COMPARISON OF VOLUME AND DISTANCE CONSTRAINT ON HYPERSPECTRAL UNMIXING

Dhinaharan Nagamalai et al. (Eds) : SIGEM, CSEA, Fuzzy, NATL - 2017
pp. 13– 23, 2017. © CS & IT-CSCP 2017 DOI : 10.5121/csit.2017.70902
COMPARISON OF VOLUME AND
DISTANCE CONSTRAINT ON
HYPERSPECTRAL UNMIXING
Zeheng Li1
and Fuxiang wang2
and Xiurui Geng3
1
School of Electronic and Information Engineering, Beihang University,
Beijing Laboratory for General Aviation Technology,
Beijing Key Laboratory for Network-based Cooperative Air Traffic
Management, Beijing, 100191, P.R.China
2
School of Electronic and Information Engineering, Beihang University,
Beijing Laboratory for General Aviation Technology,
Beijing Key Laboratory for Network-based Cooperative Air Traffic
Management, Beijing, 100191, P.R.China
ABSTRACT
Algorithms based on minimum volume constraint or sum of squared distances constraint is
widely used in Hyperspectral image unmixing. However, there are few works about performing
comparison between these two algorithms. In this paper, comparison analysis between two
algorithms is presented to evaluate the performance of two constraints under different
situations. Comparison is implemented from the following three aspects: flatness of simplex,
initialization effects and robustness to noise. The analysis can provide a guideline on which
constraint should be adopted under certain specific tasks.
KEYWORDS
Hyperspectral unmixing, volume constraint, distance constraint, relative capacity
1. INTRODUCTION
Characterized as extremely high spectral resolution and numerous narrow continuous bands,
hyperspectral remote sensing has raised extensive concerns [1] [18]. Hyperspectral images
usually consist of mixed pixels due to limited spatial resolution of sensors. Thus, hyperspectral
unmixing whose purpose is to decompose the mixed pixels into material signature (endmembers)
and the corresponding abundance fractions, has become a challenging task.
Hyperspectral data unmixing is commonly based on linear mixed model (LMM) [2]. LMM
hypothesizes that each pixel vector can be represented as the product of the endmember matrix
and abundance vector. The abundance vector need to satisfy nonnegative and sum-to-one
constraints. Based on LMM, there are mainly two groups of methods solving hyperspectral
unmixing problem. The algorithms in the first group require existing pure signatures in
hyperspectral image, such as Pixel Purity Index (PPI) [3], N-FINDR [4], Vertex Component
Analysis (VCA) [5], iterative error analysis (IEA) [6]. The algorithms in the second group can
process image without requirement of pure signatures, such as single individual evolutionary
strategy (SIE) [7], nonnegative matrix factorization (NMF) [8], minimum volume constraint NMF

