Successful performance of radiological search mission is dependent on effective utilization of mixture of signals. Examples of modalities include, e.g., EO imagery and gamma radiation data, or radiation data collected during multiple events. In addition, elevation data or spatial proximity can be used to enhance the performance of acquisition systems. State of the art techniques in processing and exploitation of complex information manifolds rely on diffusion operators. Our approach involves machine learning techniques based on analysis of joint data- dependent graphs and their associated diffusion kernels. Then, the significant eigenvectors of the derived fused graph Laplace and Schroedinger operators form the new representation, which provides integrated features from the heterogeneous input data. The families of data-dependent Laplace and Schroedinger operators on joint data graphs, shall be integrated by means of appropriately designed fusion metrics. These fused representations are used for target and anomaly detection.
KEYWORDS: Hyperspectral imaging, Data integration, Data fusion, Sensors, Satellite imaging, Earth observing sensors, Satellites, Analytical research, Target detection, Control systems
As new remote sensing modalities emerge, it becomes increasingly important to nd more suitable algorithms
for fusion and integration of dierent data types for the purposes of target/anomaly detection and classication.
Typical techniques that deal with this problem are based on performing detection/classication/segmentation
separately in chosen modalities, and then integrating the resulting outcomes into a more complete picture. In
this paper we provide a broad analysis of a new approach, based on creating fused representations of the multi-
modal data, which then can be subjected to analysis by means of the state-of-the-art classiers or detectors.
In this scenario we shall consider the hyperspectral imagery combined with spatial information. Our approach
involves machine learning techniques based on analysis of joint data-dependent graphs and their associated
diusion kernels. Then, the signicant eigenvectors of the derived fused graph Laplace operator form the new
representation, which provides integrated features from the heterogeneous input data. We compare these fused
approaches with analysis of integrated outputs of spatial and spectral graph methods.
We analyze Schroedinger Eigenmaps - a new semi-supervised manifold learning and recovery technique - for
applications in hyperspectral imagery. This method is based on an implementation of graph Schroedinger
operators with appropriately constructed potentials as carriers of expert/labeled information. In this paper, we
analyze the features of Schroedinger Eigenmaps through analysis of the potential locations and their imapct on
the classication. The imaging modalities which we shall incorporate in our analysis include multispectral and
hyperspectral imagery. For the purpose of constructing ecient methods for building the potentials we refer to
expert ground-truth data, as well as to using automated clustering techniques. We also investigate the role of
dierent sources of the barrier potential locations, and the role they play in the separation of classes.
There are intrinsic wavelet applications, by which we mean mathematical modeling of a physical phenomenon
in which wavelet theory is the most natural quantitative means of explaining the phenomenon. This is not
the same as the invaluable use of dyadic wavelets, say, as a tool with which to zoom-in or -out with regard to
multi-scale phenomena. An example of an intrinsic wavelet application is wavelet auditory modeling (WAM).
WAM is analyzed herein, and a natural excursion, one of many possibilities, is taken from WAM to applications
of finite frames. This path includes the role of the Discrete Fourier Transform (DFT) in WAM, the emergence
of DFT frames, and their use in analyzing Σ▵ quantization, which itself is a staple in audio engineering as well
as in a host of other applications.
State of the art dimension reduction and classification schemes in multi- and
hyper-spectral imaging rely primarily on the information contained in the spectral
component. To better capture the joint spatial and spectral data distribution we
combine the Wavelet Packet Transform with the linear dimension reduction method
of Principal Component Analysis. Each spectral band is decomposed by means of the
Wavelet Packet Transform and we consider a joint entropy across all the spectral
bands as a tool to exploit the spatial information. Dimension reduction is then
applied to the Wavelet Packets coefficients. We present examples of this technique
for hyper-spectral satellite imaging. We also investigate the role of various shrinkage
techniques to model non-linearity in our approach.
KEYWORDS: Chemical species, Associative arrays, Algorithm development, Interference (communication), Signal analyzers, Data modeling, Algorithms, Wavelets, Signal analysis, Reconstruction algorithms
Sparse approximation is typically concerned with generating compact representation of signals and data vectors by constructing a tailored linear combination of atoms drawn from a large dictionary. We have developed an algorithm based on simultaneous matching pursuits that facilitates the concurrent approximation of multiple signals in a common, low-dimensional representation space. The algorithm leads to an effective method of extracting signal components from collections of noisy data, and in particular is robust against jitter as well as additive noise. We illustrate its utility and compare performance in several variations by numerical examples.
Sigma-Delta (ΣΔ) schemes are shown to be an effective approach for quantizing finite frame expansions. Basic error estimates show that first order ΣΔ schemes can achieve quantization error of order 1/N, where N is the frame size. Under certain technical assumptions, improved quantization error estimates of order (logN)/N1.25 are obtained. For the second order ΣΔ scheme with linear quantization rule, error estimates of order 1/N2 can be achieved in certain circumstances. Such estimates rely critically on being able to construct sufficiently small invariant sets for the scheme. New experimental results indicate a connection between the orbits of state variables in ΣΔ schemes and the structure of constant input invariant sets.
