高光谱分解：几何、统计与稀疏回归方法概览

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"这篇文章是关于高光谱分解的概述，主要探讨了基于几何、统计和稀疏回归的三种方法在高光谱成像中的应用。文章由多位 IEEE 会员和高级会员共同撰写，发表在2012年4月的 IEEE 《应用地球观测与遥感》期刊上。高光谱成像技术通过测量场景瞬间视场中的电磁能量散射，以数百或数千个高光谱通道的高分辨率捕捉图像，相比多光谱相机具有更高的光谱分辨率。这种技术使得材料识别成为可能，广泛应用于各种需要识别场景中物质的应用，尤其是在传统光谱分析不适用的情况下。然而，由于高光谱相机的低空间分辨率、微小物质混合以及多次散射，其测量到的光谱是场景中各种物质光谱的混合体。" 高光谱分解（Hyperspectral Unmixing，HU）是遥感和图像处理领域中的一个重要概念，它涉及到从高光谱图像中分离出各个纯光谱成分，即所谓的端元（endmembers），并确定它们在图像像素中的相对贡献，这个过程被称为 abundance fractions。在本文中，作者讨论了三种不同的方法： 1. **几何方法**：这种方法通常基于物理模型，如混合线性模型（Linear Mixture Model, LMM），假设光谱可以表示为不同端元光谱的线性组合。几何方法包括最小二乘法（Least Squares）、非负最小二乘法（Non-Negative Least Squares, NNLS）等，这些算法试图找到最接近观测光谱的端元组合。 2. **统计方法**：统计方法利用统计学原理来估计端元和丰度。例如，最大似然估计（Maximum Likelihood Estimation, MLE）和贝叶斯方法（Bayesian Approach）可以用于处理不确定性，并考虑先验信息来改进结果。这些方法通常更复杂，但可以提供更好的模型稳健性和解释性。 3. **稀疏回归方法**：近年来，稀疏表示理论在高光谱分解中得到广泛应用。稀疏性假设认为大部分像素可以被少数端元稀疏地表示。L1正则化（LASSO）、压缩感知（Compressive Sensing, CS）和基于稀疏矩阵分解的方法（如非负矩阵分解Non-negative Matrix Factorization, NMF）等都属于这一类。稀疏方法可以更好地处理混合像素的复杂性，并且在数据噪声较大的情况下表现优秀。这三种方法各有优势和适用场景，选择哪种方法取决于实际问题的具体需求，如数据质量、计算资源和对结果解释性的要求。高光谱分解在环境监测、地质调查、农作物分类、军事侦察等多个领域都有重要应用。通过精确的高光谱分解，可以获取关于地表覆盖、环境污染、植被健康状况等关键信息，从而支持决策制定和科学研究。

358 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATION S AND REMOTE SENSING, VOL. 5, NO. 2, A

PRIL 2012

Fig. 6. Illustration of the concept o f simplex o f minimum volume containin g the data for three data sets. The endme mbers in the left hand side and in the middle

are identiﬁable by ﬁtting a simplex of minimum volume to the data, whereas this is not applicable to the righ t hand side data set. The former data set correspond

to a highly mix ed scenario.

II. LINEAR MIXTURE MODEL

If the mul tip le scattering among distinct endm emb e rs is neg-

ligible and the surface is partitioned according to the fractional

abundances, as illustrated in Fig. 2, then the spectrum of each

pixel is well approximated by a linear mixture o f endmember

spectra weighted by the corresponding fractional abundances

[1], [3], [29], [ 39]. In this case, the spectral measurement

channel

( is the total number o f channels)

from a given pixel, den oted by

,isgivenbythelinear mixing

model (LMM)

(1)

where

denotes the spectral measurement of endmember

at the spectral band, denotes th e frac-

tional abu ndan ce of endmember

, denotes an additive per-

turbation (e.g., noise and modeling er rors), and

denotes the

number o f endmembers. At a giv en pixel, the fractional abun-

dance

, as the name suggests, represents the fractio nal area

occupied by the

th endmem ber. Therefore, the fractional abun-

dances are subject to the following constraints:

