A Quantum Annealer for Subset Feature Selection and the Classiﬁcation of Hyperspectral Images

—Hyperspectral images showing objects belonging to several distinct target classes are characterized by dozens of spectral bands being available. However, some of these spectral bands are redundant and/or noisy, and hence, selecting highly-informative and trustworthy bands for each class is a vital step for classiﬁcation and for saving internal storage space; then the selected bands are termed a highly-informative spectral band subset. We use a Mutual Information (MI)-based method to select the spectral band subset of a given class and two additional binary quantum classiﬁers, namely a quantum boost (Qboost) and a quantum boost plus (Qboost-Plus) classiﬁer, to classify a two-label dataset characterized by the selected band subset. We pose both our MI-based band subset selection problem and the binary quantum classiﬁers as a quadratic unconstrained binary optimization (QUBO) problem. Such a quadratic problem is solvable with the help of conventional optimization techniques. However, the QUBO problem is an NP-hard global optimization problem, and hence, it is worthwhile for applying a quantum annealer. Thus, we adapted our MI-based optimization problem for selecting highly-informative bands for each class of a given hyperspectral image to be run on a D-Wave quantum annealer. After the selection of these highly-informative bands for each class, we employ our binary quantum classiﬁers to a two-label dataset on the D-Wave quantum annealer. In addition, we provide a novel multi-label classiﬁer exploiting an Error-Encoding Output Code (ECOC) when using our binary quantum classiﬁers. As a real-world dataset in Earth observation, we used the well-known AVIRIS hyperspectral image (HSI) of Indian Pine, north-western Indiana, USA. We can demonstrate that the MI-based band subset selection problem can be run on a D-Wave quantum annealer that selects the highly-informative spectral band subset for each target class in the Indian Pine HSI. We can also prove that our binary quantum classiﬁers and our novel multi-label classiﬁer generate a correct two-and multi-label dataset characterized by their selected bands and with high accuracy such as having been produced by conventional classiﬁers - and even better in some instances.


