Hyperspectral remote sensing exploits the fact that all materials reflect, absorb and emit electromagnetic energy at specific wavelengths. In comparison to a typical camera that uses red, green and blue colors as three wavelength bands, hyperspectral imaging (HSI) sensors acquire digital images in many contiguous and very narrow spectral bands that typically span the visible, near-infrared and mid-infrared portions of the spectrum. This enables to construct essentially continuous radiance spectrum for every pixel in the scene. The goal of the paper is to identify and classify materials that are present in HSI and to detect targets of interest.

Material and object detection using remotely sensed spectral information has many military and civilian applications. Detection algorithms can be divided into two classes: supervised and unsupervised. The unsupervised detection is also called “anomaly detection”. Anomalies are defined as patterns in the data that do not conform to a well defined notion of normal data behavior. In anomaly detection tasks, hyperspectral pixels have to be classified as either background (normal behavior) or anomalies. Every pixel or group of pixels, which are spectrally different in a meaningful way from the abundant (background) spectral material, are classified as anomalies. A survey of unsupervised detection algorithms is given in [1].

In a supervised detection (also called “target detection”), the main objective is to search for a specific given spectral material (target) in HSI. The spectral signature of the target is known a priori from laboratory measurements. In this paper, we focus on supervised target detection in HSI.

The paper has the following structure. Problem statement including HSI descriptions, which are used for testing the proposed algorithms, are given in Section 2. Related work are described in Section 3. Section 4 describes the needed mathematical background that is used for the constructed solution. Section 5 presents the ROTU algorithm that is based on orthogonal rotations. Classification for an unmixing algorithm is given in Section 6 and its experimental results are presented in Section 6.4.

The related algorithms, which appeared in Section 3 (also in [2]), assume that spectra of different materials are statistically independent. Weaker assumptions for the relations between spectra were considered also in [2]. A weaker assumption is the orthogonality condition expressed later in Proposition 4.3: Assume that B is an unknown background spectrum, T and P are the known spectra of the target and the mixed pixel, respectively. Assume that the parameter t satisfies the linear mixing model P = tT + (1 − t)B. If B and T are orthogonal, then t = 〈T, P 〉/‖T‖².

This proposition implies a solution if the background and the target have statistically independent spectra. Note that if 〈T, B〉 = ∊, then the variance of this estimation error is ∊ /‖T‖².

The linear unmixing algorithms, which are presented in [2], assume only linear independency between different spectral materials. But they can have big mutually correlated coefficients. We assume that the spectra of different materials have different sections in their spectrum: some sections are mutually correlated, some sections are smooth, some sections have a Gaussian random behavior and some sections have oscillatory behavior. First and second large derivatives are used as characteristic features to achieve spectral classification.

The first algorithm in [2] (Section 3), titled WDR, works well but does not detect sub-pixel targets. The second algorithm in [2] (Section 4), titled UNSP, works well but it is computationally expensive due to the need to perform eigenvectors decomposition in each pixel’s neighborhood. The UNSP method, which has to construct an orthocomplement of the linear span of the principal directions of the background of each pixel, is computationally expensive.

In this paper, we present two new methods based on the same assumption about the relation between spectra of different materials as was done in [2]. The CLUN algorithm is based on classification by learning. The learning phase finds the set of wavebands which satisfies independency between the target and the background. By using this set, we map the spectra space to another space, which provides independency between the target’s spectrum and the background’s spectra. All the computationally expensive steps in this algorithm belong to the learning phase. The target detection phase is fast. ROTU is based on random orthogonal projections. This algorithm does not require a learning phase. It is less reliable than WDR and UNSP, which were described in [2], since it generates more false alarms. On the other hand, it is faster. The ROTU algorithm assumes that the first and the second derivatives of the spectra from different materials are close to be orthogonal. In this algorithm, as opposed to the previous algorithm, several targets can be detected. This algorithm scans through all the scene and searches each pixel without taking into consideration its neighborhood.

The experiments in this paper including performance evaluation were performed on three real hyper-spatial datasets titled: “desert” (Figure 4), “city” (Figure 5) and “field” (Figure 6), which were captured by the Specim camera [3]. Their properties with a display of one waveband per dataset is given in Figures 4–6.

The description in this section, which is based on [23], is given since it is being used later in our development of the proposed methodology.

