Spectral Feature Extraction Based on Orthogonal Polynomial Function Fitting

Liwei LI, Bing ZHANG, Senior Member, IEEE, Lianru GAO, and Yuanfeng WU

Key Laboratory of Digital Earth Science, Institute of Remote Sensing and Digital Earth, Chinese Academy of Sciences, Beijing 100094, China. Corresponding Author: zb@ceode.ac.cn (Bing ZHANG)

Abstract—we propose a new method for spectral feature extraction based on Orthogonal Polynomial Function (OPF) fitting. Given a spectral signature, it is firstly divided into spectral segments by a splitting strategy. All segments are fitted by using OPF respectively. The features of input spectrum are selected from the fitting coefficients of all segments. 10 laboratory spectra of various materials are selected to validate the ability of the proposed method. The results show that our method can efficiently mine geometric structural information of spectral signatures, and compress them into a few parameters. These parameters can be used to sparsely represent the input spectra and also well discriminate different spectral signatures. The proposed method is more powerful than the inverse Gaussian function model as it can not only will fit the red-edge spectral segment but also can fit other types of spectral curves. Also, the extracted features are slightly better than the original bands at the ability of discrimination in terms of RSDPW in Euclidean space while largely reduce the number of features. Overall, the proposed method has promising prospects in hyperspectral data analysis.

Keywords—hyperspectral data, orthogonal polynomial function, fitting, feature extraction

I. INTRODUCTION Hyperspectral data record spectral absorption features of

surface materials. Thus the materials present in a spectral signature can be identified [1], [2]. However, information extraction from hyperspectral data is always difficult due to the high dimension and large redundancy [3]. The performance of most classification algorithms suffers as the number of features becomes large [4]. In addition to defying the curse of dimensionality [5], eliminating irrelevant features can also reduce system complexity, processing time of data analysis, and the cost of collecting irrelevant features.

Current spectral feature extraction methods can be classified into two categories based on whether or not explicitly using the geometric structure of spectral signatures. The first category named functional method analyzes specific wavelength positions and characteristic shapes of the absorption features in the spectral signature by using function fitting and/or derivative analysis et al. [6]-[12]. Typical examples include the use of a set of parameters (spectral absorption width, depth, symmetric type, area, etc.) for the analysis of mineral spectra [8], the use of the Inverse Gaussian Function (IGF) model for analysis of spectral red-edge area of vegetation [10], and the use of intrinsic structural information about spectral curve to construct a spectral measure to distinguish different features [11], [12]. The second category is

called the statistical method in which spectra data are treated as random vectors in the high-dimensional feature space [13]-[15]. Typical examples include Maximum Noise Fraction (MNF) transformation [13], Best-bases method [14], and Discriminative Analysis Boundary Extraction (DBFE) [15] and so on.

Compared with the statistical method which suffers a lot of uncertainty in the parameter estimation process and usually involves a large amount of calculation [16], the functional method is usually simpler and more efficient, and can directly mine the geometric structure of spectral signatures associated with the natures of materials. In addition, features extracted from the function method can be used as input of statistical method [17], [18].

Orthogonal wavelet decomposition (OWD) is a useful tool in the spectral analysis [19], [20]. Although it is natural for OWD to extract local spectral features, it shows general discriminability using large scale coefficients [19]. This characteristic inspires us to carry out spectral signature feature extraction by introducing Orthogonal Polynomial Function (OPF). The OPF is widely used in data smoothing and data interpolation [21]. In contrast to OWD, It can naturally decompose spectral curves into a series of fitting coefficients in a global manner. The coefficients can be considered as vectors in a much lower dimensional space after a linear orthogonal transformation of the original feature space, and are used as features to assist spectra analysis.

