Signal Subspace Identification in Hyperspectral Linear Mixtures

Chpater in Book Proceedings with Peer Review Second Iberian Conference, IbPRIA 2005, Estoril, Portugal, June 7-9, 2005, Proceedings, Part II

Estimation of signal subspace on hyperspectral data

Dimensionality reduction plays a crucial role in many hyperspectral data processing and analysis algorithms. This paper proposes a new mean squared error based approach to determine the signal subspace in hyperspectral imagery. The method first estimates the signal and noise correlations matrices, then it selects the subset of eigenvalues that best represents the signal subspace in the least square sense. The effectiveness of the proposed method is illustrated using simulated and real hyperspectral images.

Unsupervised hyperspectral signal subspace identification

Hyperspectral imaging sensors provide image data containing both spectral and spatial information from the Earth surface. The huge data volumes produced by these sensors put stringent requirements on communications, storage, and processing. This paper presents a method, termed hyperspectral signal subspace identification by minimum error (HySime), that infer the signal subspace and determines its dimensionality without any prior knowledge. The identification of this subspace enables a correct dimensionality reduction yielding gains in algorithm performance and complexity and in data storage. HySime method is unsupervised and fully-automatic, i.e., it does not depend on any tuning parameters. The effectiveness of the proposed method is illustrated using simulated data based on U.S.G.S. laboratory spectra and real hyperspectral data collected by the AVIRIS sensor over Cuprite, Nevada.

Hyperspectral signal subspace estimation

Given an hyperspectral image, the determination of the number of endmembers and the subspace where they live without any prior knowledge is crucial to the success of hyperspectral image analysis. This paper introduces a new minimum mean squared error based approach to infer the signal subspace in hyperspectral imagery. The method, termed hyperspectral signal identification by minimum error (HySime), is eigendecomposition based and it does not depend on any tuning parameters. It first estimates the signal and noise correlation matrices and then selects the subset of eigenvalues that best represents the signal subspace in the least squared error sense. The effectiveness of the proposed method is illustrated using simulated data based on U.S.G.S. laboratory spectra and real hyperspectral data collected by the AVIRIS sensor over Cuprite, Nevada.

Signal subspace identification in hyperspectral imagery

Terrestrial remote sensing imagery involves the acquisition of information from the Earth's surface without physical contact with the area under study. Among the remote sensing modalities, hyperspectral imaging has recently emerged as a powerful passive technology. This technology has been widely used in the fields of urban and regional planning, water resource management, environmental monitoring, food safety, counterfeit drugs detection, oil spill and other types of chemical contamination detection, biological hazards prevention, and target detection for military and security purposes [2-9]. Hyperspectral sensors sample the reflected solar radiation from the Earth surface in the portion of the spectrum extending from the visible region through the near-infrared and mid-infrared (wavelengths between 0.3 and 2.5 µm) in hundreds of narrow (of the order of 10 nm) contiguous bands [10]. This high spectral resolution can be used for object detection and for discriminating between different objects based on their spectral xharacteristics [6]. However, this huge spectral resolution yields large amounts of data to be processed. For example, the Airbone Visible/Infrared Imaging Spectrometer (AVIRIS) [11] collects a 512 (along track) X 614 (across track) X 224 (bands) X 12 (bits) data cube in 5 s, corresponding to about 140 MBs. Similar data collection ratios are achieved by other spectrometers [12]. Such huge data volumes put stringent requirements on communications, storage, and processing. The problem of signal sbspace identification of hyperspectral data represents a crucial first step in many hypersctral processing algorithms such as target detection, change detection, classification, and unmixing. The identification of this subspace enables a correct dimensionality reduction (DR) yelding gains in data storage and retrieval and in computational time and complexity. Additionally, DR may also improve algorithms performance since it reduce data dimensionality without losses in the useful signal components. The computation of statistical estimates is a relevant example of the advantages of DR, since the number of samples required to obtain accurate estimates increases drastically with the dimmensionality of the data (Hughes phnomenon) [13].

NDA waveform estimation in the low-SNR regime

This correspondence addresses the problem of nondata-aidedwaveform estimation for digital communications. Based on the unconditionalmaximum likelihood criterion, the main contribution of this correspondenceis the derivation of a closed-form solution to the waveform estimationproblem in the low signal-to-noise ratio regime. The proposed estimationmethod is based on the second-order statistics of the received signaland a clear link is established between maximum likelihood estimation andcorrelation matching techniques. Compression with the signal-subspace isalso proposed to improve the robustness against the noise and to mitigatethe impact of abnormals or outliers.

