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Determining the Intrinsic Dimension of a Hyperspectral Image Using Random Matrix Theory Kerry Cawse-Nicholson, Steven B

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Determining the Intrinsic Dimension of a Hyperspectral Image Using Random Matrix Theory Kerry Cawse-Nicholson, Steven B IEEE TRANSACTIONS ON IMAGE PROCESSING 1 Determining the Intrinsic Dimension of a Hyperspectral Image Using Random Matrix Theory Kerry Cawse-Nicholson, Steven B. Damelin, Amandine Robin, and Michael Sears, Member, IEEE Abstract— Determining the intrinsic dimension of a hyper- In this paper, we will focus on the application to hyper- spectral image is an important step in the spectral unmixing spectral imagery, although this does not preclude applicability process and under- or overestimation of this number may lead to other areas. A hyperspectral image may be visualised as a to incorrect unmixing in unsupervised methods. In this paper, × × we discuss a new method for determining the intrinsic dimension 3-dimensional data cube of size (m n p). This is a set using recent advances in random matrix theory. This method is of p images of size N = m × n pixels, where all images entirely unsupervised, free from any user-determined parameters correspond to the same spatial scene, but are acquired at p and allows spectrally correlated noise in the data. Robustness different wavelengths. tests are run on synthetic data, to determine how the results were In a hyperspectral image, the number of bands, p,islarge affected by noise levels, noise variability, noise approximation, ≈ and spectral characteristics of the endmembers. Success rates are (typically p 200). This high spectral resolution enables determined for many different synthetic images, and the method separation of substances with very similar spectral signatures. is tested on two pairs of real images, namely a Cuprite scene However, even with high spatial resolution, each pixel is often taken from Airborne Visible InfraRed Imaging Spectrometer a mixture of pure components. (AVIRIS) and SpecTIR sensors, and a Lunar Lakes scene taken It is of interest to unmix each pixel, to determine the from AVIRIS and Hyperion, with good results. abundances of certain pure substances in the image. In unsu- Index Terms— Hyperspectral, intrinsic dimension, linear pervised remote sensing, the spectra of the K pure substances mixture model, random matrix theory, unmixing. themselves are also unknown, and the presence of noise forces the dataset to full dimension p. In practice the data set will I. INTRODUCTION involve significant redundancy. If we suppose that the data are ETERMINING the number of sources in a signal is noise free, then they would be contained in a proper vector K p Dimportant for the processing of many different types subspace R of R . Various terms have been introduced in of data, including chemical unmixing [1], extracting speech the literature for similar concepts. signals in a noisy line [2], unmixing minerals [3] and unmixing Bioucas-Dias and Nascimento [7] define the Intrinsic environmental landscapes [4], among many others. Determi- Dimension of the image (ID) as the dimension of the signal nation of this number is necessary for classification methods, subspace. Chang and Du [8] define ID as the “minimum unmixing methods and target detection [5]. It is common number of parameters required to account for the observed practice for the user to select the number of endmembers properties of the data.” They also define a separate estimate, to suit the application, but this does not necessarily agree called Virtual Dimension (VD), which is the number of end- with the intrinsic dimension of the image, and an incorrect members necessary to give accurate unmixing, which may be estimation of this number may have detrimental effects on the larger than the number of so-called “idealized substances.” end results [6]. Here substances may be understood as pure targets or end- members, and could represent different objects depending on Manuscript received July 20, 2011; revised October 24,IEEE 2012; accepted the application, e.g. chemical powders, cover types such as October 25, 2012. This work was supported in the part by the Centre for High Performance Computing flagship project: Computational Research Initiative tree species, etc. By definition, VD may be dependent on in Imaging and Remote Sensing and CSIR. The associate editor coordinating the unmixing method that is used, unlike ID. Wu et al. have the review of this manuscript and approving it for publication was Dr. Brian compared estimates of VD and ID in their survey paper [6] and D. Rigling. K. Cawse-Nicholson is with the School of Computational and Applied found them to be comparable for the Airborne Visible/Infrared Mathematics, University of the Witwatersrand, Johannesburg 2000, South Imaging Spectrometer (AVIRIS) scene of Cuprite. Africa, and also with the Remote Sensing Research Unit, Meraka Institute, Bajorski [9], however, found that the concept of VD