Absorption feature based methods calculate ratios between some of the variables (spectral bands) or describe absorption features using geometric parameters (e.g. width, depth, asymmetry). In hyperspectral data, re ectance in bands adjacent to or in close proximity to each other can be highly correlated. This redundancy brings into question whether all bands are actually necessary for an eective classi cation algo- rithm (Hughes phenomena - `curse of dimensionality', Bellman, 1957; Hughes, 1968). Some algorithms have been developed to operate on the entire spectral curve shape , i.e. using every single band in the spectrum. Thus, such methods need to use some mathematical metric which is used to determine the degree of similarity between an unknown and a reference spectrum.
Three basic metrics (i.e. similarity-measures) are introduced in this section which are most commonly used in remote sensing (Keshava, 2004) but also more widely for pattern matching and classi cation problems (van der Meer, 2006). These metrics are the:
1. Euclidean distance 2. Correlation metric 3. Angular metric
3.1 Traditional methods for analysis and classi cation of hyperspectral data 45 Euclidean distance metric
The Euclidean distance is a measure often used to determine the closeness between objects, for example, as used in the K-means classi er. If the closeness between two spectra is to be determined using the Minimum Euclidean Distance (MED) measure:
M ED = ∆ = ||x − x0|| = v u u t L X i=1 (xi− x0i)2, (3.4)
where x and x0 are two spectra, e.g. a target and a reference spectrum and ∆ is the
MED between points given by the two spectra of length L (Figure 3.5).
α λ1 λ2 λ3 Δ λ1 λ2 λ3 x x ̍ x x ̍
a)
b)
Δ*Figure 3.5 Illustration of two points in a vector space given by the vectors x and x0. λ1, λ2, λ3 represent three bands of a spectrum, i.e. three elements of a vector.
a) ∆ is the minimal Euclidian distance (MED; dashed green line) that connects the endpoints of the two vectors. ∆∗ is the MED after extending the vectors (blue
arrows) which is larger than ∆. b) α is the angle between the vectors x and x0.
3.1 Traditional methods for analysis and classi cation of hyperspectral data 46 Correlation measure
Spectral correlation measure or cross correlogram spectral matching (CCSM; van de Meer and Bakker, 1997, 1998) is a measure of correlation describing the similarity between a target spectrum and a reference spectrum. To minimise eects of noise, the cross correlation is calculated at several positions or osets, between the unknown and a reference spectrum. The greatest correlation across all osets is then used as the measure of closeness between the unknown and training spectrum. The CCSM is normalised between `-1' and `1'. It is also centered around the mean of both spectra. The CCSM can be calculated using Equation 3.5
CCSM = P(x − ¯x)(x
0 − ¯x0)
pP(x − ¯x)2P(x0− ¯x0)2 , (3.5)
where x and x0 are for example a target and reference spectrum and ¯x, ¯x0 their re-
spective means. The CCSM yields a value of 1 for a perfect match (high correlation), a value of -1 for perfect inverse correlation (spectra are reverse-images of one another) and 0 for no correlation between two spectra. Van de Meer and Bakker (1998) use the root mean square error (RMSE) to create a map of correlations between an un- known spectrum's cross correlogram and the ideal cross correlogram for the reference spectrum.
Angular metric
The angle between two spectra (or vectors) is the determining measure of similarity between an unknown and a reference spectrum using an angular metric. Mathemati-
cally, a spectrum of L bands can be seen as a vector in RL, i.e. each spectral band can
be seen as a dimension in this vector space (e.g. as shown for 2 bands in Figure 3.6). The elements of a vector also de ne the coordinates of a point. The distances from the origin to the endpoint of this vector vary depending on the re ectance values (brightness) of the bands. If the re ectance values of all bands vary by the same amount, the direction of a vector with respect to the vector coordinate frame remains
3.1 Traditional methods for analysis and classi cation of hyperspectral data 47 constant. The angle between two vectors (i.e. spectra) can thus be used as a measure of similarity between spectra even if they dier in brightness, i.e. spectral magnitude (Figure 3.5a). band 1 ban d 2 0.8 0.9 0.18 0.16 α α α =α
Figure 3.6 Spectral angle concept of a 2 dimensional (2d) vector using two spectra (i.e. vectors) with two spectral bands. The black and green vector represent a dark and a bright ( ve times brighter) spectrum of the same material, respectively. The green dot represents a distance from the origin of the coordinate frame and describes the location of the green vector in this 2d space. The black vector has a dierent distance to the origin, however, the angle is the same as the green vector with respect to the coordinate frame.
