# GAUSS: Guided Encoder - Decoder Architecture for Hyperspectral Unmixing with
Spatial Smoothness

D.Y.L. Ranasinghe, H.M.H.K. Weerasooriya, G.M.R.I. Godaliyadda, M.P.B.
Ekanayake, H.M.V.R. Herath, , D. Jayasundara, L. Ramanayake, N. Senarath and
D. Wickramasinghe D.Y.L. Ranasinghe, H.M.H.K. Weerasooriya, G.M.R.I.
Godaliyadda, H.M.V.R. Herath, M.P.B. Ekanayake, D. Jayasundara, L. Ramanayake,
N. Senarath and D. Wickramasinghe are with the Department of Electrical and
Electronic Engineering, University of Peradeniya, KY 20400, Sri Lanka (email:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>and dulanthawickramasinghe001@eng.pdn.ac.lk).

###### Abstract

In recent hyperspectral unmixing (HU) literature, the application of deep
learning (DL) has become more prominent, especially with the autoencoder (AE)
architecture. We propose a split architecture and use a pseudo-ground truth
for abundances to guide the ‘unmixing network’ (UN) optimization. Preceding
the UN, an ‘approximation network’ (AN) is proposed, which will improve the
association between the centre pixel and its neighbourhood. Hence, it will
accentuate spatial correlation in the abundances as its output is the input to
the UN and the reference for the ‘mixing network’ (MN). In the Guided Encoder-
Decoder Architecture for Hyperspectral Unmixing with Spatial Smoothness
(GAUSS), we use one-hot encoded abundances generated through the k-means
algorithm as the pseudo-ground truth to guide the UN in the initial state.
Furthermore, we relieve the single-layer constraint on the MN by introducing
the UN-generated abundances, in contrast to the standard AE used for HU. In
the second stage of training, we introduce two variants on the pre-trained
network using the GAUSS method. In GAUSSblind, we have concatenated the UN and
the MN to back-propagate the reconstruction error gradients to the encoder.
Then, in the GAUSSprime, reliable abundance results of a signal processing
(SP) method were used as the pseudo-ground truth with the GAUSS
architectureȦccording to quantitative and graphical results for four
experimental datasets, the two GAUSS variants either transcended or equated
the performance of existing HU algorithms from both DL and SP domains.

###### Index Terms:

Hyperspectral unmixing, deep autoencoder, split architecture, spatial
smoothness and correlation, supervised learning

##  I Introduction

Hyperspectral (HS) imaging is extensively used in earth observation satellites
for various applications [1]: lithological mapping [2, 3, 4, 5, 6],
environmental monitoring [7, 8, 9], and agricultural activities [10, 11, 12],
since it captures more spectral information than its counterparts:
multispectral and RGB imaging. However, HS images (HSIs) are of poor spatial
resolution and are corrupted by noise and interference from atmospheric
alterations and instrumentation imperfections[13]. Therefore, identifying the
composition of a topography requires information recovery algorithms, known as
unmixing algorithms.

HS unmixing (HU) is a source separation technique that strives to extract the
constituent source signal spectra and estimate their fractional presence in
each pixel, referred to as endmembers and abundances. Over the evolution of HU
techniques, numerous unmixing algorithms have been proposed for both linear
and non-linear mixing models [14, 15, 16, 17] under signal processing (SP) and
deep learning (DL) domains. SP unmixing benefits from various algorithmic
approaches [18] such as geometric [19, 20, 21, 22, 23, 24, 25, 26, 27],
statistical [28, 29, 30], and sparse regression-based [31, 32]. In addition,
non-negative matrix factorization (NMF) [33] has been profoundly used in HU
for its concurrent extraction of endmembers and abundances with matrix
decomposition and its adherence to fundamental constraints of the HU problem
imposed by practical realities such having non-negative abundances. Variants
of the NMF algorithm are present in the literature, which promote sparsity of
the abundances [34, 35], independence of the endmembers [36], spatial-spectral
information and piece-wise smoothness [37, 38, 39, 40].

Application of DL, though nascent, is now prominent in HU due to the growing
capacity in computational power, the ability to incorporate any contextual
feature readily and the potentiality to extract information from unstructured
data [41]. DL unmixing is frequently performed through semi-supervised or
unsupervised fashion using convolutional networks [42, 43, 44, 45, 46, 47, 48,
49] , dense networks [50, 50, 51, 52, 53, 54], and recurrent networks [55]
with the assistance of other functional layers such as pooling,
regularization, and shaping. Nonetheless, most DL unmixing algorithms are
structured as autoencoders (AEs) due to their resemblance to matrix
decomposition applied in blind source separation techniques and their
unsupervised learning capabilities.

However, AEs have limitations on the architecture when used for blind
unmixing, even though AEs are widely employed in DL unmixing. In particular,

  * •

The encoder optimization is conditioned on the decoder optimization with the
end-to-end structure.

  * •

The convergence of the encoder is slower due to the absence of a direct
steering mechanism other than the decoder because of vanishing gradients.

Besides, according to the ablation studies in [42, 56, 45],

  * •

The convolutional layers merely distil abstract spatial patterns of the HSI
into the abundance maps rather than learning the association amongst the
pixels.

  * •

There is no forcing mechanism to support the spatial correlation in abundance
values since the reference for the loss function is the original HSI.

Further, with the end-to-end architecture of AE for blind unmixing,

  * •

The decoder will be limited to a single-layer network with linear activations
to ensure the resemblance to matrix decomposition [57, 58].

  * •

The non-linearity in spectral information is modelled [59, 60, 55], rather
than a non-linear mixing process.

In this regard, we propose the three-network HU architecture (HUA). The
proposed architecture is trained through two major stages. In the initial
stage, the architecture referred to as ‘Guided Encoder-Decoder Architecture
for Hyperspectral Unmixing with Spatial Smoothness’ (GAUSS) is followed. In
the second stage, two variants GAUSSblind and GAUSSprime are introduced by
altering the main architecture (GAUSS).

In GAUSS, the first two networks: the ‘approximation network’ (AN) and
‘unmixing network’ (HU), are encoders, and the ‘mixing network’ (MN) is the
final network that serves as the decoder of the HUA. The AN is introduced to
improve the association between the centre pixel and its neighbourhood in
order to improve contextual spatial information in the abundance map. The
output of the AN is compared against the actual spectral signature of the
centre pixel to ensure the convergence of the network. The estimated centre
pixel is used as the input to the UN. Furthermore, it is used as the output
reference to compute the reconstruction error for the optimization of the MN.

Next, we investigate using a pseudo-ground truth, created by first segmenting
the HSI with k-means and then converting it to a one-hot encoded
representation for abundances to optimize the unmixing process of the spectral
signatures. The pseudo-ground truth provides gradients for the UN to optimize
its weights, untied from the decoder gradients. This opposes the conventional
optimization of the unmixing process of AEs, where the unmixing optimization
is conditioned on the HS pixel or image reconstruction. Hence, the pseudo-
ground truth steers the output of the UN towards the latent space spanned by
the actual abundances.

After initializing through the GAUSS architecture in the second phase, we
train the network introducing several changes to the architecture resulting in
two variants with improved performance. First, inspired by [44], the untied MN
will be merged with the UN in the GAUSSblind method, while only allowing the
encoder’s training to optimize abundances. Further, merging the UN and the MN
in the second stage of training allowed the exclusion of the pseudo-ground
truth. Then, in the second variant, GAUSSprime, the original GAUSS
architecture is kept intact while being retrained with a more refined pseudo
ground truth generated by a specifically selected SP algorithm [34] based on
its unmixing performance.

The decoder of a typical AE architecture used in existing methods for HU is
single-layered. The introduction of pseudo ground truth to train the UN prior
to the training of the MN adds more flexibility to the decoder enabling the
addition of more layers to it. The use of a pseudo-ground truth instils the
spatial context and basic texture of the abundance map for the UN. This
training of the UN prior to MN ensures the generated abundance to be at least
an approximation abundance for the corresponding pixel. The input to the MN is
the output provided by the UN while its’ output is the reconstructed pixel.
Hence, the functionality of the decoder is to find the layer weights which
will eventually generate the spectral signature once the abundances are
provided. Therefore, because of this explicit nature of the input in the
proposed architectures, the decoder needs not to be confined to a single
layer. Thereby, with the design flexibility of the decoder, it is not bounded
to perform a merely linear unmixing. Furthermore, it provides the possibility
of unmixing following a linear or a non-linear premise.

In summary, the primary contribution of this article is twofold.

  1. 1.

Introduction of a novel architecture enabling preservation of the smoothness
and spatial correlation of the local subspace structure and improving the
neighboring pixels’ association with the centre pixel.

