An Algorithm and Heuristic based on Normalized Mutual Information for Dimensionality Reduction and Classiﬁcation of Hyperspectral images

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An Algorithm and Heuristic based on Normalized Mutual

Information for Dimensionality Reduction and

Classiﬁcation of Hyperspectral images

Elkebir Sarhrouni1, Ahmed Hammouch2and Driss Aboutajdine1

1LRIT,FSR, UMV-A

Rabat, Morocco

sarhrouni436@yahoo.fr

2LGGE, ENSET, UMV-SOUISSI

Rabat, Morocco

ABSTRACT

In the feature classiﬁcation domain, the choice of data affects widely the results. The Hy-

perspectral image (HSI), is a set of more than a hundred bidirectional measures (called

bands), of the same region (called ground truth map: GT). The HSI is modelized at a set

of N vectors. So we have N features (or attributes) expressing N vectors of measures for

C substances (called classes). The problematic is that it’s pratically impossible to invest-

gate all possible subsets. So we must ﬁnd K vectors among N, such as relevant and no

redundant ones; in order to classify substances. Here we introduce an algorithm based on

Normalized Mutual Information to select relevant and no redundant bands, necessary to

increase classiﬁcation accuracy of HSI.

Keywords: Feature Selection, Normalized Mutual information, Hyperspectral images, Clas-

siﬁcation, Redundancy.

Mathematics Subject Classiﬁcation: 68U10, 68R05.

Computing Classiﬁcation System: I.4.7, I.4.8, I.4.9.

1 Introduction

The Hyperspectral image AVIRIS 92AV3C, Airborne Visible Infrared Imaging Spectrometer,

(Perdue, 97) is a substitution of 220 images for the region ”Indiana Pine” at ”north-western In-

diana”, USA (Landgrebe, 1997). The 220 called bands are taken between 0.4µm and 2.5µm.

Each band has 145 lines and 145 columns. The ground truth map is also provided, but only

10366 pixels (49%) are labeled fro 1 to 16. Each label indicates one from 16 classes. The

zeros indicate pixels how are not classiﬁed yet, see Figure.1. The hyperspectral image AVIRIS

92AV3C contains numbers between 955 and 9406. Each pixel of the ground truth map has

a set of 220 numbers (measures) along the hyperspectral image. This numbers (measures)

represent the reﬂectance of the pixel in each band. So the pixel is shown as vector off 220

components,see Figure .2.

To classify pixels of HSI, we ﬁnd some problems realed to their high dimentions, that needs

arXiv:2210.13456v1 [cs.CV] 22 Oct 2022

Figure 1: The Ground Truth map of AVIRIS 92AV3C and the 16 classes

Figure 2: The notion of pixel vector

many cases to detect the relationship between the vectors and the classes, according to

Hughes phenomenon (Huges, 1968). other problems are related to the redundant images

(bands); they complicate the learning system and product incorrect prediction (Lei and Huan,

2004). So we must pick up only the relevant bands. Now we can identfy the problematic re-

laited to HSI as a dimentionality reduction. It is commonly reencountered when we have N

features (or attributes) that express Nvectors of measures for Csubstances (called classes).

The problematic is to ﬁnd Kvectors among N, such as relevant and no redundant ones; in or-

der to classify substances. The number of selected vectors Kmust be lower than Nregarding

the problems above. So we must choose relevant vectors, that means there ability to predicate

the classes. Indeed the bands don’t all contain the information; some bands are irrelevant

like those affected by various atmospheric effects, see Figure.7, we can show the atmospheric

effects on bands: 155, 220 and other bands; so the classiﬁcation accuracy decreases. We

can reduce the dimensionality of hyperspectral images by selecting only the relevant bands

(feature selection or subset selection methodology), or extracting, from the original bands, new

bands containing the maximal information about the classes, using any functions, logical or nu-

merical (feature extraction methodology) (Kwak and Choi, 2007; Kwak and Kim, 2006; YANG,