Hyperspectral subspace identification

Signal subspace identification is a crucial first step in many hyperspectral processing algorithms such as target detection, change detection, classification, and unmixing. The identification of this subspace enables a correct dimensionality reduction, yielding gains in algorithm performance and complexity and in data storage. This paper introduces a new minimum mean square error-based approach to infer the signal subspace in hyperspectral imagery. The method, which is termed hyperspectral signal identification by minimum error, is eigen decomposition based, unsupervised, and fully automatic (i.e., it does not depend on any tuning parameters). It first estimates the signal and noise correlation matrices and then selects the subset of eigenvalues that best represents the signal subspace in the least squared error sense. State-of-the-art performance of the proposed method is illustrated by using simulated and real hyperspectral images.

Vertex component analysis: a geometric-based approach to unmix hyperspectral data

Hyperspectral remote sensing exploits the electromagnetic scattering patterns of the different materials at specific wavelengths [2, 3]. Hyperspectral sensors have been developed to sample the scattered portion of the electromagnetic spectrum extending from the visible region through the near-infrared and mid-infrared, in hundreds of narrow contiguous bands [4, 5]. The number and variety of potential civilian and military applications of hyperspectral remote sensing is enormous [6, 7]. Very often, the resolution cell corresponding to a single pixel in an image contains several substances (endmembers) [4]. In this situation, the scattered energy is a mixing of the endmember spectra. A challenging task underlying many hyperspectral imagery applications is then decomposing a mixed pixel into a collection of reflectance spectra, called endmember signatures, and the corresponding abundance fractions [8–10]. Depending on the mixing scales at each pixel, the observed mixture is either linear or nonlinear [11, 12]. Linear mixing model holds approximately when the mixing scale is macroscopic [13] and there is negligible interaction among distinct endmembers [3, 14]. If, however, the mixing scale is microscopic (or intimate mixtures) [15, 16] and the incident solar radiation is scattered by the scene through multiple bounces involving several endmembers [17], the linear model is no longer accurate. Linear spectral unmixing has been intensively researched in the last years [9, 10, 12, 18–21]. It considers that a mixed pixel is a linear combination of endmember signatures weighted by the correspondent abundance fractions. Under this model, and assuming that the number of substances and their reflectance spectra are known, hyperspectral unmixing is a linear problem for which many solutions have been proposed (e.g., maximum likelihood estimation [8], spectral signature matching [22], spectral angle mapper [23], subspace projection methods [24,25], and constrained least squares [26]). In most cases, the number of substances and their reflectances are not known and, then, hyperspectral unmixing falls into the class of blind source separation problems [27]. Independent component analysis (ICA) has recently been proposed as a tool to blindly unmix hyperspectral data [28–31]. ICA is based on the assumption of mutually independent sources (abundance fractions), which is not the case of hyperspectral data, since the sum of abundance fractions is constant, implying statistical dependence among them. This dependence compromises ICA applicability to hyperspectral images as shown in Refs. [21, 32]. In fact, ICA finds the endmember signatures by multiplying the spectral vectors with an unmixing matrix, which minimizes the mutual information among sources. If sources are independent, ICA provides the correct unmixing, since the minimum of the mutual information is obtained only when sources are independent. This is no longer true for dependent abundance fractions. Nevertheless, some endmembers may be approximately unmixed. These aspects are addressed in Ref. [33]. Under the linear mixing model, the observations from a scene are in a simplex whose vertices correspond to the endmembers. Several approaches [34–36] have exploited this geometric feature of hyperspectral mixtures [35]. Minimum volume transform (MVT) algorithm [36] determines the simplex of minimum volume containing the data. The method presented in Ref. [37] is also of MVT type but, by introducing the notion of bundles, it takes into account the endmember variability usually present in hyperspectral mixtures. The MVT type approaches are complex from the computational point of view. Usually, these algorithms find in the first place the convex hull defined by the observed data and then fit a minimum volume simplex to it. For example, the gift wrapping algorithm [38] computes the convex hull of n data points in a d-dimensional space with a computational complexity of O(nbd=2cþ1), where bxc is the highest integer lower or equal than x and n is the number of samples. The complexity of the method presented in Ref. [37] is even higher, since the temperature of the simulated annealing algorithm used shall follow a log( ) law [39] to assure convergence (in probability) to the desired solution. Aiming at a lower computational complexity, some algorithms such as the pixel purity index (PPI) [35] and the N-FINDR [40] still find the minimum volume simplex containing the data cloud, but they assume the presence of at least one pure pixel of each endmember in the data. This is a strong requisite that may not hold in some data sets. In any case, these algorithms find the set of most pure pixels in the data. PPI algorithm uses the minimum noise fraction (MNF) [41] as a preprocessing step to reduce dimensionality and to improve the signal-to-noise ratio (SNR). The algorithm then projects every spectral vector onto skewers (large number of random vectors) [35, 42,43]. The points corresponding to extremes, for each skewer direction, are stored. A cumulative account records the number of times each pixel (i.e., a given spectral vector) is found to be an extreme. The pixels with the highest scores are the purest ones. N-FINDR algorithm [40] is based on the fact that in p spectral dimensions, the p-volume defined by a simplex formed by the purest pixels is larger than any other volume defined by any other combination of pixels. This algorithm finds the set of pixels defining the largest volume by inflating a simplex inside the data. ORA SIS [44, 45] is a hyperspectral framework developed by the U.S. Naval Research Laboratory consisting of several algorithms organized in six modules: exemplar selector, adaptative learner, demixer, knowledge base or spectral library, and spatial postrocessor. The first step consists in flat-fielding the spectra. Next, the exemplar selection module is used to select spectral vectors that best represent the smaller convex cone containing the data. The other pixels are rejected when the spectral angle distance (SAD) is less than a given thresh old. The procedure finds the basis for a subspace of a lower dimension using a modified Gram–Schmidt orthogonalizati on. The selected vectors are then projected onto this subspace and a simplex is found by an MV T pro cess. ORA SIS is oriented to real-time target detection from uncrewed air vehicles using hyperspectral data [46]. In this chapter we develop a new algorithm to unmix linear mixtures of endmember spectra. First, the algorithm determines the number of endmembers and the signal subspace using a newly developed concept [47, 48]. Second, the algorithm extracts the most pure pixels present in the data. Unlike other methods, this algorithm is completely automatic and unsupervised. To estimate the number of endmembers and the signal subspace in hyperspectral linear mixtures, the proposed scheme begins by estimating sign al and noise correlation matrices. The latter is based on multiple regression theory. The signal subspace is then identified by selectin g the set of signal eigenvalue s that best represents the data, in the least-square sense [48,49 ], we note, however, that VCA works with projected and with unprojected data. The extraction of the end members exploits two facts: (1) the endmembers are the vertices of a simplex and (2) the affine transformation of a simplex is also a simplex. As PPI and N-FIND R algorithms, VCA also assumes the presence of pure pixels in the data. The algorithm iteratively projects data on to a direction orthogonal to the subspace spanned by the endmembers already determined. The new end member signature corresponds to the extreme of the projection. The algorithm iterates until all end members are exhausted. VCA performs much better than PPI and better than or comparable to N-FI NDR; yet it has a computational complexity between on e and two orders of magnitude lower than N-FINDR. The chapter is structure d as follows. Section 19.2 describes the fundamentals of the proposed method. Section 19.3 and Section 19.4 evaluate the proposed algorithm using simulated and real data, respectively. Section 19.5 presents some concluding remarks.

