**4.1 Band grouping and band extraction: state of the art and proposed strategy**

**4.2 Hierarchical band merging**

*DOI: http://dx.doi.org/10.5772/intechopen.88507*

*4.2.1 Proposed algorithm*

Thus, each *H*ð Þ*<sup>i</sup>*

algorithm is defined below:

Then *<sup>H</sup>*ð Þ *<sup>l</sup>*þ<sup>1</sup> <sup>¼</sup> *T H*ð Þ*<sup>l</sup> ;* ^

merged: find ^

*T H*ð Þ*<sup>l</sup> ; <sup>k</sup>* � � <sup>¼</sup> *<sup>H</sup>*ð Þ*<sup>l</sup>*

A table *Ll*þ<sup>1</sup>

chy levels:

**91**

*<sup>H</sup>* <sup>¼</sup> *<sup>H</sup>*ð Þ*<sup>i</sup>* � �

say with contiguous bands with different bandwidth.

is to say *ni* ordered groups of adjacent bands from *B*.

merged bands *B*<sup>1</sup> and *B*<sup>2</sup> is *B*<sup>1</sup> ⊕ *B*<sup>2</sup> ¼ ½ � *B*1*:λmin*; *B*2*:λmax* .

the hierarchy only contains one individual original band). **Band merging: create level l + 1 from level l**:

*<sup>k</sup>* <sup>¼</sup> *argminkJTH*ð Þ*<sup>l</sup> ; <sup>k</sup>* � � � � with

*<sup>k</sup>* <sup>⊕</sup> *<sup>H</sup>*ð Þ*<sup>l</sup>*

h iÞ.

for 1≤*j*≤ ^

*Ll*þ<sup>1</sup> *<sup>l</sup> <sup>H</sup>*^ *k* ð Þ*<sup>l</sup>* � � <sup>¼</sup> *<sup>H</sup>*^

*<sup>k</sup>* <sup>þ</sup> <sup>2</sup>≤*j*<sup>≤</sup> *nl*, *<sup>L</sup><sup>l</sup>*þ<sup>1</sup>

*L<sup>l</sup>*þ<sup>1</sup> *<sup>l</sup> <sup>H</sup>*^ *k* ð Þ*l* þ 1 � � <sup>¼</sup> *<sup>H</sup>*^

for ^

values over the different bands it contains.

*4.2.2 Band merging criteria*

*<sup>k</sup>*�<sup>1</sup>; *<sup>H</sup>*ð Þ*<sup>l</sup>*

*k* � �.

<sup>0</sup> ; …; *<sup>H</sup>*ð Þ*<sup>l</sup>*

*<sup>j</sup>* is defined as a spectral domain:

*<sup>j</sup>* <sup>¼</sup> *<sup>H</sup>*ð Þ*<sup>i</sup>*

*H*ð Þ*<sup>i</sup>*

The first step of the proposed approach consists in building a hierarchy of groups of adjacent bands that are then merged. Even though it is here intended to be used to select an optimal band subset, this hierarchy of merged bands can also be a way to explore several band configurations with varying spectral resolution, that is to

*Spectral Optimization of Airborne Multispectral Camera for Land Cover Classification…*

Notations. Let *B* ¼ f g *λ<sup>i</sup>* <sup>0</sup> <sup>≤</sup>*<sup>i</sup>* <sup>≤</sup>*nbands* be the (ordered) set of original bands. Let

*i*th level of this hierarchy of merged bands. It is composed of *ni* merged bands, that

*<sup>j</sup> :λmin*; *<sup>H</sup>*ð Þ*<sup>i</sup> <sup>j</sup> :λmax* h i

Let *J*ð Þ*:* be the score that has to be optimized during the band merging process. The proposed hierarchical band merging approach is a bottom-up one. The

**Initialization**: *<sup>H</sup>*ð Þ <sup>0</sup> <sup>¼</sup> *<sup>B</sup>* (that is to say that each merged band of the first level of

Find the pair of adjacent bands at level *l* that will optimize the score if they are

*<sup>k</sup>*þ<sup>2</sup>; …; *<sup>H</sup>*ð Þ*<sup>l</sup> nl*

*<sup>l</sup>* is defined to link the different merged bands at consecutive hierar-

*k*ð Þ *<sup>l</sup>*þ<sup>1</sup>

*<sup>l</sup> <sup>H</sup>*ð Þ*<sup>l</sup> j* � � <sup>¼</sup> *<sup>H</sup>*ð Þ *<sup>l</sup>*þ<sup>1</sup>

*k*ð Þ *<sup>l</sup>*þ<sup>1</sup>

*<sup>j</sup>*�<sup>1</sup> .

