ELECCION DEL DISEÑO

The fine spectral resolution of hyperspectral images enables tracking even smallest changes and features in the reflectance spectrum. This property allows to study the reflectance spectra in a similar way like it is done in spectroscopy. Based on the fact that each material has its own unique spec- tral signature it allows recognition of these materials and use in applications like mineral mapping, target detection or classification. However, unlike in laboratory based spectroscopy one can not assure the measured spectrum comes from only one pure material. Therefore, while studying hyperspectral reflectance spectra, the fact that these spectra might consist of a mixture of reflectance from different materials appearing in the pixel should be taken into account. In this section the spectral mixing as well as some basic defi- nitions and concepts to deal with this problem will be discussed.

2.3.1 Spectral Mixing Model

The relatively low GSD of hyperspectral images, acquired from airborne of spaceborne platform, results usually in contribution of more than only one material spectrum in one measured spatial pixel. This means that in this pixel a mixture of spectra is recorded and the process is usually referred to as spectral mixing. Spectral mixing in most of the cases can be modelled as a linear system of equations

24 CHAPTER 2. HYPERSPECTRAL IMAGING

a) SNRp≈ 0 b) SNRp ≈ 40dB

c) SNRp≈ 30dB d) SNRp ≈ 20dB

Figure 2.13: Comparison of spatial effects caused by additive Gaussian noise with different SNR level. The SNR ≈ 0 is achieved by filtering and down-sampling the original image with GSD = 0.2m to GSD = 2m.

where yj ∈ Rm stands for the measured spectrum from the jth hyperspectral

pixel with m spectral bands, A ∈ R[m×n] _{is the mixing matrix containing n}

unique linearly independent spectra and xj ∈ Rnis a vector representing pro-

portions of each element from A in the pixel yj. This model equation (2.16)

for pixel synthesis represents the linear mixing model (LMM) (Bioucas-Dias

et al.,2013;Drake et al.,1999;Horwitz et al., 1971;Keshava,2003;Keshava and Mustard,2002). In other words, the LMM states that the radiance mea- sured by the sensor can be represented as the integrated sum of the spectral radiance of all objects within the captured pixel. Wherein, the proportions

of the ai spectra for each ith material is in linear relation to the geometric

proportions in the pixel

yj = n X i=1 aixji where xji ∝ area(ai) GSD2_y j . (2.17)

Based on the above considerations, the proportions vector xj satisfies the

following positivity and sum to one constrains    xji > 0, ∀i = 1, · · · , n, Pn i=1xji = 1. (2.18)

Although, most of the mixing process is linear, a fraction of radiance re- ceived by the hyperspectral sensor might have undergone a nonlinear mix- ing. While the linear mixing occurs at sensor the nonlinear mixing is usually caused by physical interactions of the incident light e.g. multiple scattering at either macroscopic level [Figure 2.14 e)] or microscopic level [Figure 2.14 f)] (Bioucas-Dias et al.,2013;Halimi et al.,2011;Heylen et al.,2011b;Keshava,

2003; Keshava and Mustard, 2002; Yokoya et al., 2014).

2.3.2 Spectral Endmember

Until now we have referred to the unique, linearly independent material spec- trum as a final element of the mixing matrix in the LMM. This assumption is sufficient for understanding of basic concepts, need to be specified for fur-

SUN SENSOR

(a) Material borders

SUN SENSOR

(b) shadowed area

SUN SENSOR

(c) small discrete ob- jects

SUN SENSOR

(d) mixture of materi- als

SUN SENSOR

(e) macroscopic mul- tiple scattering

SUN SENSOR

(f) microscopic multi- ple scattering

Figure 2.14: Possible mixing scenarios

ther consideration of this thesis. For this purpose we introduce the concept of spectral endmember. Spectral endmember is assumed to represent pure

material at macroscopic level and varies depending on applications (Bioucas-

Dias et al., 2013; Halimi et al., 2011; Heylen et al., 2011b; Keshava, 2003;

Keshava and Mustard,2002; Ma et al., 2014). In this section we present two points of view for endmember namely physical and geometrical.

Physical Approach - Material Endmember

Despite the fact that a material endmember is uniform at macroscopic scale, it can sometimes consist of other materials or chemical elements from which each separately can have its unique spectral features. The size of the macro- scopic scale is highly dependent on the application and sometimes a definition of soil endmember will be sufficient, whereas in other case, a division to clay, sand, sill and loan or even consisting minerals and moisture levels might be necessary.

The spectra of endmembers are usually measured using spectrometer, in 26

the laboratory or in the field, and converted to the reflectance spectra. A collection of spectral endmembers are called spectral libraries.

Signal Processing Approach - Data Cloud

Each hyperspectral pixel consists of m spectral bands, this means that the

pixel yj can be considered as a vector in m-dimensional Euclidean space.

All the hyperspectral pixels build a m dimensional data cloud. Taking into

account the constrains from the equation (2.18) all vectors xj will build

a simplex hence the data cloud spans all pixels, yj will be a simplex too.

The geometrical definition of an endmember assumes that the pure spectra or endmembers occupy extremities of the m dimensional, i.e., vertices of

the data cloud simplex (Keshava and Mustard, 2002; Nascimento and Dias,

2005). The graphical interpretation of that concept using real data is shown

in the Figure 2.15.