In this paper we consider the problem of joint segmentation of hyperspectral images in the Bayesian framework. The proposed approach is based on a Hidden Markov Modeling (HMM) of the images with common segmentation, or equivalently with common hidden classification label variables which is modeled by a Potts Markov Random Field. We introduce an appropriate Markov Chain Monte Carlo (MCMC) algorithm to implement the method and show some simulation results.
The most significant recent advances in remote sensing has been the development of hyperspectral sensors and software to analyze the resulting image data. Over the past decade hyperspectral image analysis has matured into one of the most powerful and fastest growing technologies in the field of remote sensing. The "hyper" in hyperspectral means "over" as in "too many" and refers to the large number of measured wavelength bands. Hyperspectral images are spectrally overdetermined, which means that they provide ample spectral information to identify and distinguish spectrally unique materials. Hyperspectral imagery provides the potential for more accurate and detailed information extraction than possible with any other type of remotely sensed data. However the huge amount of data in hyperspectral images make its information exploitation difficult and image processing tools (classification, segmentation, comprising and coding) are needed to summarize the information included in these data. This paper will introduce a segmentation method for hyperspectral images. Several unsupervised and supervised algorithms have been developed for segmentation of multispectral images. However, these algorithms fail to deliver high accuracies for classifying hyperspectral images [1,2,3,4,5]. In this paper we consider the problem of joint segmentation of hyperspectral images in the Bayesian framework.
The proposed approach is based on a Hidden Markov Modeling (HMM) of the images with common segmentation, or equivalently with common hidden classification label variables which is modeled by a Potts Markov Random Field. We introduce an appropriate Markov Chain Monte Carlo (MCMC) algorithm to implement the method and show some simulation results. This approach has previously been considered by [6] for multispectral images.
In that work, the pixels of the same region in different images are assumed independent. This independence assumption is a valid hypothesis for multispectral images.
However in hyperspectral images the pixel values in each channel are not independent. This work is then an extension to that work by considering a Markov model for these pixels along each channel.
This paper is organized as follows: In the next two sections first we introduce our method for segmentation of hyperspectral images in the Bayesian framework. Then we propose an appropriate MCMC Gibbs sampling particularly designed for this segmentation task. Finally, in the last section we present some simulation results to show the performances of the proposed methods.
Let g i (r) be the observed value of the pixel r, r ∈ Z 2 , in the spectral band i of a hyperspectral image. We model the observations by
where f i (r) is the unknown perfect value of g i (r) and ε i (r) is a noise. Note that if we consider images, the pixels r belong to a finite lattice S, and we will note S the number of pixels of this lattice. In the following we also use the notations
where g i = {g i (r), r ∈ S} and g = {g i , i = 1, . . . , M } and a similar definition for f and . We introduce a label variable z(r) for the regions and consider the region labels as common feature between all images. Thus the hidden variable z = {z(r), r ∈ S} represent a common classification of the images for different bands. The main result of this paper is estimation of joint segmentation label z.
Assuming independent noises ε i among the different observations we have
Assuming ε i centered, white and Gaussian ε i ∼ N (0, σ 2 ε i I), and S the number of pixels of an image, we have:
where σ 2 ε i ∼ IG(α ε i 0 , β ε i 0 ) with unknown fixed parameters α ε i 0 and β ε i 0 (inverse gamma is conjugate prior for a random variance in the Gaussian case).
To assign p(f i |z, .) we first define the sets of pixels which are in the same class:
We assume that all the pixels f ik of an image f i which are in the same class k will be Gaussian with a random mean m ik and a random variance σ 2 i k , i.e.
With these notations we have :
and
where 1 is a vector with all components equal to 1, σ 2 i k ∼ IG(α i0 , β i 0 ), with unknown fixed parameters, and m ik is an autoregressive of order 1, AR(1), for each class k i.e.
where η ik ∼ N (0, σ 2 i 0 ), φ k and σ 2 i 0 are unknown fixed parameters. Therefore
The assumption of (7) is the main difference of this paper with [6], i.e. in hyperspectral images the pixel values of a class k, in each channel, are not independent.
Using the relation (1) and the density p(f i |z, m ik , σ 2 i k ) and p(ε i ), we can calculate p(g i |z, .), i.e.
Finally we have to assign P (z). As we introduced the hidden variable z for finding statistically homogeneous regions in images, it is natural to define a spatial dependency on these labels. The simplest model to account for this desired local spatial dependency is a Potts Markov Random Field model:
where S is the set of pixels, δ(0) = 1, δ(t) = 0 if t = 0, V(r) denotes the neighborhood of the pixel r (here we consider a ne
