Allan Aasbjerg Nielsen

A test statistic in the complex Wishart distribution and its application to change detection in polarimetric SAR data

When working with multilook fully polarimetric synthetic aperture radar (SAR) data, an appropriate way of representing the backscattered signal consists of the so-called covariance matrix. For each pixel, this is a 3/spl times/3 Hermitian positive definite matrix that follows a complex Wishart distribution. Based on this distribution, a test statistic for equality of two such matrices and an associated asymptotic probability for obtaining a smaller value of the test statistic are derived and applied successfully to change detection in polarimetric SAR data. In a case study, EMISAR L-band data from April 17, 1998 and May 20, 1998 covering agricultural fields near Foulum, Denmark are used. Multilook full covariance matrix data, azimuthal symmetric data, covariance matrix diagonal-only data, and horizontal-horizontal (HH), vertical-vertical (VV), or horizontal-vertical (HV) data alone can be used. If applied to HH, VV, or HV data alone, the derived test statistic reduces to the well-known gamma likelihood-ratio test statistic. The derived test statistic and the associated significance value can be applied as a line or edge detector in fully polarimetric SAR data also.

Multivariate Alteration Detection (MAD) and MAF Postprocessing in Multispectral, Bitemporal Image Data: New Approaches to Change Detection Studies

Abstract This article introduces the multivariate alteration detection (MAD) transformation which is based on the established canonical correlations analysis. It also proposes using postprocessing of the change detected by the MAD variates using maximum autocorrelation factor (MAF) analysis. The MAD and the combined MAF/MAD transformations are invariant to linear scaling. Therefore, they are insensitive, for example, to differences in gain settings in a measuring device, or to linear radiometric and atmospheric correction schemes. Other multivariate change detection schemes described are principal component type analyses of simple difference images. Case studies with AHVRR and Landsat MSS data using simple linear stretching and masking of the change images show the usefulness of the new MAD and MAF/MAD change detection schemes. Ground truth observations confirm the detected changes. A simple simulation of a no-change situation shows the accuracy of the MAD and MAF/MAD transformations compared to principal components based methods.

Multiset canonical correlations analysis and multispectral, truly multitemporal remote sensing data

This paper describes two- and multiset canonical correlations analysis (CCA) for data fusion, multisource, multiset, or multitemporal exploratory data analysis. These techniques transform multivariate multiset data into new orthogonal variables called canonical variates (CVs) which, when applied in remote sensing, exhibit ever-decreasing similarity (as expressed by correlation measures) over sets consisting of 1) spectral variables at fixed points in time (R-mode analysis), or 2) temporal variables with fixed wavelengths (T-mode analysis). The CVs are invariant to linear and affine transformations of the original variables within sets which means, for example, that the R-mode CVs are insensitive to changes over time in offset and gain in a measuring device. In a case study, CVs are calculated from Landsat Thematic Mapper (TM) data with six spectral bands over six consecutive years. Both Rand T-mode CVs clearly exhibit the desired characteristic: they show maximum similarity for the low-order canonical variates and minimum similarity for the high-order canonical variates. These characteristics are seen both visually and in objective measures. The results from the multiset CCA R- and T-mode analyses are very different. This difference is ascribed to the noise structure in the data. The CCA methods are related to partial least squares (PLS) methods. This paper very briefly describes multiset CCA-based multiset PLS. Also, the CCA methods can be applied as multivariate extensions to empirical orthogonal functions (EOF) techniques. Multiset CCA is well-suited for inclusion in geographical information systems (GIS).

CFAR edge detector for polarimetric SAR images

Finding the edges between different regions in an image is one of the fundamental steps of image analysis, and several edge detectors suitable for the special statistics of synthetic aperture radar (SAR) intensity images have previously been developed. In this paper, a new edge detector for polarimetric SAR images is presented using a newly developed test statistic in the complex Wishart distribution to test for equality of covariance matrices. The new edge detector can be applied to a wide range of SAR data from single-channel intensity data to multifrequency and/or multitemporal polarimetric SAR data. By simply changing the parameters characterizing the test statistic according to the applied SAR data, constant false-alarm rate detection is always obtained. An adaptive filtering scheme is presented, and the distributions of the detector are verified using simulated polarimetric SAR images. Using SAR data from the Danish airborne polarimetric SAR, EMISAR, it is demonstrated that superior edge detection results are obtained using polarimetric and/or multifrequency data compared to using only intensity data.

Kernel Maximum Autocorrelation Factor and Minimum Noise Fraction Transformations

This paper introduces kernel versions of maximum autocorrelation factor (MAF) analysis and minimum noise fraction (MNF) analysis. The kernel versions are based upon a dual formulation also termed Q-mode analysis in which the data enter into the analysis via inner products in the Gram matrix only. In the kernel version, the inner products of the original data are replaced by inner products between nonlinear mappings into higher dimensional feature space. Via kernel substitution also known as the kernel trick these inner products between the mappings are in turn replaced by a kernel function and all quantities needed in the analysis are expressed in terms of this kernel function. This means that we need not know the nonlinear mappings explicitly. Kernel principal component analysis (PCA), kernel MAF, and kernel MNF analyses handle nonlinearities by implicitly transforming data into high (even infinite) dimensional feature space via the kernel function and then performing a linear analysis in that space. Three examples show the very successful application of kernel MAF/MNF analysis to: 1) change detection in DLR 3K camera data recorded 0.7 s apart over a busy motorway, 2) change detection in hyperspectral HyMap scanner data covering a small agricultural area, and 3) maize kernel inspection. In the cases shown, the kernel MAF/MNF transformation performs better than its linear counterpart as well as linear and kernel PCA. The leading kernel MAF/MNF variates seem to possess the ability to adapt to even abruptly varying multi and hypervariate backgrounds and focus on extreme observations.