14 Computer Science & Information Technology (CS & IT)
(MVC-NMF) [10], iterative constrained endmember method (ICE) [11], robust nonnegative
matrix factorization [19], minimum volume simplex analysis(MVSA) [20].
2. RELATED WORK
Since there is no requirement on pure signatures for the algorithms in the second group, they have
been widely used in hyperspectral unmixing [9] [10] [11]. Among them, NMF with minimum
volume constraint (VC) [10] considers simplex volume composed of the unknown endmembers
during endmember extraction. VC adopts the commonly used formula to measure the ‘volume’ of
simplex enclosed by endmembers. Though VC can efficiently restrain simplex volume, it often
involves massive computation. On the other hand, ICE [11] imposes sum of squared distances
constraint (SSD) on the original objective function, which can generally achieve satisfied result.
Though algorithms based on VC and SSD have been proposed for several years, there hasn’t been
any further analysis and comparison of these two methods. Thus, this paper mainly gives a
detailed comparison of these two methods under various situations. The comparison analysis aims
to give an instruction on how to choose the constraint under certain situation. To achieve this, the
comparison includes flatness of simplex analysis, initialization analysis and robustness to noise
analysis.
3. HYPERSPECTRAL UNMIXING ALGORITHM
In this section, we’ll briefly introduce linear mixing model and unmixing algorithms based on VC
and SSD.
3.1. Linear Mixture Model (LMM)
LMM [2], [12] assumes that the hyperspectral data is a linear combination of endmember spectra,
with the weights being proportions. Mathematically, the model is given as:
x As ε= + ,
subjected to:
1
0, 1, , 1
M
k k
k
s k M s
=
≥ = =∑L ， , (1)
where, x is L -dimensional vector ( L is the number of bands) which is one of the pixel in image,
s denotes corresponding abundance, ε represents possible errors. In real data processing,
abundance should satisfy two constraints called as nonnegative constraint and sum-to-one
constraint, as shown in (1).
The matrix involving all pixels in image is shown as Equation(2)
X AS= +Θ , (2)
where 1 2[ , , ]X x x xN= L represents hyperspectral data which is assumed to be composed with
material signatures 1 2[ , , ]A a a aM= L and abundance fractions 1 2[ , , ]S s s s N= L . N is the
number of pixel. M is the number of endmembers. Θ is the error matrix.
3.2. Constraints on Endmembers
For real data, there are substantial local minimum problem due to non-convexity of unconstrained
objection function. As shown in

Computer Science & Information Technology (CS & IT) 15
Figure 1, the red polyline indicates real simplex for hyperspectral data. Meanwhile, blue and
black polyline is the solution obtained with different initial value. Since there must be
corresponding abundance if endmembers enclose all scatters, it is necessary to put certain
constraint on endmembers, as shown in (3):
21
( , ) || || ( )
2
A S X AS AFf Jλ= − + (3)
where, ( )AJ is the constraint added to endmember, λ is regularization factor to tradeoff the
reconstruction error and constraint.
Before minimizing (3), several preprocessing steps will be taken to remove noises and reduce the
dimension of original data, which aims to reduce computation complexity. Then, appropriate
optimal strategy is used to minimize (3) and update A and S iteratively: first, given endmember
matrix A , calculate the abundance matrix S by the optimal strategy. Then update A by fixed
S in the same way. After several iterations, (3) will approach its minimum value with A and S
well-decided.
Bandb
Band a
Figure 1. Endmember extraction by unconstraint algorithms. There exist many local minimum solutions
due to the non-convex property. Although many solutions have relatively minor linear square error, the
obtained points are still far from scattering, which cannot be regard as appropriate endmembers.
The volume and distance constraints are briefed as follows:
1. Minimum Volume Constraint
VC minimize volume [15] of simplex in its model. In VC, the expression of ( )AVJ is as for
those.
2 11
( ) det ( )
2( 1)!
A
M A
T
M
VJ
 
=  
−  
(4)
After adding volume constraint, volume of simplex will be compressed as small as possible.
Meanwhile, the hyperspectral data reconstructed by the extracted endmembers and corresponding
abundance matrix can also close to real data. Therefore, we can get relatively accurate solution
for endmember and abundance.
2. Sum of Squared Distance Constraint
In ICE [11], the constraint which minimizes SSD among several endmembers on hyperplane is
adopted and given as equation (5). Like VC, SSD can also efficiently control the shape of simplex
during the iteration by minimizing the distance between any two endmembers.
1
1 1
( ) ( ) ( )A a a a a
M M
T
D k l k l
k l k
J
−
= = +
= − −∑ ∑ , (5)
Where, M is the number of endmembers, ak and al are any two endmember vectors.

