper. 2 Kernel Feature Spaces Mercer kernel-based learning algorithms [3] make use of the following idea: via a nonlinear mapping : Rd F, x (x) (1) the data x1, . . . , xN Rd is mapped into a potentially much higher dimensional feature space F. For a given learning problem one now considers the same algorithm in F instead of in Rd, that is, one works with (x1),...,(xN) F. Consider a symmetric kernel function k(x, y). If k : C C R is a continuous kernel of a positive integral operator in a Hilbert space L2(C) on a compact set C Rd, i.e. L2(C) : k(x,y)(x)(y)dxdy 0, (2) C then there exists a space F and a mapping : Rd F, such that by Mercer's theorem [4] NF k(x, y) = (x), (y) = ii(x)i(y), (3) i=1 where , denotes an inner product, the i's are the orthonormal eigenfunctions of the kernel and NF [3]. In this case (x) = [ 11(x), 22(x), . . . ]T , (4) can potentially be realized. In some cases, it may be desirable to realize this mapping. This issue has been addressed in [5]. Define the (N N) Gram matrix, K, also called the affinity, or kernel matrix, with elements Kij = k(xi, xj), i, j = 1, . . . , N . This matrix can be diagonalized as ET KE = , where the columns of E contains the eigenvectors of K and is a diagonal matrix containing the non-negative eigenvalues ~ 1, . . . , ~ N , ~ 1 ~N. In [5], it was shown that the eigenfunctions and eigenvalues of (4) can ~ be approximated as j j (xi) Neji, j , where e N ji denotes the ith element of the jth eigenvector. Hence, the mapping (4), can be approximated as (xi) [ ~1e1i,..., ~NeNi]T. (5) Thus, the mapping is based on the eigenspectrum of K. The feature space data set may be represented in matrix form as NN = [(x1), . . . , (xN )]. Hence, = 1 2 ET . It may be desirable to truncate the mapping (5) to C-dimensions. Thus, T only the C first rows of are kept, yielding ^ . It is well-known that ^ K = ^ ^ is the best rank-C approximation to K wrt. the Frobenius norm [6]. The most widely used Mercer kernel is the radial-basis-function (RBF) k(x, y) = exp -||x - y||2 . (6) 22 3 Function Approximation using Parzen Windowing Parzen windowing is a kernel-based density estimation method, where the resulting density estimate is continuous and differentiable provided that the selected kernel is continuous and differentiable [7]. Given a set of iid samples {x1,...,xN} drawn from the true density f (x), the Parzen window estimate for this distribution is [7] N ^ 1 f (x) = W N 2 (x, xi), (7) i=1 where W2 is the Parzen window, or kernel, and 2 controls the width of the kernel. The Parzen window must integrate to one, and is typically chosen to be a pdf itself with mean xi, such as the Gaussian kernel 1 W2 (x, xi) = exp , (8) d -||x - xi||2 (22) 2 22 which we will assume in the rest of this paper. In the conclusion, we briefly discuss the use of other kernels. Consider a function h(x) = v(x)f (x), for some function v(x). We propose to estimate h(x) by the following generalized Parzen estimator N ^ 1 h(x) = v(xi)W N 2 (x, xi). (9) i=1 This estimator is asymptotically unbiased, which can be shown as follows 1 N Ef v(xi)W N 2 (x, xi) = v(z)f (z)W2 (x, z)dz = [v(x)f (x)] W2(x), i=1 (10) where Ef () denotes expectation with respect to the density f(x). In the limit as N and (N) 0, we have lim [v(x)f (x)] W2(x) = v(x)f(x). (11) N (N )0 Of course, if v(x) = 1 x, then (9) is nothing but the traditional Parzen estimator of h(x) = f (x). The estimator (9) is also asymptotically consistent provided that the kernel width (N ) is annealed at a sufficiently slow rate. The proof will be presented in another paper. Many approaches have been proposed in order to optimally determine the size of the Parzen window, given a finite sample data set. A simple selection rule was proposed by Silverman [8], using the mean integrated square error (MISE) between the estimated and the actual pdf as the optimality metric: 1 d+4 opt = X 4N -1(2d + 1)-1 , (12) where d is the dimensionality of the data and 2 = d-1 , where are the X i Xii Xii diagonal elements of the sample covariance matrix. More advanced approximations to the MISE solution also exist. 4 The Laplacian PDF Distance Cost functions for clustering are often based on distance measures between pdfs. The goal is to assign memberships to the data patterns with respect to a set of clusters, such that the cost function is optimized. Assume that a data set consists of two clusters. Associate the probability density function p(x) with one of the clusters, and the density q(x) with the other cluster. Let f (x) be the overall probability density function of the data set. Now define the f -1 weighted inner product between p(x) and q(x) as p, q f p(x)q(x)f-1(x)dx. In such an inner product space, the Cauchy-Schwarz inequality holds, that is, p, q 2 q, q . Based on this discussion, an information theoretic distance f p, p f f measure between the two pdfs can be expressed as p, q D f L = - log 0. (13) p, p q, q f f We refer to this measure as the Laplacian pdf distance, for reasons that we discuss