may CSIR, Pretoria 0001, South Africa (e-mail: [email protected]). S. B. Damelin is with the Department of Mathematics, Wayne Country Day be misleading in certain circumstances, since the value may School, Goldsboro, NC 27530 USA (e-mail: [email protected]). change whenProof the image is shifted and rotated. Bajorski [9] A. Robin is with the School of Computational and Applied Mathematics, instead defines the Effective Dimensionality (ED) as “the University of the Witwatersrand, Johannesburg 2000, South Africa (e-mail: [email protected]). dimensionality of the affine subspace giving an acceptable M. Sears is with the School of Computer Science, University of the Witwa- approximation to all pixels.” Schlamm et al. [10] define an tersrand, Johannesburg 2000, South Africa (e-mail: [email protected]). Inherent Dimension K , where “the entire spectral image can Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. lie in the same K -dimensional hyperplane.” These authors Digital Object Identifier 10.1109/TIP.2012.2227765 also claim that the Inherent Dimension is not equivalent to 1057–7149/$31.00 © 2012 IEEE 2 IEEE TRANSACTIONS ON IMAGE PROCESSING VD, or the number of spectral signatures present in the image. including chemistry and signal array processing, and are In contrast, the Spanning Dimension is defined as the “min- applied to hyperspectral imagery. imum number of basis vectors required to span the space” From the real and synthetic experiments described in [6], of pixel observations [10]. Schlamm et al. define many the authors determined that the best method for hyperspectral other types of dimensions which may apply to hyperspec- imagery was HFC, and the methods with the strictest noise tral imagery, to display the confusion around the term. assumptions (i.e. that the noise is Gaussian and i.i.d.) per- These authors also define their own intrinsic dimension formed the worst. On the other hand, HFC was shown to be as “the smallest number of parameters needed to con- sensitive to user-defined values. tain all of the variability in the data through a mapping function.” This differs from Bioucas-Dias and Nascimento’s definition. B. Linear Mixture Model The many different definitions of Intrinsic Dimension may A common model used in remote sensing to separate be confusing, and although some of the above definitions mixed signals is the linear mixture model, introduced by are closely connected, some definitions may lead to different Horwitz et al. [17]. This model assumes that the measurement results, and so it is core for this paper to state a formal in each pixel is made up of a convex linear combination of definition for ID that is independent of the method used to pure components. Kritchman and Nadler [1] define the set of calculate it. × =[ ,..., ] pixel observations by the (p N)matrixX x1 x N First, we define some generic notation that will be used in and assume that for each pixel indexed by i, the observed this paper. Underlined variables, such as x represent vectors, ∈ Rp measurement vector xi is a realisation of a random random variables are denoted by a tilde, x˜, and subscript vector x˜,sothat th indices xi refer to measurement x for the i sample. ≤ ≤ ∈ Rp ˜ = ˜ + ξ˜, For each pixel i,1 i N,letxi be the observed x V u (1) spectral measurement. Assuming the measurement may be decomposed into signal and noise, write x = s + ξ ,where where u˜ is a (K × 1) random vector subject to the constraints i i i T s represents the information in pixel i and ξ represents the u˜ ≥ 0 and 1 u˜ = 1, and represents the proportions of pure i i noise. We introduce the following definition. components; V is a (p×K ) deterministic matrix, with columns Definition 1 (Intrinsic Dimension): The Intrinsic Dimen- corresponding to the spectral measurements characterizing the [ ,..., ] pure components present in the whole scene; ξ˜ is a random sion (ID) of a dataset x1 x N is the dimension, K ,of { , ≤ ≤ } vector in Rp representing the noise; and K is the total number the vector subspace spanned by si 1 i N . Definition 1 is considered to be equivalent to the Bioucas- of pure components. { ,..., } It is commonly assumed that ξ˜ follows a Gaussian dis- Dias and Nascimento definition [7], since s1 s N spans the signal subspace, and therefore K is the dimension of tribution N (0,),where is the noise correlation matrix. the signal subspace. This is also equivalent to the spanning This model allows the noise to be spectrally but not spatially dimension defined in [10]. correlated. This assumption has been successfully used in chemical unmixing [1] and in many methods described in the survey paper by Wu et al. [6]. The matrix must be estimated A. Literature Review from the data using a noise approximation method such as Some of the most popular methods to calculate the ID Meer’s method [18]. of a hyperspectral image include Maximum Orthogonal- Complements Algorithm (MOCA) [11], Harsanyi–Farrand– C. Background on Random Matrix Theory Chang (HFC) [8] and HySime [12].
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