II. SPECTRAL EXTRACTION BASED ON OPF FITTING

A. Orthogonal Polynomials Function Fitting In mathematics, an orthogonal polynomial sequence is a

family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. Legendre polynomial is selected in our approach, and the graphs of the first five polynomials are shown in figure 1:

Fig. 1. First five terms of the Legendre polynomial

Given a set of data points (xi, yi), and generates a function that "best" fits the data for a given maximum degree of polynomial. The generated polynomial will be of the form in the equation (1):

g x b p xjj

m

j( ) ( )==∑

0 (1)

Where pj is a set of m+1 Legendre polynomials, bj is a set of m+1 constants that can be gotten by least squares technique.

B. Feature extraction process The feature extraction process includes three steps:

Splitting strategy: an input spectral signature is firstly divided into spectral segments through a splitting strategy. A binary tree is used to segment each spectral profile. Each segment is named as Lij, i refers to a level index and is within the range [1, 2BandNum-1]. j refers to a segment index and is within the range [1, 2i]. At the top level, the original bands are treated as a single segment, which is named as L11.

Fitting strategy: Given a predefined OPF, the only parameter is the polynomial order. The number of fitting parameters is the order plus one. A large order can be used to fit segments in the low Level. Each coefficient captures a part of the global structure of spectral signature. A relatively small order can be used to fit segments in the high level. Each coefficient captures a part of local structure of spectral signature. In the real application, the number of coefficients required is always smaller than the number of original bands because of the inner smooth property of spectral signatures.

Selection strategy: Two selection strategies are designed. For the representation purpose, all fitting coefficients of each segment are firstly sorted in descending order, and then select the first N significant coefficients as features. N can be set to a fixed number or adaptively determined by setting maximum fitting error. The extracted features work like a sparse representation of the spectral segment. For the discrimination purpose, the sequence of significant sparse features may be not the same for the different spectral segment. A simple way is to select all coefficients in the stacked vector as extracted features. In order to facilitate the distinction, the features without sparsity constraints are called normal features.

III. EXPERIMENTAL DATA AND ANALYSIS To verify the effectiveness of the proposed feature

extraction method, we carried out experiments by using laboratory spectra.

A. Data and Experiment Setup Select 10 laboratory spectra from the USGS spectral library

for analysis. The band range is 400nm-2500nm and the wavelength number is 420. The first 5 spectra are vegetation, including aspenlf1.spc, aspenlf2.spc, blackbru.spc, bluespru.spc and cheatgra.spc respectively. The second 5 spectra are minerals including a-alunit.spc, a-chlori.spc, a-illite.spc, a-jarosi.spc, and a-smecti.spc. These spectra are numbered in order as s1, s2, s3… s10, and the corresponding spectral curves are shown in Fig. 2.

Two groups of experiment are setup as shown in table 1. The first group is to validate the representational ability of the

extracted features using L1 segment of s1. Normal features extracted by OPF with the order of 4/19/39 (L1O4F5, L1O19F20, and L1O39F40) are compared in terms of their ability of spectral curve representation. And then normal features and sparse features both with the number of 20 (L1O19F20 and L1O19SF20) are compared. Finally, features of IGF and normal features with the number of 5 (IGF4 and L1O4F5) are compared in terms of their fitting ability on the red-edge spectral segment (680nm-780nm).

The second group is to validate the discrimination ability of normal features using all 10 spectra. Two types of normal features with the number of 40 are extracted using different strategies (L1O19F20, L3O4F20). Relative Spectral Discriminatory Power (RSDPW) in Euclidean space is used to test the discrimination of the obtained features [22]. RSDPW is calculated as follows: assume that m( , ) is any given distance measure , Let d be the feature of a reference spectral signature. Suppose that s and t are the features of any pair of spectral signatures. RSDPW calculates two ratios the ratio of m(s,t;d) to m(t,s;d) and the ratio of m(t,s;d) to m(s,t;d),and selects the larger of them as the discriminatory power of m( , ).

