INTRODUCTION

Hyper spectral imaging spectrometers collect data in the form of an image cube that represents reflected energy from the Earth’s surface materials, where each pixel has the resultant mixed spectrum of the reflected source radiation [1]. The mixed spectrum phenomenon causes a mixed pixel problem because the intrinsic scale of spatial variation in land cover (LC) due to the heterogeneous and fragmented landscapes [2] is usually finer than the scale of sampling imposed by the image pixels (for example, MODIS data at 250 m to 1 km spatial resolution) resulting in mixed pixels. Mixed pixels thus are a mixture of more than one distinct object and exist for one of 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 happens independently of the spatial resolution of the sensor [3].

Commonly used approaches to mixed pixel classification have been linear spectral unmixing [4], supervised fuzzy-c means classification [5], ANN (artificial neural networks) [6,7] and Gaussian mixture discriminant analysis [8], etc. which use 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 a systematic combination of the component’s reflectance spectra in the mixture (called endmembers). The combination of these endmembers is linear if the component of interest regarding a pixel appears 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 multiply 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 [9] and require non-linear mixture model for unmixing the components of interest. In such cases, LMM have mostly failed in modeling a mixed pixel [10-12] and non-linear models have been found to be appropriate as evident from various studies [2], including vegetation and canopy discrimination [13] water quality assessment [12,14], etc.

If there are M spectral bands and N classes, then associated with each pixel is a M-dimensional vector y whose components are the gray values corresponding to the M bands. Let E = [e₁, …e_n-1, en, e_n+1. . , eN] be a M × N matrix, where {e_n} is a column vector representing the spectral signature (endmember) of the nth target material. For a given pixel, the abundance or fraction of the nth target material present in a pixel is denoted by αn, and these values are the components of the N-dimensional abundance vector α. Assuming LMM [15], the observation vector y is related to E by

where η accounts for the measurement noise. We further assume that the components of the noise vector η are zero-mean random variables that are i.i.d. (independent and identically distributed). Therefore, the covariance matrix of the noise vector is σ²I, where σ² is the variance, and I is M × M identity matrix. Two constraints imposed on the abundances in equation (1) are the non-negativity and sum-to-one given as

and

This allows proportions of each pixel to be partitioned between classes. A non-linear mixture model (NLMM) is expressed as:

where, f is an unknown non-linear function that defines the interaction between E and α. Theory and experiments demonstrate that we will get the fractions of endmembers wrong by using a linear model when spectral mixing actually is non-linear [10,11]. Non-linear effects are an area of active research in particular applications where LMM generally results in poor accuracy [12].

In this context, 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. [2] applied a MLP (milti-layer perceptron) model to decompose AVHRR imagery, and it was superior to the linear unmixing model and a fuzzy c-means classifier. Another popular ANN model—ARTMAP—introduced to identify the life form components of the vegetation mixture [13] using Landsat data could capture non-linear effects, performing better than LMM [16]. ART MMAP, an extension of ARTMAP was designed specifically for mixture analysis with enhanced interpolation function and it provides better prediction of mixture information than ARTMAP [17]. A regression tree has also been used as a non-linear unmixing model [7]. All of these methods 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 a fraction that accounts for the non-linear mixture of the class types. We call this model the Hybrid Mixture Model (HMM). HMM is carried out in three stages: (i) Endmembers are extracted from the image itself using an iterative N-FINDR algorithm; (ii) the endmembers are used in the linear unmixing model for abundance estimation; (iii) the abundance values along with the actual ground proportions are used to refine the abundance estimates using MLP for the individual classes to account for the non-linear nature of the mixing classes of interest.

This paper is structured in six sections. Methods for automatic endmember extraction, linear unmixing and MLP are discussed in Section 2 followed by the description of HMM in Section 3. Data preparation is dealt with in Section 4 with the experimental results and discussion in Section 5. Section 6 concludes with model limitations.