高光谱图像解混：最新算法与现状解析

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高光谱图像处理是遥感领域中的关键技术，其主要目标是从复杂的数据集中解析出包含多种物质成分的混合光谱信号，将其分解为一组具有纯正光谱特征的基元（endmembers）和相应的丰度（abundances），即各基元在混合物中所占的比例。这种技术对于环境监测、矿物探测、植被分类等应用至关重要。本文档重点讨论了一种新颖的最小体积包围（Minimum-Volume Enclosing Algorithm, MVEA）方法，该方法由Eligius M. T. Hendrix等人在2012年的IEEE Transactions on Geoscience and Remote Sensing期刊上发表。MVEA算法旨在解决高光谱数据中自动或半自动提取基元面临的挑战。传统的策略通常假设输入数据集中每个不同的物质类型至少有一个纯光谱签名，并通过搜索高维空间中的最纯光谱来识别基元。这种方法面临的一个关键问题是高维数据中找到纯基元的复杂性，以及如何有效地构建一个能包容所有可能基元的简单多面体（simplex）。 MVEA算法的创新之处在于它提出了一种优化策略，旨在找到能够最小化包围所有潜在基元的简单多面体体积的方法。这样做的好处在于，较小的体积意味着更准确地识别了真正的纯基元，避免了误识别和混杂。通过这种方式，算法可以提高光谱解混的精度和稳定性，减少后续分析中的不确定性。算法的核心步骤可能包括数据预处理、基元候选检测、多面体构建、体积优化和丰度估计等。在具体实施时，可能涉及线性代数、凸优化和概率统计等数学工具。为了实现MVEA，研究人员可能会使用迭代方法或者利用梯度下降等优化算法来寻找最优的基元组合和丰度值。论文中提供的参考文献不仅提供了算法的详细描述，还可能包含了其他相关研究的对比分析，以便读者理解当前领域的前沿进展和挑战。通过阅读这篇论文，读者可以了解到如何将MVEA应用于实际的高光谱图像解混任务，并且可以进一步探索如何改进或扩展这一方法，以适应更复杂的场景和数据。这是一篇深入探讨高光谱图像处理中基于最小体积包围法的基元识别和丰度估计的重要研究，为高光谱数据的准确解析提供了新的思路和技术支持。

2746 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 7, JULY 2012

for an abundance estimation process such as the one conducted

under the assumption that pure pixels are present in the input

data. Recently, a similar approach has been developed using se-

quential linear programming to solve the MVES problem [40],

[44]. As will be shown, standard available nonlinear optimi-

zation algorithms can be used to solve the same problem.

The remainder of this paper is organized as follows.

Section II describes the method and its underlying principles.

Section III provides a detailed experimental assessment of the

performance of the method using a database of synthetic hyper-

spectral scenes generated using fractals and a real hyperspectral

data set collected by AVIRIS. In all cases, the performance is

compared to that of state-of-the-art techniques for endmember

extraction which assume the presence of pure pixels [OSP,

N-FINDR, VCA, spatial–spectral endmember extraction

(SSEE), AMEE, SPP, and RBSPP] and also to that of

techniques that do not incorporate such assumption (MVES

and MVSA). Section IV concludes with some remarks and

hints at plausible future research.

II. Minimum-Volume Simplicial Enclosure Method

A. Principle of Volume Reduction

Linear spectral unmixing methods [3] exploit the idea that

the observations x in m-dimensional space (m = 224 spectral

bands for the AVIRIS imaging spectrometer) are in reality com-

binations of only n endmembers. This means that observations

in the space E = {x = Ea|a ∈ R



=1}⊂R

can be

represented by points in V = {z = Va|a ∈ R



=1}⊂

n−1

, which is much lower in dimension, i.e., n  m, where

V =[v

,...,v

] is an (n − 1) × n matrix of the so-called

vertices v

. The transformation is based on some observations

from linear algebra and works as follows.

