Archetypal Analysis for Sparse Representation-
Based Hyperspectral Sub-pixel Quantification
Lukas Drees, Ribana Roscher, and Susanne Wenzel
Abstract
The estimation of land cover fractions from remote sensing
images is a frequently used indicator of the environmental
quality. This paper focuses on the quantification of land cover
fractions in an urban area of Berlin, Germany, using simulat-
ed hyperspectral
EnMAP
data with a spatial resolution of 30 m
× 30 m. We use constrained sparse representation, where
each pixel with unknown surface characteristics is expressed
by a weighted linear combination of elementary spectra with
known land cover class. We automatically determine the
elementary spectra from image reference data using arche-
typal analysis by simplex volume maximization, and combine
it with reversible jump Markov chain Monte Carlo method. In
our experiments, the estimation of the automatically derived
elementary spectra is compared to the estimation obtained by
a manually designed spectral library by means of reconstruc-
tion error, mean absolute error of the fraction estimates, sum
of fractions, R
2
, and the number of used elementary spectra.
The experiments show that a collection of archetypes can be
an adequate and efficient alternative to the manually de-
signed spectral library with respect to the mentioned criteria.
Introduction
The estimation of the degree of imperviousness as an indica-
tor of environmental quality is subject of current research
towards a time and cost efficient monitoring of urban areas
[1]. Due to increasing land consumption in cities in the recent
years, which has negative effects on the natural
water
cycle,
the monitoring of land use in those areas is important [2].
Remote sensing data, such as imaging spectroscopy, builds
a valuable basis to comprehensively map urban areas and
quantify the imperviousness based on the spectral informa-
tion (e.g., [3], [4]). Especially, hyperspectral imagery is a
suitable source for mapping of such areas, because it offers
a high spectral separability of different materials. However,
generally, the temporal and spatial resolution is limited in
comparison to sensors with lower spectral resolution. These
limitations are partially overcome with the launch of mis-
sions such as Environmental Mapping and Analysis Program
(
EnMAP
), which increases the availability of hyperspectral
data and the temporal resolution [5]. Nevertheless, due to its
spatial resolution, the provided data is mainly characterized
by spectrally mixed pixels, which demands sophisticated
sub-pixel quantification approaches in order to estimate the
fraction of various land cover classes in each pixel.
In this context, several approaches have been developed
comprising regression approaches [6], [7], probabilistic classifi-
cation methods [8], [9], [10], and the usage of spectral libraries
for spectral mixture analysis [11], [12]. An overview of a wide
variety of unmixing approaches can, for example, be found in
[13]. While the latter approach needs a spectral library contain-
ing elementary spectra of known materials, the first two ap-
proaches also require mixed spectra for learning an appropri-
ate model. These mixed spectra can be derived from the image
using information about known mixed pixels, or from syntheti-
cally mixed pixel, as it has been presented in [8] and [6].
When using spectral libraries, a critical step is the extrac-
tion of the elementary spectra. A manual extraction is time-
consuming and requires human expert-knowledge and there-
fore, automatic extraction techniques have been an active
field of research during the past decade (e.g., [13], [14]). Most
of the algorithms rely on the assumption that the elementary
spectra lie on a convex hull or a convex polytope enclosing
the data distribution (e.g., [15], [16]). Based on this assump-
tion, all data samples can be reconstructed by a non-negative
linear combination of the elementary spectra. A promising
approach from this group is the so-called archetypal analysis,
which searches for extreme points (also known as archetypes)
in the data distribution (e.g., [17], [18], [19], [20]). Archetypal
analysis has already been successfully applied in the field of
sport analytics [21], plant phenotyping [22], or text analysis
[23]. A valuable extension to archetypal analysis is presented
by [24], in which extreme points are extracted in the kernel
space, enabling an efficient nonlinear unmixing. Besides the
actual determination of elementary spectra, other challenges
exist which need to be tackled: The number of elementary
spectra is unknown beforehand and thus, a suitable amount
of spectra needs to be extracted to make the set representative
enough, but also small enough to keep the sub-pixel quantifi-
cation robust and efficient. Moreover, many extraction tech-
niques depend on the initialization and thus, a strategy needs
to be defined to ensure a stable result (e.g., [25], [26]).
In this paper, we address the challenge of automatically
finding a representative set of elementary spectra, including
the automatic determination of the number of elements, for
sub-pixel quantification. We perform the sub-pixel quantifica-
tion using a freely available simulated
EnMAP
scene (Figure
1) of an urban area in Berlin, Germany [27]
1
, aiming at the
estimation of a fraction map containing the classes
impervi-
ous surface, vegetation, soil, and water
. In order to determine
the class fractions, we use sparse representation with non-
negativity, L
0
-sparsity and sum-to-one constraint. We exploit
archetypal analysis to extract elementary spectra in a fully
automatic and unsupervised way. Moreover, we perform
archetypal analysis by simplex volume maximization (SiVM),
which states an efficient selection method [28], [29].
The main contribution of this work is the combination of
archetypal analysis with reversible jump Markov chain Monte
Carlo method (
rjMCMC
, [30]) to obtain a spectral library of high
representational power, yet a small number of elementary
spectra. Using this approach, we are able to determine the
University of Bonn, Bonn, Germany
(
).
Photogrammetric Engineering & Remote Sensing
Vol. 84, No. 5, May 2018, pp. 279–286.
0099-1112/18/279–286
© 2018 American Society for Photogrammetry
and Remote Sensing
doi: 10.14358/PERS.84.5.279
1.
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
May 2018
279