Centre pixel estimation through the neighborhood has enabled the injection of
spectral correlation and smoothness of the given local environment to be
inhibited into the unmixing process. This generative process ensures the
functional continuity among the pixels thus ensuring spectral smoothness and
continuity while subjected to the constraints of the given environment. Hence,
it has the added advantage of imposing a degree of smoothness that is inherent
to the given environment.

  2. 2.

Introduction of a pseudo-ground truth at the end of the UN decouples the
encoder abundance map generation process and the decoder mixing for end member
generation allowing the encoder and decoder to be optimized separately.

    1. (a)

Decoupling the dependency of abundance map generation of encoder from the
decoder optimization enables the unmixing network to find the solution space
separately. While the addition of a pseudo ground truth steers the learning
process of the encoder towards an optimum solution, through successive
training rounds more refined pseudo-ground truth can be injected to further
refine the unmixing solution.

    2. (b)

Improved flexibility of the decoder releases the single layer constraint which
binds the performance of the decoder to be linear, therefore allowing the
possibility of modeling the underlying non-linearity of the mixing process.

  3. 3.

A more complex and novel synthetic data set has been created for comparison
purposes which incorporates fast varying irregularities, and pure regions to
emulate the environment of a real HSI. The process of generation of the
synthetic data set is explained in section IV - B.

Further expanding on the capabilities of the GAUSS initializer, we proposed
two variants: GAUSSblind and GAUSSprime, with modifications to the
architecture in the former and to the pseudo-ground truth in the latter based
upon the flexibility introduced by the initial training stage. The
introduction of the two variants established a framework to interpret
variances in unmixing performances of different HU algorithms for various HS
datasets and a simulated dataset with reasonable spatial complexity. The
remainder of the paper is as follows. In Section II, the concept of the GAUSS
initializer and the variants are delineated followed by Section III with
implementation details for the HUA. Finally, the experiments and the result
comparison with competing algorithms are presented in Section IV with
explanations for the exceptional performance of the two variants ascribing to
the flexibility offered by GAUSS and the effective combination of different
strategies for a multitude of datasets.

##  II Problem formulation and the principles of the solution

Figure 1: Moore neighbourhood graph for the approximation network where the
centre pixel is in yellow and its neighbourhood in orange Figure 2:  (a) GAUSS
Initializer \- AN training to associate neighborhood spatial information with
center pixels. (b) GAUSS Initializer \- Sequentially trained; UN with pseudo
ground truth and MN with reconstruction loss. (c) GAUSSblind \- Sequentially
trains UN then MN using the same reconstruction loss. (d) GAUSSprime \-
Sequentially trained; UN with signal processing based abundances and MN using
reconstruction loss.

###  II-A Notation

Scalars are denoted by both lower ($x$) and uppercase ($X$) normal fonts.
Boldface lowercase fonts ($\mathbf{x}$) are for 1-D arrays, and 1-D arrays are
ordered as column vectors. Then, boldface uppercase fonts ($\mathbf{X}$) are
for both 2-D and 3-D arrays.

Consider a HSI, $\mathbf{H}$ $\in\mathbb{R}_{+}^{w\times h\times B}$, with a
width and a height of $w$ and $h$ in pixels, respectively, and $B$ spectral
bands. The HS datacube can be rearranged as a matrix of size $B\times N$,
where $N$ is the total number of observations (pixels) and is the product of
$w$ and $h$. Accordingly, the HSI can be denoted as $\mathbf{Y}$
$=\big{[}\mathbf{y}_{1}~{}\cdots~{}\mathbf{y}_{j}~{}\cdots~{}\mathbf{y}_{N}\big{]}\in\mathbb{R}_{+}^{B\times
N}$ and each column $\mathbf{y}_{j}\in\mathbb{R}_{+}^{B}$ is the spectrum of
the $j$th pixel. The spectral signature of the pixel at the ($m$,$n$) location
of $\mathbf{H}$ corresponds to the $j$th column vector of $\mathbf{Y}$, such
that $j=\\{w(m-1)+n~{}|~{}1<m\leq h,~{}1<n\leq w\\}$ where $m$ and $n$ is the
row and column position of the pixel, respectively.

###  II-B Mixture Models

The linear mixture model (LMM) depends on the assumption that the incident
light is reflected from the macroscopic surface only once and is measured by
the sensing device of the imaging spectrometer without scattering. For its
simplicity, LMM is the most frequently used model for HU in the literature
[14]. The luxury of the LMM is that a linear combination of endmember spectra
can be written for each observation as

$\mathbf{y}_{j}=\sum_{k=1}^{K}\mathbf{S}_{kj}\mathbf{a}_{k}+\mathbf{e}_{j}$
(1)

where $\mathbf{S}_{kj}$ is the fractional composition of the $k$th endmember
in the $j$th, $\mathbf{a}_{k}\in\mathbb{R}_{+}^{B}$ is the spectrum of the
$k$th endmember of the HSI, $\mathbf{e}_{j}\in\mathbb{R}_{+}^{B}$ is an
additive disruption due to noise and modeling errors, and $K$ is the
predetermined number of endmembers in the HSI. The elements in
$\mathbf{y}_{j}$ and $\mathbf{a}_{k}$ are non-negative as the HSI is
represented using the reflectance values because it inherits atmospheric
corrections and sensor independencies. Due to physical implications, the
abundance non-negativity constraint (ANC) $\mathbf{S}_{kj}\geq 0$ and the
abundance sum-to-one constraint (ASC) $\sum_{k=1}^{K}\mathbf{S}_{kj}=1$ are
imposed in 1 to assure that the fractional compositions are non-negative and
the HSI only consists of the predetermined endmembers. Under the LMM, the HSI
can be represented using the matrix notation as

$\mathbf{Y}=\mathbf{A}\times\mathbf{S}+\mathbf{E}$ (2)

where $\mathbf{A}\in\mathbb{R}_{+}^{B\times K}$ is the endmember matrix whose
columns are the spectra of each of the $K$ endmembers,
$\mathbf{S}\in\mathbb{R}_{+}^{K\times N}$ is the abundance matrix with the
fractional composition of each of the $N$ pixels is organized as a column
vector, and $\mathbf{E}\in\mathbb{R}^{B\times N}$ is the noise matrix.
Accordingly, the extraction of the endmember spectra ($\mathbf{A}$) and their
abundances ($\mathbf{S}$) are the functions of HU.

When the assumptions of the LMM cannot be satisfied, a non-linear interaction
between the signatures will occur. Several models have been introduced to
capture the non-linear behaviour of the mixing process [61]. Intimate mixtures
exploit the fact that various particles are in very close vicinity; hence,
non-linear interactions between the constituents at microscopic and
macroscopic levels. Due to the multipath and scattering, interactions of the
photons from different sources give rise to non-linearity. Bi-linear mixture
models such non-linear interactions considering that order of interactions is
two. Moreover, polynomial mixture models consider the within component
interactions as well.

###  II-C Neighbourhood approximation

While the endmember spectra contain information about the spectral response of
the underlying materials in the HSI, the spatial attributes of that image are
stored in the abundance matrix. Therefore, in literature, to increase spatial
smoothness and spatial correlation of the fractional composition, several
regularization functions have been introduced in the objective function [62,
63, 35, 37, 38, 39, 40] to optimize HU. However, given that the spectral
signature of a given HSI is the product of the endmember spectra and its
fractional abundances, the spatial correlation and smoothness could be
increased using the spectral signatures of the neighbouring pixels.

If the centre pixel can be approximated from its immediate neighbourhood, then
the spatial correlation and smoothness shall be established in spectral
signatures, thereby in abundance values. Consider the pixel at the ($m$,$n$)
location of the HSI ($\mathbf{H}$) and the set ($\mathcal{S}$) of immediate
neighbourhood pixels as illustrated in Fig. 1. The set can be defined as,
$\mathcal{S}=\big{\\{}\mathbf{H}_{i,j}~{}|~{}\mathbf{H}_{i,j}\in\mathbb{R}_{+}^{B},i\in\\{m-1,m+1\\},~{}j\in\\{n-1,n+1\\},~{}i,j\in\mathbb{Z}^{+}\big{\\}}$
and the cardinality of the set is equal to the number of neighbourhood pixels
considered during the construction of the set and it is denoted by
$|\mathcal{S}|$.

Then following the pixel to column position transition explained in Section
II-A, by rearranging the vectors in $\mathcal{S}$ as a single column vector
denoted by $\mathbf{n}_{j}$, the following relationship can be formulated
between the centre pixel ($\mathbf{y}_{j}$) and its neighbourhood.