next. It can be seen that the distance DL is zero if and only if the two densities are equal. It is non-negative, and increases as the overlap between the two pdfs decreases. However, it does not obey the triangle inequality, and is thus not a distance measure in the strict mathematical sense. We will now show that the Laplacian pdf distance is also a cost function for clus- tering in a kernel feature space, using the generalized Parzen estimators discussed in the previous section. Since the logarithm is a monotonic function, we will derive the expression for the argument of the log in (13). This quantity will for simplicity be denoted by the letter "L" in equations. Assume that we have available the iid data points {xi}, i = 1,...,N1, drawn from p(x), which is the density of cluster C1, and the iid {xj}, j = 1, . . ., N2, drawn from q(x), the density of C2. Let h(x) = f - 12 (x)p(x) and g(x) = f - 12 (x)q(x). Hence, we may write h(x)g(x)dx L = . (14) h2(x)dx g2(x)dx We estimate h(x) and g(x) by the generalized Parzen kernel estimators, as follows N1 N2 ^ 1 1 h(x) = f - 12 (xi)W f - 12 (xj )W N 2 (x, xi ), ^ g(x) = 2 (x, xj ). (15) 1 N2 i=1 j=1 The approach taken, is to substitute these estimators into (14), to obtain N N 1 1 1 2 h(x)g(x)dx f - 12 (xi)W f - 12 (xj )W N 2 (x, xi ) 2 (x, xj ) 1 N2 i=1 j=1 N 1 1 ,N2 = f - 12 (xi)f - 12 (xj ) W N 2 (x, xi )W2 (x, xj )dx 1N2 i,j=1 N 1 1 ,N2 = f - 12 (xi)f - 12 (xj )W N 22 (xi, xj ), (16) 1N2 i,j=1 where in the last step, the convolution theorem for Gaussians has been employed. Similarly, we have N 1 1 ,N1 h2(x)dx f - 12 (xi)f - 12 (xi )W N 2 22 (xi, xi ), (17) 1 i,i =1 N 1 2 ,N2 g2(x)dx f - 12 (xj)f - 12 (xj )W N 2 22 (xj , xj ). (18) 2 j,j =1 Now we define the matrix Kf , such that Kf = K ij f (xi, xj ) = f - 1 2 (xi)f - 12 (xj )K(xi, xj ), (19) where K(xi, xj ) = W22 (xi, xj) for i, j = 1, . . . , N and N = N1 + N2. As a consequence, (14) can be re-written as follows N1,N2 Kf (xi, xj) L = i,j=1 (20) N1,N1 K K i,i =1 f (xi, xi ) N2,N2 j,j =1 f (xj , xj ) The key point of this paper, is to note that the matrix K = Kij = K(xi, xj), i, j = 1, . . . , N , is the data affinity matrix, and that K(xi, xj) is a Gaussian RBF kernel function. Hence, it is also a kernel function that satisfies Mercer's theorem. Since K(xi, xj) satisfies Mercer's theorem, the following by definition holds [4]. For any set of examples {x1,...,xN} and any set of real numbers 1,...,N N N ijK(xi, xj) 0, (21) i=1 j=1 in analogy to (3). Moreover, this means that N N N N ijf - 12 (xi)f - 12 (xj )K(xi, xj) = ijKf (xi, xj) 0, (22) i=1 j=1 i=1 j=1 hence Kf (xi, xj ) is also a Mercer kernel. Now, it is readily observed that the Laplacian pdf distance can be analyzed in terms of inner products in a Mercer kernel-based Hilbert feature space, since Kf (xi, xj) = f (xi), f (xj) . Consequently, (20) can be written as follows N1,N2 f (xi), f (xj) L = i,j=1 N1,N1 i,i =1 f (xi), f (xi ) N2,N2 j,j =1 f (xj ), f (xj ) 1 N1 N2 N i=1 f (xi ), 1 N j=1 f (xj ) = 1 2 1 N1 N1 N2 N2 N f (xi), 1 f (xi ) 1 f (xj ), 1 f (xj ) 1 i=1 N1 i =1 N2 j=1 N2 j =1 m1 , m2 = f f = cos (m , m ), (23) ||m 1f 2f 1f ||||m2f || where m Ni i = 1 f N f (xl), i = 1, 2, that is, the sample mean of the ith cluster i l=1 in feature space. This is a very interesting result. We started out with a distance measure between densities in the input space. By utilizing the Parzen window method, this distance measure turned out to have an equivalent expression as a measure of the distance between two clusters of data points in a Mercer kernel feature space. In the feature space, the distance that is measured is the cosine of the angle between the cluster mean vectors. The actual mapping of a data point to the kernel feature space is given by the eigendecomposition of Kf , via (5). Let us examine this mapping in more detail. 1 Note that f 2 (xi) can be estimated from the data by the traditional Parzen pdf estimator as follows N 1 1 f 2 (xi) = W (xi, xl) = di. (24) N 2 f l=1 Define the matrix D = diag(d1, . . . , dN ). Then Kf can be expressed as Kf = D- 12 KD- 12 . (25) Quite interestingly, for 2 = 22, this is in fact the Laplacian data matrix. 1 f The above discussion explicitly connects the Parzen kernel and the Mercer kernel. Moreover, automatic procedures exist in the density estimation literature to opti- mally determine the Parzen kernel given a data set. Thus, the Mercer kernel is also determined by the same procedure. Therefore, the mapping by the Laplacian matrix to the kernel feature space can also be determined automatically. We regard this as a significant result in the kernel based learning theory. As an example, consider Fig. 1 (a) which shows a data set consisting of a ring with a dense cluster in the middle. The MISE kernel size is opt = 0.16, and the Parzen pdf estimate is shown in Fig. 1 (b). The data mapping given by the corresponding Laplacian matrix is shown in Fig. 1 (c) (truncated to two dimensions for visualization purposes). It can be seen that the data is distributed along two lines radially from the origin, indicating that clustering based on the angular measure we have derived makes sense. The above analysis can easily be extended to any number of pdfs/clusters. In the C-cluster case, we define the Laplacian pdf distance as C-1 pi, pj L = f . (26) i=1 j=i C pi, pi p f j , pj f In the kernel feature space, (26), corresponds to all cluster mean vectors being pairwise as orthogonal to each other as possible, for all possible unique pairs. 