In equation (3), the first term intends to decrease reconstruction error, and the second term is used
to limit the overall ‘volume’ of simplex contrasted by endmembers. During the optimization
process, we can control the tradeoff between spectral reconstruction accuracy and the
distance/volume constraint ( )AJ via λ .
Furthermore, since abundance must satisfy sum-to-one constraints, we can adjust equation (5) by
adding 1M and 1N to endmember matrix A and original hyperspectral matrix X respectively.
To control the influence of sum-to-one constrains, we introduce a regulation factor α to 1M and
1N , as shown in Equation (6):
1 1
A X
A X
T T
M Nα α   
← ←   
   
(6)
4. SIMPLEX PATTERN AND PARAMETER SELECTION
Since algorithm’s performance fluctuated significantly with the variation of simplex pattern,
analysis of simplex pattern should be significant. Two constraints mentioned above differ in most
situations. It’s necessary to give a comparative analysis to decide which constraint is more
operative given certain simplex pattern.
4.1. Analysis of Simplex Pattern
As both VC and SSD can restrict simplex of endmember closing to original hyperspectral
scattering, it is necessary to analyze the equivalence for these two constraints. We mainly do
analysis in following two situations: First, for regular or quasi-regular simplex, there is a one-to-
one correspondence between volume and distance for simplex, as show in Figure 2(a). In this
situation, the VC and SSD have a similar performance.
However, when the simplex is not regular, namely non-regular simplex, the relationship between
volume and distance of simplex is indefinite, as shown in Figure 2(b). In this case, it is necessary
to analysis which one is better. Since in real hyperspectral data set, simplex for endmembers are
not always regular, analysis for non-regular case is very important. For convenience, we define η
as shown in (7) to measure degree of flatness in simplex.
=
θ
η
π
(7)
where, θ represents the maximum generalized angle of simplex.
Because of the inequivalence in non-regular simplex, flatness analysis is given to identify the
performance for each constraint. Additionally, to further differentiate VC and SSD, random
initialization and anti-noise analysis are implemented to compare algorithm performance.

(a) regular simplex (b) non-regular simplex
Figure 2. Relationship between volume and distance for regular/non-regular simplex. In the case of regular
simplex, the relationship between two constraints is definite, while in the case of non-regular, the
relationship is indefinite.
4.2. Parameter Selection
To present a fair comparison, we need to guarantee all variables except for the constraint item to
be the same during unmixing process. For further details, (a) the update rule involved in these two
constraints is fixed to quadratic programming method and steepest descent method respectively.
(b) The regulation factor λ is carefully chosen to ensure similar ratio between constraint value
and reconstruction error in each case.
To decide Dλ and Vλ for each constraint, we need to find out the relationship between volume
V and sum of squared distance SSD for regular simplex. The relationship satisfies following
equation:
3
( ) MV
c M d
SSD
−
= (8)
where, M is the number of endmember, c is a variable only related to M , d is the distance
between any two endmembers.
Thus, in order to ensure equivalence of these two constraints, we need to make:
3
1 1
/ ( )
D
M
V V SSD c M d
λ
λ −
= = (9)
5. EXPERIMENT
In this part, we analyse application range of these two algorithms. Then we apply these two
algorithms to real hyperspectral data unmixing and compare the performance. As simplex pattern,
initial value and SNR are the most important factors in hyperspectral unmixing, we mainly
conduct the comparative analysis in these three aspects.
5.1. Comparison Criterion
In the process of comparing VC and SSD, a suitable comparison metric is needed to measure the
unmixing performance. Since, endmember data and abundance map can be transferred to
corresponding vector, we adopt angle distance(AD) which measures angular difference between
two vectors as criterion, as shown in equation (10). For spectral endmember and abundance map,
we refer to angle distance as spectral angle distance(SAD) and abundance angle distance(AAD)
separately.

