Suppose that a hyperspectral sensor with M spectral bands measures solar electromagnetic radiation reflecting from the N distinct substances. Each pixel of the measured hyperspectral image cube can be described by the following M x N linear mixing model [1][2][3][4][5][6][7]: where x[n] == [ Xl [n], ... ,XM [n] ]T is the nth observed pixel vector comprising M spectral bands, A == [a1, ... , aN ] E IR M x N denotes the signature matrix whose ith column vector a, is the ith endmember, s[n] == [ si [n], ... ,sN[n] ]T E IR N is an abundance vector comprising N fractional abundances, and L is the total number of observed pixel vectors. The goal of hyperspectral unmixing is to estimate A and s[n] from the the observed pixels x[n]. Our convex analysis formulation for hyperspectral unmixing is based on the following assumptions [1]: (A1) Intensities of all the abundance vectors are non-negative, i.e.,sdn] 2: 0 for all i and n. (A2) Abundance fractions are proportionally distributed for each observed pixel, Le., L:~1 sdn] == 1 for all n. (A3) min{L, M} 2: N and the endmember signatures are linearly independent, i.e., A is of full column rank. algorithms requiring the pure pixel assumption may be computationally less complex, such as pixel purity index (PPI) [5], N-finder (N-FINDR) [6], and vertex component analysis (YCA) [7]. For instance, N-FINDR [6] (proposed by Winter) is based on a criterion that the volume of a simplex formed by the purest pixels (or endmember estimates) is the maximum, and found such purest pixels by inflating the volume of a simplex inside the data set.In this paper, we provide a convex analysis and optimization perspective to hyperspectral unmixing problems, which have the intuitive ideas from Craig's and Winter's criteria, respectively. The endeavor is not only motivated by the prevalence of convex optimization techniques in signal processing, but also by the fact that some convex analysis concepts, such as affine hull and convex hull, are quite a good match with the nature of hyperspectral unmixing (i.e., the non-negativity and full additivity of abundances). Using convex analysis, we formulate two optimization problems for hyperspectral unmixing using Craig's and Winter's criteria, and prove their optimal solutions to be identical when pure pixels exist. We also demonstrate how to use alternating linear programming to approximate the formulated problems. Finally, some Monte Carlo simulation results are presented, which show a good consistency with our analytical results.
ABSTRACTIn hyperspectral remote sensing, unmixing a data cube into spectral signatures and their corresponding abundance fractions plays a crucial role in analyzing the mineralogical composition of a solid surface. This paper describes a convex analysis perspective to (unsupervised) hyperspectral unmixing. Such an endeavor is not only motivated by the recent prevalence of convex optimization in signal processing, but also by the nature of hyperspectral unmixing (specifically, non-negativity and full additivity of abundances) th...