$\centering\mathbf{y}_{j}\approx\mathbf{W_{n}}\mathbf{n}_{j}\@add@centering$
(3)

where $\mathbf{W_{n}}\in\mathbb{R}^{B\times B|\mathcal{S}|}$ approximates the
centre pixel using its neighbourhood pixel vector $\mathbf{n}_{j}$. To learn
the spatial correlation across the HSI, $\mathbf{W_{n}}$ can be trained using
the following objective function,

$\centering\operatorname*{arg\,min}_{\mathbf{W_{n}}}{\|\mathbf{Y}-\mathbf{W_{n}}\,\mathbf{N}\|}^{2}_{F}\@add@centering$
(4)

Here, $\mathbf{N}$ is the matrix constructed by concatenating the neighbour
vectors of each centre pixel, and $\mathbf{Y}$ is the matrix form of the HSI
and $\|\cdot\|_{F}$ is the Frobenius norm. Then by performing HU on pixels
generated from the trained $\mathbf{W_{n}}$ in (4), the spatial correlation
and smoothness could be brought forward to fractional abundances by the AN
displayed in Fig. 2.

###  II-D Controlled abundance estimation

Generally, DL unmixing is performed using the AE architecture with the pixel
reconstruction error as the objective function to optimize. However, a
drawback of using the AE for unmixing is that it precludes direct optimization
of abundances in contrast to non-DL methods. Moreover, a conventional AE
architecture may suffer from the vanishing gradient problem as the network
becomes deeper and deeper. That is because gradients descending through back-
propagation may exponentially decrease when they reach the layers in the
encoder. In addition, the unmixing process is dependent on the decoder
performance of the AE because of the back-to-back connection between the
encoder and the decoder.

Consequently, to avoid the dependency on the endmember signature optimization
from the abundance optimization, the proposed work considers an additional
objective function to train the network for abundance optimization. We create
a reference for the abundance maps to train the UN, first by segmenting the
HSI with k-means and converting the result to its one-hot encoded
representation. This reference for the abundances is referred to as the
pseudo-ground truth and is described in Section III-C. Further, the injection
of a pseudo-ground truth facilitated by the split architecture provides a
mechanism to reinforce the abundance optimization. It allows for a more
refined pseudo-ground truth from unsupervised SP algorithms as proposed in the
GAUSSprime variant (see Section III-D) for subsequent training for better
accuracy.

Each pixel’s fractional abundances can be considered a probability
distribution with the ASC and ANC. Each abundance value for the corresponding
endmember is the individual probability. This definition allows us to
reconsider the unmixing process as a classification using categorical-cross
entropy as the objective function. When the UN learns hidden representations
of the HSI by mapping the spectral signatures onto the probability space using
the pseudo-ground truth, pixels that are not comprised of pure pixels, i.e.
mixed pixels, will be forced towards a particular endmember. However, since
the UN is a transformation on the pixel signature and the signature of a mixed
pixel is different from a pure signature, the fractional abundances can be
driven towards pure abundances without losing the generalization of the UN. As
these two objectives behave as opposite forces, the fractional abundances will
be at an equilibrium rather than driven by a single cost function.

###  II-E Expanding decoder flexibility

AEs for HU are omnipresent because they resemble the matrix decomposition used
in blind source separation. Since the encoder task is designed to transform
the pixel or image of the HS dataset into a latent representation, there is no
limitation on the number of layers in the encoder. However, unlike the
encoder, the decoder of the AE does not emulate a process; instead performs
the matrix multiplication between the abundances and endmembers. Therefore,
the endmembers of the HS data are extracted by the weights of the decoder.
Hence, the decoder architecture is bound to a single layer decoder to ensure
the output of the encoder will be the abundances.

By adopting the pseudo-ground truth mechanism, the UN is trained to produce
abundance values at its output in our proposed architecture independent of the
decoder architectural form. Therefore, since it is guaranteed that the input
of the MN is the abundances of the centre pixel, the single-layered decoder
constraint can be released without using recurrent layers as in [55]. Because
the output of the decoder is compared against the pixel signature, the decoder
requirement is to find the optimal combination of weights that will map the
abundances to spectral signatures. Further, the decoder can model the non-
linear mixing process with the proposed change.

Therefore, in the GAUSS architecture, the pixel’s neighbourhood is fed to the
AN, and its output is compared against the actual centre pixel and the input
to the UN. The UN will generate the abundances of the centre pixel at its
output which is subsequently used as the input to the MN. After training,
endmembers are generated by feeding an identity matrix
($\mathbf{I}\in\mathbb{R}^{K\times K}$) to the MN, which represents pure pixel
feeds because, with the modifications to the decoder architecture, the
endmembers can no longer be extracted from the weights of the MN.

##  III Methodology

This section elaborates the proposed architecture for HU with spatial
smoothness. The proposed method is primarily based on the traditional AE model
used in standard HU practice under DL. However, the proposed method can be
viewed as three separate networks instead of the standard AE model in HU due
to the method of training adapted in this work. The architecture of the
aggregated network is illustrated in Fig. 2(a). The AN is used to impose
spatial smoothness on the HSI as discussed in Section II-C. The network
approximates the centre pixels $\mathbf{Y}\in\mathbb{R}_{+}^{B\times N}$ using
their neighbourhood pixels $\mathbf{N}\in\mathbb{R}_{+}^{B|\mathcal{S}|\times
N}$ given by

$\mathbf{\hat{Y}}=f_{n}(\mathbf{N})$ (5)

with $f_{n}\colon\mathbb{R}_{+}^{B|\mathcal{S}|\times
N}\to\mathbb{R}_{+}^{B\times N}$. The network learns the parameters for
approximation by minimizing the average reconstruction error between the input
$\mathbf{Y}$ and the reconstructed spectral signatures $\mathbf{\hat{Y}}$ from
the neighbourhood pixels. Therefore, after the learning process is terminated,
the smoothness matrix ($\mathbf{W_{n}}$) in (4) and the spectral signatures
with improved spatial characteristics can be realized by

Smoothness transitions $\displaystyle\colon
f_{n}(\mathbf{N})\Rightarrow\mathbf{W_{n}}\mathbf{N}$ (6) Smoothed pixels
$\displaystyle\colon\mathbf{\hat{Y}}.$ (7)

Next, to learn the hidden representation of the HSI, the UN is trained, which
transforms the smoothed spectral signatures ($\mathbf{\hat{Y}}$) from the AN
to a low dimensional representation
($\mathbf{\hat{L}}\in\mathbb{R}_{+}^{K\times N}$) given by

$\centering\mathbf{\hat{L}}=f_{e}(\mathbf{\hat{Y}}),\@add@centering$ (8)

with $f_{e}\colon\mathbb{R}_{+}^{B\times N}\to\mathbb{R}_{+}^{K\times N}$. The
output from the UN is fed to the MN, which decompresses the hidden
representation to reconstruct the smoothed spectral signatures at its output.
The decoder function, therefore, can be represented as

$\centering\mathbf{\hat{\hat{Y}}}=f_{d}(\mathbf{\hat{L}})\@add@centering$ (9)

where $f_{d}\colon\mathbb{R}_{+}^{K\times N}\to\mathbb{R}_{+}^{B\times N}$,
and $\mathbf{\hat{\hat{Y}}}$ is the output of the decoder network. After the
successful termination of the learning process for HU, the unmixing results
can be extracted by$\colon$

Abundance estimation
$\displaystyle\colon\mathbf{\hat{L}}^{*}\Rightarrow\mathbf{\hat{S}}$ (10)
Endmember estimation $\displaystyle\colon
f_{d}^{*}(\mathbf{I_{K}})\Rightarrow\mathbf{\hat{A}}$

where $\mathbf{I_{K}}\in\mathbb{R}_{+}^{K\times K}$ is the identity matrix and
* denotes the optimal solution at the end of the training process.

###  III-A Encoder

The encoder combines the AN and UN to improve spatial characteristics and
learn hidden representations in this work. For each pixel of the HSI, we first
consider its immediate neighbourhood and reconstruct the centre pixel from the
set of neighbourhood pixels. Then, the output of the AN is sent to the UN to
learn the unmixing process and estimate abundances. The architecture of the
encoder is given in Fig. 2(a), and fully connected layer (FCNL) architecture
is used in constructing the encoder.

