Two Linear Unmixing Algorithms to Recognize Targets Using Supervised Classification and Orthogonal Rotation in Airborne Hyperspectral Images

1. Introduction

2. Problem Statement

Figure 1. Exemplary points that were taken from the area pointed by the arrow (full image is given in 2nd figure in Section 2.1).

Figure 2. Zoom on the exemplary point in Figure 1. Left: The marked pixel contains the target with the spectrum T as a whole pixel. Center: The marked pixel is a mixed pixel with the spectrum P. Right: The marked pixel is a pure background pixel with the spectrum B.

Figure 3. Spectra of the background, the mixed pixel and the target from Figures 1 and 2.

2.1. Motivation and Research Approach

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.

Figure 4. The dataset “desert” is an hyperspectral image of a desert place taken by an airplane flying 10,000 feet above sea level. The resolution is 1.3 m/pixel, 286 × 2,640 pixels per waveband with 168 wavebands.

Figure 5. The dataset “city” is an hyperspectral image of a city taken by an airplane flying 10,000 feet above sea level. The resolution is 1.5 m/pixel, 294 × 501 pixels per waveband with 28 wavebands.

Figure 6. The dataset “field” is an hyperspectral image of a field taken by an airplane flying 9,500 feet above sea level. The resolution is 1.2 m/pixel, 286 × 300 pixels per waveband with 50 wavebands.

3. Related Work

• Linear approach: Under the linear mixing model, where the number of endmembers and their spectral signatures are known, hyperspectral unmixing is a linear problem, which can be addressed, for example, by the ML setup [8] and by the constrained least squares approach [9]. These methods do not supply sufficiently accurate estimates and do not reflect the physical behavior. Distinction between different material’s spectra is conditioned generally by the distinction in the behavior of the first and second derivatives and not by a trend.
• Independent component analysis (ICA) is an unsupervised source separation process that finds a linear decomposition of the observed data yielding statistically independent components [10,11]. It has been applied successfully to blind source separation, to feature extraction and to unsupervised recognition such as in [12], where the endmember signatures are treated as sources and the mixing matrix is composed by the abundance fractions. Numerous works including [13] show that ICA cannot be used to unmix hyperspectral data.
• Geometric approach: Assume a linear mixing scenario where each observed spectral vector is given by r = x+n=Mγa+n, γa = s, where r is an L-vector (L is the number of bands), M = [m₁, m₂, . . . m_p] is the mixing matrix (m_i denotes the ith endmember signature and p is the number of endmembers present in the sensed area), s ≜ γa (γ is a scale factor that models illumination variability due to a surface topography), a = [a₁, a₂, . . . a_p]^T is the abundance vector that contains the fractions of each endmember ((·)^T denotes a transposed vector) and n is the system’s additive noise. Owing to physical constraints, abundance fractions are nonnegative and satisfy the so-called positivity constraint ${\sum }_{k=1}^p{a}_k=1$. Each pixel can be viewed as a vector in a L-dimensional Euclidean space, where each channel is assigned to one axis. Since the set {a ∈ 𝕉^p : ${\sum }_{k=1}^p{a}_k=1$, a_k > 0 for all k} is a simplex, then the set S_x ≜ {x ∈ 𝕉^L : x = Ma, ${\sum }_{k=1}^p{a}_k=1$, a_k > 0 for all k} is also a simplex whose vertices correspond to endmembers.
  Several approaches [14,15,16] exploited this geometric feature of hyperspectral mixtures. The minimum volume transform (MVT) algorithm [16] determines the simplex of a minimal volume that contains the data. The method presented in [17] is also of MVT type, but by introducing the notion of bundles, it takes into account the endmember variability that is usually present in hyperspectral mixtures.
  The MVT type approaches are complex from computational point of view. Usually, these algorithms first find the convex hull defined by the observed data and then fit a minimum volume simplex to it. Aiming at a lower computational complexity, some algorithms such as the pixel purity index (PPI) [15] and the N-FINDR [18] still find the maximum volume simplex that contains the data cloud. They assume the presence of at least one pure pixel of each endmember in the data. This is a strong assumption that may not be true in general. In any case, these algorithms find the set of most of the pure pixels in the data.
• Extending subspace approach: A fast unmixing algorithm, termed vertex component analysis (VCA), is described in [19]. The algorithm is unsupervised and utilizes two facts: 1. The endmembers are the vertices of a simplex; 2. The affine transformation of a simplex is also a simplex. It works with projected and unprojected data. As PPI and N-FINDR algorithms, VCA also assumes the presence of pure pixels in the data. The algorithm iteratively projects data onto a direction orthogonal to the subspace spanned by the endmembers already detected. The new endmember’s signature corresponds to the extreme projection. The algorithm iterates until all the endmembers are exhausted. VCA performs much better than PPI and better than or comparable to N-FINDR. Yet, its computational complexity is between one and two orders of magnitude lower than N-FINDR.
  If the image is of size approximately 300 × 2, 000 pixels, then this method, which builds linear span in each step, is too computationally expensive. In addition, it relies on “pure” spectra which are not available all the time.
• Statistical methods: In the statistical framework, spectral unmixing is formulated as a statistical inference problem by adopting a Bayesian methodology where the inference engine is the posterior density of the random objects to be estimated as described for example in [20,21,22].

