Torbjørn Eltoft

On the multivariate Laplace distribution

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.

Homomorphic wavelet-based statistical despeckling of SAR images

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.

Estimation of the Equivalent Number of Looks in Polarimetric Synthetic Aperture Radar Imagery

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.

Modeling the amplitude statistics of ultrasonic images

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.

Application of the Matrix-Variate Mellin Transform to Analysis of Polarimetric Radar Images

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.

Independent vector analysis: an extension of ICA to multivariate components

In this paper, we solve an ICA problem where both source and observation signals are multivariate, thus, vectorized signals. To derive the algorithm, we define dependence between vectors as Kullback-Leibler divergence between joint probability and the product of marginal probabilities, and propose a vector density model that has a variance dependency within a source vector. The example shows that the algorithm successfully recovers the sources and it does not cause any permutation ambiguities within the sources. Finally, we propose the frequency domain blind source separation (BSS) for convolutive mixtures as an application of IVA, which separates 6 speeches with 6 microphones in a reverberant room environment.

Classification With a Non-Gaussian Model for PolSAR Data

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 multivariate K-distribution. Based on this new distribution for the sample covariance matrix, termed as the K -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.

The Rician inverse Gaussian distribution: a new model for non-Rayleigh signal amplitude statistics

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 /spl chi//sup 2/-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.

Characterization of Marine Surface Slicks by Radarsat-2 Multipolarization Features

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.

Independent component analysis for texture segmentation

Independent component analysis (ICA) of textured images is presented as a computational technique for creating a new data dependent filter bank for use in texture segmentation. We show that the ICA filters are able to capture the inherent properties of textured images. The new filters are similar to Gabor filters, but seem to be richer in the sense that their frequency responses may be more complex. These properties enable us to use the ICA filter bank to create energy features for effective texture segmentation. Our experiments using multi-textured images show that the ICA filter bank yields similar or better segmentation results than the Gabor filter bank.