Spectral Angle Mapper (SAM)
The spectral angle mapper (Kruse et al., 1993) is one of the most commonly used classical hyperspectral classi ers (e.g. Rowan and Mars, 2003; Dennison et al., 2004; Park et al., 2007; Baissa et al., 2011; Murphy et al., 2012). SAM calculates the simi- larity of two spectra in a high dimensional space using the spectral angle α (Equation 3.6). The main reason for SAM being very popular for classi cation of hyperspectral data is that it uses the entire re ectance curve and is not limited to spectra which exhibit speci c absorption features. This means that SAM can be used to classify materials which have no diagnostic absorption features, including spectra where, due to eects of noise important features are suppressed, as long as the overall curve shape is more or less preserved. In addition, SAM is relatively insensitive to dier-
3.1 Traditional methods for analysis and classi cation of hyperspectral data 48 ences in albedo / brightness and topographic variability in a scene. This makes it ideally suited for classi cation of scenes with variable illumination conditions using an independent spectral library. Such libraries are acquired under controlled conditions of illumination. The brightness of a spectrum does not in uence the spectral angle, i.e. the norm of a vector does not cause a change in angle between two vectors. The spectral angle α is calculated using Equation 3.6:
α = cos−1 ~x · ~x0 ||~x|| · ||~x0|| , (3.6)
which can also be written as
α = cos−1 L P i=1 xi· x0i s L P i=1 xi· s L P i=1 x0 i , (3.7)
where x and x0 are two spectra (e.g. target and reference spectrum) and ||x||, ||x0||
are their respective norms (i.e. vector length). The dimensionality of the spectrum (e.g. number of spectral bands) is given by L. The spectral angle α is bound between
values of -1 and 1 (or 0 and π), however, only values between 0 and π
2 are applicable,
since re ectance cannot take negative values.
SAM is often implemented by applying a user-speci ed angular threshold (e.g. Den- nison et al., 2004, Hecker et al., 2008). This means that if the angle between the target and reference spectrum is smaller than the de ned threshold, the pixel is clas- si ed as belonging to the class of the reference spectrum (Figure 3.7). For example, a spectral angle of zero would indicate that target and reference spectrum are the same, although dierences in albedo are not accounted for by SAM. Because for any particular material, the optimal threshold for SAM is rarely known a priori, and the threshold severely eects classi cation accuracy, other implementations can be used to improve scene classi cation (Murphy et al., 2012). For example, a `minimum angle' criterion can be applied, whereby a target vector is compared to all reference spectra in a spectral library. A class label is then assigned by comparing the angles of all
3.1 Traditional methods for analysis and classi cation of hyperspectral data 49 target-reference combinations and selecting the class of the reference spectrum which has the smallest angle with the unknown (target) spectrum. This approach assumes that the mineral is present in the spectral library. Using the minimum angle always yields a classi cation, as there will always be a smallest angle between an unknown and a reference spectrum. Using a threshold, however, pixels may be unclassi ed in cases where the spectral angles between unknown and reference spectra are larger than the threshold. There is no general recommendation for the use of either ap- proach as it is dependent on the application and the assumptions made to solve a speci c problem. band 1 band 2 α1 αt α2 Reference spectrum Unknown spectrum 1 Unknown spectrum 2 Threshold
Figure 3.7 Schematic representation of SAM in two dimensions (i.e. two spectral bands). A reference spectrum (green vector) is obtained from a spectral library and compared against two unknown spectra (blue and red arrow) using a xed threshold (black dashed line). Spectral angles α1 and α2 are calculated between
each reference-target spectrum combination. These spectral angles are compared against αt which is the threshold angle.