Hyperspectral Image Classification Using Multinomial Logistic Regression and Non-local Prior on Hidden Fields Le Sun, Hiuk Jae Shim, Byeungwoo Jeon College of Information and Communication Engineering, SKKU, Suwon, Korea Email: {sunlecncom, bjeon}@skku.edu Yuhui Zheng, Yunjie Chen School of Computer and Software NUIST, Nanjing, China Email: zhengyh@vip.126.com Liang Xiao, Zhihui Wei School of Computer Science and Engineering, NJUST, Nanjing, China Email: xiaoliang@mail.njust.edu.cn Abstract—In this paper, we present a supervised hyperspec- tral image segmentation method based on multinomial logistic regression and a convex formulation of a marginal maximum a posteriori (MAP) segmentation with non-local total variation prior on the hidden fields under Bayesian framework. It not only exploits the basic assumption that samples within each class approximately lie in a lower dimensional subspace, but also sidesteps the discrete nature of the image segmentation problems by modeling spatial prior with vectorial non local means on the hidden fields. Alternating direction method of multipliers (ADMM) is finally extended to solve the proposed model. The proposed algorithm is validated by real hyperspectral data set. Keywords—sparse logistic regression, hyperspectral classifica- tion (HC), hidden fields, non-local total variation I. I NTRODUCTION In partitioning the hyperspectral image (HSI) scene such that pixels belonging to the same partition region share similar attributes, supervised hyperspectral classification has been a fundamental technique in a large number of hyperspectral image applications. Recently, a multitude of techniques have been developed for supervised HSI classification. Sparse multinomial logistic regression (SMLR) based techniques are among the state-of- the-art discriminative ones. The logistic regression via variable splitting and augmented Lagrangian (LORSAL) algorithm [1], which is able to learn a sparse regressor with Laplace prior distribution of SMLR, has become a new approach to deal with larger data sets and number of classes efficiently. These ideas have been successfully applied to HSI classification [2]. However, the SMLR approaches only exploits the spectral information. Spectral-spatial techniques, based on composite kernels (CKs) (e.g. [3]), joint sparse representation (e.g. [4]) and Markov random fields (MRFs) (e.g. [5]) and so on, tend to be be able to improve the performance of hyperspectral classification. Those existing methods mostly focus on the integer optimization problems. In [6], [7], the idea of using a hidden set of real-valued fields opens a new door to sidestep the discrete nature of the image segmentation problem. Inspired by these ideas, in our previous work, we proposed a new framework [8], [9] for hyperspectral image classification to sidestep the discrete nature of image segmentation by using spectral-spatial information with an adaptive TV regularization on the real-valued hidden fields and extended ADMM to achieve the segmentation. Then, J.Bioucas-Dias and F. Condessa etc. [10], [11] systematically further formulated the problem in Bayesian framework and introduced the Segmentation via the Constrained Split Aug- mented Lagrangian Shrinkage Algorithm (SegSALSA) to infer the hidden set of real-valued random fields, thus, converting the original segmentation optimization into a convex problem. In this paper, under Bayesian framework, we extend the segmentation formulation to include a generalization of vec- torial non-local total variation prior on the real-valued hidden fields, thus deriving a more general non-local algorithm for hyperspectral image segmentation. The non-local prior is con- structed from the structure of hidden fields based on Euclidean distance and spectral angle distance, akin to [12]. The paper is organized as follows. Section II describes the hidden fields and marginal MAP (MMAP) formulation. Section III presents the non local total variation prior on the hidden field. Section IV formulates our optimization problem and presents the proposed algorithm. Section V illustrates experimental results on real hyperspectral data, Section VI concludes with some remarks. II. PROBLEM FORMULATION Firstly, we make some notations on HSI classification prob- lem. Let S≡{1, 2, ..., N } denote integer indexes of N pixels of a hyperspectral image and x ≡ (x 1 , x 2 , ..., x N ) ∈ R L×N be a hyperspectral image with N samples of L features (bands), each x i is a L-dimensional vector, and y ≡ (y 1 ,y 2 , ..., y N ) ∈ K N an image of class labels, where K≡{1, ..., K} denotes a set of K class labels and each y i =[y (1) i ,y (2) i , ..., y (K) i ] is a 1-of-K encoding of the classes (y (j) i ∈{0, 1}, for j ∈K and K is the number of classes). Under Bayesian perspective, the MAP classification is for- mulated as:  y = arg max y∈K N p(y|x) = arg max y∈K N {p(x|y)p(y)} , (1) where p(y|x) is the posterior probability, p(x|y) is the observation model, and p(y) is the prior on the discrete