Peter Bajorski

Second Moment Linear Dimensionality as an Alternative to Virtual Dimensionality

In an effort to assess the effective linear dimensionality of hyperspectral images, a notion of virtual dimensionality (VD) has been developed, and it is being used in many papers including those published in the IEEE Transactions on Geoscience and Remote Sensing. The ever-spreading use of VD warrants its thorough investigation. In this paper, we investigate the properties of VD, and we show that VD does not have a desirable property of being invariant to translations and rotations. We show specific examples when VD gives entirely misleading results. We also propose a new method (called second moment linear dimensionality), which is a natural alternative to VD because it is based on the positive aspects of VD without suffering from its pitfalls. We implement this new methodology to calculate the dimensionalities of an artificial data set and two images-AVIRIS and HyMap images.

Target Detection Under Misspecified Models in Hyperspectral Images

In evaluating the performance of detectors such as the orthogonal subspace projection (OSP) detector, it is often assumed that the model under which the detector is constructed is the correct model. However, in practice, the ability to identify all background endmembers might be limited. Consequently, the OSP detector would use only a subset of all background endmembers. In this paper, we consider consequences of using such incomplete information. To this end, we contrast between two detectors, one that uses all endmembers and another one that uses only some endmembers. The remaining endmembers might be unknown or deliberately discarded if we believe that this would improve the detector performance. We provide formulas for detection power of the two detectors. We also calculate their relative efficiency, which is a useful tool for comparison of detectors without directly calculating the detection power. We then show some theorems that help us in understanding relationships between the power detection of the two detectors. We also provide numerical results demonstrating the consequences of our theorems in specific scenarios. We show when the detector using more information is more powerful, but we also show examples of the opposite to be true. Practical consequences of using a given number of endmembers in the OSP detector are evaluated using an AVIRIS hyperspectral image. The results show that using too many endmembers is not beneficial for the OSP detector, and the optimal number of endmembers to be used depends on the size of the target and the desired false alarm level.

Statistical Inference in PCA for Hyperspectral Images

Principal component analysis (PCA) is a popular tool for initial investigation of hyperspectral image data. There are many ways in which the estimated eigenvalues and eigenvectors of the covariance matrix are used. Further steps in the analysis or model building for hyperspectral images are often dependent on those estimated quantities. It is therefore important to know how precisely the eigenvalues and eigenvectors are estimated, and how the precision depends on the sampling scheme, the sample size, and the covariance structure of the data. This issue is especially relevant for applications such as difficult target or anomaly detection, where the precision of further steps in the algorithm may depend on the reliable knowledge of the estimated eigenvalues and eigenvectors. The issue is also relevant in the context of small sample sizes occurring in local types of detectors (such as RX) and in investigations of individual components and small multi-material clusters within a hyperspectral image. The sampling properties of eigenvalues and eigenvectors are known to some extent in statistical literature (mostly in the form of asymptotic results for large sample sizes). Unfortunately, those results usually do not apply in the context of hyperspectral images. In this paper, we investigate the sampling properties of eigenvalues and eigenvectors under three scenarios. The first two scenarios consider the type of sampling traditionally used in statistics, and the third scenario considers the variability due to image noise, which is more appropriate for hyperspectral imaging applications. For all three scenarios, we show the precision associated with the estimated eigenvalues, eigenvectors, and intrinsic dimensionality in six images of various sizes. In a broader context, we show an example of the correct statistical inference (construction of confidence intervals and bias-adjusted estimates) that can be implemented in other imaging applications.

Simplex projection methods for selection of endmembers in hyperspectral imagery

It is well known that performance of subpixel target detection algorithms depends on the choice of endmembers used to characterize the background and the target in hyperspectral imagery. In this paper, we investigate how well a set of endmembers characterizes a given set of spectra. We are assuming a fully constrained linear mixing model, and analyze the resulting residuals. To facilitate geometric and intuitive interpretation, we formulate the resulting constrained least-squares estimation problem in terms of projections on low-dimensional simplexes. Consequently, we define a family of simplex projection methods (SPM) for endmember selection. We give numerical results for two known endmember-selection procedures -the pixel purity index (PPI) and the maximum distance (MaxD) methods. Then we compare these results to those for a simple version of SMP, called the Farthest Pixel Selection (FPS) method. This new class of techniques promises to give better descriptions of the target and background regions than do current methods, which in turn should lead to more precise detection (with lower false alarm rates) of low-visibility small (subpixel) targets