Model-Based Satellite Image Fusion

A method is proposed for pixel-level satellite image fusion derived directly from a model of the imaging sensor. By design, the proposed method is spectrally consistent. It is argued that the proposed method needs regularization, as is the case for any method for this problem. A framework for pixel neighborhood regularization is presented. This framework enables the formulation of the regularization in a way that corresponds well with our prior assumptions of the image data. The proposed method is validated and compared with other approaches on several data sets. Lastly, the intensity-hue-saturation method is revisited in order to gain additional insight of what implications the spectral consistency has for an image fusion method.

Visualization and unsupervised classification of changes in multispectral satellite imagery

The statistical techniques of multivariate alteration detection, minimum/maximum autocorrelation factors transformation, expectation maximization and probabilistic label relaxation are combined in a unified scheme to visualize and to classify changes in multispectral satellite data. The methods are demonstrated with an example involving bitemporal LANDSAT TM imagery.

Spectral Mixture Analysis: Linear and Semi-parametric Full and Iterated Partial Unmixing in Multi- and Hyperspectral Image Data

As a supplement or an alternative to classification of hyperspectral image data linear and semi-parametric mixture models are considered in order to obtain estimates of abundance of each class or end-member in pixels with mixed membership. Full unmixing based on both ordinary least squares (OLS) and non-negative least squares (NNLS), and the partial unmixing methods orthogonal subspace projection (OSP), constrained energy minimization (CEM) and an eigenvalue formulation alternative are dealt with. The solution to the eigenvalue formulation alternative proves to be identical to the CEM solution. The matrix inversion involved in CEM can be avoided by working on (a subset of) orthogonally transformed data such as signal maximum autocorrelation factors, MAFs, or signal minimum noise fractions, MNFs. This will also cause the partial unmixing result to be independent of the noise isolated in the MAF/MNFs not included in the analysis. CEM and the eigenvalue formulation alternative enable us to perform partial unmixing when we know one desired end-member spectrum only and not the full set of end-member spectra. This is an advantage over full unmixing and OSP. The eigenvalue formulation of CEM inspires us to suggest an iterated CEM scheme. Also the target constrained interference minimized filter (TCIMF) is described. Spectral angle mapping (SAM) is briefly described. Finally, semi-parametric unmixing (SPU) based on a combined linear and additive model with a non-linear, smooth function to represent end-member spectra unaccounted for is introduced. An example with two generated bands shows that both full unmixing, the CEM, the iterated CEM and TCIMF methods perform well. A case study with a 30 bands subset of AVIRIS data shows the utility of full unmixing, SAM, CEM and iterated CEM to more realistic data. Iterated CEM seems to suppress noise better than CEM. A study with AVIRIS spectra generated from real spectra shows (1) that ordinary least squares in this case with one unknown spectrum performs better than non-negative least squares, and (2) that although not fully satisfactory the semi-parametric model gives better estimates of end-member abundances than the linear model.

Computerised image analysis of biocrystallograms originating from agricultural products

Abstract Procedures are presented for computerised image analysis of biocrystallogram images, originating from biocrystallization investigations of agricultural products. The biocrystallization method is based on the crystallographic phenomenon that when adding biological substances, such as plant extracts, to aqueous solutions of dihydrate CuCl2, biocrystallograms with reproducible dendritic crystal structures are formed during crystallisation. The morphological features found in the structures are traditionally applied for visual ranking or classification, e.g. in comparative studies of the effects of farming systems on crop quality. The circular structures contain predominantly a single centre from where ramifications expand in a zonal structure. In previous studies primarily texture analysis was applied, and the images analysed and classified by means of a circular region-of-interest (ROI), i.e. the region specified for analysis. In the present study the objective was to examine how the discriminative information relevant for classification purposes is distributed over the zonal structure, and how the information is affected by the varying location of the crystallisation centre. The texture analysis procedures were applied to a so-called degradation series of 33 images, including seven groups representing discrete ‘treatment levels’. The biocrystallograms were produced over seven consecutive days, on the basis of a single carrot extract degrading while stored at 6°C. This degradation is known to induce systematic changes in morphological features over a number of successive days. The biocrystallograms were scanned at 600 dpi, with 256 grey levels. Eight first-order statistical parameters were calculated for four resolution scales, and 15 second-order parameters for five scales, giving a total of 107 observations for each image. Classification of an individual image was performed by means of stepwise discriminant analysis. Four main types, and several subtypes and sizes of ROI were examined. The 33 images as well as a subset of 21 images were examined. When imposing a restriction on the centre location in the subset, thereby reducing the within-group variance, the scores were markedly improved. Classifications of the total set and the subset showed scores up to 84.8 and 100%, respectively. A number of parameters showed a monotonic relationship with degradation day number. Multiple linear regressions based on up to eight parameters indicated strong relationships, with R2 up to 0.98. It is concluded that the procedures were able to discriminate the seven groups of images, and are applicable for biocrystallization investigations of agricultural products. Perspectives for the application of image analysis are briefly mentioned.

Change detection in polarimetric SAR data and the complex Wishart distribution

When working with multi-look fully polarimetric synthetic aperture radar (SAR) data an appropriate way of representing the backscattered signal consists of the so-called covariance matrix. For each pixel this is a 3/spl times/3 Hermitian, positive definite matrix which follows a complex Wishart distribution. Based on this distribution a test statistic for equality of two such matrices and an associated asymptotic probability for obtaining a smaller value of the test statistic are given and applied to change detection in polarimetric SAR data. In a case study EMISAR L-band data from 17 April 1998 and 20 May 1998 covering agricultural fields near Foulum, Denmark are used. The derived test statistic can be applied as a line or edge detector in fully polarimetric SAR data also.