(2)

i.e., the fractional abundance vector

(the notation indicates vector transposed) is in the standard

-simplex (or unit -simplex). In HU jargon, the

nonnegativity and the sum-to-o ne constraints are termed a bun -

dance nonnegativity constraint (ANC) and abunda nce sum con-

straint (A SC), respectively. Researchers may sometimes expect

that the abundance fractions su m to less than one since an algo -

rithm m ay not be able to account for every m aterial in a pixel;

it is not clear whether it is better to relax the constraint or to

simply consider that part of the modeling e rror.

Let

denote a -dimensional column vector, and

denote the spectral signatu re of the

th endmember. Expression (1) can then be written as

(3)

Although the typ e of spe ctral quantity (radiance, reﬂectance, etc.) is impor-

tant when processing data, speciﬁcation is not necessary to derive the math e-

matical approaches.

where is the mixing matrix containing

the signatures of the endmembers present in the covered area,

and

. Assuming that the columns of are

afﬁnely independent, i.e.,

are linearly independent, then the set

i.e., the convex hull of the columns of ,isa -simplex

. Fig. 5 illustrates a 2-simplex for a hypothetical mixing

matrix

containing three endmembers. The points in green de-

note spectral vectors, whereas the points in red are vertices of

the simplex and corresp ond t o the e nd members. Note that the

inference of the mixing matrix

is equivalent to identifying

the vertices of the simplex

. This geometrical point of view,

exploited by many unmixing algorithms, will be fu rt her devel-

oped in Section IV-B.

Since m any algorithms adopt either a geometrical or a sta-

tistical framework [34], [36], they are a focus of this paper. To

motivate these two dir ectio ns, let us consid er the th ree d ata sets

shown in Fig. 6 generated under the linear model given in (3)

where the noise is assumed to be negligible. The spectral v ec-

tors generated according to (3) are in a simplex whose vertices

correspond to the endmembers. The l eft hand side data set con-

tains pure pixels, i.e, for any of the

endmembers th ere is at

least one pixel containing only the correspondent material; the

data set in the middle does not contain pure pixels but contains

at least

spectral vectors on each facet. In both data sets

(left and middle), the endm em bers may by inferred by ﬁtting a

minimum volume (MV) simplex to the data; this rather simple

and yet powerful idea, introduced by Craig in his seminal work

[81], underlies several geometrical based unmixing algorithms.

A sim ilar idea was introduced in 1989 by Perczel in the area of

Chemometrics et al. [82].

The MV simplex show n in the right hand side example of

Fig. 6 is smaller than the true one. T his s ituation corresponds

to a highly mixed data set where there are no spectral vectors

near the facets. For these classes of problems, w e usually re-

sort to the statistical framework in which the estimation of the

mixing matrix and of the fractio nal abundances are formulated

as a statistical inference problem by adop tin g suitable proba-

bility models for the variables and p arameters involved, namely

for the fractional abundances and for the mixing matri x.

BIOUCAS-DIAS et al.: HYPERSPECTRAL UNMIXING OVERVIEW: GEOMETRICAL, STATISTICAL, AND SPARSE REGR

ESSION-BASED APPROACHES 359

Fig. 7. Illustration of a bad ly-conditioned mixing matrices and noise (repre-

sented by uncertainty r egions) centered on clean spectral v ectors represented

by green circles.

Fig. 8. Signal- to- no ise-ratio spectral distr ibu tion (SNR-SD) for the data sets

SudP5SNR40, SusgsP5SNR40, and Rcuprite. The ﬁrst two are simulated and

contain

endmembers and the third is a subset of the AV IRIS Cuprite data

set.