cos ( )
|| |||| ||
x x
x x
T
SAD −
= 1 1 2
1 2
(10)
where, 1x and 2x are two transformed vectors. AAD is calculated in the similar way.
5.2. Analysis on Synthetic Data
For synthetic data, we pick several spectra from spectral library as endmembers. Then we create
hyperspectral data by multiplying normalized endmember data with abundance map generated
according to dirichlet distribution. Additionally, Gaussian noise with certain level is also added to
data.
1. Flatness Analysis
In this experiment, we reconstruct hyperspectral data with two bands and three endmembers. To
demonstrate unmixing capability, we increase the degree of flatness by certain value at each
experiment.
Then, we implement unmixing algorithm based on VC and SSD on hyperspectral data. The initial
values for both algorithms are randomly set. We compare experimental result of endmember
extraction with chosen spectra from library and compute SAD. The result is shown in figure3.
According to Figure 3, we can see that SAD of VC based algorithm is becoming increasingly
bigger compared to SSD with increase of degree of flatness. Whereas the resulting endmember
based on VC and SSD are similar when the simplex approaches to the regular form. Therefore,
we can draw the conclusion that SSD is better than VC when scattering is flat. In real data
processing, scattering on hyperplane is usually non-regular, so algorithm based on SSD can
handle most of these cases according to the result above.
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
0
1
2
3
4
5
6
7
Flatness Degree
SAD
SSD Constraint Volume Constraint
Figure 3 SAD of VC and SSD based algorithm.
2. Random Initialization Analysis
Normally, it is necessary to give an initial value. In this study, we used PPI or N-FINDR to obtain
a relatively suitable initial value for following iterations. For abundance matrix, it is often
initialized as random value. However, due to the complexity of real data set, these methods do not
always perform well on finding well-conditioned initial values. Consequently, the result may be
trapped into some local minima. Thus, random initial value analysis can give us a view that which
one is more susceptible to ill initial conditions. We carry out the same random initialization on
both VC and SSD algorithms, then we compare the result with original true endmember value.

We use random values following Gaussian normal distribution as initial value of endmember and
abundance matrices. We conduct 50 comparison experiments with different initial values. Then
we compute SAD of resulting endmember data and endmember data in spectral library, as shown
in Figure 4.
We can see that endmember extraction result based on SSD is obviously closer to original
endmember data. However, some of the result based on VC absolutely deviate from original
endmember data. In several cases, though VC-based algorithm also can fit original hyperspectral
data well, the evaluated endmember data completely differs from original endmember and
consequently abundance matrix obtained by volume constraint is completely wrong.
0 5 10 15 20 25 30 35 40 45 50
0
2
4
6
8
10
12
14
16
Experimets
SAD
SSD Constraint Volume Constraint
Figure 4 Solution between two constrained algorithms with random initial values. Simulated hyperspectral
data consists of 5 bands and 6 endmembers.
3. Robustness Analysis for Noise
Since real data consists of much noise, original unmixing algorithm is sensitive to noise for the
sake of fitting every individual data sample. However, unmixing algorithms based on VC and
SSD can be applied to unmixing hyperspectral image by minimizing ‘volume’ of simplex. Thus,
reconstructed data will not inevitably approximate all data samples. As a result, these algorithms
show strong anti-noise capacity. However, as these two constraints are considered to be
inequivalent in many cases, noise-sensitivity may be different with each other. Furthermore, we
need to evaluate the suitable degree of SNR for VC and SSD.
We create 50 50× hyperspectral data including 5 bands and 6 endmembers. Then, we add white
noise with different levels to synthetic data. The SNR is 10db, 15db, 20db, 30db. During the
process of unmixing based on two constraints, we use identical iteration method with same upper
limit construction error. We implement two constrained algorithms under each SNR ratio with the
same initial value following Gaussian distribution during each experiment.
0 5 10 15 20 25 30 35 40 45 50
2
4
6
8
10
12
14
16
Experimets
SAD
SSD Constraint
Volume Constraint
(a) SAD with 10db SNR
0 5 10 15 20 25 30 35 40 45 50
2
4
6
8
10
12
14
16
Experimets
SAD
SSD Constraint
Volume Constraint
(b) SAD with 15db SNR