According to [23], a binary classification is frequently performed by using a real-valued function f : X ⊆ 𝕉ⁿ → 𝕉 in the following way: the input x = (x₁, ..., x_n)^T is assigned to a positive class if f(x) ≥ 0, otherwise, to a negative class. We consider the case where f(x) is a linear function of x with the parameters w and b such that (3) f ( x ) = 〈 w ⋅ x 〉 + b = ∑ i = 1 n w i x i + bwhere (w, b) ∈ 𝕉ⁿ × 𝕉 are the parameters that control the function. The decision rule is given by sgn(f(x)). w is assumed to be the weight vector and b is the threshold, then according to [23].

Definition 3.1 A training set is a collection of training examples (data) (4) S = ( ( x 1 , y 1 ) , … , ( x l , y l ) ) ⊆ ( X × Y ) lwhere l is the number of examples, X ⊆ 𝕉ⁿ and Y = {−1, 1} is the output domain.

The Rosenblatt’s Perceptron algorithm ([23,24], pages 12 and 8, respectively) creates an hyperplane 〈w ·x〉+b = 0 with respect to the training set S. It creates the best linear separation between positive and negative examples via minimization of measurement function of “margin” distribution λ_i = y_i(〈w, x_i〉 + b). The classification for (x_i, y_i) is correct when λ_i > 0.

The Perceptron algorithm is guaranteed to converge only if the training data are linearly separable. A procedure that does not suffer from this limitation is the Linear Discriminant Analysis (LDA) via Fisher’s discriminant functional [23]. The aim is to find the hyperplane (w, b) on which the projection of the data is maximally separated. The cost function (the Fisher’s function) to be optimized is: (5) F = m 1 − m − 1 σ 1 2 + σ − 1 2where m_i and σ_i are the mean and the standard deviation, respectively, of the function output values P_i = {〈w · x_j〉 + b : y_j = i} for the two classes P_i, i = 1, −1. From [23] we get the following definition.

Definition 3.2 The dataset S in Equation 4 is linearly separable if the hyperplane 〈w · x〉 + b = 0, which is obtained via the LDA algorithm ([23]), correctly classifies the training data. It means that λ_i = y_i(〈w, x_i〉 + b) > 0, i = 1, . . . , l. In this case, b is the separation threshold. If λ_i < 0, then the dataset is linearly inseparable.

Definition 3.3 The vector x ∈ 𝕉ⁿ is isolated from the set P = {p₁, ..., p_k} ⊆ 𝕉ⁿ if the training set S = ((x, 1), (p₁, −1), ..., (p_k, −1)) is linearly separable according to Definition 3.2. In this case, the absolute value of b is the separation threshold.

Suppose that we have a set S = {x₁, .., x_n} of n samples. First, we want to partition the data into exactly two disjoint subsets S₁ and S₋₁. Each subset represents a cluster. The solution is based on the K-means algorithm ([25]). K-means maximizes the function J(e) (Equation (6)) where e is a partition. The value of J(e) depends on how the samples are grouped into clusters and on the number of clusters (see [25]). (6) J ( e ) = tr ( S W − 1 S B )where S W = ∑ i = 1 l ∑ x ∈ S i ( x − m i ) ( x − m i ) T is called “within-cluster scatter matrix” ([25]), l is the number of classes, S_i are the classes and m_i are the center of each class. S_B is called “between-cluster scatter matrix” ([25]), where S B = ∑ i = 1 l n i ( m i − m ) ( m i − m ) T, n_i is the cardinality of the class i and m is the center for the dataset.

Definition 3.4 Let (w, b) be the best separation for the set S = {x₁, ..., x_n} ⊆ 𝕉ⁿ via K-means and Fisher’s discriminant analyses [23,24]. (w, b) is called the Fisher’s separation and b the Fisher’s threshold for the data P.

When is a dataset separable? One criterion is when m₁ −m₋₁ > max(diam(P₁), diam(P₋₁)), where the notation in Equation (5) is used. Another criterion is:

Definition 3.5 ([25]) A dataset is separable if from Equation (6), J(e₁) < J(e₂) where e₁ is the partition and the number of classes is 1 and e₂ is the best partition into two classes. If J(e₁) ≥ J(e₂) then the dataset is inseparable and Fisher’s separation is incorrect.

Denote by pr_W (y) the orthogonal projection of the vector y ∈ 𝕉ⁿ on the subspace W ⊂ 𝕉ⁿ.