4.1 Connection to the Ng et al. [2] algorithm Recently, Ng et al. [2] proposed to map the input data to a feature space determined by the eigenvectors corresponding to the C largest eigenvalues of the Laplacian ma- trix. In that space, the data was normalized to unit norm and clustered by the C-means algorithm. We have shown that the Laplacian pdf distance provides a 1It is a bit imprecise to refer to Kf as the Laplacian matrix, as readers familiar with spectral graph theory may recognize, since the definition of the Laplacian matrix is L = I - Kf . However, replacing Kf by L does not change the eigenvectors, it only changes the eigenvalues from i to 1 - i. 0 0 (a) Data set (b) Parzen pdf estimate (c) Feature space data Figure 1: The kernel size is automatically determined (MISE), yielding the Parzen estimate (b) with the corresponding feature space mapping (c). clustering cost function, measuring the cosine of the angle between cluster means, in a related kernel feature space, which in our case can be determined automati- cally. A more principled approach to clustering than that taken by Ng et al. is to optimize (23) in the feature space, instead of using C-means. However, because of the normalization of the data in the feature space, C-means can be interpreted as clustering the data based on an angular measure. This may explain some of the success of the Ng et al. algorithm; it achieves more or less the same goal as cluster- ing based on the Laplacian distance would be expected to do. We will investigate this claim in our future work. Note that we in our framework may choose to use only the C largest eigenvalues/eigenvectors in the mapping, as discussed in section 2. Since we incorporate the eigenvalues in the mapping, in contrast to Ng et al., the actual mapping will in general be different in the two cases. 5 The Laplacian PDF distance as a risk function We now give an analysis of the Laplacian pdf distance that may further motivate its use as a clustering cost function. Consider again the two cluster case. The overall data distribution can be expressed as f (x) = P1p(x) + P2q(x), were Pi, i = 1, 2, are the priors. Assume that the two clusters are well separated, such that for xi C1, f (xi) P1p(xi), while for xi C2, f(xi) P2q(xi). Let us examine the numerator of (14) in this case. It can be approximated as p(x)q(x) dx f (x) p(x)q(x) p(x)q(x) 1 1 dx + dx q(x)dx + p(x)dx. (27) C f (x) f (x) P1 P2 1 C2 C1 C2 By performing a similar calculation for the denominator of (14), it can be shown to be approximately equal to 1 . Hence, the Laplacian pdf distance can be written P1P1 as a risk function, given by 1 1 L P1P2 q(x)dx + p(x)dx . (28) P1 C P2 1 C2 Note that if P1 = P2 = 1 , then L = 2P 2 e, where Pe is the probability of error when assigning data points to the two clusters, that is Pe = P1 q(x)dx + P2 p(x)dx. (29) C1 C2 Thus, in this case, minimizing L is equivalent to minimizing Pe. However, in the case that P1 = P2, (28) has an even more interesting interpretation. In that situation, it can be seen that the two integrals in the expressions (28) and (29) are weighted exactly oppositely. For example, if P1 is close to one, L p(x)dx, while P C e 2 q(x)dx. Thus, the Laplacian pdf distance emphasizes to cluster the most un- C1 likely data points correctly. In many real world applications, this property may be crucial. For example, in medical applications, the most important points to classify correctly are often the least probable, such as detecting some rare disease in a group of patients. 6 Conclusions We have introduced a new pdf distance measure that we refer to as the Laplacian pdf distance, and we have shown that it is in fact a clustering cost function in a kernel feature space determined by the eigenspectrum of the Laplacian data matrix. In our exposition, the Mercer kernel and the Parzen kernel is equivalent, making it possible to determine the Mercer kernel based on automatic selection procedures for the Parzen kernel. Hence, the Laplacian data matrix and its eigenspectrum can be determined automatically too. We have shown that the new pdf distance has an interesting property as a risk function. The results we have derived can only be obtained analytically using Gaussian ker- nels. The same results may be obtained using other Mercer kernels, but it requires an additional approximation wrt. the expectation operator. This discussion is left for future work. Acknowledgments. This work was partially supported by NSF grant ECS- 0300340.