*<sup>k</sup>*þ<sup>1</sup>; *<sup>H</sup>*ð Þ*<sup>l</sup>*

*k,L<sup>l</sup>*þ<sup>1</sup> *<sup>l</sup> <sup>H</sup>*ð Þ*<sup>l</sup> j* � � <sup>¼</sup> *<sup>H</sup>*ð Þ *<sup>l</sup>*þ<sup>1</sup> *j*

At the end, the value of a pixel in a merged band is defined as the mean of its

Several optimization scores *J* were examined. (In the algorithm described in Section 4.2.1, this score is aimed to be minimized.) They can be either

Thus, the merged band *B*<sup>1</sup> ⊕ *B*<sup>2</sup> obtained when merging two such adjacent

*j* n o

1≤*j*≤*ni*

is the

<sup>0</sup> <sup>≤</sup>*<sup>i</sup>* <sup>&</sup>lt;*nlevels* be the hierarchy of merged bands. *<sup>H</sup>*ð Þ*<sup>i</sup>* <sup>¼</sup> *<sup>H</sup>*ð Þ*<sup>i</sup>*

#### *4.1.1 State of the art*

Band grouping and clustering. In the specific case of hyperspectral data, adjacent bands are often very correlated to each other. Thus, band selection encounters the question of the clustering of the spectral bands of a hyperspectral data set. This can be a way to limit the band selection solution space. Band clustering/grouping has sometimes been performed in association with individual band selection. For instance, [15] first groups adjacent bands according to conditional mutual information and then performs band selection with the constraint that only one band can be selected per cluster. Su et al. [66] performs band clustering applying k-means to band correlation matrix and then iteratively removes the too inhomogeneous clusters and the bands too different from the representative of the cluster to which they belong. Martínez-Usó et al. [22] first clusters 'correlated' features and then selects the most representative feature of each group, according to mutual information. Chang et al. [40] performs band clustering using a more global criterion taking specifically into account the existence of several classes: simulated annealing is used to maximise a cost function defined as the sum, over all clusters and over all classes, of the sum of correlation coefficients between bands belonging to a same cluster. Bigdeli et al. and Prasad et al. [38, 68] perform band clustering, but not for band extraction: a multiple SVM classifier is defined, training one SVM classifier per cluster. Bigdeli et al. [68] has compared several band clustering/grouping methods, including k-means applied to the correlation matrix or an approach considering the local minima of mutual information between adjacent bands as cluster borders. Prasad and Bruce [38] proposes another band grouping strategy, starting from the first band of the spectrum and progressively growing it with adjacent bands until a stopping condition based on mutual information is reached.

Band extraction. Specific band grouping approaches have been proposed for spectral optimization. De Backer et al. [30] defines spectral bands by Gaussian windows along the spectrum and proposes a band extraction optimizing score based on a separability criterion (Bhattacharyya error bound) thanks to a simulated annealing. [34] merges bands according to a criteria based on mutual information. Jensen and Solberg [69] merges adjacent bands decomposing some reference spectra of several classes into piece-wise constant functions. Wiersma and Landgrebe [70] defines optimal band subsets using an analytical model considering spectra reconstruction errors. Serpico and Moser [52] proposes an adaptation of his steepest ascent algorithm to band extraction, also optimizing a JM separability measure. Minet et al. [26] applies genetic algorithms to define the most appropriate spectral bands for target detection. Last, some studies have also studied the impact of spectral resolution [71], without selecting an optimal band subset.

#### *4.1.2 Proposed approach*

The approach proposed in this study consists in first building a hierarchy of groups of adjacent bands. Then, band selection is performed at the different levels of this hierarchy.

Thus, it is here intended to use the hierarchy of groups of adjacent bands as a constraint for band extraction and a way to limit the number of possible combinations, contrary to some existing band extraction approaches such as [52] that extract optimal bands according to JM information using an adapted optimization method or [26] that directly use a genetic algorithm to optimize a wrapper score.