PSWT Based Linear Predictive Coding and Development of a Parallel Multiple Subsequence Structure for DWT Computation

During 1990's the Wavelet Transform emerged as an important signal processing tool with potential applications in time-frequency analysis and non-stationary signal processing.Wavelets have gained popularity in broad range of disciplines like signal/image compression, medical diagnostics, boundary value problems, geophysical signal processing, statistical signal processing,pattern recognition,underwater acoustics etc.In 1993, G. Evangelista introduced the Pitch- synchronous Wavelet Transform, which is particularly suited for pseudo-periodic signal processing.The work presented in this thesis mainly concentrates on two interrelated topics in signal processing,viz. the Wavelet Transform based signal compression and the computation of Discrete Wavelet Transform. A new compression scheme is described in which the Pitch-Synchronous Wavelet Transform technique is combined with the popular linear Predictive Coding method for pseudo-periodic signal processing. Subsequently,A novel Parallel Multiple Subsequence structure is presented for the efficient computation of Wavelet Transform. Case studies also presented to highlight the potential applications.

Convergence analysis of chip- and fractionally spaced LMS adaptive multiuser CDMA detectors

This paper analyzes the convergence behavior of the least mean square (LMS) filter when used in an adaptive code division multiple access (CDMA) detector consisting of a tapped delay line with adjustable tap weights. The sampling rate may be equal to or higher than the chip rate, and these correspond to chip-spaced (CS) and fractionally spaced (FS) detection, respectively. It is shown that CS and FS detectors with the same time-span exhibit identical convergence behavior if the baseband received signal is strictly bandlimited to half the chip rate. Even in the practical case when this condition is not met, deviations from this observation are imperceptible unless the initial tap-weight vector gives an extremely large mean squared error (MSE). This phenomenon is carefully explained with reference to the eigenvalues of the correlation matrix when the input signal is not perfectly bandlimited. The inadequacy of the eigenvalue spread of the tap-input correlation matrix as an indicator of the transient behavior and the influence of the initial tap weight vector on convergence speed are highlighted. Specifically, a initialization within the signal subspace or to the origin leads to very much faster convergence compared with initialization in the a noise subspace.

