Performance Analysis of SUnSAL

In Remote Sensing (RS) cameras, used for earth observation, are generally mounted on satellite or on aero plane. Due to very high altitude of Hyperspectral Cameras (HSCs) the spatial resolution of images taken by such camera is very poor, in order of 4 m by 4m to 20m by 20m. So a single pixel from image taken by HSC may contain more than one materials and it is not possible to know about the materials present in single pixel. HSC measures the reflectance of object in the wavelength of range from 0.4 to 2.5um at 200 bands with spectral resolution of 10nm. High spectral resolution enables the accurate estimation of number of materials present in scene, known as endmembers, their spectral signature and fractional proportion within pixel, known as abundance map. This process is known as Hyperspectral Unmixing (HU). Due to large data size, environmental noise, endmember variability, not availability of pure endmembers HU is a challenging task. HU enables various application like an agricultural assessment, environmental monitoring, change detection, mineral exploitation, ground cover classification, target detection and surveillance. There are three approaches to solve this task: Geometrical, statistical and sparse regression. First two methods are Blind Source Separation (BSS) techniques. Third approach is based on sparsity and considered as semi-blind approach because it assumes the availability of spectral library. Spectral library contains the spectral signatures of various materials measured on the earth surface using advance Spectro radiometers. In sparse unmixing a mixed pixel is represented in the form of linear combination of a number of spectral signature known in advance and available in standard library. In this paper, mathematical steps for Spectral Unmixing using variable Splitting and Augmented Lagrangian (SUnSAL) are simplified. performance of SUnSAL is evaluated with the help of standard and publically available synthetic data base

MINVO Basis: Finding Simplexes with Minimum Volume Enclosing Polynomial Curves

This paper studies the problem of finding the smallest $n$-simplex enclosing a given $n^{\text{th}}$-degree polynomial curve. Although the Bernstein and B-Spline polynomial bases provide feasible solutions to this problem, the simplexes obtained by these bases are not the smallest possible, which leads to undesirably conservative results in many applications. We first prove that the polynomial basis that solves this problem (MINVO basis) also solves for the $n^\text{th}$-degree polynomial curve with largest convex hull enclosed in a given $n$-simplex. Then, we present a formulation that is \emph{independent} of the $n$-simplex or $n^{\text{th}}$-degree polynomial curve given. By using Sum-Of-Squares (SOS) programming, branch and bound, and moment relaxations, we obtain high-quality feasible solutions for any $n\in\mathbb{N}$ and prove numerical global optimality for $n=1,2,3$. The results obtained for $n=3$ show that, for any given $3^{\text{rd}}$-degree polynomial curve, the MINVO basis is able to obtain an enclosing simplex whose volume is $2.36$ and $254.9$ times smaller than the ones obtained by the Bernstein and B-Spline bases, respectively. When $n=7$, these ratios increase to $902.7$ and $2.997\cdot10^{21}$, respectively.Comment: 25 pages, 16 figure