A. Characterization of the Spectral Unmixing Inverse Problem

Given the data set

containing

-dimensional spectral vectors, the linear H U problem is, with

reference to the linear model (3), t he estimation of the mixing

matrix

and of the fractional abundances vectors corre-

sponding to pixels

.Thisisoftenadifﬁcult inverse

problem, because the spectral signatures tend to b e strongly cor-

related, y ielding badly-conditioned mixing matrices an d, thus,

HU estimat es can be highly sensitive to noise. This scenario is

illustrated in Fig. 7, where endmembers

and are very

close, thus yielding a badly-conditioned matrix

,andtheef-

fect of noise is represented by uncertainty regions.

To characterize the linear HU inverse problem, we use the

signal-to-noise-ratio (SNR)

where and are, respectively, the signal (i.e., )

and noise correlation m atrices and

denotes expected value.

Besides SNR, we introduce the signal-to-noise-ratio spectral

distribution (S NR - SD ) deﬁned as

(4)

where

is the eigenvalue-eigenvector couple of

ordered by decreasing value of .Theratio

yields the signal-to-noise ratio (SNR) alon g th e signal direc-

tion

. Therefore, we m ust have for

, in order to obtain acceptable unmixing results.

Otherwise, there are direction s in the signal subspace signiﬁ-

cantly corrupted by noise.

Fig. 8 plot s

, in the interval ,for

the following data sets:

• SudP5SNR40: sim ulated; mixing matrix

sampled

from a uniform ly distributed random variable in the in -

terval [0, 1];

; ; fractional abundances dis-

tributed unifo rmly on the 4 -un it simplex;

• SusgsP5SNR40: sim ulated; mixing matrix

sampled

from the United States Geological Survey (USGS) spec-

tral library;

; ; fractional abundances dis-

tributed unifo rmly on the 4 -un it simplex;

• Rcuprite: real; subset of the well-kn own AVIRIS cuprite

data cube

with size 250 lines by 191 columns by 188

bands (noisy bands were removed).

The signal and noise correlation matrices were obtained with the

algorithms and code distributed with HySim e [83]. From those

plots, we read that, for SudP5SN R40 data set,

for and for , indicating that

the SNR is high in the signal subspace. For SusgsP5SNR40,

the singular values of the mixing matrix decay faster due to

the high correla tio n o f the USGS spectral signatures. Never th e-

less the “big picture” is similar to that of SudP5SNR40 data set.

The Rcuprite data set yields the more difﬁcult inverse problem

because

has “close to convex shape” slowly ap-

proaching the value 1. T his is a clear indication of a badly-con-

ditioned inverse problem [84].

III. S

IGNAL SUBSPACE IDENTIFICATION

The number of endmem bers present in a given scene is, very

often, much smaller than the number of bands

. Therefore, as-

suming that the linear model is a good approximation, spectral

vectors l ie in or very close to a low-dimensional linear subspace.

The identiﬁcation of this subspace enables low-dimensional yet

accurate representation of spectral vectors, thus yielding gains

in computational time and complexity, data storage, and SNR. It

is usually advantageous and sometimes necessary to operate on

data represented in the signal subspace. Therefore, a signal sub-

space identiﬁcati o n algorithm is required as a ﬁrst processing

step.

Unsupervised subspace identiﬁcation has been approached in

many ways. Band selection or band extraction, as the name sug-

gests, exploits the high correlation existing between adjacent

bands to select a few spectral components among those with

higher SNR [85], [86]. Pro jection techniques seek for the best

subspaces to represent data by optimizing objective functions.

For example, principal component analysis (PCA) [87] mini-

mizes sums of squares; singular value decomp osition (SVD)

[88] maximizes power; projections on the ﬁrst

eigenvectors

of the em pirical correlation matrix maximize l ikelihood, if the

noise i s additive and white and the subspace dimension is know n

to be

[88]; maximum noise fraction (MN F) [89] and noise ad-

justed principal components (NAPC) [90] minimize the ratio of

noise power to signal power. NAPC is mathematically equiva-

lent to MNF [90] and can be interpreted as a sequence of two

http://speclab.cr.usgs.gov/spectral-lib.html

http://aviris.jpl .n asa.gov/data/free\_data.html

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