In our work, the fractional abundances are considered probabilities, and
learning hidden representations of the HSI is similar to performing a multi-
class classification with the pseudo-ground truths. In our work, the encoder
is designed to use spatial information and optimize the estimation of
fractional abundances without interference from the decoder optimization. To
optimize the AN, we use the reconstruction error of the centre pixel from its
neighbourhood as given by

$\mathcal{J}_{\text{smooth}}=\frac{1}{N}\|\mathbf{Y}-\mathbf{\hat{Y}}\|_{F}^{2}$
(11)

where $\|\cdot\|_{F}$ is the Frobenius norm. Then, to abide by the ANC and
ASC, we apply the following scaling and standardization at the final layer of
the UN defined as follows,

$\hat{l}_{kj}=\frac{e^{-z_{kj}}}{\sum_{k=1}^{K}e^{-z_{kj}}}$ (12)

where $z_{kj}$ is the output of the $k$th node at the final layer of the
encoder prior activation and $\hat{l}_{kj}$ is the estimated fractional
abundance for the $k$th endmember of the $j$th column in $\mathbf{\hat{L}}$.
The standardization to probability distributions given in 12 can be realized
through Softmax activation function of the output layer of the encoder. Then
to optimize the unmixing process of the encoder, the categorical cross-entropy
is used as below,

$\displaystyle\mathcal{J}_{\text{unmixing}}=-\frac{1}{N}\sum_{\begin{subarray}{c}j,k=1\end{subarray}}^{N,K}l_{kj}\log\hat{l}_{kj}+(1-l_{kj})\log(1-\hat{l}_{kj})$
(13)

where $l_{kj}$ are abundance values for $k$th endmember of the $j$th column of
the pseudo-ground truth used to expedite optimization of the UN.

###  III-B Decoder

An MLP-based network is adopted for the decoder in our proposed scheme, with
its structure reported in Fig. 2. In this work, the decoder is designed to
reconstruct the input to the UN, which comprises smoothed pixels generated
from the AN of the proposed architecture. As discussed in Section III-A, with
a pseudo-ground truth mechanism for abundance estimation, the decoder
architecture is not restricted to a single-layered linear network; therefore,
it can learn non-linear relationships as well.

The decoder of the proposed network learns non-linearity relationships between
abundances and endmember signatures during the mixing process because of the
introduced extension in the layers. Nevertheless, as mentioned in Section
III-B, by extending the decoder layers, the extraction of the endmember
signatures cannot be done through the network weights because the unmixing
process is now not limited to a single layer rather laid out over a single
network with multiple layers. However, since the decoder is trained by using
abundances generated from the encoder network, the decoder can reconstruct the
corresponding spectral signature for any abundance vector in the low
dimensional space learned by the encoder. Hence, by feeding this one-hot
encoded vector to the decoder at the end of weight training, the spectral
information of the corresponding endmember can be extracted. For example, to
find the endmember signature of the $k$th endmember, we will construct a one-
hot vector $\mathbf{o_{k}}$ such that $\mathbf{o}_{k}=\big{[}0,\cdots
0,1,0\cdots,0\big{]}$ where the unity entry is at the $k$th position of the
vector. The vector format implies the existence of a pure pixel containing
only that end member. Hence, the resulted spectral signature of the MN is
simply the corresponding endmember signature. Then by concatenating these one-
hot vectors of each endmember, we can construct an identity matrix, when given
to the decoder, which will output the endmember signatures of the HSI in a
single attempt as presented in 10 for endmember estimation.

To train the decoder network, we diverge from the conventional practice of
using the mean squared error between the encoder input $\mathbf{Y}$ and the
decoder output of each pixel $\hat{\mathbf{Y}}$ defined as

$\mathcal{J}_{\text{reconstruction}}=\frac{1}{NB}\big{\|}\mathbf{Y}-\mathbf{\hat{\hat{Y}}}\big{\|}_{F}^{2}$
(14)

for the HSI. Rather, we adopt the pixel spectral information divergence (pSID)
proposed by [64] as the reconstruction error to optimize the mixing process,
which is defined by,

$\displaystyle\mathcal{J}_{\text{mixing}}=$ $\displaystyle
D(\mathbf{\hat{Y}}\|\mathbf{\hat{\hat{Y}}})+D(\mathbf{\hat{\hat{Y}}}\|\mathbf{\hat{Y}})$
(15) $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{\begin{subarray}{c}j,b=1\end{subarray}}^{N,B}\Big{\\{}q_{bj}\log\Big{(}\frac{q_{bj}}{\hat{q}_{bj}}\Big{)}+\hat{q}_{bj}\log\Big{(}\frac{\hat{q}_{bj}}{q_{bj}}\Big{)}\Big{\\}}$

where
$\mathbf{q}_{j}=\nicefrac{{\mathbf{\hat{y}_{j}}}}{{\|\mathbf{\hat{y}}_{j}\|_{1}}}$
and
$\mathbf{\hat{q}}_{j}=\nicefrac{{\mathbf{\hat{\hat{y}}}_{j}}}{{\|\mathbf{\hat{\hat{y}}}_{j}\|_{1}}}$
are the probability distribution vectors of the spectral signature of the
$j$th pixel of $\mathbf{\hat{Y}}$ and $\mathbf{\hat{\hat{Y}}}$, respectively.
Accordingly, $\mathbf{q}_{bj}$ and $\mathbf{\hat{q}}_{bj}$ represent the
probability values of the $b$th band of the $j$th pixel. Then, $\|\cdot\|_{1}$
is 1-norm of the corresponding spectral signature. In 15, $D(\cdot\|\cdot)$ is
called the relative entropy between the smoothed pixels at the UN input and
reconstructed pixels at the decoder output. The pSID defined by 15 can be used
to measure the spectral similarity between corresponding pixel vectors in
$\mathbf{\hat{Y}}$ and $\mathbf{\hat{\hat{Y}}}$.

###  III-C Pseudo-ground truth preparation for abundances

In the original GAUSS architecture, one-hot encoded representation is used as
the pseudo-ground truth to guide the UN of the encoder. To construct the one-
hot encoded representation of the HSI, we first segment the HSI using the
k-means algorithm on the spectral signatures. Further, the injection of the
pseudo-ground truth enables greater flexibility for the decoder structure and
multiple opportunities for subsequent abundance generation steering through
refined pseudo-ground truths as proposed below.

###  III-D Variants of the GAUSS algorithm

The GAUSS architecture is the foundation that uses the one-hot encoder
abundances as the pseudo-ground truth. Then, we propose two architecture
variants: GAUSSblind and GAUSSprime, for HU that are pre-trained using the
GAUSS method. First, following [44], we will connect the decoder to the UN and
remove the pseudo-ground truth. Then, we will train the network, similar to
the conventional DL unmixing method, without allowing the decoder to train.
This strategy will allow the unmixing process to optimize abundance results
according to pixel reconstruct error defined by 15. After its completion, the
decoder of the HUA will be trained while the encoder training is withheld. For
reference, this training method is called ‘GAUSSblind’ (see Fig. 2(b)).

Second, analogous to using unsupervised HU algorithms for endmember weight
initialization, we consider abundance results of a SP algorithm with reliable
unmixing performance as a pseudo-ground truth for HUA. This training strategy
is referred to as the ‘GAUSSprime’ method (see Fig. 2(c)). The GAUSSprime
method is a combination of both GAUSS and GAUSSblind strategies because and
facilitates a hybrid modus operandi by fusing the best of the abilities of the
other two methods. However, the output of the UN will be compared against the
new pseudo-ground truth. Further, in this scheme, the encoder and the decoder
will also be trained successively like in the GAUSSblind strategy.
Nonetheless, the subsequent refinement in abundance optimization steering
through the insertion of abundance maps generated via high-performing SP was
enabled by using the pseudo-ground truth mechanism. Accordingly, we use the
abundance results of the L$\nicefrac{{1}}{{2}}$ NMF algorithm for all the
datasets to maintain consistency even though it could be any contextual driver
for abundance generation.

###  III-E Implementation and training of the network

The HUA was developed in TensorFlow v2 using the Google Colaboratory using
default settings for optimization parameters, and the detailed architecture of
the HUA is given in Table I. Then the Adam optimizer was used to optimize the
three networks, and for the input-output dataset preparation, we used a batch
size of 32 for each HSI dataset.

In all three training strategies, the encoder and the decoder are successively
trained, and the section was trained for 25 epochs because the GAUSSblind and
GAUSSprime methods are used with the network pre-trained by the GAUSS method.
Accordingly, in both GAUSSblind and GAUSSprime, the HUA was first trained for
25 epochs under the GAUSS method, and another 25 epochs either with the
GAUSSblind and GAUSSprime method totalled 50 epochs. Since the total number of
epochs for the GAUSS method is different from that for the other two methods,
the pre-trained AE was further trained for additional 25 epochs with the GAUSS
method to enable fair comparison amongst the proposed three algorithms.

The pseudo-ground truth used for the GAUSSprime method was the abundance
results from the L$\nicefrac{{1}}{{2}}$ NMF algorithm considering its
abundance performance for all three real datasets.