Definition 4.1 Given a vector x = (x₁, x₂, ..., x_n). Denote mode_ɛ(x) ≜ argmax_p(card{i ∈ [1, .., n] : |p − x_i| < ɛ}) where p is scalar. mode_ɛ(x) is called the mode with ɛ-error of the vector x.

Let q be a positive number. A vector x is called q − ɛ − sparse if the cardinality of its support {i : |x_i| > ɛ} is less or equal to q.

The norm of the linear operator L, L : 𝕉ⁿ → 𝕉ⁿ, is ‖L‖ ≜ sup_‖x‖=1 ‖Lx‖.

Definition 4.2 Let G be a vector space and assume that there is a finite subset ℑ ⊂ G. The set ℑ is ɛ-dense if for any element x ∈ G there is s ∈ ℑ such that ‖x − s‖ < ɛ.

Definition 4.3 Let G be a vector space and assume that the set ℑ is ɛ-dense in G. Let V G r ( s ) = {x ∈ G : ‖x − s‖ < r}. ℑ is homogeneous if for any s₁, s₂ ∈ ℑ and r > ɛ, | V G r ( s 1 ) ∩ ℑ | = | V G r ( s 2 ) ∩ ℑ | holds. In this notation, V G r ( s ) is called the r − neighborhood of s.

For example, the integers set 𝕑 is ɛ-dense in the real numbers set 𝕉 when ɛ = 1. The set of rational numbers 𝕈 is ɛ-dense in 𝕉 for any ɛ. 𝕑 and 𝕈 are homogeneous.

Given two vector v₁ and v₂ in 𝕉ⁿ. Assume that S₁ and S₂ are two linear operators in 𝕉ⁿ and L is a linear span of {v₁, v₂}. We say that S₁ and S₂ are equivalent in relation to v₁ and v₂, denoted by S₁ ∼ S₂, if S₁(L) = S₂(L).

The goal of this section is to generate the conditions for which the vectors provide a solution for the unmixing problem given by Equation (2).

Lemma 4.1 Let B be an unknown background spectrum, T and P are the known spectra of the target and of the mixed pixel, respectively. The parameter t satisfies the linear mixing model P = tT +(1−t)B. If for any fixed positive value γ > 0, 0 < ɛ < 1 , S₁ = {i : |B(i)| > γ} and S₂ = {i : |T (i)| >γ/ɛ} we have that card(S₂ \ S₁) ≥ card(S₁ ∩ S₂), then t = mode_ɛζ(Definition 4.1) where ζ is a vector whose coordinates are { P T 1 } and T₁ = {T |‖T ‖ >γ/ɛ}.

Proof: Let ξ = P T 1 = t T + ( 1 − t ) B T 1 = t + ( 1 − t ) B T 1 and let ζ = B T 1, then ξ = t + (1 − t) ζ. Let W = (S₂\S₁) and V = (S₂ ∩ S₁), then |W| ≥ |V|, W ∩ V = ∅, {ζ(i)}_i∈W ∩{ζ(i)}_i∈V = ∅ and {ζ(i)} ={ζ(i)}_i∈W ∪ {ζ(i)}_i∈V. If i ∈ W then |ζ(i)| = B ( i ) T ( i ) < γ/(γ/ɛ) = ɛ and card({i : |ζ(i)| ≥ ɛ}) ≤ |V| ≤ |W|. Thus, mode_ɛ(ζ) = 0 and mode_ɛ(ξ) = t.

¿From Lemma 4.1 we get Proposition 4.2.

Proposition 4.2 Let B be an unknown background spectrum and T and P be the known spectra of the target and of the mixed pixel, respectively. The parameter t satisfies the linear mixing model P = tT + (1 − t)B. Let T₁ = {T |‖T ‖ > γ} and ξ is a vector with the coordinates { P T 1 }. Assume that B and P are ɛ − sparce − independent. Then, mode_ɛ/γ (ξ) = t.

Proposition 4.3 Assume that B is the unknown background spectrum where T and P are the known spectra of the target and of the mixed pixel, respectively. Assume that the parameter t satisfies the linear mixing model P = tT + (1 − t)B. If B and T are orthogonal, then t = 〈T, P 〉/‖T ‖².

Proof: 〈T, P 〉 = 〈T, t · T + (1 − t)B〉 = t · 〈T, T 〉 = t · ‖T ‖².