*Spectral Optimization of Airborne Multispectral Camera for Land Cover Classification… DOI: http://dx.doi.org/10.5772/intechopen.88507*

## **4.2 Hierarchical band merging**

The first step of the proposed approach consists in building a hierarchy of groups of adjacent bands that are then merged. Even though it is here intended to be used to select an optimal band subset, this hierarchy of merged bands can also be a way to explore several band configurations with varying spectral resolution, that is to say with contiguous bands with different bandwidth.

### *4.2.1 Proposed algorithm*

Notations. Let *B* ¼ f g *λ<sup>i</sup>* <sup>0</sup> <sup>≤</sup>*<sup>i</sup>* <sup>≤</sup>*nbands* be the (ordered) set of original bands. Let *<sup>H</sup>* <sup>¼</sup> *<sup>H</sup>*ð Þ*<sup>i</sup>* � � <sup>0</sup> <sup>≤</sup>*<sup>i</sup>* <sup>&</sup>lt;*nlevels* be the hierarchy of merged bands. *<sup>H</sup>*ð Þ*<sup>i</sup>* <sup>¼</sup> *<sup>H</sup>*ð Þ*<sup>i</sup> j* n o 1≤*j*≤*ni* is the *i*th level of this hierarchy of merged bands. It is composed of *ni* merged bands, that is to say *ni* ordered groups of adjacent bands from *B*.

Thus, each *H*ð Þ*<sup>i</sup> <sup>j</sup>* is defined as a spectral domain:

$$H\_j^{(i)} = \left[ H\_j^{(i)}.\lambda\_{min}; H\_j^{(i)}.\lambda\_{max} \right]\_i$$

Thus, the merged band *B*<sup>1</sup> ⊕ *B*<sup>2</sup> obtained when merging two such adjacent merged bands *B*<sup>1</sup> and *B*<sup>2</sup> is *B*<sup>1</sup> ⊕ *B*<sup>2</sup> ¼ ½ � *B*1*:λmin*; *B*2*:λmax* .

Let *J*ð Þ*:* be the score that has to be optimized during the band merging process.

The proposed hierarchical band merging approach is a bottom-up one. The algorithm is defined below:

**Initialization**: *<sup>H</sup>*ð Þ <sup>0</sup> <sup>¼</sup> *<sup>B</sup>* (that is to say that each merged band of the first level of the hierarchy only contains one individual original band).

**Band merging: create level l + 1 from level l**:

Find the pair of adjacent bands at level *l* that will optimize the score if they are merged: find ^ *<sup>k</sup>* <sup>¼</sup> *argminkJTH*ð Þ*<sup>l</sup> ; <sup>k</sup>* � � � � with

$$\begin{aligned} T(H^{(l)},k) &= \left[ H\_0^{(l)}; \ldots; H\_{k-1}^{(l)}; H\_k^{(l)} \oplus H\_{k+1}^{(l)}; H\_{k+2}^{(l)}; \ldots; H\_{n\_l}^{(l)} \right] \\ \text{Then } H^{(l+1)} &= T\left( H^{(l)}, \hat{k} \right). \end{aligned}$$

A table *Ll*þ<sup>1</sup> *<sup>l</sup>* is defined to link the different merged bands at consecutive hierarchy levels:

$$\begin{aligned} \text{for } 1 \le j \le \hat{k}, \, L\_l^{l+1} \left( H\_j^{(l)} \right) = H\_j^{(l+1)} \\\\ L\_l^{l+1} \left( H\hat{k}^{(l)} \right) = H\hat{k}^{(l+1)} \end{aligned}$$

$$\begin{aligned} L\_l^{l+1} \left( H\hat{k}^{(l)} + \mathbf{1} \right) = H\hat{k}^{(l+1)} \end{aligned}$$

$$\text{for } \hat{k} + 2 \le j \le n\_l, \, L\_l^{l+1} \left( H\_j^{(l)} \right) = H\_{j-1}^{(l+1)}.$$

At the end, the value of a pixel in a merged band is defined as the mean of its values over the different bands it contains.

#### *4.2.2 Band merging criteria*

Several optimization scores *J* were examined. (In the algorithm described in Section 4.2.1, this score is aimed to be minimized.) They can be either

supervised or unsupervised, depending whether classes are considered or not at this step.

#### *4.2.2.1 Correlation between bands*

Between band correlation (either the classic normalized correlation coefficient or mutual information) (see **Figure 11**) measures the dependence between bands. So a first band merging criterion intends to merge adjacent bands considering how they are correlated to each other. Thus, it tries to obtain consistent groups of adjacent correlated bands.