In this paper, we introduce the homomorphic /spl Gamma/-WMAP (wavelet maximum a posteriori) filter, a wavelet-based statistical speckle filter equivalent to the well known /spl Gamma/-MAP filter. We perform a logarithmic transformation in order to make the speckle contribution additive and statistically independent of the radar cross section. Further, we propose to use the normal inverse Gaussian (NIG) distribution as a statistical model for the wavelet coefficients of both the reflectance image and the noise image. We show that the NIG distribution is an excellent statistical model for the wavelet coefficients of synthetic aperture radar images, and we present a method for estimating the parameters. We compare the homomorphic /spl Gamma/-WMAP filter with the /spl Gamma/-MAP filter and and the recently introduced /spl Gamma/-WMAP filter, which are both based on the same statistical assumptions. The homomorphic /spl Gamma/-WMAP filter is shown to have better performance with regard to smoothing homogeneous regions. It may in some cases introduce a small bias, but in our studies it is always less than that introduced by the /spl Gamma/-MAP filter. Further, the speckle removed by the homomorphic /spl Gamma/-WMAP filter has statistics closer to the theoretical model than the speckle contribution removed with the other filters.

In this paper, we introduce a new statistical distribution for modeling nonRayleigh amplitude statistics. We call the new model the Rician Inverse Gaussian (RilG) distribution. The theoretical basis of the model is briefly presented, and we give an EM-type algorithm for estimating its parameters from data. Finally, we include some modeling examples, where we have tested the ability to fit to histograms of linear and log-compressed medical ultrasound data.