(a)(b)(c)(d) Figure 3: RGB images of the datasets (a). Simulated (b). Samson
(c). Jasper-Ridge (d). Urban

##  IV Experiments and Discussions

###  IV-A Performance criteria

The performance of the abundance estimation is measured by the average root
mean square error (aRMSE), average abundance information divergence (aAID),
and average abundance angle distance (aAAD). Then, the accuracy of endmember
estimation from the HU algorithm is evaluated using the average spectral angle
distance (aSAD) and the average spectral information divergence (aSID), which
are defined as follows,

aRMSE
$\displaystyle=\sqrt{\frac{1}{NK}\big{\|}\mathbf{S}-\mathbf{\hat{S}}\big{\|}_{F}^{2}}$
(16) aAAD
$\displaystyle=\frac{1}{N}\sum_{i=j}^{N}\mathrm{cos}^{-1}\Big{(}\frac{\langle\mathbf{s}_{j},\mathbf{\hat{s}}_{j}\rangle}{\|\mathbf{s}_{j}\|\|\mathbf{\hat{s}}_{j}\|}\Big{)}$
(17) aAID $\displaystyle=\frac{1}{N}\sum_{\begin{subarray}{c}j=1\\\
k=1\end{subarray}}^{N,K}\Big{\\{}s_{kj}\log\Big{(}\frac{s_{kj}}{\hat{s}_{kj}}\Big{)}+\hat{s}_{kj}\log\Big{(}\frac{\hat{s}_{kj}}{s_{kj}}\Big{)}\Big{\\}}$
(18) aSAD
$\displaystyle=\frac{1}{K}\sum_{k=1}^{K}\mathrm{cos}^{-1}\Big{(}\frac{\langle\mathbf{a}_{k},\mathbf{\hat{a_{k}}}\rangle}{\|\mathbf{a}_{k}\|\|\mathbf{\hat{a}}_{k}\|}\Big{)}$
(19) aSID $\displaystyle=\frac{1}{K}\sum^{K,B}_{\begin{subarray}{c}k=1\\\
b=1\end{subarray}}\Big{\\{}p_{bk}\log\Big{(}\frac{p_{bk}}{\hat{p}_{bk}}\Big{)}+\hat{p}_{bk}\log\Big{(}\frac{\hat{p}_{bk}}{p_{bk}}\Big{)}\Big{\\}}$
(20)

where $\mathbf{S}$ and $\mathbf{\hat{S}}$ represents the ground truth and
estimated abundances. Then, $\mathbf{s}_{j}$ and $\mathbf{\hat{s}}_{j}$ are
the ground truth and estimated abundances at $j$th column or pixel, and
$s_{kj}$ and $\hat{s}_{kj}$ are the ground truth and estimated abundance value
for $k$th endmember of the $j$th pixel. Next, $\mathbf{a}_{k}$ and
$\mathbf{\hat{a}}_{k}$ are ground truth and extracted endmember for the $k$th
source, and $p_{bk}$ and $\hat{p}_{bk}$ are probability value for the $b$th
spectral band of the $k$th ground truth and extracted endmembers,
respectively. The probability vector for the $k$th endmember is computed as
$\mathbf{p}_{k}=\nicefrac{{\mathbf{a}_{k}}}{{\|\mathbf{a}_{k}\|}}$ and
$\mathbf{\hat{p}}_{k}=\nicefrac{{\mathbf{\hat{a}}_{k}}}{{\|\mathbf{\hat{a}}_{k}\|}}$
for the ground truth and extracted signatures, respectively.

TABLE I: Network parameters of the autoencoder for unmixing

Network | Layer description | Output shape | Bias state
---|---|---|---
Encoder |  Approximation network | Input layer |  [(None, $|\mathcal{S}|\times B$)] | N/A
Dense layer 1 |  (None, $\lfloor|\mathcal{S}|\times\nicefrac{{B}}{{2}}\rfloor$) | TRUE
Dense layer 2 |  (None, $\lfloor|\mathcal{S}|\times\nicefrac{{B}}{{8}}\rfloor$) | TRUE
Dense layer 3 |  (None, $B$) | TRUE
Unmixing network | Input layer |  [(None,$B$)] | N/A
Dense layer 1 |  (None, $\lfloor\nicefrac{{B}}{{2}}\rfloor$) | FALSE
Dense layer 2 |  (None, $\lfloor\nicefrac{{B}}{{8}}\rfloor$) | FALSE
Dense layer 3 |  (None, $K$) | FALSE
Decoder | Input layer |  [(None, $K$)] | N/A
Dense layer 1 |  (None, $\lfloor\nicefrac{{B}}{{4}}\rfloor$) | FALSE
| Dense layer 2 |  (None, $B$) | FALSE
$\lfloor\cdot\rfloor$ represents the floor operation.

###  IV-B Experimental setting

To evaluate the performance of GAUSSin a complex environment an algorithm that
is capable of generating simulated datasets with a sufficient amount of
complexity was designed. Compared with the simulated data sets in the
literature, the dimensions were selected to be $100\times 100$ with a spectral
resolution of 198 bands.

For the generation of abundances, initially, a zeros tensor of size $100\times
100\times 4$ was segmented into sixteen $25\times 25\times 4$ tensors. Next,
the algorithm randomly selects one from the segmented tensors and initiates
the assignment of abundance fractions. This is done through a superpixel of
size $3\times 3$. Thereafter, a random selection of a center pixel is done and
checked for occupancy. If the pixel is not already occupied then the
possibility of it being on the outer edge is investigated. If the pixel is not
a part of the outer edge then the superpixel of the above-mentioned dimension
is constructed and the abundance fractions are assigned while satisfying the
sum to one constraint.If the selected center pixel is at an edge,then
necessary alterations are done to the superpixel to get adapted according to
the nature of the edge. However, following this procedure will not make the
dataset realistic which contains pure regions and sparse sections as remote
sense HSIs. To achieve this, some of the $25\times 25$ were allotted to be
pure or to be to maintain a constant abundance fraction variation. This
process makes sharp edges. To resolve this issue, initially, the sharp edges
were extended irregularly, and then the entire image was smoothened through
mean filtering. The generated image is shown in Fig. 3.

To generate the synthetic image next endmember assignment was carried out. For
this, initially, four endmembers were chosen from the United States Geological
Survey (USGS)111https://www.usgs.gov/labs/spectroscopy-lab/science/spectral-
library endmember library [65] was utilized. The spectral signatures of
”Ilmenite, Montmorillonite, Limestone, and Tree” were chosen as the candidate
endmembers. Then, as the spectral signatures extracted contained bands which
were not calibrated preprocessing was carried out for the removal of not
calibrated bands. Furthermore, as the spectral signatures of different
elements were acquired from different sensors there was a slight deviation in
the wavelengths present in each spectral signatures. To overcome this issue
cubic interpolation was used to generate spectral signature containing the
same wavelengths.

Moreover, a set of three real HS
datasets222http://lesun.weebly.com/hyperspectral-data-set.html: Samson,
Jasper-Ridge, and Urban were used for quantitative evaluation purposes. A
precise ground truth for the endmembers and abundances for each dataset is
available on each website. The Samson dataset consists of three endmembers
with 156 spectral bands covering 401 nm to 889 nm wavelengths, while Jasper-
Ridge comprises four sources with 198 spectral bands ranging from 380 nm to
2500 nm. The urban dataset has 162 spectral bands spanning the region from 400
nm to 2500 nm and five endmembers. The spatial resolutions of the datasets, in
units of pixels $\times$ pixels, are 95$\times$95, 100$\times$100 and
307$\times$307, respectively.

In order to compare the results of the proposed algorithm, we have chosen
eight different classical HU algorithms. In addition to DL architectures such
as CNNAEU [44], DAEN [52], [57], uDAS [51] and EGU-Net[66], renowned signal
processing algorithms which leverage different NMF and other geometric
frameworks namely, L$\nicefrac{{1}}{{2}}$ NMF [34], R-CoNMF [26], MLNMF [67],
SULoRA [68] and VCA-FCLS [19] were used for this purpose. The proposed and
comparison algorithms were tested on the real hyperspectral datasets as well
as the simulated dataset incorporating the optimum parameters suggested by the
authors in their work. The RGB images of the datasets are illustrated in Fig.
3.

