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N-FINDR: an algorithm for fast autonomous spectral end-member

determination in hyperspectral data,

Michael E. Winter*, Department of Earth Sciences, University of Queensland (Australia) and Technical

Research Associates, Inc. (USA)

ABSTRACT

The analysis of hyperspectral data sets requires the determination of certain basis spectra called "end-members". Once these

spectra are found, the image cube can be "unmixed" into the fractional abundance of each material in each pixel. There exist

several techniques for accomplishing the determination of the end-members, most of which involve the intervention of a

trained geologist. Often these end-members are assumed to be present in the image, in the form of pure, or unmixed, pixels.

In this paper a method based upon the geometry of convex sets is proposed to find a unique set of purest pixels in an image.

The technique is based on the fact that in N spectral dimensions, the N-volume contained by a simplex formed of the purest

pixels is larger than any other volume formed from any other combination of pixels. The algorithm works by "inflating" a

simplex inside the data, beginning with a random set of pixels. For each pixel and each end-member, the end-member is

replaced with the spectrum of the pixel and the volume is recalculated. If it increases, the spectrum of the new pixel replaces

that end-member. This procedure is repeated until no more replacements are done. This algorithm successfully derives end-

members in a synthetic data set, and appears robust with less than perfect data. Spectral end-members have been extracted

for the AVIRIS Cuprite data set which closely match reference spectra, and resulting abundance maps match published

mineral maps.

Keywords: end-member, hyperspectral, autonomous

1. INTRODUCTION

Hyperspectral data represents a challenge from a data-processing point of view, as it can consist of hundreds of bands. A

necessary first step is to reduce the complexity of the image by a dimensionality reduction, which compresses the image data

to a few meaningful bands. The most widely used methods for reducing the dimensionality of the data are orthogonal

subspace projections (OSPs) and unmixing the image based on a set of component spectra (end-members). OSPs reduce the

dimensionality of the data by fmding the combinations of bands, which best represent the image in some manner. While

OSPs reduce dimensionality, the resulting images have a mathematical rather than physical relationship with the original

image. The principal advantage of unmixing an image using end-members it that offers the reduction of the complexity of

the data set based on a physical set of component spectra. The resulting images are the abundance of the corresponding

substance for that pixel.

In many cases end-member spectra for an image are unknown. The image may be classified using laboratory spectra,

however this requires that the image be converted to reflectance. Moreover, the selection of end-members can be non-

unique. A more optimal approach is to determine the end-member spectra based solely on the information contained within

the image itself.

A perfect algorithm for the determination of end-members would find end-members directly from the image regardless of

composition or noise, without any a priori knowledge. However, due to the mathematical complexity of the problem and

imperfections in any real data (the influence of atmosphere, sensor noise etc), a more realistic goal is to determine

recognizable spectra, and produce useful abundance maps. Recognizable spectra allow the determination of the real end-

member from a spectral library, while the abundance maps show roughly the relative amount of each end-member in each

pixel.

Generally, algorithms which derive end-members directly from an image fall into two broad categories: those which assume

that the end-members themselves are present in the image in the form of pure, or unmixed, pixels, and those which derive

the spectra of the end-members analytically (for example "factor analysis", Harmon, 1967). Pure pixel based techniques,

which include this algorithm, the selection of extreme points of an n-dimensional scatter plot, and the "convex-cone"

*

Correspondence:

Email: ; Telephone: +61 7 3876-7319

266

Part of

the SPIE Conference on lmacjincj Spectrometry V • Denver, Colorado • July 1999

SPIE Vol. 3753 • 0277-786X1991$

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method (Ifarraguerri, 1997), leverage the simplification that the end-members themselves are present in the image. This

allows a reduction in the scope of the problem by restricting the possible characteristics of the end-members to discrete points

in spectral-feature-space.

The N-FINDR algorithm (Winter, 1999) is essentially an automated technique for finding the purest pixels in an image. The

goal of the algorithm is to duplicate the successful non-automated technique of selecting extreme points of an n-dimensional

scatter plot. Although this method has its critics (e.g. Craig 1990; 1994) it remains the most widely used and successfully

applied method for determining end-members without a priori information. N-FINDR attempts to find the simplex of

maximum volume that can be inscribed within the hyperspectral data set using a simple non-linear inversion. The convex

nature of hyperspectral data allows this operation to be performed in a quick and relatively straightforward manner.

Subsequent optimizations of the algorithm (not detailed here) have allowed the use of this algorithm at speeds approaching

those required by real-time applications.

This work focuses on the performance of the algorithm on both synthetic and real data sets. To illustrate the viability of the

algorithm under realistic circumstances, areal distributions of the minerals alunite, kaolinite and calcite are mapped using the

AVIRIS Cuprite Nevada sample image.

2. THE ALGORITHM

Most methods of determining end-members autonomously from an image make use of the geometric nature of hyperspectral

data (for example, see Craig I 994, Boardman, 1 995).

Data

analysis problems can often be viewed geometrically, with each

observation occupying a point in a space spanned by all ofthe data (cf. Rawlings, 1988).

Generally, the spectra for a given pixel in an image is assumed to be a linear combination ofthe end-member spectra:

joy —eIkckJ

k

+

,

where p is the i-th band ofthej-th pixel, elk

(1)

is

the i-th band ofthe k-th end-member, CkJ

is

the mixing proportions for the

j-th pixel from the k-th end-member, and t'

is

gaussian random error (assumed to be small). Since the pixel compositions are

assumed to be percentages, the mixing proportions are should sum to one:

kj

k

(2)

The mixing proportions can be visualized as "abundance maps" which depict the fractional composition of the end-member

material as a gray level image. Any distribution of data which follows the mixture model outlined in Equations 1 and 2

forms a simplex (Lay, 1982) in data space. A simplex is the simplest geometric object which spans a given space, for

example a triangle in two dimensions and a tetrahedron in three dimensions.

If CkJ

' 1 for

any end-member contribution in a pixel, the other end-member contributions must be near zero, and the pixel

can be classified as pure. These pure pixels define the vertices of the N-dimensional scatter plot of the data. Moreover, these

pure pixels define a simple of maximum volume which can be inscribed within the data set. This procedure "inflates" a

simplex within the data set in order to determine these pixels. A necessary assumption is that there exists at least one pure

pixel per end-member species within the image.

2.1 Pre-processing

In order for the volume to be determined the dimensionality of the image must be reduced to be one less than the number of

end-members. This is accomplished through an orthogonal subspace projection. OSPs act to compress the information in an

image based on a mathematical criteria: maximum power for the singular value decomposition (Rawlings, 1988), maximum

variation for the principal components transform, and noise weighted variation for the Maximum Noise Fraction (MNF)

transform (Green, 1988). In theory, the MNF transform offers the best performance, however for the

purposes of this

algorithm all methods appear to work equally well.

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