Image Processing Reference

In-Depth Information

reflectance mix linearly in proportion to area within the instantaneous field of view

(IFOV) (Singer and McCord
1979
; Singer
1981
; Johnson et al.
1983
). This obser-

vation has made possible the development of a systematic methodology for SMA

(Adams et al.
1986, 1989
; Smith et al.
1990
; Gillespie et al.
1990
) that has proven

successful for a variety of quantitative applications with multispectral imagery (e.g.

Adams et al.
1995
; Pech et. al.
1986
; Smith et al.
1990
; Elmore et al.
2000
; Roberts

et al.
1998
). The linear mixing model can be expressed as:

n

n

∑

∑

(14.2)

R

=

fR

+

e

and 0

≤

f

≤

1

i

k

ik

i

k

k

=

1

k

=

1

where
R
is the spectral reflectance for band
i
for each pixel;
n
is the number of

endmembers;
f
is the fraction of endmember
k
;
i
R
is the reflectance of endmem-

ber
k
in band
i
; and
e
the difference between measured and modeled digital num-

ber in band
i
. A unique solution for Eq.
14.2
is obtained by minimizing the residual

error,ε
i
, in a least-square solution, given the constraints on
f
.

Endmembers can be selected from either a spectral library derived from labora-

tory or field measurement, or from representative homogeneous pixels in a satellite

image. If a limited number of distinct spectral endmem-

bers is known
a priori
it is possible to define a “mixing

space” within which mixed pixels can be described as

linear mixtures of the endmembers. Given sufficient spec-

tral resolution, a system of linear mixing equations can be

defined and the best fitting combination of endmember

fractions can be estimated for the observed reflectance

spectra. The strength of the SMA approach lies in the fact

that it explicitly takes into account the physical processes

responsible for the observed radiances and therefore

accommodates the existence of mixed pixels.

A comparative SMA of Landsat ETM+ imagery for a diverse collection of 28

cities highlights several consistencies in the spectral properties of these cities

(Small
2004
). The analysis demonstrates that all of these cities have similar triangular

mixing spaces in which urban reflectance can be accurately represented as linear

mixtures of high albedo substrate, vegetation, and dark surfaces. An analysis of

high-resolution IKONOS imagery for 14 cities yields very similar results suggesting

that a simple three-endmember spectral mixture model provides a multiscale physical

representation of the reflectance of urban mosaics. Areal vegetation fraction esti-

mates for these urban areas agree with high-resolution vegetation fraction measure-

ments to within 10% in New York (Small
2001, 2004
).

Several recent studies have used physical rather than statistical classifications of

urban land cover in individual cities with some degree of success (e.g., Kressler and

Steinnocher
2001
; Rashed et al.
2002
; Small
2001, 2004
; Wu and Murray
2003
; Lu

and Weng
2004
). SMA is quite useful in the analysis of Landsat TM and SPOT

images of urban areas, however it is important to acknowledge that with the significant

spectral mixture

analysis provides

a physically based

methodology to

quantify spectrally

heterogeneous

urban reflectance

within mixed

pixels