Note that if 〈T, B〉 = ∊, then the variance of this estimation error is ∊ /‖T‖². “Sparse-independency” and orthogonality are strong conditions. Theorem 4.1 shows that “ɛ-sparse-ergodic independency” also provides a solution for the linear unmixing problem.

Theorem 4.1 Given ɛ > 0, B is an unknown background spectrum, T and P are the known spectra of the target and of the mixed pixel, respectively. The parameter t satisfies the linear mixing model P = tT + (1 − t)B. Assume B and P are ɛ-sparse-ergodic independent (see Definition 4.5). T_i denotes the ith coordinate of the vector T. Let T₁ = {T_i : |T_i| > ɛ} and ξ is a vector with the coordinates { P T 1 }. Then, mode_ɛ(ξ) = t.

Proof: Let ξ = { P T 1 } and ζ = { B T 1 }. ξ = t + (1 − t)ζ , then we only need to prove that mode_ɛ(ζ) = 0. Assume that p ≠ 0 is a real number. Let Ω = {i|B_i = pT_i} and Λ = {i|B_i < ɛ² ∧ T_i > ɛ}. The ɛ-sparse-ergodic independency (see Definition 4.5) of B and T follows from the fact that card(Ω) < card(Λ), because if W = ⊕_i∈Ω e_i𝕉, then W is a dependent basic subspace of T and B and dim(W) = card(Ω). Note that {i|B_i < ɛ² ∧ T_i > ɛ} = {i|ζ_i < ɛ} and {i|B_i = pT_i} = {i|ζ_i = p}. From the condition card({i|ζ_i = p}) < card({i|ζ_i < ɛ}) follows that mode_ɛ(ζ) = 0.

Now we show how to obtain the “ɛ-sparse-ergodic independency” (see Definition 4.5).

Theorem 4.2 Assume that {v₁, v₂} ⊂ 𝕉ⁿ, 〈v₁, v₂〉 = 0, ɛ > 0. Let Λ = {S₁, S₂, ..., S_r} be the ɛ-dense and homogeneous set of isometrics operators in 𝕉ⁿ. Then,

w₁ ≜ [v₁, S₁(v₁), S₂(v₁), ..., S_r(v₁)] and w₂ ≜ [v₂, S₁(v₂), S₂(v₂), ..., S_r(v₂)] are ɛ-sparse-ergodic independent.

The rest of this section proves Theorem 4.2.

Proposition 4.4 For any pair of orthogonal vectors {v₁, v₂} ⊂ 𝕉² such that 〈v₁, v₂〉 = 0, for any linear operator S : 𝕉² → 𝕉¹ where ‖S‖ = 1 and for any real t ≠ 0, the inequality P [S(v₂) = t · S(v₁)] < P [S(v₂) = 0] holds when P is probability.

Proof: Any linear operator S : 𝕉² → 𝕉¹ with ‖S‖ = 1 can be represented as S(x) = 〈s, x〉 where s = (s₁, s₂) is a vector with ‖s‖ = 1.

Consider S ^ = ( s 1 s 2 − s 2 s 1 ) = ( cos ( φ ) sin ( φ ) − sin ( φ ) cos ( φ ) )to be a rotation of 𝕉². This rotation splits the linear operator S into a product of the rotation operator with the orthogonal projection such that S = Ŝ ○ Pr₁ where Pr₁(x₁, x₂) ≜ (x₁, 0).

We can choose the basis {e₁, e₂} of 𝕉² such that v₂ = (a, 0) and v₁ = (0, b). In this representation, S ( v 2 ) S ( v 1 ) = − a ⋅ sin ( φ ) b ⋅ cos ( φ ) = ( − a / b ) tan ( φ ). For any t ∈ 𝕉, the solution of t = (−a/b) tan(φ) exists.

In addition, tan ′ ( φ ) = 1 cos 2 ( φ ) has a minimum in zero. Hence, the size of the set {φ : | tan(φ)| < ɛ} is more than the size of the set {φ : | tan(φ) − t| < ɛ} for t ≠ 0.

Corollary 4.1 Let {S₁, S₂..., S_r} be ɛ-dense and homogeneous set (Definitions 4.2 and 4.3) of the linear operators 𝕉² → 𝕉¹ and {v₁, v₂} is orthogonal in 𝕉². Then, the vector ς with the coordinates ς i = S i ( v 1 ) S i ( v 2 ), i = 1, …, r, has mode_ɛ(ς) = 0.