Such measure inspired from [40] can be defined by the next function in equation 9 (intended to be minimized):

$$J\left(H^{(l)}\right) = \sum\_{i=1}^{n\_l} \sum\_{\substack{b\_1 = H\_i^{(l)} \ \lambda\_{\text{min}} \ b\_2 = H\_i^{(l)} \ \lambda\_{\text{min}}}}^{H\_i^{(l)} \ \lambda\_{\text{max}}} (1 - c(b\mathbf{1}, b\mathbf{2})) \tag{9}$$

*4.2.2.3 Separability*

**Figure 12.**

*4.2.3 Results*

**93**

merging can be defined by equation 11 as

*DOI: http://dx.doi.org/10.5772/intechopen.88507*

*constant reconstructed spectra for these merged bands (Pavia data set).*

bands will have a little impact on class separability.

Another criterion to merge adjacent band is their contribution to separability between classes. Possible separability measures are the Bhattacharyya distance (*B-distance*) or the Jeffries-Matusita distance [35, 52] already used as FS score in 3. At a level of the band merging hierarchy, the best set of merged bands is the one that maximizes class separability. So a possible criterion *J* (to minimize) for band

*On the left, examples of merged bands superimposed on the original reference spectra. On the right, piece-wise*

*Spectral Optimization of Airborne Multispectral Camera for Land Cover Classification…*

**Figure 13** shows results on Pavia data set for the three criteria described in the previous section. The separability-based criterion tends to lead to more different results than the other ones. The different criteria do not consider the same parts of the spectrum as having to be kept at fine resolution. For instance, correlation or spectra reconstruction criteria tend to fast merge bands between number 30 and 32, while separability tends to preserve them at fine resolution. On the opposite, separability tends to fast merge some bands in the red-edge domain, while the other criteria keep this domain at fine resolution. This can be understood considering the underlying criteria; indeed adjacent bands are not very correlated to each other in this domain, and the slope of spectra is strong for vegetation classes; thus they cannot be merged easily according to correlation or spectra approximation error band merging criteria. On the opposite, the only interesting information for classification (e.g. for class separability) is the fact there is a slope there and thus the values of the bands before and after this domain. Thus, merging these red-edge

As the hierarchy of merged bands can also be a way to explore several band configurations with varying contiguous bands with different spectral resolution, the different band configurations corresponding to the different levels were evaluated using a classification quality measure. Thus, for each level, a classification was performed using a support vector machine (SVM) classifier with a radial basis function (rbf) kernel and evaluated. Its Kappa coefficient was considered.

Such results are presented on **Figure 14**. It can be seen that some spectral configurations made it possible to obtain better results than at original spectral resolution. Configurations obtained using the correlation coefficient are generally less good than for the two other criteria. Except for Pavia, the spectra piece-wise approximation

*J H*ð Þ*<sup>l</sup>* ¼ �*JM H*ð Þ*<sup>l</sup>* (11)

where *c b*ð Þ <sup>1</sup>*; b*<sup>2</sup> is the correlation score between bands *b*<sup>1</sup> and *b*2.

#### *4.2.2.2 Spectra approximation error*

Band merging can also use the method as described in [69] to decompose some reference spectra of several classes into piece-wise constant functions (**Figure 12**). Adjacent bands are then merged trying to minimize the reconstruction error between the original and the piece-wise constant reconstructed spectra.

Such measure is defined by the next function (see equation 10) for a set *sj*<sup>1</sup>≤*<sup>j</sup>* <sup>≤</sup>*ns* of *ns* spectra:

$$J\left(H^{(l)}\right) = \sum\_{j=1}^{n\_s} \sum\_{i=1}^{n\_l} \sum\_{b=H\_i^{(l)} \cdot \lambda\_{\text{min}}}^{H\_i^{(l)} \cdot \lambda\_{\text{max}}} |s\_j(b) - mean\left(s\_j, H\_i^{(l)}\right)| \tag{10}$$

where *mean sj; H*ð Þ*<sup>l</sup> i* � � denotes the mean of spectra *sj* over spectral domain *<sup>H</sup>*ð Þ*<sup>l</sup> i* .