I. INTRODUCTION
A QUANTUM Annealer (QA) is a computing machine configured as a graph network G = (E, V ), at each vertex of which particles are residing, and its edges define the interaction strengths among these particles which are in quantum states ups or downs.For a D-Wave quantum annealer, the graph G has a specific network topology named Pegasus, in which only certain edges are connected.In particular, the interaction among the particles is constrained [1], [2].
A D-Wave quantum annealer works as a metaheuristic process which is dedicated to tackle specific classes of optimization problems, e.g., QUBO problems.There are theoretical studies that a D-Wave quantum annealer can solve these QUBO problems faster than a conventional annealer (even for NP problems) [3], [4].However, currently, there are no indications of computational advantages for real-world problems.For practical applications, several studies are devoted to benchmark and assess a D-Wave quantum annealer for an operational planning and feature extraction from remotelysensed images [5], [6].
For a real-world dataset in Earth observation, remotely sensed images differ in their image content representations due to the diverse satellite platforms with their different types of sensors.When we want to use a D-Wave quantum annealer with an Earth observation dataset, some of the challenges are the proper choice of appropriate remotely sensed images specified by their image content representations such as their spatial information, polarization states, spectral bands, and the embedding of a given dataset in the topology of a D-Wave quantum annealer.Here, we consider hyperspectral images (HSIs), and a selection of their highly-informative band subset is a very vital procedure in Earth observation.Hence, we use a Mutual Information (MI)-based optimization method to select the highly-informative band subset, and more importantly, we can easily embed and optimize the MI-based optimization method in the Pegasus topology of a D-Wave quantum annealer.Therefore, HSIs are one of the most proper datasets in Earth observation for a D-Wave quantum annealer than others.In particular, HSIs became an important field of study to classify or identify objects in a ground scene such as roads, land cover, or agriculture since each object is characterized by a high-dimensional vector of the different spectral bands within the given full wavelength range.Due to the rich information content of the spectral bands, some of these bands carry more discriminatory information than others.Hence, some studies are focused on extracting highlyinformative features or a dimensionality reduction of HSIs, for instance, by using deep learning networks or Principal Component Analysis (PCA) [7], [8].On the other hand, some researchers focused on how to select a highly-informative band subset by using the concept of information theory; in particular, MI-based methods which provide a measure of independence between several spectral bands.Moreover, these MI-based methods are based on prior or reference knowledge of the spectral signatures of objects; such knowledge can be obtained in specific spectral-signature databases of common ground targets [9], [10].
In this paper, we use an Indian Pine HSI of Indian Pine composed of l = 16 distinct classes.Each class is characterized by n = 200 bands (features); thus, the number of subsets of the features of a given class, e.g., y S = 1, is 2 n combinations.A way to find the best feature subset of this given class is to try all combinations.This is clearly computationally expensive for a large number of these features.
Hence, in the first part of this study, we introduce an MIbased subset feature selection problem as a global optimization problem for the Indian Pine HSI.Then we propose to optimize this MI-based band subset selection problem on a D-Wave quantum annealer.As a first step, we map the MI-based band subset selection problem to a QUBO-based band subset selection problem.This is our first problem mapping step.Secondly, we optimize this mapping problem on a D-Wave quantum annealer; quantum optimization [11], [12]; this part was strongly motivated by a feature selection tutorial offered by D-Wave [2].
In the second part of this study, we use binary quantum classifiers, namely a quantum boost (Qboost) and a quantum boost plus (Qboost-Plus) classifier, in contrast to an adaptive boost (Adaboost) classifier [13], [14].We first apply these quantum classifiers to a two-label dataset of the Indian Pine HSI, and secondly, we provide a novel multi-label classifier via an ECOC when using our binary quantum classifiers [15], [16]; each resulting class is discriminated by the selected bands in the first part of our study.We also benchmarked and assessed these binary quantum classifiers and the novel multi-label classifier with respect to conventional classifiers, a Decision Tree Classifier (DTC), a Support Vector Machine (SVM), and an Adaboost classifier.
Our contribution in this paper is then an attempt to benchmark and assess a D-Wave quantum annealer for Earth observation data and to recognize the challenges that are encountered with real-world datasets and future quantum annealers or devices.Towards these goals, we are employing a D-Wave quantum annealer for feature selection and classification of the Indian Pine HSI as a machine learning technique; our contribution consists of a three-step approach, 1) Feature selection on a D-Wave quantum annealer: the MI-based band subset selection, 2) Binary classification on a D-Wave quantum annealer: the binary quantum classifiers to a two-label dataset characterized by those selected bands.3) Multi-label classification on a D-Wave quantum annealer: the ECOC generates a multi-label dataset when we are using our binary quantum classifiers.Moreover, the D-Wave quantum annealer may prove relevant even if we are not intending to demonstrate its advantage over a conventional annealer.
The paper is structured as follows: We introduce the basics of hyperspectral imaging in Section II.We present the basics of information theory and MI-based band subset selection problem in Section III.In Section IV, we discuss the fundamentals of a QUBO problem and demonstrate the problem mapping of an MI-based problem to a QUBO-based problem.We introduce the basics of a D-Wave quantum annealer and optimize the QUBO-based band subset selection problem for the Indian Pine HSI on a D-Wave quantum annealer (see Section V, quantum optimization).Finally, we apply the binary quantum classifiers and the novel multi-label classifier to the two-and multi-label dataset in Section VI, and in Section VII, respectively.We then draw a conclusion in Section VIII.

II. INTRODUCTION TO HYPERSCPECTRAL IMAGING
A hyperspectral imaging sensor mounted on a satellite or aircraft measures the electromagnetic spectrum ranging from the visible to the near infrared wavelengths; for instance, the Imaging Spectroscopy and the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) sensor measures 224 continuous spectral bands ranging 400 nm to 2500 nm at 10 nm intervals [17].
As a real-world dataset of HSIs, we consider an Indian Pine HSI obtained by the AVIRIS sensor (see Fig. 1).This low-noise Indian Pine image having the spectral bands of X = {band1, . . ., band200} elements is a high-dimensional dataset.However, not all of these spectral bands are informative for characterizing a specific class; in other words, some bands of X are redundant or noisy.
It is advantageous to select a highly-informative band subset of the given spectral bands for a given class.Hence, we employ an MI-based band subset selection problem as a global optimization problem.