Proposition 4.5 Given a pair of orthogonal vectors {v₁, v₂} ⊂ 𝕉ⁿ, 〈v₁, v₂〉 = 0 and ɛ > 0. Assume that S₁, S₂, ..., S_r is an ɛ-dense and homogeneous set of isometrics operators in 𝕉ⁿ. Then, the vector ς, which has the coordinates ς i , α = χ i α, i = 1, . . . , n, α = 1, . . . , r, has mode_ɛ(ς_i,α) = 0 where χ i α is the ith coordinate of the vector χ α = ( ( S α ( v 1 ) ) 1 ( S α ( v 2 ) ) 1 , … , ( S α ( v 1 ) ) n ( S α ( v 2 ) ) n ).

Proof: Denote Λ = {S₁, S₂, ..., S_r}. The set of χ coordinates is divided into the union of the subsets τ_i where τ_i = {ς_i,α}_Sα∈Λ α = 1, . . . , r. We have to show that mode_ɛ(τ_i) = 0, i = 1, . . . , n. Assume that θ_i is a projection to coordinate i, such that for x = (x₁, ..., x_n), θ_ix = x_i. Assume that L is a linear span of {v₁, v₂}. Define R_α = θ_i ○ S_α where R_α: 𝕉² → 𝕉¹. Our assumption was that Λ = {S₁, S₂, ..., S_r} is ɛ-dense and homogeneous. Therefore, {R₁, R₂, ..., R_r} is ɛ-dense and homogeneous too. Then, Corollary 4.1 implies mode_ɛ(τ_i) = 0.

Proof: proof of Theorem 4.2: Let L be the linear span of w₁ and w₂ from 𝕉^nr = ⊕_S⊂Λ 𝕉ⁿ. If the theorem is not true, then there is basic subspace W in ⊕_S⊂Λ 𝕉ⁿ such that dim(W) = q and dim(Pr_w(L)) = 1. It means that w₁|_W = s and w₂|_W = t · s, t ≠ 0, such that dim(W) > card{i|(w₂)(i) = 0 and (w₁)(i) ≠ 0}. This means that a vector with the coordinates ς i = ( w 2 ) ( i ) ( w 1 ) ( i ), i =1, . . . , nr, has mode_ɛ(ς_i) = t ≠ 0. This contradicts Proposition 4.5.

Generally speaking, spectra of different materials have different sections in them: some sections are mutually correlated, some sections are smooth and some have sections with oscillatory behavior. Large first and second derivatives are used as characteristic features for a spectral classification.

Some section in the spectrum can be autocorrelated. Let D = (D₁, D₂) be a pair of the first and the second derivatives operator. Our assumptions imply that D(T) and D(B) are more independent than T and B. The D operator reduces the correlation between spectra and makes them more independent.

Let x be a spectrum. We denote by a d − spectrum the vector Dx. If two d − spectra of different materials are statistically independent (or orthogonal), then estimating t can be obtained by using Proposition 4.3. Statistical independency is a strong condition. Sometimes we have sparse − independency for the d − spectra. The sparse-independency condition yields its estimation from Proposition 4.2. In general, we have ɛ deviation from orthogonality and the condition of sparse − independency never holds.

The unmixing algorithm has the following steps:

Step 1: The operator D is applied to T and P. We assume that |corr(DT, DB)| < ɛ.

Step 2: Assume that Λ = {S₁, S₂, ..., S_r} is random isometric rotations in 𝕉ⁿ, W : 𝕉ⁿ ↣ 𝕉^2nr = ⊕_S⊂Λ 𝕉ⁿ is the embedding of 𝕉ⁿ into 𝕉^2rn such that (8) w T ≜ W T = [ D T , S 1 D T , S 2 D T , … , S r D T ] , (9) w B ≜ W B = [ D B , S 1 D B , S 2 D B , … , S r D B ] , (10) w P ≜ W P = [ D P , S 1 D P , S 2 D P , … , S r D P ] .

Step 3: Construct the vector ς with the coordinates (11) ς α , i = ( w P ) i ( w T ) i , α = 1 , … , r ,     i = 1 , … , 2 n .

Step 4: Estimation of t, denoted by t̂, is computed by t ^ = mode ɛ ( ( w P ) i ( w T ) i ), i = 1, …, 2rn. B is estimated by B̂ = (P − t̂T)/(1 − t̂).