**Figure 11.** *Examples of groups of bands superimposed on the band correlation matrix (for Pavia data set).*

*Spectral Optimization of Airborne Multispectral Camera for Land Cover Classification… DOI: http://dx.doi.org/10.5772/intechopen.88507*

#### **Figure 12.**

*On the left, examples of merged bands superimposed on the original reference spectra. On the right, piece-wise constant reconstructed spectra for these merged bands (Pavia data set).*

#### *4.2.2.3 Separability*

Another criterion to merge adjacent band is their contribution to separability between classes. Possible separability measures are the Bhattacharyya distance (*B-distance*) or the Jeffries-Matusita distance [35, 52] already used as FS score in 3.

At a level of the band merging hierarchy, the best set of merged bands is the one that maximizes class separability. So a possible criterion *J* (to minimize) for band merging can be defined by equation 11 as

$$J\left(H^{(l)}\right) = -j\mathbf{M}\left(H^{(l)}\right) \tag{11}$$

#### *4.2.3 Results*

**Figure 13** shows results on Pavia data set for the three criteria described in the previous section. The separability-based criterion tends to lead to more different results than the other ones. The different criteria do not consider the same parts of the spectrum as having to be kept at fine resolution. For instance, correlation or spectra reconstruction criteria tend to fast merge bands between number 30 and 32, while separability tends to preserve them at fine resolution. On the opposite, separability tends to fast merge some bands in the red-edge domain, while the other criteria keep this domain at fine resolution. This can be understood considering the underlying criteria; indeed adjacent bands are not very correlated to each other in this domain, and the slope of spectra is strong for vegetation classes; thus they cannot be merged easily according to correlation or spectra approximation error band merging criteria. On the opposite, the only interesting information for classification (e.g. for class separability) is the fact there is a slope there and thus the values of the bands before and after this domain. Thus, merging these red-edge bands will have a little impact on class separability.

As the hierarchy of merged bands can also be a way to explore several band configurations with varying contiguous bands with different spectral resolution, the different band configurations corresponding to the different levels were evaluated using a classification quality measure. Thus, for each level, a classification was performed using a support vector machine (SVM) classifier with a radial basis function (rbf) kernel and evaluated. Its Kappa coefficient was considered.

Such results are presented on **Figure 14**. It can be seen that some spectral configurations made it possible to obtain better results than at original spectral resolution. Configurations obtained using the correlation coefficient are generally less good than for the two other criteria. Except for Pavia, the spectra piece-wise approximation

error merging criterion tends to lead to the best results. But for Pavia, the classifica-

*Kappa (in %) reached by a rbf SVM for the different band configurations of the hierarchy (x-axis = number of merged bands in the spectral configuration corresponding to the hierarchy level): for Pavia (top), Indian Pines*

*Spectral Optimization of Airborne Multispectral Camera for Land Cover Classification…*

To optimize spectral configuration for a limited number of merged bands, a greedy approach was first used: it performed band selection at the different levels of the hierarchy of merged bands, paying no attention at results obtained at the

The feature selection (FS) score to optimize was the JM separability measure.

Obtained results on Pavia data set are presented on **Figure 15**: five merged bands

(as in [27]) were selected at each level of the hierarchy of merged bands. The positions of the selected merged bands do not change a lot when climbing the hierarchy, except when reaching the lowest spectral resolution configurations. At some levels of the hierarchy, the position of some selected merged bands can also move and then come back to its initial position when climbing the hierarchy. Thus, it can be possible to use the selected bands at a level *l* to initialize the algorithm at the next level *l* þ 1. This modified method will be presented in

tion Kappa reached using the different criteria remained very similar.

previous level. Thus a set of merged bands was selected at each level of

It was optimized at each level of the hierarchy using an SFFS incremental

**4.3 Band selection within the hierarchy**

*DOI: http://dx.doi.org/10.5772/intechopen.88507*

*(middle) and Salinas (bottom) data sets.*

*4.3.1 Greedy algorithm*

optimization heuristic [44].

the hierarchy.

**Figure 14.**

*4.3.1.1 Results*

Section 4.3.2.

