Introduction

Hyperspectral imaging spectrometers collect images in the form of image cube that represents reflected energy from the Earth’s surface materials, where each pixel has the resultant mixed spectrum from reflected source radiation [1]. The mixed spectrum phenomenon causes mixed pixel problem because the intrinsic scale of spatial variation on the Earth’s land cover (LC) is finer than the scale of sampling imposed by the image pixels. In other words, mixed pixels are a mixture of more than one distinct object, and exist for one of the two reasons. Firstly, if the spatial resolution of the sensor is not high enough to separate different LC types, these can jointly occupy a single pixel, and the resulting spectral measurement will be a composite of the individual spectra that reside within a pixel. Secondly, mixed pixels can also result when distinct LC types are combined into a homogeneous mixture. This occurs independent of the spatial resolution of the sensor [2].

Commonly used approach to mixed pixel classification has been linear spectral unmixing [3], supervised fuzzy-c means classification [4], artificial neural networks [5 and 6], and Gaussian mixture discriminant analysis [7], etc. which uses a linear mixture model (LMM) to estimate the abundance fractions of spectral signatures lying within a pixel. LMM assumes that the reflectance spectrum of a mixture is systematic combination of the component reflectance spectra in the mixture (called endmembers). The combination of these endmembers is linear if component of interest in a pixel appear in spatially segregated patterns. If, however, the components are in intimate association, the electromagnetic spectrum typically interacts with more than one component as it is multiple scattered, and the mixing systematics between the different components are highly non-linear. In other words, non-linear mixing occurs when radiance is modified by one material before interacting with another one under the assumption that incident solar radiation is scattered within the scene itself and that these interaction events may involve several types of ground cover materials [8] and require non-linear mixture model (NLMM) for unmixing the components of interest.

If K is the number of spectral bands in the data set, and P, the number of distinct classes of objects in the physical scene, then associated with each pixel is a K-dimensional vector  whose components are the gray values corresponding to the K bands. If E = [e₁, e₂, …e_p, …, e_P], where {e_P} is a column vector representing the spectral signature of the p^th target material or category; the column vectors of the  matrix E are the endmembers. For a given pixel, the abundance fraction of the pth target material present in a pixel is denoted by α_P, and these values are the components of the P-dimensional abundance vector α. A nonlinear mixture model is then expressed as:

                                                                      (1)

where, f is an unknown nonlinear function that defines the interaction between E and α, and  is a noise vector. In this context, artificial neural network (ANN) based NLMMs outperform the traditional linear unmixing models. ANNs have been widely studied as a promising alternative to accomplish the difficult task of estimating fractional abundances of endmembers. Atkinson et al., (1997) [9] applied a MLP (milti-layer perceptron) model to decompose AVHRR imagery and was superior to the linear unmixing model and a fuzzy c-means classifier. Another popular ANN model - ARTMAP was first introduced to identify the life form components of the vegetation mixture [10] using Landsat imagery which could capture non-linear effects and thus performed better than LMM [11]. ART MMAP, an extension of ARTMAP was designed specifically for mixture analysis with enhanced interpolation function and provided better prediction of mixture information than ARTMAP [12]. Regression tree has also been used as non-linear unmixing models [6]. All of these methods are stand alone and work on the data directly when endmembers are known a priori. The objective of this paper is to develop an automated procedure to unmix hyperspectral imagery for obtaining fraction that accounts for the non-linear mixture of the class types. We call this model as automatic linear-nonlinear mixture model (AL-NLMM) as shown in the block diagram in Fig. 1. This paper is structured in six sections. Methods for automatic endmember extraction, linear unmixing and MLP are discussed in section II followed by the description of AL-NLMM in section III. Data preparation is dealt in section IV with the experimental results and discussion in section V. Section VI concludes with model limitations.

Fig. 1. Block diagram of the proposed AL-NLMM method for spectral mixture analysis.