III. INFORMATION THEORY AND MUTUAL INFORMATION-BASED BAND SUBSET SELECTION
We select a highly-informative band subset for each class of the Indian Pine HSI; for instance, we consider the spectral bands X = {X 1 , . . ., X 200 } = {band1, . . ., band200} of a given class y S and find its most informative band subset.To find the highly-informative band subset for that specific class, we employ information theory; information is a function of probabilities.Hence, we represent band X i and its corresponding class y S as probabilities.We derived the probabilities for the band X i and its class y S by dividing them into ten bins in a histogram.The probability is then defined as where X n and y m represent the number of occurrences of band X i and its class y S in the n -th or m -th bin, respectively.Their joint probability is defined in the same way.
For the selection of the band subset, we exploit Mutual Information (MI) which measures independence between band X i and its class y S .It is defined by and by Conditional Mutual Information (CMI) which is a measure of the dependence between band X i and its class y S given another band X j .The CMI can then be written as where E is the entropy that is a measure of the uncertainty of a random variable [18].These band subset selection techniques expressed by both Eq. ( 2) and Eq. ( 3) are named after a MI-based band subset selection problem which became popular in machine learning due to its strong mathematical foundation rooted in information theory.
In the next sections, we pose the MI-based band subset selection problem as a global optimization problem.First, we map our MI-based band subset selection problem to a QUBO problem, and the QUBO problem to a QUBO-based band subset selection problem.Finally, we optimize the QUBObased band subset selection problem on a D-Wave quantum annealer.

A. Mapping of a Mutual Information-based problem to a QUBO problem
In this part, we consider and pose the MI-based band subset selection problem as a global optimization problem [11], [12].Moreover, the maximization over the subsets {X i } can be written as where X i represents the bands of a given class y S of the Indian Pine HSI (see Fig. 1).
Let us consider the band data X = {band1, . . ., band200} of a given class of Alfalfa or simply y S = 1 as an example case.We assume that Eq. ( 4) is maximized when we use the subset X S = {X 1 , X 2 } = {band1, band2}.We can express this result in a matrix form such that  here x1 = 1, x2 = 1, and x3 = • • • = xn = 0. On the other hand, we can interpret this matrix form that the xn 's are for selecting a highly-informative band subset.Hence, we can express the MI-maximization problem expressed by Eq. ( 4) alternatively as where T represents a transpose operation, and We can even transform this maximization problem to a minimization problem by multiplying it by "−1".As a result, we have where This form of the minimization problem over binary variables x is called a Quadratic Unconstrained Binary Optimization (QUBO) problem.
The MI-based band subset selection problem is therefore equivalent to a QUBO problem when we write "−I(X i ; y S )" and "−I(X i ; y S |X j )" in the Q matrix, and minimize the Q matrix over the binary variables.

B. Mapping the QUBO problem to the QUBO-based subset band selection problem
To select a highly-informative band subset characterizing each class of the Indian Pine image (Fig. 1), we employ the QUBO problem described by Eq. ( 7) with an additional constraint where k is the number of bands (band subset) of interest, and n = 200 is the total number of given bands.Hence, we define the QUBO-based band subset selection problem as where γ is a Lagrange multiplier.As an experiment for selecting the most informative band subset for the specific class of an Indian Pine HSI, we consider the band subsets with three elements (k = 3).

V. QUANTUM OPTIMIZATION: USING A D-WAVE
QUANTUM ANNEALER

A. D-Wave quantum annealer
We selected a highly-informative band subset characterizing the specific class of the Indian Pine HSI by optimizing the QUBO-based band subset selection problem in the form of Eq. ( 9).We optimized this optimization problem on a D-Wave quantum annealer, and we even benchmark the D-Wave quantum annealer with respect to its conventional version.
A D-Wave Quantum Annealer is a quantum annealer for the special class of optimization problems, in particular, QUBOlike problems.Such a quantum annealer is a metaheuristic process evolving slowly enough from its initial energy H i to its final energy H f in the form of a QUBO problem.The evolution process is expressed by X, Ẑ are Pauli-x and -z matrices, H i is the initial Hamiltonian of a system for a given time function of λ(t) = 0, and H f is the QUBO problem with λ(t) = 1 [1]- [3].
The hardware of the D-Wave quantum annealer has a specific graph topology G = (V, E) named Pegasus; its vertices represent binary variables x, and its edges define interaction strengths among the binary variables.However, the connectivity of these binary variables in the Pegasus topology is very constrained; in particular, only the certain binary (x, y) = (x 1 , y 1 ), . . ., (x S , y S ); {x S represents the three selected bands for a given class y S (Table I).}2: y ∈ {−1, +1} S ; {S is the size of the input dataset, and y represents the two-label of the Indian Pine HSI (Table III).w S = w S /sum(w S ).variables are allowed to interact with others through the edges [19].
In addition, the performance of a D-Wave quantum annealer strongly depends on mapping the binary variables of our QUBO problem expressed by Eq. ( 9) to the Pegasus topology.As it is possible to map (embed) our QUBO problem to the Pegasus topology as efficiently as possible, we employed a technique called minor-embedding which is offered by the company D-Wave systems [2], [19].