**95**

#### **Figure 13.**

*Hierarchies of merged bands obtained for different criteria for Pavia data set: Spectra piece-wise approximation error (top), between band correlation (middle) and class separability (bottom). X-axis corresponds to the band numbers/wavelengths. y-axis corresponds to the level in the band merging hierarchy (bottom, finest level with original bands; top, only a single merged band). Vertical black lines are the limits between merged bands: the lower the hierarchy, the more the merged bands are. Reference spectra of the classes are displayed in colour.*

*Spectral Optimization of Airborne Multispectral Camera for Land Cover Classification… DOI: http://dx.doi.org/10.5772/intechopen.88507*

#### **Figure 14.**

*Kappa (in %) reached by a rbf SVM for the different band configurations of the hierarchy (x-axis = number of merged bands in the spectral configuration corresponding to the hierarchy level): for Pavia (top), Indian Pines (middle) and Salinas (bottom) data sets.*

error merging criterion tends to lead to the best results. But for Pavia, the classification Kappa reached using the different criteria remained very similar.

#### **4.3 Band selection within the hierarchy**

#### *4.3.1 Greedy algorithm*

To optimize spectral configuration for a limited number of merged bands, a greedy approach was first used: it performed band selection at the different levels of the hierarchy of merged bands, paying no attention at results obtained at the previous level. Thus a set of merged bands was selected at each level of the hierarchy.

The feature selection (FS) score to optimize was the JM separability measure. It was optimized at each level of the hierarchy using an SFFS incremental optimization heuristic [44].

#### *4.3.1.1 Results*

Obtained results on Pavia data set are presented on **Figure 15**: five merged bands (as in [27]) were selected at each level of the hierarchy of merged bands. The positions of the selected merged bands do not change a lot when climbing the hierarchy, except when reaching the lowest spectral resolution configurations. At some levels of the hierarchy, the position of some selected merged bands can also move and then come back to its initial position when climbing the hierarchy.

Thus, it can be possible to use the selected bands at a level *l* to initialize the algorithm at the next level *l* þ 1. This modified method will be presented in Section 4.3.2.

The merged band subsets selected at the different levels of the hierarchy were evaluated according to a classification quality measure. As in the previous section, the Kappa coefficient reached by a rbf SVM was considered. Results for Pavia and Indian Pines data sets can be seen in **Figure 16**. At each level of the hierarchy, 5 bands were selected for Pavia, and 10 bands for Indian Pines. It can be seen that these accuracies remain very close to each other whatever the band merging criterion used, and no band merging criterion tends to really be better than the other ones. Results obtained using merged bands are generally better than using the

*Kappa (in %) reached for rbf SVM classification for merged band subsets selected at the different levels of the hierarchy for Pavia and Indian Pines data sets using the greedy FS algorithm (x-axis = number of merged bands*

*Spectral Optimization of Airborne Multispectral Camera for Land Cover Classification…*

*DOI: http://dx.doi.org/10.5772/intechopen.88507*

The previous merged band selection approach is greedy and computing time expensive. So an adaptation of the SFFS heuristic was proposed to directly take into account the band merging hierarchy in the band selection process. As for the hierarchical band merging algorithm, a bottom-up approach was chosen. Contrary to the greedy approach, this one uses the band subset selected at the previous lower level when performing band selection at a new level of the hierarchy of merged

*4.3.2 Taking into account the band merging hierarchy during selection*

*in the spectral configuration corresponding to the hierarchy level).*

original bands.

**97**

**Figure 16.**

*4.3.2.1 Proposed algorithm*

bands. This algorithm is described below:

#### **Figure 15.**

*Pavia data set: selected bands at the different levels of the hierarchy using the greedy approach for hierarchies of merged bands obtained using different band merging criteria: spectra piece-wise approximation error (top), between band correlation (middle) and class separability (bottom); x-axis corresponds to the band numbers/ wavelengths; y-axis corresponds to the level in the band merging hierarchy (bottom—finest level with original bands; and top—only a single merged band).*

*Spectral Optimization of Airborne Multispectral Camera for Land Cover Classification… DOI: http://dx.doi.org/10.5772/intechopen.88507*

#### **Figure 16.**

*Kappa (in %) reached for rbf SVM classification for merged band subsets selected at the different levels of the hierarchy for Pavia and Indian Pines data sets using the greedy FS algorithm (x-axis = number of merged bands in the spectral configuration corresponding to the hierarchy level).*

The merged band subsets selected at the different levels of the hierarchy were evaluated according to a classification quality measure. As in the previous section, the Kappa coefficient reached by a rbf SVM was considered. Results for Pavia and Indian Pines data sets can be seen in **Figure 16**. At each level of the hierarchy, 5 bands were selected for Pavia, and 10 bands for Indian Pines. It can be seen that these accuracies remain very close to each other whatever the band merging criterion used, and no band merging criterion tends to really be better than the other ones. Results obtained using merged bands are generally better than using the original bands.