We introduce a new statistical distribution for modeling nonRayleigh amplitude statistics. We call the new model the Rician Inverse Gaussian (RiIG) distribution. The theoretical basis of the model is briefly presented, and we give an EM-type algorithm for estimating its parameters from data. Finally, we include some modeling examples, where we have tested the ability to fit to histograms of segments of single-look SAR data.

Generally, the contrast between water and land in SAR images decreases with decreasing incidence angle. Thus, surface water detection by intensity thresholding requires high incidence angle data. In this work we demonstrate a texture based surface water detector that produces accurate results, independently of the incidence angle.

We propose a new clustering algorithm using Renyi's entropy as our similarity metric. The main idea is to assign a data pattern to the cluster, which among all possible clusters, increases its within-cluster entropy the least, upon inclusion of the pattern. We refer to this procedure as differential entropy clustering. Not knowing the true number of clusters in advance, initially a number of small clusters are "seeded" randomly in the data set, labeling a small subset of the data. Thereafter all remaining patterns are labeled by differential entropy clustering. Subsequently, we identify the "worst cluster" by a quantity we name as the between-cluster entropy. Its members are re-clustered, again by differential entropy clustering, reducing the overall number of clusters by one. This procedure is repeated until only two clusters remain. At each step we store the current labels, thus producing a hierarchy of clusters. The between-cluster entropy also enables us to select our final set of clusters in other cluster hierarchy. We demonstrate the clustering algorithm when applied both to artificially created data sets and a real data set.

The use of multiple antennas at the transmitter and receiver results in enormous capacity increase. However, the performance of many current space-time architectures like Vertical Bell Laboratories Layered Space-Time (V-BLAST) are not optimum in the maximum likelihood (ML) sense. We present a new algorithm, which aims at bridging this gap for the V-BLAST system. Both optimum detection and original V-BLAST detection are special cases of this algorithm. In addition, any in-between performance-complexity can be achieved by suitably choosing two parameters. Moreover, with little increase in computational power, a very high improvement in error performance over the original V-BLAST algorithm is achieved.

We study time series prediction capabilities of a new type of artificial neural networks, called dynamical-functional artificial neural networks (D-FANNs). These are two-layer neural systems in which the synaptic weights are "functions" rather than numbers, and where the action of a synapse on a signal passing through it takes place in the form of a scalar product in L/sup 2/ between the functional weight and the signal. The functional weights of the first layer of a D-FANN are modeled as the impulse responses of a set of linear time-invariant differential dynamical systems. We consider only the special discrete case when these impulse responses are the set of discrete signals corresponding to the discrete cosine transform. The time series prediction performance of this D-FANN model is demonstrated on a speech signal with high dynamics.