B. Quantum optimization for the band subset selection
Quantum optimization is an optimization of our QUBObased band subset selection problem on a D-Wave quantum annealer.We performed our experiment in a classical annealer and a D-Wave quantum annealer.Both of these annealers selected the same band subset for each class of the Indian Pine HSI; we shown these selected band subsets in Table I while k = 3 in Eq. (9).
To prove that we selected the highly-informative band subset for each class on a D-Wave quantum annealer, we performed the scene classification for our Indian Pine HSI by using a Decision Tree Classifier (DTC) and a Support Vector Machine (SVM) shown in Fig. (2) as a proof-of-concept.
In addition, we discovered that we needed at least a 10dimensional parameter to reach the same accuracy as our proof-of-concept method when we apply the PCA for the dimensionality reduction and conventional classifiers (the DT end for 14: end for 15: Optimize QUBO problem on a D-Wave quantum annealer 16: An optimal estimator weight vector: α * .17: PREDICT: given the test band set (x 1 , . . .x t ); and the SVM) for the multi-class classification of the Indian Pine HSI (see Table .I) [8].For this scenario, we present the classification accuracy of the test dataset in Table .II.
These findings lead to the conclusion that our QUBO-based band subset selection method identified the highly-informative band subset, and it even helped to reduce a storage space and the computational load for training the given classifiers.

VI. A CASE STUDY OF A BINARY QUANTUM CLASSIFIER ON A D-WAVE QUANTUM ANNEALER FOR HYPERSPECTRAL IMAGE
We have the Indian Pine HSI with 16 classes, where each class is characterized by three highly-informative bands selected by our QUBO-based band subset selection method shown in Table I.In this section, we analyse binary quantum classifiers, namely a quantum boost (Qboost) classifier, and a quantum boost plus (Qboost-Plus) classifier, for a two-label dataset of the Indian Pine HSI created as the binary output of a D-Wave quantum annealer; for instance, Alfalfa and Corn-notill, or Corn-mintill and Corn-notill, etc. Further, we benchmarked the classification accuracy of our binary quantum classifiers with respect to conventional binary classifiers, such as a DTC, an SVM, and an Adaboost classifier.
We considered first two types of boosting algorithms, a Qboost and an Adaboost algorithm (classifier).The Qboost classifier is a quantum version of an Adaboost classifier.Here, we use two types of terminology for these classifiers, a strong classifier C and a weak classifier c i .The strong classifier leverages many weak classifiers to achieve its high classification accuracy; the weak classifier is a classifier that classifies a given dataset better than random guessing [20].

A. Basics of an Adaboost classifier
An Adaboost classifier is an algorithm for finding an optimal estimator weight of many weak classifiers so that the classifier C is maximized [21]; (11) where (x S , y S ) represents a training dataset, and α i ∈ [0, +1] is the estimator weight that is continuous-valued.Here, sign(f < 0, and sign(f (x S )) = 0 otherwise.The loss of the Adaboost classifier is defined as an exponential loss In contrast, a Qboost classifier is an algorithm for finding an optimal estimator weight which takes only binary numbers α i ∈ {0, +1}, and its loss is defined by a squared loss denoted as L 2 .Hence, the Qboost classifier is equivalent to a subset selection algorithm among many weak classifiers to approximately maximize the accuracy of the strong classifier.
In next section, we delve into the Qboost classifier in more detail.
In general, these boosting algorithms start with assigning identical weights w S to our dataset.The weak classifiers classify these datasets, and if the data is misclassified then the weight of that data is increased (boosted).This procedure is repeated until no further improvement in the classification accuracy can be seen.A DTC with a depth of one is considered as a weak classifier; sometimes, it is called a decision stump classifier.We already presented the steps for boosting a weight w S and the weak classifier in Algorithm 1.