#### *4.3.2 Taking into account the band merging hierarchy during selection*

#### *4.3.2.1 Proposed algorithm*

The previous merged band selection approach is greedy and computing time expensive. So an adaptation of the SFFS heuristic was proposed to directly take into account the band merging hierarchy in the band selection process. As for the hierarchical band merging algorithm, a bottom-up approach was chosen. Contrary to the greedy approach, this one uses the band subset selected at the previous lower level when performing band selection at a new level of the hierarchy of merged bands. This algorithm is described below:

Let *<sup>S</sup>*ð Þ*<sup>l</sup>* ¼ f*<sup>S</sup>* ð Þ*l <sup>i</sup>* g1≤*i* ≤*p* be the set of selected merged bands at level *l* of the hierarchy. (NB: the same number *p* of bands is selected at each level of the hierarchy.) **Initialization**: standard SFFS band selection algorithm is applied to the base

error') can be seen both for the greedy FS algorithm and for the hierarchy aware one in **Figure 18**: obtained results remain very close, whatever the optimization

*Spectral Optimization of Airborne Multispectral Camera for Land Cover Classification…*

(see **Table 4**), while the proposed hierarchy aware algorithm is really faster.

Both algorithms lead to equivalent results considering classification performance

Hyperspectral imagery consists of hundreds of contiguous spectral bands, but only a subset of well-chosen bands is generally sufficient for a specific classification

*Computing times and best kappa coefficients reached on Pavia (for a 5-band subset) and Indian Pines (for a 10-band subset) data sets for band merging criterion 'spectra piece-wise approximation error'.*

*Kappa (in %) reached for rbf SVM classification for merged band subsets selected at the different levels of the hierarchy (built for band merging criterion 'spectra piece-wise approximation error') for Pavia and Indian*

*Pines data sets, using the hierarchy aware band selection algorithm.*

algorithm.

*DOI: http://dx.doi.org/10.5772/intechopen.88507*

**5. Conclusion**

**Table 4.**

**99**

**Figure 18.**

level *H*ð Þ <sup>0</sup> of the hierarchy.

**Iterations over the levels of the hierarchy:**

Generate *S*ð Þ *<sup>l</sup>*þ<sup>1</sup> from *S*ð Þ*<sup>l</sup>* : *<sup>S</sup>*ð Þ *<sup>l</sup>*þ<sup>1</sup> f*L<sup>l</sup>*þ<sup>1</sup> *<sup>l</sup> S* ð Þ*l i* <sup>g</sup><sup>1</sup>≤*i*≤*<sup>p</sup>* Remove possible duplications from *S*ð Þ *<sup>l</sup>*þ<sup>1</sup> . **if** #*S*ð Þ *<sup>l</sup>*þ<sup>1</sup> <*p*,

find

$$\mathfrak{s} = \operatorname\*{argmax}\_{\mathbf{b} \in H^{(l+1)}} \mathbf{s}^{(l+1)} \mathbf{J} \left( \mathbf{S}^{(l+1)} \mathbf{U} \mathbf{b} \right),$$

$$\mathbb{S}^{(l+1)} \leftarrow \{ \mathbb{S}^{(l+1)}; \imath \}$$

### **endif**

Question *<sup>S</sup>*ð Þ *<sup>l</sup>*þ<sup>1</sup> : find band *<sup>s</sup>*<sup>∈</sup> *<sup>S</sup>*ð Þ *<sup>l</sup>*þ<sup>1</sup> such that *<sup>S</sup>*ð Þ *<sup>l</sup>*þ<sup>1</sup> \f g*<sup>s</sup>* maximizes FS score, i.e. *<sup>s</sup>* <sup>¼</sup> *argmaxz* <sup>∈</sup>*S*ð Þ *<sup>l</sup>*þ<sup>1</sup> *J S*ð Þ *<sup>l</sup>*þ<sup>1</sup> \f g*<sup>s</sup>* .

*<sup>S</sup>*ð Þ *<sup>l</sup>*þ<sup>1</sup> *<sup>S</sup>*ð Þ *<sup>l</sup>*þ<sup>1</sup> \f*s*<sup>g</sup>

Then apply classic SFFS algorithm until #*S*ð Þ *<sup>l</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>p</sup>*.