4. Mathematical Preliminaries

5. Random Orthogonal Transformation Algorithm for Unmixing (ROTU)

5.1. The Unmixing Algorithm

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

$$w_T\triangleq WT=[DT,S_{1}DT,S_{2}DT,\dots ,S_{r}DT],$$

(8)

$$w_B\triangleq WB=[DB,S_{1}DB,S_{2}DB,\dots ,S_{r}DB],$$

(9)

$$w_P\triangleq WP=[DP,S_{1}DP,S_{2}DP,\dots ,S_{r}DP].$$

(10)

Step 3: Construct the vector ς with the coordinates

$${\varsigma }_{\alpha ,i}=\frac{{(w_P)}_i}{{(w_T)}_i},\alpha =1,\dots ,r,\hspace{0.17em}\hspace{0.17em}i=1,\dots ,2n.$$

(11)

Step 4: Estimation of t, denoted by t̂, is computed by $\widehat{t}={\mathit{mode}}_{\varepsilon }\left(\frac{{(w_P)}_i}{{(w_T)}_i}\right)$, 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

$${\varsigma }_{\alpha ,i}=\frac{{(S_{\alpha }{D}_{1}B)}_i}{{(S_{\alpha }{D}_{1}T)}_i},\alpha =1,\dots ,r,\hspace{0.17em}\hspace{0.17em}i=1,\dots ,2n,$$

(12)

is mode_ɛ(ς) = 0.

It follows that ${\mathit{mode}}_{\varepsilon }\left({\left\{\frac{{(w_P)}_i}{{(w_T)}_i}\right\}}_{i=1}^{2rn}\right)$ 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${\varsigma }_k=\frac{{(w_P)}_k}{{(w_T)}_k}$, 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.

Figure 7. The “unmixing histogram” for Figure 3. The x-axis is the b partition of [0, 1], ɛ = 0.01. The y-axis is the number of coordinates that belongs to each interval in the b partition

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.

Figure 8. The solid line corresponds to the real spectrum of the background and the dotted line corresponds to the estimated spectrum of the background

5.2. Experimental Results

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.

Figure 9. Random spectral vector signatures.

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.

Figure 10. The “unmixing histogram” from Definition 5.1 for different t values. The horizontal and the vertical axes are the b partition of [0, 1], ɛ = 0.01. The cardinality of the coordinate’s set belongs to each interval in the b partition. The boxes inside the graphs is the estimated results from the application of the ROTU algorithm.

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.

Figure 11. The spectra T and B of different materials from the database of signatures, where the x- and y-axes are the wavebands and their values, respectively

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.

Figure 12. Left: The “unmixing histogram” from Definition 5.1. Right: Comparison between T and its estimate T̂.

Figure 15. Left: The “unmixing histogram” from Definition 5.1. Right: Comparison between B and its estimation B̂

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).

Figure 16. Left: The scene from Figure 4. Right: Points that contain a target in above pixel when t = 1.

Figure 17. Left: The scene from Figure 5. Right: Points that contain a target in a sub-pixel when t > 0.5.

6. Classification for an Unmixing (CLUN) Algorithm

6.3. Implementations of the CLUN Algorithm

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.

Figure 18. Left: The source image. Right: The result from the application of the projection separators. The white points contain targets with t ≥ 0.7.

Figure 19. Left: The source image. Right: The result from the application of the projection separators. The white points contain a targets with t ≥ 0.7.

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

• d − support local − characteristic is the set of weak classifiers.
• d − support global − characteristic is the set of weak classifiers.
• Assume that Γ = {ξ_i,j} is the set of weak classifiers where ⟨ξ_i,j, X⟩ = X_i − X_j. This set of weak classifiers works well if the vectors from Φ and Ω have a low entropy. It means that many coordinates have equal values and random differences between some coordinates provide sparse and independent behavior.

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 20. Left: The red line is the target’s spectrum and the black line measures the background (it was taken from the scene in Figure 4). Right: The same two vectors after the projection of the separating mapping. The x-axis represents the wavelengths and the y-axis represents the wavelengths amplitude before (left) and after (right) the application of the projection.

Figure 21. Representation of 20 sensors (“weak classifiers”) with small supports. They were obtained as shown in Section 6.1. The x-axis represents the wavelengths index and the y-axis represents the values of the sensor-function.

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 22. Left: The target’s spectrum. Right: The principal components of the background’s spectra. The spectra was taken from the “city” image (Figure 5).

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

Figure 23. Left: This plot represents 200 d-support global-characteristic sensors, d = 4. They were obtained as shown in Section 6.1. Right: The spectrum of the target (red) and the PCA of the background (black) after the application of the projection separators. The x-axis represents the wavelengths and the y-axis represents the wavelengths amplitude before (left) and after (right) the application of the projection.

6.4. Experimental Results for Sub- and Above Pixel Target Recognition

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.

Figure 24. Left: The source image. Right: The white points contain a target with t > 0.5 and corr > 0.8.

7. Conclusions

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