The vectors w_T , w_B and w_P are related by w_P = t · w_T + (1 − t) · w_B, where w_B is unknown. Proposition 4.5 and Theorems 4.1 and 4.2 yield that {w_T, w_B} are ɛ-sparse-ergodic independent (Definition 4.5) and the mode_ɛ(ς) of the vector ς, which have the coordinates (12) ς α , i = ( S α D 1 B ) i ( S α D 1 T ) i , α = 1 , … , r ,     i = 1 , … , 2 n ,is mode_ɛ(ς) = 0.

It follows that mode ɛ ( { ( w P ) i ( w T ) i } i = 1 2 r n ) estimates t. As the number of operators is increased then this estimate becomes more accurate.

Definition 5.1 Equations (8–11) denote the vector ζ = (ζ_k) with the set of coordinates ς k = ( w P ) k ( w T ) k, k = 1, . . . , 2rn. Assume ɛ > 0 and b = {0, ɛ, 2ɛ, 3ɛ, 4ɛ, . . . , 1 = mɛ} is a partition of [0, 1]. Then, the vector H such that H_i = card{k|iɛ − ɛ/2 < ζ_k ≤ iɛ + ɛ/2}, i = 1, . . . , m, is an histogram of ζ_k related to the partition b. This histogram is called the “unmixing histogram”.

Figure 7 illustrates the “unmixing histogram” for B and P from Figure 3.

In Figure 7, we can see the distribution of the values of the vector ζ on the interval [0, 1], where the partition step is 0.01. This histogram shows that the mode of this distribution is 0.4 = 40 · 0.01. Thus, t is estimated by t̂ = mode_ɛ(ξ) = 0.4, where the vector ξ is computed by Equation (11). As long as t̂ is known, we can estimate the spectrum of the background B̂ = (P − t̂T)/(1 − t̂). The comparison between B and B̂ is shown in Figure 8.

1. Experiment with random spectral signatures. Assume the vectors B and T were randomly generated. B is unknown. The vector P was computed by P = tT + (1 − t)B for t = 0.1, 0.3, 0.7, 0.9. Figure 9 displays the plots of P , T and B for t = 0.7. The vector P corresponds to the spectrum of the mixed pixel, T corresponds to the known target spectrum and B corresponds to an unknown background spectrum.

The ROTU algorithm, which estimates t, is applied to the pair P and T with different t values. Then, the estimations of t̂and t are compared. Once t̂ is obtained then the unknown B is estimated by B̂ = (P − t̂T)/(1 − t̂).

The “unmixing histogram” from Definition 5.1 is displayed in Figure 10 for several t values.

From Figure 10 we get that the estimations for the true values of t by t̂ are:

t = 0.3 vs. t̂ = 0.3, t = 0.7 vs. t̂ = 0.7, t = 0.9 vs. t̂ = 0.9 and t = 0.1 vs. t̂ = 0.1. In all these cases, t̂ = t. It means that the reliability of our algorithm on these datasets is 100%.

2. Experiment with Spectra of Real Materials. The spectra B and T of two materials were taken from a database of signatures. They are presented in Figure 11. P is computed by P = tT + (1 − t)B for t = 0.3, 0.55, 0.7, 0.8.

P and T are known while B is unknown. The ROTU algorithm is applied to the pair P and T for t = 0.3, 0.55, 0.7, 0.8. The ROTU algorithm estimates t by t̂. Once t̂ is obtained, then the unknown B can be estimated from B̂ = (P − t̂T)/(1 − t̂).

The “unmixing histograms” from Definition 5.1 and the comparison between B and its estimate B̂ are given below. In each of the Figures 12–15, the horizontal axes of the left images represent the b partition of [0, 1], ɛ = 0.01. The vertical axes of the left images represent the cardinality of the coordinate’s set that belongs to each interval in the b partition.

Case 1. True value is t = 0.3 where its estimate is t̂= 0.28.

Case 2. True value is t = 0.7 where its estimate is t̂= 0.69.

Case 3. True value is t = 0.49 where its estimate is t̂= 0.45.

Case 4. True value is t = 0.8 where its estimate is t̂= 0.82.

3. Experiment with sub- and above-pixel’s target recognition. We present the results from the application of the ROTU algorithm to two different types of scenes: (1). “Desert” (see Figure 4) has targets that occupy more than a pixel. It is shown in Figure 16. “Field” (see Figure 6) has a subpixel target. It is shown in Figure 17. In each figure, the parameter t, which was introduced in Equation (7), means the portion of the target’s material in the detected pixel (fractional abundance).