TABLE II: Unmixing performance comparison for the four datasets: Simulated,
Samson, Jasper-ridge, and Urban. The best-performing algorithms are ranked up
to the third, and the ranks are superscript.

|  | Proposed methods | Deep learning methods | Signal processing methods
---|---|---|---|---
|  |  GAUSSblind |  GAUSSprime | CNNAEU | DAEN | DAEU | EGUnet | uDAS |  | $L_{\nicefrac{{1}}{{2}}}$
---
NMF
R-CoNMF | MLNMF | SULoRA |  | VCA
---
FCLS
Simulated | aRMSE | 0.2725 |  0.13121 | 0.2453 | 0.2550 |  0.23713 | 0.2783 | 0.2349 | 0.2419 |  0.16182 | 0.2896 | 0.3724 | 0.2874
aAAD | 0.6077 |  0.34141 | 0.8532 | 0.7386 | 0.8297 | 0.6844 | 0.8430 |  0.69093 |  0.42562 | 0.8240 | 0.7732 | 0.8168
aAID | 1.7655 | 0.2496 | 3.4171 | 1.5346 | 2.8457 |  2.39813 | 7.1713 | 2.5311 |  0.98262 | 3.3080 | 3.4135 | 3.9947
aSAD | 0.1325 | 0.2459 | 0.1316 | 0.0301 | 0.0756 | 0.1059 | 0.3111 |  0.00081 |  0.02693 | 0.0984 | 0.5475 |  0.00092
aSID |  0.18742 | 0.2739 | 0.3764 | 0.5461 | 0.3246 | 0.2332 | 0.4308 | 0.3436 |  0.00031 |  0.24673 | 1.2544 | 0.3433
|  |  |  |  |  |  |  |  |  |  |  |  |
Samson | aRMSE |  0.13482 |  0.04671 | 0.3724 | 0.3909 | 0.3902 |  0.15733 | 0.3027 | 0.2689 | 0.3330 | 0.3424 | 0.3553 | 0.3996
aAAD | 0.3899 |  0.07241 | 0.9073 | 0.7774 | 0.8384 |  0.25182 | 0.5793 |  0.49913 | 1.0979 | 0.6533 | 0.7054 | 0.8825
aAID | 5.7561 |  4.47363 | 7.6865 | 6.7775 |  5.55612 |  1.04551 | 4.7794 |  4.0999 2 | 10.7096 | 6.4476 | 6.4119 | 8.0004
aSAD |  0.03452 |  0.02111 | 0.1769 | 0.1413 | 0.1155 |  0.06033 | 0.0697 | 0.0811 | 0.2451 | 0.0897 | 0.0990 | 0.1364
aSID | 0.1326 |  0.05843 | 1.5040 | 1.5217 | 2.4887 |  0.04862 |  0.03281 | 0.1953 | 0.4725 | 0.4758 | 0.3954 | 0.6746
|  |  |  |  |  |  |  |  |  |  |  |  |
Jasper-ridge | aRMSE |  0.12243 |  0.04831 | 0.2888 | 0.2227 | 0.2623 | 0.4651 |  0.11902 | 0.3393 | 0.3570 | 0.4135 | 0.3494 | 0.4903
aAAD | 0.6732 |  0.14891 | 0.7186 |  0.50013 | 0.6239 | 1.2312 |  0.21772 | 0.8437 | 1.0048 | 1.0811 | 0.8596 | 1.2592
aAID | 12.8246 | 12.4485 |  6.87173 |  4.54972 | 7.4464 | 15.9190 |  1.58401 | 9.6208 | 8.6153 | 11.3338 |  10.93633 | 19.8816
aSAD |  0.05131 |  0.09652 | 0.1552 | 0.1328 | 0.1634 | 0.1447 | 0.1062 | 0.1677 |  0.14223 | 0.2241 | 0.1893 | 0.2092
aSID | 0.2126 |  0.14872 | 0.2635 |  0.20923 | 0.2710 | 1.3000 |  0.04411 | 0.8390 | 0.2914 | 1.1527 | 0.8440 | 1.3342
|  |  |  |  |  |  |  |  |  |  |  |  |
Urban | aRMSE |  0.09121 |  0.19212 | 0.3203 | 0.2454 | 0.3777 | 0.5097 | 0.3425 | 0.4231 | 0.3330 | 0.4135 | 0.3187 |  0.20613
aAAD |  0.73212 | 2.9351 | 0.9947 |  0.75833 | 1.0989 | 1.4478 | 1.0362 | 1.2346 | 1.0979 | 1.0811 | 0.9746 |  0.67861
aAID | 14.8739 | 11.5265 |  10.29483 |  7.90271 | 12.2434 | 27.6442 | 14.9865 | 20.8305 | 10.7096 | 11.3338 | 14.2694 |  7.94992
aSAD |  0.00431 | 0.5814 | 0.2070 | 0.1753 | 0.1878 | 0.1351 |  0.01582 | 0.1258 | 0.2451 | 0.2241 | 0.1141 |  0.09563
aSID |  0.02311 |  0.43543 | 0.9303 | 1.0081 | 1.2864 | 1.3433 | 0.7105 | 0.9319 | 0.4725 | 1.1527 | 0.5245 |  0.12082

###  IV-C Experiments on simulated data

When comparing the performance of existing algorithms, signal processing-based
approaches perform better than DL unmixing methods. As a whole compared to
other DL methods the proposed methods have superior performance.

The conventional unmixing algorithms used in the analysis are geometry-based
methods, and their derivation is based on ideal HSIs. The simulated dataset
resonates with the requirement of those algorithms. The presence of pure or
marginally-pure pixels favouring unmixing algorithms that exploit underlying
geometric frameworks to prevail over DL unmixing algorithms is reflected
through these results.

However, the results of GAUSSprime method are pre-eminent in abundance
estimation. Notably, the estimated abundances of GAUSSprime have managed to be
better than its pseudo-ground truth abundances from the L$\nicefrac{{1}}{{2}}$
NMF algorithm. This result can be taken as evidence for the fact that the
supervisory inputs used in the GAUSSprime method have contrived the abundance
estimation to be accurate even for a complex environment.

With regard to end member extraction, SP based L$\nicefrac{{1}}{{2}}$ NMF and
RCo-NMF provide best results for aSAD and aSID error matrices respectively
while the GAUSSblind has achieved the second finest result for aSID.

Figure 4: Groundtruth and abundance maps for Simulated, Samson, Jasper Ridge
and Urban data sets from GAUSSprime GAUSSblind uDAS and L$\nicefrac{{1}}{{2}}$
NMF  Figure 5: Real endmember and signatures extracted for Simulated, Samson,
Jasper Ridge and Urban data sets from GAUSSprime GAUSSblind uDAS and
L$\nicefrac{{1}}{{2}}$ NMF  TABLE III:  Result variances after 5 consecutive
trials

|  |  GAUSSprime |  GAUSSblind
---|---|---|---
Simulated | aRMSE |  4.3*10-5 |  1.6*10-4
aAAD |  2.1*10-4 |  7.2*10-5
aAID |  9.6*10-5 |  2.5*10-5
aSAD |  6.4*10-5 |  1.4*10-6
aSID |  7.3*10-4 |  1.7*10-5
|  |  |
Samson | aRMSE |  6.35*10-5 |  4.46*40-7
aAAD |  2*10-3 |  5*10-3
aAID |  6.23*10-5 |  8.22*10-5
aSAD |  8.07*10-5 |  2.26*10-5
aSID |  5*10-5 |  7*10-4
|  |  |
Jasper-ridge | aRMSE |  7.9*10-5 |  3*10-4
aAAD |  2*10-4 |  1*10-3
aAID |  4.2*10-5 |  6.3*10-5
aSAD |  1.2*10-4 |  3*10-4
aSID |  2.1*10-4 |  6.9*10-3
|  |  |
Urban | aRMSE |  2.3*10-4 |  3.9*10-4
aAAD |  1.3*10-2 |  2.8*10-2
aAID |  5.3*10-3 |  4.7*10-3
aSAD |  7.8*10-3 |  8*10-6
aSID |  1.6*10-2 |  9.2*10-5

###  IV-D Experiments on real data

####  IV-D1 Samson dataset

It is imperative to understand the Samson dataset to analyze the performance
of the two methods. It has three principal endmembers, and the boundaries of
these endmembers are almost non-overlapping. The performance of the GAUSSblind
and the GAUSSprime variants is contributed by the one-hot encoded pseudo
ground truth in the initial stage of the training. Similarly, by the
L$\nicefrac{{1}}{{2}}$ NMF pseudo ground truth in the GAUSSprime variant and
the combined training of the encoder and decoder in the GAUSSblind variant
during the latter stages. Consequently, the pre-training of the HUA under the
GAUSS architecture has driven the optimization towards an optimal solution
space that further optimizes the reconstruction error gradients.