0 5 10 15 20 25 30 35 40 45 50
0
2
4
6
8
10
12
14
16
Experimets
SAD
SSD Constraint
Volume Constraint
(c) SAD with 20db SNR
0 5 10 15 20 25 30 35 40 45 50
0
2
4
6
8
10
12
14
Experimets
SAD
SSD Constraint
Volume Constraint
(d) SAD with 30db SNR
Figure 5 SAD for two constraints with different SNR
We can see from
Figure 5, SAD for VC between extracted and real endmember value change little with the decline
of SNR. On the contrary, SAD for SSD fluctuates with the SNR significantly. Thus, algorithm
based on VC is more robust than SSD in the sense of noise robustness.
5.3. Analysis on Real Data
After finishing the analysis above, we can conclude that SSD is better than VC when hyper
scattering with high degree of flatness or under ill-conditioned initial values. While VC is better
in the sense of robustness to noise. However, above-mentioned experiments are based on
synthetic data. In this experiment, we will utilize real data(AVIRIS data) to identify these two
algorithms.
The used AVIRIS data over Cuprite, Nevada totally contains 400*350 pixels and 50 bands. We
do some pre-processes to raw data to reduce computation complexity before iteration. Firstly, we
utilize principal component analysis (PCA) [16] to reduce data dimension and select principal
band numbers. Secondly, we need to find good initial endmember and abundance value to ensure
algorithms can extract real ground objects efficiently. We utilize endmember data extracted by N-
FINDR as initial endmember matrix. Since hyperspectral data can be regarded as the product of
endmember matrix and abundance matrix, we use unconstrained NMF algorithm to calculate
corresponding abundance matrix S as initial value by fixing endmember abundance A .In
addition, experiment shows that we can achieve much better results by assigning regulation factor
Vλ as 0.15 for VC and Dλ as 0.01 for SSD.
Table 1 SAD among different algorithm
N-FINDR Volume Constraint SSD Constraint
Alunite 4.43 5.45 4.00
Kaolinite 3.28 5.29 5.35
Andradite 4.41 5.03 4.35
Nontronite 4.14 7.58 4.19
Muscovite 6.16 2.34 4.71
Chalcedony 3.75 6.86 3.57
Average 4.36 5.43 4.46
Then, we begin to do iteration for two constraint algorithms until it satisfies terminating
condition. we can find out best matching mineral obtained by two algorithms via comparing with
each mineral reflectance in spectral library [17].

As shown in Figure 6, it’s the endmember extraction result based on SSD. Solid line and dashed
line represent extraction result obtained by SSD and best matched data in spectral library
respectively. The reflectance is normalized to [0, 1]. From several subgraph results, endmembers
extracted by SSD are very close to the spectral of real data, like Alunite and Muscovite. The
closer they are, the smaller SAD is. Table 1 represents SAD for different algorithms. For some
minerals, SSD gives a better solution. For other minerals, VC performs better.
Figure 6 Extracted endmember by SSD.
6. CONCLUSION
In this paper, we analyse VC and SSD algorithm from flatness of simplex, anti-noise and
initialization to discriminate these two algorithms. We aim to provide a guidance on which
constraint is more suitable under some special conditions.
First, we do analysis for flatness and conduct three comparative experiments using synthetic data.
For the pattern of scattering, SSD is better than VC when the scattering is flat. Whereas these two
algorithms’ performance resemble each other while the simplex is regular. As for initialization,
endmember extracted under SSD is closer to original data in random initialization. For anti-noise
performance, VC is more robust in different level of noise.
Eventually, on real data, similar solutions can be achieved for these two constrains with well-
conditioned initial value. Quantitively, for some minerals, SAD of SSD is smaller, like
Chalcedony in Table 1. Yet, for other minerals, like Muscovite, VC works better. Thus, VC and
SSD both work similarly in hyperspectral unmixing task.
According to what mentioned above, relatively practical instruction on how to choose constraints
can be attained.

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