Antipodal random sequences with prescribed second-order statistics: application to Compressive Sensing and UWB system based on DS-CDMA

It is usual to hear a strange short sentence: «Random is better than...». Why is randomness a good solution to a certain engineering problem? There are many possible answers, and all of them are related to the considered topic. In this thesis I will discuss about two crucial topics that take advantage by randomizing some waveforms involved in signals manipulations. In particular, advantages are guaranteed by shaping the second order statistic of antipodal sequences involved in an intermediate signal processing stages. The first topic is in the area of analog-to-digital conversion, and it is named Compressive Sensing (CS). CS is a novel paradigm in signal processing that tries to merge signal acquisition and compression at the same time. Consequently it allows to direct acquire a signal in a compressed form. In this thesis, after an ample description of the CS methodology and its related architectures, I will present a new approach that tries to achieve high compression by design the second order statistics of a set of additional waveforms involved in the signal acquisition/compression stage. The second topic addressed in this thesis is in the area of communication system, in particular I focused the attention on ultra-wideband (UWB) systems. An option to produce and decode UWB signals is direct-sequence spreading with multiple access based on code division (DS-CDMA). Focusing on this methodology, I will address the coexistence of a DS-CDMA system with a narrowband interferer. To do so, I minimize the joint effect of both multiple access (MAI) and narrowband (NBI) interference on a simple matched filter receiver. I will show that, when spreading sequence statistical properties are suitably designed, performance improvements are possible with respect to a system exploiting chaos-based sequences minimizing MAI only.

Direction Finding Estimators of Cyclostationary Signals in Array Processing for Microwave Power Transmission

A solar power satellite is paid attention to as a clean, inexhaustible large- scale base-load power supply. The following technology related to beam control is used: A pilot signal is sent from the power receiving site and after direction of arrival estimation the beam is directed back to the earth by same direction. A novel direction-finding algorithm based on linear prediction technique for exploiting cyclostationary statistical information (spatial and temporal) is explored. Many modulated communication signals exhibit a cyclostationarity (or periodic correlation) property, corresponding to the underlying periodicity arising from carrier frequencies or baud rates. The problem was solved by using both cyclic second-order statistics and cyclic higher-order statistics. By evaluating the corresponding cyclic statistics of the received data at certain cycle frequencies, we can extract the cyclic correlations of only signals with the same cycle frequency and null out the cyclic correlations of stationary additive noise and all other co-channel interferences with different cycle frequencies. Thus, the signal detection capability can be significantly improved. The proposed algorithms employ cyclic higher-order statistics of the array output and suppress additive Gaussian noise of unknown spectral content, even when the noise shares common cycle frequencies with the non-Gaussian signals of interest. The proposed method completely exploits temporal information (multiple lag ), and also can correctly estimate direction of arrival of desired signals by suppressing undesired signals. Our approach was generalized over direction of arrival estimation of cyclostationary coherent signals. In this paper, we propose a new approach for exploiting cyclostationarity that seems to be more advanced in comparison with the other existing direction finding algorithms.

New developments on VCA unmixing algorithm

Hyperspectral sensors are being developed for remote sensing applications. These sensors produce huge data volumes which require faster processing and analysis tools. Vertex component analysis (VCA) has become a very useful tool to unmix hyperspectral data. It has been successfully used to determine endmembers and unmix large hyperspectral data sets without the use of any a priori knowledge of the constituent spectra. Compared with other geometric-based approaches VCA is an efficient method from the computational point of view. In this paper we introduce new developments for VCA: 1) a new signal subspace identification method (HySime) is applied to infer the signal subspace where the data set live. This step also infers the number of endmembers present in the data set; 2) after the projection of the data set onto the signal subspace, the algorithm iteratively projects the data set onto several directions orthogonal to the subspace spanned by the endmembers already determined. The new endmember signature corresponds to these extreme of the projections. The capability of VCA to unmix large hyperspectral scenes (real or simulated), with low computational complexity, is also illustrated.

The VISTA system for compression of television signal bandwidths.

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Optical frequency conversion, pulse compression and signal copying using triangular pulses

We propose techniques of optical frequency conversion, pulse compression and signal copying based on a combination of cross-phase modulation using triangular pump pulses and subsequent propagation in a dispersive medium.