Next, the GAUSSprime has performed the best in terms of abundance estimation
with the exception of AID. Moreover, using the L$\nicefrac{{1}}{{2}}$ NMF
abundances as the pseudo-ground truth, the GAUSSprime method has produced
results better than the L$\nicefrac{{1}}{{2}}$ NMF algorithm. The reason is
that the network had benefited from pre-training the network using the GAUSS
model and L$\nicefrac{{1}}{{2}}$ NMF method, which has learned different
properties of the latent space of the HSI. Further, the fact that GAUSSprime
continues to outperform L$\nicefrac{{1}}{{2}}$ NMF is a clear evidence of
optimal combining, and that it is better than the GAUSSblind is because
reconstruction optimization is not as effective as L$\nicefrac{{1}}{{2}}$ NMF
steering due to the specific landscape.

While GAUSSprime stands on top with regard to SAD, the performance of uDAS is
higher when considering SID. Furthermore, the endmember performance of the
GAUSSprime method has better results than that of the L$\nicefrac{{1}}{{2}}$
NMF algorithm, which produced the pseudo-ground truths for the GAUSSprime
method. Therefore, as the GAUSSprime method has outperformed
L$\nicefrac{{1}}{{2}}$ NMF algorithm in both abundance and endmember
performances, the performance increment should be from combining the best
attributes of the GAUSS initializer and the L$\nicefrac{{1}}{{2}}$ NMFṪhe
injection of conventional SP as a supervisory input has managed to generate a
significant performance boost in HU compared to pure DL methods. This
validates the combining of the two domains which has been done in the
GAUSSprime architecture which is made possible by the introduction of the
supervisory ground truth for UN training by the GAUSS
architectureĊonsequently, the decoder of the GAUSSprime method has benefited
from using the AN in the HUA and the pseudo-ground truth from the
L$\nicefrac{{1}}{{2}}$ NMF algorithm. Since the AN improves the association
between the neighboring pixels and the centre pixel, the sparsity in the HSI
resulted from the direct impact of the L$\nicefrac{{1}}{{2}}$ NMF algorithm
will be complemented. As the output of the encoder is fed to the decoder and
its output is compared against the pixels that promote sparsity in the image,
the decoder benefits from the properties used by other unmixing algorithms.
This combination of different aspects of algorithms via dedicated networks
signifies the use of the AN and a pseudo-ground truth, as proposed in this
work.

####  IV-D2 Jasper-ridge dataset

The Jasper-ridge dataset has four endmembers, and their boundaries are not as
non-overlapping as in the Samson dataset.

GAUSSprime has been able to achieve the best results for abundance estimation
except for the aAID error matric. The much superior ground truth in the latter
stages and the added effectiveness of the AN’s spatial smoothing mechanic
might have helped the proposed GAUSSblind and GAUSSprime to perform better
than the other DL methods. Here, the pseudo-ground truth from the
L$\nicefrac{{1}}{{2}}$ NMF abundances is considered superior to the more
rudimentary one-hot encoded maps used in conditioning the network through
GAUSS initializer. The GAUSS initializer input information on pure or
primarily pure abundances while the L$\nicefrac{{1}}{{2}}$ NMF algorithm has
information about abundances with lower fractions. Therefore, using the two
pseudo-ground truths in consecutive training stages has complemented each
other in the GAUSSprime method, which reaffirms the benefits of using
supervisory data to train the encoder, initially proposed for the first
training stage (GAUSS architecture) with the k-means algorithm.

In the Jasper-ridge dataset, the road endmember is challenging to extract
since it is scarce compared to other endmembers and heavily correlates with
the dirt endmember, despite having high reflectance values. As a result, most
algorithms have not successfully extracted the road endmember, especially
conventional algorithms and ultimately its abundance values. However,
according to Fig. 5, all two methods in the proposed work have extracted a
valid signature for the road while GAUSSblind GAUSSprime and uDAS algorithm
have had the highest success. This achievement of GAUSS variants was made
possible by the AN and the pseudo-ground truth. Because the AN has managed to
decorrelate the pure endmembers to conform to their high correlation, and as a
result, the decoder has been allowed to extract the road endmember accurately
as opposed to the comparative algorithms.

####  IV-D3 Urban dataset

For the urban dataset, the GAUSSblind method has produced the best results for
estimating the abundance and endmembers. Even though DAEN and VCA-FLCS have
good performance in terms of aAID, GAUSSblind provides the best results for
the rest matrices. Since the GAUSS variants perform unmixing pixelwise, the
benefits of the spatial features in the urban HSI should be extracted by AN,
which feeds its output to the UN. Also, the abundance maps for
L$\nicefrac{{1}}{{2}}$ NMF in Fig. 4 have much lower abundance values for most
of the endmembers, and the maps have significantly deviated from the ground
truths. Since there is a marked difference between the ground truth and the
L$\nicefrac{{1}}{{2}}$ NMF abundances, the UN in GAUSSprime has not been able
to improve using a conventional algorithm with inferior performance like with
the Jasper-ridge dataset. Nonetheless, it should be noted that the pseudo-
ground truth of the GAUSSprimem̀ethod could be any reliable SP algorithm, and
the L$\nicefrac{{1}}{{2}}$ NMF algorithm was used to maintain consistency
among the datasets.

The GAUSSblind method has the best performance for endmember extraction.
Therefore, it can be said that the decoder for the GAUSSblind method has
already been initiated around the decoder’s optimal solution after the initial
training stage following the GAUSS structure. According to the aSAD metric,
the endmember performance is as improved as the abundance error metric. Hence,
the proposed AN and the one-hot encoded pseudo-ground truth have been
conducive to endmember estimation.

According to Table II, the method with the least reconstruction error for each
dataset has the best performing figures for unmixing results in Table II.
Therefore, though the authors have chosen the L$\nicefrac{{1}}{{2}}$ NMF
algorithm as the conventional algorithm to construct a pseudo-ground truth for
the training of the UN, the selection of a suitable unsupervised algorithm can
also be based on the reconstruction error of the HSI for that algorithm.
However, in this work, the L$\nicefrac{{1}}{{2}}$ NMF algorithm was used to
maintain consistency in the comparison and remove the effect of using
different algorithms in constructing the pseudo-ground truth.

###  IV-E Summary

Even though the performance of existing SP and DL is competitive, the proposed
GAUSSprime and GAUSSblind is comparatively superior as a whole when
considering the overall results.

For sparse environments like Samson and Jasper Ridge, the performance of
GAUSSprime is considerably high compared to all other methods. This might be
due to the presence of many pure pixels which complement the usage of k-means
in the stage of training and L$\nicefrac{{1}}{{2}}$ NMF in the latter stages.
Further explanation can be found in section III - D.

While considering environments that are complex with sudden and scattered
variations of abundances like the Urban dataset, the results have scattered
among DL and SP domains equally. However, in terms of the urban dataset, the
performance of the gauss blind method is noteworthy.

The consistency of performance of the introduced algorithms is assured through
summarizing the results for five consecutive trials in Table II. The variances
of the obtained results are illustrated in Table III

TABLE IV: Ablation study for the GAUSSblind method

|  | aRMSE | aAAD | aAID | aSAD | aSID
---|---|---|---|---|---|---
Samson | AE | 0.4150 | 0.7826 | 6.4259 | 0.1416 | 1.5271
AN + AE | 0.4386 | 0.7470 | 6.1648 | 0.1113 | 1.2928
AN + pGT | 0.4030 | 0.7241 | 6.2504 | 0.1140 | 0.0634
AE + AN + pGT | 0.2150 | 0.4350 | 5.9902 | 0.0947 | 0.0725
AE + AN + GAUSS | 0.1348 | 0.3899 | 5.7561 | 0.0345 | 0.1326
initializer + pGT |  |  |  |  |
|  |  |  |  |  |
Urban | AE | 0.1793 | 0.7823 | 15.7651 | 0.0217 | 0.0288
AN + AE | 0.1158 | 0.8328 | 15.1476 | 0.0145 | 0.0286
AN + pGT | 0.1073 | 0.8876 | 15.0938 | 0.0074 | 0.3674
AE + AN + pGT | 0.0988 | 0.8743 | 15.0076 | 0.0106 | 0.0302
AE + AN + GAUSS | 0.0912 | 0.7321 | 14.8739 | 0.0043 | 0.0231
initializer + pGT |  |  |  |  |

TABLE V: Ablation study for the GAUSSprime method

|  | aRMSE | aAAD | aAID | aSAD | aSID
---|---|---|---|---|---|---
Samson | AE + pGT | 0.4186 | 0.6975 | 5.4374 | 0.0857 | 0.8212
AN + AE + pGT | 0.1989 | 0.6183 | 4.1830 | 0.0218 | 0.0657
AE + AN + GAUSS | 0.0467 | 0.0724 | 4.4736 | 0.0211 | 0.0584
initializer + pGT |  |  |  |  |
|  |  |  |  |  |
Urban | AE + pGT | 0.3426 | 3.0756 | 11.7028 | 0.7006 | 0.5257
AN + AE + pGT | 0.2342 | 2.9588 | 11.9285 | 0.6283 | 0.4769
AE + AN + GAUSS | 0.1921 | 2.9351 | 11.5265 | 0.5814 | 0.4354
initializer + pGT |  |  |  |  |

Here, AE refers to the AutoEncoder, AN refers to the Approximation Network,
and pGT refers to the Pseudo ground truth.

