Spatially-smooth piece-wise convex endmember detection

Abstract

A new hyperspectral endmember detection method that represents endmembers as distributions, autonomously partitions the input data set into several convex regions, and simultaneously determines endmember distributions and proportion values for each convex region is presented. Spectral unmixing methods that treat endmembers as distributions or hyperspectral images as piece-wise convex data sets have not been previously developed. Piece-wise Convex Endmember detection, PCE, can be viewed in two parts, the first, the Endmember Distributions detection (ED) algorithm, estimates a distribution for each endmember rather than estimating a single spectrum. By using endmember distributions, PCE can incorporate an endmember's inherent spectral variation and the variation due to changing environmental conditions. ED uses a new sparsity-promoting polynomial prior while estimating abundance values. The second part of PCE partitions the input hyperspectral data set into convex regions and estimates endmember distributions and proportions for each of these regions. The number of convex regions is determined autonomously using the Dirichlet process. PCE is effective at handling highly-mixed hyperspectral images where all of the pixels in the scene contain mixtures of multiple endmembers. Furthermore, each convex region found by PCE conforms to the Convex Geometry Model for hyperspectral imagery. This model requires that the proportions associated with a pixel be non-negative and sum-to-one. Algorithm results on hyperspectral data indicate that PCE produces endmembers that represent the true ground truth classes of the input data set. The algorithm can also effectively represent endmembers as distributions, thus, incorporating an endmember's spectral variability.

Hyperspectral Unmixing Overview: Geometrical, Statistical, and Sparse Regression-Based Approaches

“…This approach has been taken in [149]- [153]. The latter methods are Bayesian and will therefore be discussed in the next section.…”

A survey of methods incorporating spatial information in image classification and spectral unmixing

“…A joint optimization procedure was conducted to estimate endmember spectra, fractional abundances, and fuzzy membership values assigned to each pixel denoting the field to which it belonged. An improved algorithm was developed to add spatial constraints on fuzzy membership values in order to guide neighbouring pixels to share the same set of endmembers (Zare et al 2010). …”

Context dependent spectral unmixing.

“…However, it may be the case that multiple sets of endmembers, defining several overlapping convex regions, can better describe the hyperspectral image. This issue has been addressed in [49][50][51][52][53], where the linear mixing model has been extended to multiple sets of endmembers. Each endmember set is found using the convex geometry model resulting in a piece-wise convex representation of the hyperspectral data.…”

“…These include a multiple model endmember detection algorithm based on spectral and spatial information [50], a spatially-smooth piece-wise convex endmember detection algorithm [51], a competitive agglomeration piece-wise convex multiple model endmember detection algorithm [52], and a piece-wise convex spatial-spectral unmixing algorithm using possibilistic and fuzzy clustering [53].…”

“…The spatially-smooth piece-wise convex endmember detection algorithm (Spatial P-COMMEND) [51] The FLICM algorithm adds a fuzzy factor term, G, to the objective function that influences the updates of the membership values by incorporating spatial information. The G term, adapted for use in the Spatial P-COMMEND algorithm, is defined as: 15) where j is the center pixel in the local window under consideration, N j is the neighborhood around the center pixel (such as a 3 × 3 window), and d jk is the Euclidean distance in pixel space of the image indices between pixels j and k. Therefore, the G term scales the influence of neighboring pixels based on their distances to the center pixel in the index space.…”