Let V = {v₁, ..., v_s} be a set of multi-pixels spectra that were randomly chosen. We assume that the majority of the pixels from the scene are background pixels. Hence, after the application of PCA to V we get information about the background and not about the targets. Assume that Ω = {w₁, ..., w_p} are the principal components of the PCA of V and Φ is a set of the known targets spectra. In our approach, Φ = {T} has only one vector, which is the vector of the target’s signature. Our goal is to separate between the two sets Ω and Φ via the sensor building algorithm (SBA) that is discussed next. Algorithm 1: Outline of SBA

Let T be the target’s spectrum: Construct the function (13) F ( ξ ) = 〈 ξ , T 〉 − W ‖ ( 〈 ξ , w 1 〉 , 〈 ξ , w 2 〉 , … , 〈 ξ , w p 〉 ) ‖such that ξ maximizes F (ξ) and W is a weight coefficient.

Detailed description of the “weak classifier” collection:

Choose an integer r > 1000 and a value for ∊ > 0.

Assume P is the current pixel’s spectrum. Proposition 4.1 contains the following two properties where Ξ is defined in Definition 4.6.

We assume that we have the same conditions as in Section 2. We are given the target’s spectrum T and the mixed pixel spectrum P . Their relationship is described by Equation (2) using the background’s average spectrum B, which is the mix of the background’s spectra without the presence of the target, such that B = ∑ k = 1 M a k B k and (14) P = t B + ( 1 − t ) T = t ∑ k = 1 M c k B k + ( 1 − t ) T .Our goal is to estimate t, denoted by t̂, which will satisfy Equation (14) provided that B and T have some independent features. Once t̂ is found, the estimation of an unknown background average spectrum B, denoted by B̂, is computed by B̂ = (P − t̂T)/(1 − t̂).

The CLUN algorithm contains the following steps:

Apply the sensor building algorithm (SBA - Algorithm 1).

Let {P₁, . . . , P_ω} be the spectra dataset from all the pixels in the scene. Denote Y_i = ‖Ξ (X_i)‖. There are two possibilities for the dataset {Y₁, . . . , Y_ω} ⊆ 𝕉⁺:

The set {Y₁, . . . , Y_ω} is separable according to Definition 3.5;

From Definition 3.5, a dataset is separable if J(e₁) < J(e₂) where e₁ is the partition and the number of classes is 1 and e₂ is the best partition for the two classes. The function J(e) measures the separation quality between the clusters. If J(e₁) ≥ J(e₂), then the dataset is inseparable and Fisher’s separation is incorrect.

In the first case, (w, b) is the best separation between two clusters of the set S = {Y₁, . . . , Y_ω} ⊆ 𝕉⁺ according to Definition 3.4 and b is the Fisher’s threshold for this separation. Then, {i|Y_i > b} is the set of pixels that contains the targets. In the other case, there are no pixels that contain the targets.

Assume P_i is the spectrum of the current pixel. Assume ‖Ξ(X_i)‖ > b. Then, according to our assumption P_i contains the target’s spectrum as a linear component. It means that i is a suspicious pixel. corr(Ξ(P), T₁) is close to 1, then t = ‖Ξ(P)‖/‖T₁‖ is the portion that contains T in the current pixel.

We present the results from the application of the CLUN algorithm to hyperspectral images where the target once occupies more than a pixel and once it occupies a subpixel. In Figures 18 and 19, the parameter t, which was introduced in Equation (7), is the portion of the target in the current pixel.

To separate between Ω and Φ, we utilize the following sets types Γ:

d − support local − characteristic is the set of weak classifiers.

Figures 20 and 21 illustrate the separation between Ω and Φ. In this example, we used the d−support local − characteristic as the set of weak classifiers to achieve this separation.

Figure 22 illustrates the other type of situation. The spectrum was taken from the scene “City” (Figure 5). Here the features, which are located in small segments, have noise. Hence, global features sensors are required to separate between the target’s spectrum and the background’s spectrum.

Figure 23 shows how the wavelengths after the projection by the application of PCA.

We present the results from the application of the CLUN algorithm to three different types of scenes (Figures 4–6). In the first scene (Figure 18), the target occupies more than a pixel. The other two scenes (Figures 19 and 24) contain subpixel targets. In each figure, the parameter t, which was introduced in Equation (7), is the portion of the target in each pixel.