B. A Qboost classifier for a two-label dataset of the Indian Pine HSI
Moving towards the Qboost classifier, after having stopped boosting the weight of our dataset, the Qboost classifier selects the weak subset classifier so that the classification accuracy of the strong classifier is maximized.We executed the weak subset classifier selection algorithm on a D-Wave quantum annealer as shown in Algorithm 2. Below, we explain the derivation of Algorithm 2 in detail.More importantly, the Qboost classifier exploits the weight boosting by solving the weak subset classifier selection problem on a D-Wave quantum annealer.
For the two-label dataset of the Indian Pine HSI, we define the training band dataset as (x 1 , y 1 ), . . ., (x S , y S ), the test band dataset as (x 1 , . . ., x t ), and the strong classifier, C(x S ) ∈ {−1, +1}, which is a binary classifier in the form of [13], [20], (13) where α i ∈ {0, 1} is the estimator weight, and c i (x S ) is the weak classifier; we chose DTCs as our weak classifiers.
Recent papers on theoretical studies [13], [14] and a practical application for the remote sensing [22], are proposed to formulate the loss of the strong classifier as a squared loss L 2 : where α * represents the optimal estimator weight vector, S is the size of the training band dataset, and λ N i=1 α 0 i represents a 0-norm term.By expanding the squared loss function, we have which is in the form of a QUBO problem while we define Then we can write We optimized this problem on a D-Wave quantum annealer to select the weak subset classifier in its quadratic form.
Then we obtained the optimal estimator weight vector α * , and the strong classifier for the test band dataset becomes where (x 1 , . . .x t ) are from the test band dataset of the Indian Pine HSI, and T is derived experimentally to increase the classification accuracy of the strong classifier C(x t ) [13], [14].We already the procedures of the Qboost classifier in Algorithm 1 and Algorithm 2. Secondly, we chose the DTC, SVM, and Qboost classifiers as weak classifiers instead of only a DTC.This method is sometimes called an ensemble method.By exploiting Eq. ( 13) and Eq. ( 15), we again formulated weak classifiers such that

C(x
where c 1 (x S ), c 2 (x S ), c 3 (x S ) represent the DTC, SVM, and Qboost classifiers, respectively.In this scenario, we have h = [c 1 (x S ), c 2 (x S ), c 3 (x S )] in Algorithm 2, and this ensemble method is called a Qboost-Plus classifier [2].

C. Benchmarking Qboost and Qboost-Plus for the two-class classification
We run our experiment in several scenarios for the twolabel dataset of the Indian Pine HSI by using the DTC, SVM, Qboost, Qboost-Plus, and Adaboost classifier.These scenarios are: 1) DTC for the two-label dataset of the Indian HSI.
2) SVM for the two-label dataset of the Indian HSI.
3) Qboost with 30 weak classifiers for the two-label dataset of the Indian HSI; the weak classifiers are the DT classifiers with the depth three.4) Qboost-Plus for the two-label dataset of the Indian HSI; the weak classifiers are a DTC, an SVM, and a Qboost classifier.5) Adaboost with 30 weak classifiers for the two-label dataset of the Indian HSI; the weak classifiers are the decision stump classifiers.All above scenarios used for benchmarking are the twolabel classification of the Indian Pine HSI, and we present the classification accuracy of our experiment in Table III.We even compared the boosting algorithms, the Qboost-Plus and the Adaboost classifier.Their results demonstrate that the Qboost-Plus classifier performs the same as the Adaboost classifier and even better in some instances.
In this part, we selected the most highly-informative band of the Indian Pine HSI by using our QUBO-based band subset  IV).Construct two sets, G i and G i .G i consists of all labels for which M [:, i] == 1, and G i is the complement set.

9:
Fit a quantum binary classifier C i to distinguish G i from G i by using Algorithm 2. 10: end for 11: TESTING: 12: Given an unlabeled data x t .13: Evaluate the trained quantum binary classifiers C b (x t ) = {C 1 (x t ), C 2 (x t ), . . ., C 24 (x t )} by employing the step 17 of Algorithm 2. 14: Compute an Euclidean/Hamming distance: , M [j, :]), j = 1, 2, . . ., l. 15: return argmin j d j ; {codewords (a label) for the unlabeled data x t }. 16: STOP ALGORITHM.selection method.Further, we leveraged these selected bands to benchmark our Qboost Qboost-plus algorithms with to the classical classifiers.Our quantum classifiers clearly outperform the conventional classifiers for most of the binary instances of the Indian pine HSI.