In noisy environments, a constrained least-squares (CLS) approach is presented to restore images blurred by a Gaussian impulse response, where instead of choosing a global regularization parameter, each point in the signal has its own associated regularization parameter. These parameters are found by constraining the weighted standard deviation of the wavelet transform coefficients on the finest scale of the inverse signal by a function r which is a local measure of the intensity variations around each point of the blurred and noisy observed signal. Border ringing in the inverse solution is proposed decreased by manipulating its wavelet transform coefficients on the finest scales close to the borders. If the noise in the inverse solution is significant, wavelet transform techniques are also applied to denoise the solution. Examples are given for images, and the results are shown to outperform the optimum constrained least-squares solution using a global regularization parameter, both visually and in the mean squared error sense.

The authors have studied the significance of non-Gaussian signal statistics in some synthetic aperture radar (SAR) images of the ocean surface. The study consisted of calculating the amplitude histogram of the returned echoes from the images and comparing these with the Rayleigh- and K/spl nu/-distributions, corresponding to the Gaussian and non-Gaussian statistics, respectively. The images used were some C-band SAR data from the Canadian airborne SAR collected during the NORCSEX'88 campaign and some ERS-1 data collected during the NORCSEX'91 campaign. The analysis of the NORCSEX'88 data included studies of the dependency of the signal statistics on incidence angle and meteorological and imaging conditions. It was found, specifically at small incidence angles, that there was a significant deviation from Gaussian statistics. It was also found that when the wind was blowing against the waves, the deviation from Gaussian statistics was more pronounced than when the wind was blowing in the same direction as the waves were propagating. The study also showed a correlation between the signal statistics and the width of the SAR image spectra. At low incidence angles, this agrees with the interpretation that non-Gaussian statistics may be related to strong widebanded scattering events. However, since non-Gaussian statistics also were observed at incidence angles as high as 50/spl deg/, it is evident that the modulation of the scattering cross section by the long waves is also an important factor. In addition, the analysis of the ERS-1 data showed that to account for the width of the SAR image spectra, an azimuth smearing term, due to short scene coherence time, had to be included. This was in the present work done by modeling the short-coherence-time-smearing as a Gaussian low-pass filter. By this procedure, the authors were able to obtain realistic estimates for the average scene coherence time of the SAR scenes.

We propose in this paper a new unsupervised neural network which is capable of clustering a set of experimental data according to a given generic interpoint similarity measure, and then assign to each new input its appropriate cluster label. The network is able to do this for clusters of any shape, and without knowing in advance the number of clusters to be created. We call this new two-layer network a cluster-detection-and-labeling (CDL) network. In the CDL network the concept of similarity and closeness with regard to distance are combined. Specifically, clusters are represented by a set of prototypes, and the similarities between an input vector and these prototypes are calculated as the inner products of these vectors compared to some thresholds. These thresholds, which depend on the distance between the input vector and the prototype, are calculated in a separate threshold calculating unit. During clustering, the data are cycled through the network several times. At the end of each cycle the clusters are evaluated, and only those with more than a specified number of samples are retained. The others are fed back to be reclustered by an updated network. This process terminates according to a suitable criterion, such as when a prespecified portion of the data are classified. The performance of the CDL network has been compared with that of the winner-take-all (WTA) network for several different cluster structures, since the latter is widely used in cluster analysis applications. These studies demonstrate that the new network performs well for all the tested cluster shapes, also for those cases where the WTA network completely fails.

In this paper, illumination control has been studied as a means of enhancing image features. Such features are points, edges, and shading patterns, which provide important cues for the interpretation of an image of a scene and the recognition of objects present in it. Based on approximate expressions for the reflectance map of Lambertian and general surfaces, a rigorous discussion on how intensity gradients are dependent on the direction of the light is presented. Subsequently, three criteria for the illumination of convex-shaped cylindrical surfaces are given. Two of these, the contrast equalization criterion and the max-min equalization criterion, are developed for optimal illumination of convex polyhedrons. The third, denoted shading enhancement, is applicable for the illumination of convex curved objects. Examples illustrate the merit of the criteria presented.