Let E

−

be the matrix E

−

=[e

− e

,...,e

− e

Its columns span the linear space E

−

 such that one can

write E as E = {x = e

+ E

−

y|y ∈ R

n−1

}. More precisely, let

Γ be a matrix where its columns form an orthonormal basis

of E

−

. The number of columns of Γ is equal to the rank

of E

−

. The rank is n − 1 if the columns of E are afﬁne

independent. This means that the space E can be described as

E := {x = b +Γd|d ∈ R

n−1

}, with b as an arbitrary given m-

vector in E and Γ as an arbitrary m × (n − 1) orthonormal

matrix, such that Γ = E

−

, i.e., its columns have length 1

and are orthogonal with respect to each other. Now, given a

choice for b and Γ, let the vertices v

be solutions of

Γv

= e

− b, i =1,...,n. (2)

Then, in matrix notation, we have E =ΓV + b1

, where 1

is the all-one vector of appropriate dimension. For each abun-

dance vector a (keep in mind that its elements sum to one),

we have a unique point x ∈ E as well as z ∈ V related as

x = Ea =(ΓV + b1

)a =Γz + b.

Two questions arise at this point: 1) What choices are conve-

nient for b and Γ? and 2) what are the consequences for model

(1), which includes white noise, if we have such a vector b

and orthonormal basis matrix Γ? To start with the latter, let us

consider the so-called score vector z in (n − 1)-dimensional

space

z = Va+ ξ (3)

where Γξ is the projection of  on the space Γ. It can be de-

rived that ξ is white noise. Due to Γ being orthonormal, Γ

Γ is

the unit matrix, and the projection leads to ξ =(Γ

Γ)

−1

 =

. Notice that the variance–covariance matrix of ξ is Γ

Γ,

i.e., the unit matrix. The relation of (3) with the original model

(1) is

(x − b)=Ea − b +  =ΓVa+ 

=Γ(Va+ ξ)+ζ =Γz + ζ (4)

where ζ is the projection of  on the orthoplement of Γ.So

in fact,  is decomposed into  =Γξ + ζ. The consequence

of these algebraic observations and the underlying stochastic

model is that, given a support vector b and matrix Γ, the original

model (1) in m-dimensional space reduces to model (3) in

(n − 1)-dimensional space. The observations X =[x

,...,x

]

that, without noise, would lay in the simplex constituted by the

so-called convex hull of E, conv(E) ⊂ E, have a one-to-one

relation with their representation Z =[z

,...,z

] in simplex

S =conv(V ). Given these observations, a desirable objective

is to ﬁnd out the endmember representation V and the original

endmember signals in E; this process is called endmember

identiﬁcation.

In [41], a matrix factorization approach is used that simulta-

neously minimizes least squares and the simplex volume for

endmember generation. For volume reduction, it is good to

add a direction without variation to t he principal components;

zero variation gives zero volume. In turn, the least squares

idea looks for directions with maximum variation. Miao and

Qi [41] use weights to deal with these two objectives. In

this paper, we follow a similar strategy based on estimating

the number of interior/exterior points by weighting these two

terms. Speciﬁcally, we investigate a procedure which takes a

hierarchical approach by ﬁrst estimating the subspace E in

which the n endmembers are lying by means of PCA ( a similar

approach was adopted in [45]).

The choice of using PCA means that space E = b + Γ

is estimated by taking b =

x =(1/r)



and Γ=C =

,...,c

n−1

] is an m × (n − 1) matrix of principal compo-

nents; we have the biggest variation in direction c

, the second

biggest in direction c

, etc. Following this procedure, the (un-

known) scores z

are estimated from observations x

by taking

=(C

−1

− x)=C

− x). Then, the method

that we investigate minimizes (in that subspace) the volume of

the resulting simplex such that it encloses the scores z

of the

observed bands of the pixels x

B. Principle of MVES

The problem of ﬁnding the MVES of a set of points z

,k =

1,...,r in (n − 1)-dimensional space can be written as a so-

called nonlinear optimization problem [46] with (n − 1) × n

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