VII. A NOVEL MULTI-LABEL CLASSIFIER FOR THE INDIAN PINE HSI ON A D-WAVE QUANTUM ANNEALER
In the prior section, we exhibited that our quantum binary classifiers (Qboost and QboostPlus) classify the two-label of the Indian Pine HSI with high accuracy due to the binary output of a D-Wave quantum annealer.However, the Indian Pine HSI has 16 classes, and the quantum binary classifiers are needed to extend for the multi-label classification.Hence, we propose a novel technique for the multi-label classification via an Error-Correcting Output Code (ECOC), and namely, we leverage an ECOC technique to classify the multi-label of the Indian Pine HSI when using our binary quantum classifiers [15], [16], [23].
An ECOC technique (see Algorithm 3 for a detailed procedure):

Classes
A coding matrix M with l × b elements.We run our experiment for the multi-label of the Indian Pine HSI via the ECOC by using the DTC, SVM, Qboost, Qboost-Plus, and Adaboost classifier.Furthermore, we presented the classification accuracy of our experiment in Table V.We compared also the classification accuracy and the confusion matrix of the Qboost-Plus with one of the Adaboost classifier (see Fig. 3).Their results again demonstrate that the Qboost-Plus classifier beats the Adaboost classifier when we leverage the ECOC technique for a multi-label classification case.More importantly, we provided a novel multi-label classifier via the ECOC technique when applying a quantum computing device yielding binary outputs.

VIII. DISCUSSION AND CONCLUSION
In the first part of this paper, we used an MI-based band subset selection technique as a global optimization approach for a real-world problem of the Indian Pine hyperspectral dataset on a D-Wave quantum annealer.We first mapped Besides, the feature selection method saves storage space and reduces the computational load for the training process.
In the second part of our article, we first tested a binary classification for the Indian Pine HSI due to the binary output of our D-Wave quantum annealer.We proposed to employ two binary quantum classifiers, Qboost and Qboost-Plus, to our two-label dataset.Secondly, we provided an ECOC for the multi-label classification of the Indian Pine HSI when applying our binary quantum classifiers.Here, the classes are characterized by the bands selected during the first part of our study.We benchmarked these binary quantum classifiers and the novel multi-label classifier in comparison to conventional classifiers that are a Decision Tree Classifier, a Support Vector Machine and an Adaboost classifier.Our binary quantum classifiers and our novel multi-label classifier even outperform these conventional classifiers for most instances of the two-and multi-label dataset.
In the end, we realized how to leverage a quantum annealing device to extract knowledge and support real-world optimization problems in comparison to conventional machine learning techniques.In addition, we conceived strategies for formulating and embedding real-world problems to the topology of a D-Wave machine.
We must note however that our method is not intended to compete with a conventional method, but we intended to find a proper dataset in Earth observation to evaluate an existing quantum algorithm on a D-Wave quantum annealer or the future quantum computers since the choice and the size of a dataset play a vital role in quantum computers.
In terms of a future work, we will design a hybrid quantumclassical network for Earth observation datasets which exploits both quantum computers (a quantum annealer and gate-based quantum computer) and a conventional computer.Such a hybrid network will be independent of the choice and size of datasets.
Soronzonbold Otgonbaatar received his B.Sc. and M.Sc.degrees in theoretical physics from the National University of Mongolia, Mongolia, and the University of Siegen, Germany.Currently, he is pursuing his PhD degree in German Aerospace Center (DLR), Oberpfaffenhofen, Germany.His research interest spans over computational science, artificial intelligence (i.e. machine learning), quantum computing and algorithms, and quantum machine learning with practical applications in remote sensing data.

Fig. 2 .
Fig. 2. Left: Ground truth, Right: Classification of the l = 16 classes characterized by three highly-informative spectral bands shown in Table.I by using a Support Vector Machine (SVM).