Yiming and PEDERSEN., 1997) . Here we propose an algorithm based on mutual information,

and normalised mutual information fo reducing dimensionality. This will be as bellow: pick up

the relevant bands ﬁrst, and avoiding redundancy second. We illustrate the principea of this

algorithm using synthetic bands for the scene of HIS AVIRIS 92AV3C(Landgrebe, 1997) , see

Figure.5. Then we validate its effect by applying it to real datat of HSI AVIRIS 92AV3C. Here

each pixel is shown as a vector of 220 components. Figure.2 shows the vector pixels notion

(Kern ´

eis, 2007). Reducing dimensionality means selecting only the dimensions caring a lot of

information regarding the ground truth map (GT).

In the literature, we can cite some methods related to the dimension reduction; but this action

is accompanied by transformation of the multivariate data. Well-known methods include prin-

cipal component analysis, factor analysis, projection pursuit, independent component analysis

(ICA). Principal Component Analysis, or PCA (Oja, J., L. and Vigario, 95), is widely used in

signal processing, statistics, and neural computing (Homayouni, 2005; Homayouni, 1998; Hy-

varinen, 99). The PCA search another space with lower diemnsion, and when the data will be

clearly separated. The independent component analysis ICA search also a space with lower

diemntion, and which the sources of data are separated (Hyvarinen, 99; Comon, 1994; Hyvari-

nen and Karhunen, 2001); so which desired action is minimizing the statistical dependence

of the components and the components didn’t necessary orthogonal. Now due to the dimen-

tionality of Hyperspectral Image, the question proposed is: can we ﬁnd some principal com-

ponents, or independant component among the 220 bands, without transformation of data?

The response is to colculate the separation matrix using normalized mutual information as a

cost fonction. We cite here that some works (Comon, 1994; Linsker, 1992; Barlow, 1961; Lee,

M.Girolami, Bell and Sejnowski, 2000) had already use the mutual information ( but not nor-

malized form of MI) and neural network, to separate blinded sources.

2 Principle of Feature Selection with Mutual Informationl

2.1 Deﬁnition of Mutual Information

The MI is a measure of information contained in both tow ensembles of random variables A

and B:

I(A, B) = Xp(A, B)log2

p(A, B)

p(A).p(B)

Let us consider the ground truth map (GT), and bands as ensembles of random variables, the

MI between GT and each band calculates their interdependence. Fano (Lei and Huan, 2004)

has demonstrated that as soon as mutual information of already selected features Xand the

classe Chas high value, the error probability of classiﬁcation decreases, according to the

formula bellow: H(C/X)−1

Log2(Nc)≤Pe≤H(C/X)

Log2

with : H(C/X)−1

Log2(Nc)=H(C)−I(C;X)−1

Log2(Nc)

and :

Pe≤H(C)−I(C;X)

Log2

=H(C/X)

Log2

Here the conditional entropy H(C/X)is calculated between the ground truth map (i.e. the

classes C) and the subset of bands candidates X.Ncis the number of classes. So when the

features Xhave a higher value of mutual information with the ground truth map, (is more near

to the ground truth map), the error probability will be lower. But it’s impractical to compute this

conjoint mutual information I(C;X), regarding the high dimensionality (Lei and Huan, 2004).

Figure.6. shows the MI between the GT and synthetic bands. The ﬁgure.7 indicates the MI

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AnAlgorithmandHeuristicbasedonNormalizedMutualInformationforDimensionalityReductionandClassicationofHyperspectralimagesElkebirSarhrouni1,AhmedHammouch2andDrissAboutajdine11LRIT,FSR,UMV-ARabat,Moroccosarhrouni436@yahoo.fr2LGGE,ENSET,UMV-SOUISSIRabat,MoroccoABSTRACTInthefeatureclassicationdomain,the...

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An Algorithm and Heuristic based on Normalized Mutual Information for Dimensionality Reduction and Classiﬁcation of Hyperspectral images

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