Does virtual dimensionality work in hyperspectral images

The effective dimensionality (ED) of hyperspectral images is often viewed as the dimensionality of an affine subspace defined by linear combinations of spectra of materials present in the image. That affine subspace is expected to give an acceptable approximation to all pixels. At this point, there is no precise definition of ED. In an effort to assess ED, a notion of virtual dimensionality (VD) has been developed, and it is being used in many papers including those published in TGARS. The ever- spreading use of VD warrants its thorough investigation. In this paper, we investigate properties of VD, and we show that VD largely depends on the average value of all spectra rather than on ED. We show specific examples when VD would give entirely misleading results. We also explain fallacies associated with justifications for VD.

Nonparametric Two-Sample Comparisons of Changes on Ordinal Responses

Abstract We discuss two-armed clinical trials in which an ordinal response is observed on entry and at follow-up. In certain subject areas, such data are often reduced to the counts of patients who have deteriorated, remained the same, or improved. We develop asymptotic distribution theory to compare this approach to the use of nonparametric tests based on poststratification of the follow-up responses according to the entry levels. The theory for the latter approach is established by a general result for two-sample U statistics based on bivariate data. Application to data from clinical trials and evaluation of asymptotic relative efficiencies and powers for a variety of contexts illustrate the increased sensitivity of poststratified Wilcoxon tests.

Geometric basis-vector selection methods and subpixel target detection as applied to hyperspectral imagery

In this paper, we compare three basis-vector selection methods as applied to subpixel target detection. This is a continuation of previous research in which a similar comparison was performed based on an AVIRIS image. Our goal is to find out to what extent our previous observations apply more broadly to other images, more specifically, a HYDICE image used in this paper. Our target detection approach is based on generating a radiance target region using a physical model to generate radiance spectra as observed under a wide range of atmospheric, illumination., and viewing conditions. The advantage of this approach is that the resulting target detection is invariant to those changing conditions. For the purpose of target detection, we use a structured model to describe each image spectra as a linear combination of the target and background basis-vectors, and then we apply a matched subspace detector. Finally, we find ROC curves to describe the relationship between the detection rate (DR) and the false alarm rate (FAR). Due to a large number of cases considered, we use summary metrics to represent our results. The obtained results are quite different from those obtained in (Bajorski et al., 2004) for the AVIRIS image. The best method for generating the background basis vectors in the AVIRIS image was the MaxD method, while the SVD method proved to be best for the HYDICE image used in this paper. Further research is needed to find out the reasons for these differences. It is not surprising that different methods are optimal for different types of data. However, it would be useful to be able to recognize the optimal method without assuming knowledge of the targets in the image

Statistics in Civil Engineering

Statistical methods for the civil engineer. Practical examples clearly explain the underlying mathematical concepts. Readers learn to write their own programs and learn about useful commercial software.Offers examples based on real applications.-- Applies statistical techniques to civil engineering practice.

Generalized Detection Fusion for Hyperspectral Images

The purpose of this paper is to introduce a general type of detection fusion that allows combining a set of basic detectors into one more versatile detector. The fusion can be performed based on the spectral information contained in a pixel, the global characteristics of the background and target spaces, as well as spatial local information. As an example of generalized fusion, we introduce a new class of detectors called the directional segmented matched filters (DSMFs). We then concentrate on the more basic type of fusion that does not use the spatial local information. Our goal is to build a theoretical foundation for the future more sophisticated detectors. Within this setup, we define max-min and min-max types of fusion, which turn out to be equivalent to the geometric approach to continuum fusion already introduced in the literature. Nevertheless, this new framework allows natural formulation of other types of approaches, such as discrete fusion, without the continuity assumption. This new formalism also allows formulation of a general theorem about the relationship between the max-min and min-max detectors. We also provide experimental results that demonstrate the benefits of the new approach. We compare two new detectors with the global matched filter using two different targets in an AVIRIS (Airborne Visible/Infrared Imaging Spectrometer) image. The results show that various forms of the DSMFs dominate depending on the target type.

Effect of pH and oxygen on biofilm formation in acute otitis media associated NTHi clinical isolates

Biofilms occur in animal models of acute otitis media (AOM) and in children with recurrent AOM (rAOM) and chronic otitis media with effusion (OME). We therefore studied the ability of nontypeable Haemophilus influenzae (NTHi) strains from children to form biofilms in vitro under conditions we presumed occurred in the middle ear during AOM, rAOM, and OME.