Fig. 2. 10 selected spectral signatures for the test

TABLE I. EXPERIMENTAL DESIGN SHEET. THERE IS A TOTAL OF 2 TEST GROUPS; THE PARAMETERS CORRESPOND TO THE NUMBERS OF TESTED SPECTRA, THE FITTING FUNCTION, THE SELECTED SPECTRAL SEGMENT,

THE NUMBER OF FEATURES AND THE NAME OF FEATURES

No. Spectra Fitting function

Spectral segment

Features number

Feature names

1

s1 OPF with order 4 L1 5 L1O4F5

s1 OPF with order 19 L1 20 L1O19F20

s1 OPF with order 39 L1 40 L1O39F40

s1 OPF with order 19 L1 20 L1O19SF20

s1 IGF Red-edge bands 4 IGF4

s1 OPF with order 4

Red-edge bands 5 L1O4F5

2

s1-s10 OPF with order 19 L1 20 L1O19F20

s1-s10 OPF with order 4 L3 20 L3O4F20

s1-s10 OPF with order 0 L2BandNum-1 420 s1

Wavelength Number

Ref

lect

ence

B. Result Analysis Fig. 3 corresponds to the profiles of sparse features

L1O19SF20 and normal features L1O4F5, L1O19F20, and L1O39F40. The extracted features show a gradual decreasing fluctuation trend. The trend of profiles indicates that the energy of s1 mainly lies in the low frequency bandwidth. For both normal and sparse features, the low-order features are a subset of high-order features, because the features are extracted via inner product of the s1 and the corresponding OPF. The features can be regarded as the new coordinates of the original spectral bands in the OPF base. From the OPF used, the first feature, which is most prominent, is linearly proportional to the mean value of s1.

Fig. 4 corresponds to the reconstruction result of the spectrum s1 by these features. For the normal features, the accuracy of reconstruction improves with the increase in the number of features. The reconstruction result of L1O4F5 can roughly reflect the overall fluctuation structure of s1, the reconstruction result of L1O19F20 further approximates the fluctuation range of s1, and the reconstruction result of L1O39F40 can well characterize the three main absorption valleys in s1. Meanwhile, the reconstruction result of L1O19SF20 is close to L1O39F40 and is much better than L1O19F20. This proves the advantage of sparse features at the ability of spectral representation.

Fig. 3. The profiles of sparse features L1O19SF20 and normal features

L1O4F5, L1O19F20, and L1O39F40

Fig. 4. The profiles of reconstruction results by sparse features L1O19SF20

and normal features L1O4F5, L1O19F20, and L1O39F40

Fig. 5. The profiles of the red-edge segment of spectrum s1 after being fitted

by normal features L1O4F5 and IGF4

Fig. 5 corresponds to the profiles of the red-edge segment of spectrum s1 after being fitted by normal features L1O4F5 and IGF4. Fig. 7 corresponds to the profiles of reconstruction error of IGF4 and normal features L1O4F5. The reconstruction results of both features are very similar to the original curve, indicating that both are able to characterize the structural characteristics of the red edge bands. Profile of red-edge bands has good smooth continuity, and is structurally similar to the unilateral area of IGF, allowing the IGF to play a role. OPF can fit geometric structures of more complex curves and is more flexible and universal than the IGF in terms of curve fitting.

Table 2 corresponds to comparisons of 420 bands and L3O4F20/ L1O19F20 in terms of RSDPW based on s1. Values of RSDPW refer to the Euclidean distances between selected spectrum and s1 based on certain features. Given a pair of spectra, the higher RSDPW means the greater discriminatory power. From the obtained RSDPW, both L1O19F20 and L3O4F20 are slightly better than the original bands in the discrimination of input spectra. Also RSDPWs from the three groups of features show a similar statistical distribution. This indicates the extracted features mostly work as an isometric transformation of the original bands. The largest value of RSDPW is between s2 and s8, which means that s2-s8 have the largest difference among all 9 pairs of distances between s1 and other spectra. Although the extracted features only show slight advantages over the original bands in terms of RSDPW, they are much compact, and also can be extracted very fast, so the extracted features may have merits of improving the efficiency and effectiveness of hyperspectral image classification.