##  V Ablation study

An ablation study has been conducted to further illustrate the effectiveness
of GAUSSblind and GAUSSprime architectures on HU. Samson and Urban data sets
were used for this study, and the findings are tabulated in Table IV and Table
V. Furthermore, It is evident that all quantitative error metrics show the
effectiveness of the two variants.

##  VI Conclusion

In this paper, we proposed two algorithms for HU. In this work, we discuss
split training of the encoder-decoder with an additional network (AN)
preceding the UN and the use of a pseudo-ground truth for abundances for
supervised learning of the UN. Then, the AN approximates centre pixels from
their neighborhood to improve the association between nearby pixels and
spatial correlation; and performs decorrelation of pixel and endmember
impurities. In addition, the use of the pseudo-ground truth accelerated the
encoder training. By releasing the single-layer constraint on the decoder it
enables the possibility of modeling the underlying non-linearity of the mixing
process in remote sensing and enabling subsequent training procedures with
more refined abundance maps as pseudo-ground truths for improved performance.
Using the pseudo-ground truth created with k-means segmentation resulted in
abundance binarization, but it was conducive to estimating pure endmembers of
the HSIwithin the initial training stage.

Based on the initial training architecture GAUSS two variants were
established. In the GAUSSblind, we combine the encoder and decoder to back-
propagate the reconstruction error gradients to both the decoder and the
encoder. Then, in the GAUSSprime we experimented with using the abundance
results of a SP-based unmixing method as the pseudo-ground truth with the
combined encoder-decoder architecture, which was made possible due to the
introduced pseudo-ground truth mechanic that decoupled and enabled such
abundance steering. Furthermore, the use of L$\nicefrac{{1}}{{2}}$ NMF allows
the pathway to further expedite on using more refined pseudo-ground truth from
the SP and DL domains.

The proposed variants mostly outperformed or produced results competitive with
existing unmixing algorithms from DL and SP domains. In comparison with the
existing DL and SP approaches for HU the performance of the proposed models
can be distinguished for improved performance in terms of the considered error
matrices. Besides, we discussed the effect of the dataset on the performance
of the unmixing method and the selection of the unmixing algorithm out of the
proposed methods for an unseen hyperspectral dataset. Meanwhile, the same
technique could be applied to deciding on an algorithm to create the pseudo-
ground truth.

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|  D.Y.L. Ranasinghe received the B.Sc. (Eng.) degree in electrical and
electronic engineering from the University of Peradeniya, Sri Lanka, in 2020.
Immediately after that, he joined the School of Engineering, Sri Lanka
Technological Campus, Padukka, Sri Lanka, as a Research Assistant. He is
currently working as a Research Assistant with the University of Peradeniya
under a research grant from the International Development Research Centre
(IDRC), Canada. He has published in IEEE JSTARS, IEEE Access, and numerous
IEEE conferences. His research interests include hyperspectral and
multispectral imaging, remote sensing, signal and image processing, and deep
learning.
---|---
|  H.M.H.K. Weerasooriya received the B.Sc. degree in electrical and
electronic engineering with a first-class honors from the University of
Peradeniya, Peradeniya, Sri Lanka in 2020. He is currently an Instructor with
the Department of Electronic and Electrical Engineering, University of
Peradeniya. He is presently involved in the researches on hyperspectral
imaging for remote sensing and agriculture applications, and he has numerous
publications in IEEE conferences. His research interests include image
processing, signal processing, communication, machine learning and deep
learning.
---|---
|  G.M.R.I. Godaliyadda (Senior Member, IEEE) obtained his B.Sc. Engineering
degree in Electrical and Electronic Engineering from the University of
Peradeniya, Sri Lanka, in 2005, and Ph.D. from the National University of
Singapore in 2011. Currently, he is attached to the University of Peradeniya,
Faculty of Engineering, Department of Electrical and Electronic Engineering as
a Senior Lecturer. His current research interests include image and signal
processing, pattern recognition, computer vision, machine learning, smart
grid, bio-medical and remote sensing applications and algorithms. He is a
Senior Member of the IEEE. He is a recipient of the Sri Lanka President’s
Award for Scientific Publications for 2018 and 2019. He is the recipient of
multiple grants through the National Science Foundation (NSF) for research
activities. His previous works have been published in IEEE-TGRS and several
other IEEE-GRSS conferences including WHISPERS and IGARSS. He also has
numerous publications in many other IEEE transactions, Elsevier and IET
journals and is the recipient of multiple best paper awards from international
conferences for his work.
---|---
|  H.M.V.R. Herath (Senior Member, IEEE) received the B.Sc.Eng. degree in
electrical and electronic engineering with 1st class honours from the
University of Peradeniya, Peradeniya, Sri Lanka, in 1998, M.Sc. degree in
electrical and computer engineering with the award of academic merit from the
University of Miami, USA in 2002, and Ph.D. degree in electrical engineering
from the University of Paderborn, Germany in 2009. In 2009, he joined the
Department of Electrical and Electronic Engineering, University of Peradeniya,
as a Senior Lecturer. His current research interests include hyperspectral
imaging for remote sensing, multispectral imaging for food quality assessment,
Coherent optical communications and integrated electronics. Prof. Herath was a
member of one of the teams that for the first time successfully demonstrated
coherent optical transmission with QPSK and polarization multiplexing. He is a
member of the Institution of Engineers, Sri Lanka and The Optical Society. He
is a Senior Member of the IEEE. He was the General Chair of the IEEE
International Conference on Industrial and Information Systems (ICIIS) 2013
held in Kandy, Sri Lanka. His previous works have been published in IEEE-TGRS
and several other IEEE-GRSS conferences including WHISPERS and IGARSS. He
received the paper award in the ICTer 2017 conference held in Colombo Sri
Lanka. Prof. Herath is a recipient of Sri Lanka President’s Award for
scientific research in 2013.
---|---
|  M.P.B. Ekanayake (Senior Member, IEEE) received his B.Sc. Engineering
degree in Electrical and Electronic Engineering from University of Peradeniya,
Sri Lanka, in 2006, and Ph.D. from Texas Tech University in 2011. Currently,
he is attached to the University of Peradeniya as a Senior Lecturer. His
current research interests include applications of signal processing and
system modeling in remote sensing, hyperspectral imaging, and smart grid. He
is a Senior Member of the IEEE. He is a recipient of the Sri Lanka President’s
Award for Scientific Publications in 2018 and 2019. He has obtained several
grants through the National Science Foundation (NSF) for research projects.
His previous works have been published in IEEE-TGRS and several other IEEE-
GRSS conferences including WHISPERS and IGARSS. He also has multiple
publications in many IEEE transactions, Elsevier and IET journals and has been
awarded several best paper awards in international conferences.
---|---
|  D. Jayasundara received a first-class honors degree in Electrical and
Electronic Engineering from University of Peradeniya (2021). He’s currently
working as an instructor at Department of Engineering Mathematics. His
research interests include Mathematics, Deep Learning, Digital Communication
and Signal Processing. At present, he is involved in research associated with
Hyper-Spectral Imaging for Remote sensing and Applications of Multi-Spectral
Imaging.
---|---
|  L. Ramanayake graduated with a second-class honors (upper division) degree
in Electrical and Electronic Engineering from University of Peradeniya (2021).
His research interest includes Machine Learning, Spectral Imaging, High
Voltage Engineering and Signal Processing. Currently, he is involved in
research associated with Hyper-Spectral Imaging for Identification of Probable
Mineral Deposits and Application of Multispectral Imaging.
---|---
|  N. Senarath received a second-class (upper division) honors degree in
Electrical and Electronic Engineering from University of Peradeniya (2021).
He’s currently working as an instructor at Department of Engineering
Mathematics. His research interests include Mathematics, Deep Learning,
Digital Communication and Signal Processing. At present, he is involved in
research associated with Hyper-Spectral Imaging for Remote sensing and
Applications of Multi-Spectral Imaging.
---|---
|  D. Wickramasinghe , graduated with a second-class honors (upper division)
degree in Electrical and Electronic Engineering University of Peradeniya
(2021). His research interests include Machine Learning, Multi-Spectral
Imaging and Signal Processing. Currently he is involved in researches related
to Hyper-Spectral Imaging for Identification of Probable Mineral Deposits,
Potable Water Quality Parameter assessing and Identifying Sugar Adulteration
in Black Tea through Multispectral Imaging.
---|---