There is a natural formulation of the Classical Uniform Sampling Theorem in the setting of Euclidean space, and in the context of lattices, as sampling sets, and unit cells E, e.g., the Voronoi cell. For sampling at the Nyquist rate, the sampling function corresponds to the since function, and it is an integral over E. The set E is a tile for Euclidean space under translation by elements of the reciprocal lattice. We have a constructive, implementable non-uniform sampling theorem in the context of uniformly discrete sampling sets and sets E, corresponding to the unit cells of the uniform sampling result. The set E has the property that the translates by the sampling set of the polar set of E is a covering of Euclidean space. The theorem depends on the theory of frames, and can be viewed as a modest generalization of a theorem of Beurling. The application herein is to fast magnetic resonance imagine by direct signal reconstruction from spectral data on spirals.
This paper deals with the analysis of time series with respect to certain known periodicities. In particular, we shall present a fast method aimed at detecting periodic behavior inherent in noise data. The method is composed of three steps: (1) Non-noisy data are analyzed through spectral and wavelet methods to extract specific periodic patterns of interest. (2) Using these patterns, we construct an optimal piecewise constant wavelet designed to detect the underlying periodicities. (3) We introduce a fast discretized version of the continuous wavelet transform, as well as waveletgram averaging techniques, to detect occurrence and period of these periodicities. The algorithm is formulated to provide real time implementation. Our procedure is generally applicable to detect locally periodic components in signals s which can be modeled as s(t) equals A(t)F(h(t)) + N(t) for t in I, where F is a periodic signal, A is a non-negative slowly varying function, and h is strictly increasing with h' slowly varying, N denotes background activity. For example, the method can be applied in the context of epileptic seizure detection. In this case, we try to detect seizure periodics in EEG and ECoG data. In the case of ECoG data, N is essentially 1/f noise. In the case of EEG data and for t in I,N includes noise due to cranial geometry and densities. In both cases N also includes standard low frequency rhythms. Periodicity detection has other applications including ocean wave prediction, cockpit motion sickness prediction, and minefield detection.
Pyramidal structures are defined which are locally a combination of low and highpass filtering. The structures are analogous to but different from wavelet packet structures. In particular, new frequency decompositions are obtained; and these decompositions can be parameterized to establish a correspondence with a large class of Cantor sets. Further correspondences are then established to relate such frequency decompositions with more general self- similarities. The role of the filters in defining these pyramidal structures gives rise to signal reconstruction algorithms, and these, in turn, are used in the analysis of speech data.
A redundant wavelet filtering method is used in conjunction with spectrogram computations to address a component of the problem of predicting epileptic seizure activity. It is shown that spectrograms of seizure episodes exhibit multiple chirps consistent with the relatively simple almost periodic behavior of the observed time series. Scalograms corresponding to a redundant (non-dyadic) wavelet analysis are used to provide finer information about these chirps, including their evolution in preseizure intervals. Detection of the origin of such periodicities are useful in the prediction problem.
KEYWORDS: Data modeling, Autoregressive models, Brain, Signal detection, Electroencephalography, Statistical analysis, Wavelets, Signal processing, Continuous wavelet transforms, Algorithm development
A wavelet-based technique WISP is used to discriminate normal brain activity from brain activity during epileptic seizures. The WISP technique is used to exploit the noted difference in frequency content during the normal brain state and the seizure brain state so that detection and localization decisions can be made. An AR-Pole statistic technique is used as a comparative measure to base-line the WISP performance.
A new Riesz product is formulated in terms of waveletpacket pyramidal tree structures. A path within such a structure is determined by a lacunarity criterion and a sequence (H(epsilon (j))) of filters, where each H(epsilon (j)) is one or the other element of a prescribed subband coding pair (H0, H1) of FIR filters. The Riesz product associated with a given path is a continuous radon measure. Path based criteria are given to determine singular and absolutely continuous pairs of such measures. The measures have full support, and their approximants exhibit fractal behavior. These properties can be used to design a secure transmission scheme in communications theory.
We consider the effect of the quantization noise introduced by coding at subbands. We demonstrate that significant noise reduction is achieved by using wavelet frames and their associated filter banks in a subband signal processing system.
A theory of wavelet packets is developed for nonlinear operators consisting of a composition, generalizing a sigmoidal operation, followed by convolutions with filter pairs H0 and H1. The pyramidal wavelet packet structure is defined by bit reversal trees. The reconstruction theorem, from which the original signal is obtained from frequency localized data at other nodes of the three, requires fixed point theory as well as conditions on H0 and H1 resembling those defining quadrature mirror filter pairs. Applications will be to biological systems and neural networks where such nonlinearities occur.
In this paper we introduce the concept of a local Hilbert space frame and develop theory for the representation and reconstruction of signals using local frames. The theory of global frames is due to Duffin and Schaeffer. Local frames are defined with respect to a global frame and a particular element from a Hilbert space H. For any signal f* (epsilon) H, H may be decomposed into two signal dependent subspaces: a finite dimensional one which essentially contains the signal f* and one to which the signal is essentially orthogonal. The frame elements associated with the former subspace constitute the local frame around f*.
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