IV. CONCLUSION This paper proposes a new spectral feature extraction

method based on orthogonal polynomial fitting. The core of the method is to use the fitting coefficients to extract the geometric structural information of spectral signatures. To make better use of the fitting ability of OPF while alleviating its weakness in capture local curve structures, three strategies called splitting, fitting and selection are proposed. Laboratory spectra are selected for experimental analysis. The results show that the feature extraction can be efficiently carried out through the inner product of input spectral segment and OPF base. The extracted features can be used to sparsely represent the input spectra and also well discriminate different spectral

Feature Number

valu

e

Wavelength Number

Ref

lect

ence

Feature Number

valu

e

TABLE II. COMPARISONS OF 420 BANDS AND L3O4F20/ L1O19F20 IN TERMS OF RSDPW BASED ON S1; S2 TO S10 REPRESENT THE INPUT SPECTRA. THE VALUES IN RED ARE THE FIRST FIVE LARGEST ONES.

Comparisons of 420 bands (upper right) and L3O4F20 (lower left) in terms of RSDPW based on s1

s2 s3 s4 s5 s6 s7 s8 s9 s10

s2 - 5.49 10.76 11.29 57.83 56.28 63.79 55.05 47.09

s3 5.40 - 1.95 2.05 10.52 10.23 11.60 10.01 8.56

s4 10.88 2.01 - 1.04 5.37 5.22 5.92 5.11 4.37

s5 11.29 2.09 1.03 - 5.12 4.98 5.64 4.87 4.16

s6 59.31 10.98 5.45 5.25 - 1.02 1.10 1.05 1.22

s7 57.50 10.64 5.28 5.09 1.03 - 1.13 1.02 1.19

s8 65.52 12.13 6.02 5.80 1.10 1.13 - 1.15 1.35

s9 56.53 10.46 5.19 5.00 1.04 1.01 1.15 - 1.16

s10 48.35 8.95 4.44 4.28 1.22 1.18 1.35 1.16 -

Comparisons of 420 bands (upper right) and L1O19F20 (lower left) in terms of RSDPW based on s1

s2 s3 s4 s5 s6 s7 s8 s9 s10

s2 - 5.49 10.76 11.29 57.83 56.28 63.79 55.05 47.09

s3 5.37 - 1.95 2.05 10.52 10.23 11.60 10.01 8.56

s4 10.82 2.01 - 1.04 5.37 5.22 5.92 5.11 4.37

s5 11.26 2.09 1.04 - 5.12 4.98 5.64 4.87 4.16

s6 58.98 10.98 5.44 5.23 - 1.02 1.10 1.05 1.22

s7 57.04 10.62 5.26 5.06 1.03 - 1.13 1.02 1.19

s8 65.07 12.11 6.00 5.77 1.10 1.14 - 1.15 1.35

s9 56.15 10.45 5.18 4.98 1.05 1.016 1.15 - 1.16

s10 48.00 8.93 4.43 4.26 1.22 1.18 1.35 1.16 -

signatures. The OPF is more powerful than the inverse Gaussian function model in curve fit. Sparse features are more powerful than normal features in the ability of representation. The normal features are slightly better than the original bands at the ability of discrimination in terms of RSDPW while largely reduce the number of features.

Further studies include: introducing more orthogonal polynomial types and splitting strategies; studying the robustness of OPF based methods under real image spectra; making comparison of OPF based method with the orthogonal wavelet based methods.

ACKNOWLEDGMENT This research was supported by the Key Research Program

of the Chinese Academy of Sciences under Grant No. KZZD-EW-TZ-18 and National Programs (Grant No.Y2B001101A). Special thanks to Prof. Antonio Plaza for meaningful discussions and suggestions in our work.

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