Fig. 3 .
Fig. 3.The confusion matrix for the l = 16 labels of the Indian Pine HSI.
Datcu (Fellow, IEEE) received the M.S. and Ph.D. degrees in electronics and telecommunications from the University Politehnica of Bucharest (UPB), Bucharest, Romania, in 1978 and 1986, respectively, and the habilitation a Diriger Des Recherches degree in computer science from the University Louis Pasteur, Strasbourg, France, in 1999.Since 1981, he has been a Professor with the Department of Applied Electronics and Information Engineering, Faculty of Electronics, Telecommunications and Information Technology, UPB.Since 1993, he has been a Scientist with the German Aerospace Center (DLR), Wessling, Germany.His research interests include explainable and physics-aware artificial intelligence, smart radar sensors design, and quantum machine learning with applications in Earth Observation.He has held Visiting Professor appointments with the University of Oviedo, Spain, the University Louis Pasteur and the International Space University, both in Strasbourg, France, University of Siegen, Germany, University of Innsbruck, Austria, University of Alcala, Spain, University Tor Vergata, Rome, Italy, University of Trento, Italy, Unicamp, Campinas, Brazil, China Academy of Science, Shenyang, China, Universidad Pontificia de Salamanca, Campus de Madrid, Spain, University of Camerino, Italy, and the Swiss Center for Scientific Computing, Manno, Switzerland.From 1992 to 2002, he had an Invited Professor Assignment with the Swiss Federal Institute of Technology (ETH Zurich), Switzerland.Since 2001, he had been initiating and leading the Competence Center on Information Extraction and Image Understanding for Earth Observation, Paris Institute of Technology, ParisTech, France, a collaboration of DLR with the French Space Agency (CNES).He has been a Professor holder of the DLR-CNES Chair at ParisTech.He has initiated the European frame of projects for image information mining and is involved in research programs for information extraction, data mining and knowledge discovery, and data science with the ESA, NASA, and in a variety of national and European projects.He is the Director of the Research Center for Spatial Information, UPB.He is a Senior Scientist and the Data Intelligence and Knowledge Discovery Research Group Leader with the Remote Sensing Technology Institute, DLR, and a delegate in the DLR-ONERA Joint Virtual Center for AI in Aerospace.He is a member of the ESA Working Group Big Data from Space and Visiting Professor with ESA's Phi-Lab.Dr. Datcu was the recipient of the National Order of Merit with the rank of Knight, for outstanding international research results, awarded by the President of Romania, in 2008, and the Romanian Academy Prize Traian Vuia for the development of the SAADI image analysis system and his activity in image processing in 1987.He was awarded the Chaire D'excellence Internationale Blaise Pascal 2017 for data science in earth observation and the 2018 Ad Astra Award for Excellence in Science.He has served as a Co-organizer for international conferences and workshops and as a Guest Editor for the IEEE and other journals.He is representative of Romanian in the Earth Observation Program Board (EO-PB).

TABLE I SELECTION
OF THE BEST BAND SUBSET FOR EACH CLASS OF THE INDIAN PINE HSI BY USING THE QUBO-BASED BAND SUBSET SELECTION ON A D-WAVE QUANTUM ANNEALER.
Table.I by using a Support Vector Machine (SVM).

TABLE II CLASSIFICATION
ACCURACY OF THE DTC AND THE SVM.

TABLE V CLASSIFICATION
ACCURACY OF THE DTC, SVM, Qboost CLASSIFIER, Qboost-Plus CLASSIFIER, AND Adaboost CLASSIFIER FOR THE ALL 16 LABELS OF THE INDIAN PINE HSI; {1, 2, ..., 16} REPRESENTS THE ALL 16 LABELS (FIG.1).BY A BOLD FONT WE NOTED THE HIGHEST ACCURACY VALUE OF THE QBOOST-PLUS CLASSIFIER WITH RESPECT TO THE ADABOOST CLASSIFIER.We assign unique b-bits (codewords) to each class of the Indian Pine HSI such that b > log 2 l where l = 16 is a number of classes; the classes are represented by a so-called coding matrix M ∈ {0, 1} l×b (see TableIV), and M ∈ {0, 1} S×b for a training dataset with size S.In our case, each class is represented by b = 24 codewords generated randomly.We train each column of the coding matrix M by quantum binary classifiers C b = {C 1 , C 2 , . . ., C 24 }.For an unlabeled input x t , we evaluate C b (x t ) = {C 1 (x t ), C 2 (x t ), . . ., C 24 (x t )}, and then we assign C b (x t ) to the closest codewords in the coding matrix M by using an Euclidean/Hamming distance.