A New Maximum Simplex Volume Method Based on Householder Transformation for Endmember Extraction

With the advancement in technology, hyperspectral images have potential applications in the field of remote sensing due to their high spectral resolution. Despite the hyperspectral image providing abundant information, its analysis suffers from the problem of high dimensionality. Hence, Dimensionality Reduction (DR) is an essential task in all hyperspectral image analysis. Band Selection, which is one of the DR techniques, is still a challenging issue even though many algorithms have been developed. To provide remedy for this issue, this paper explores a novel approach for band selection using K-means clustering on statistical feature in hyperspectral images. The proposed method of clustering based band selection for DR is simple and accurate. A reliable estimate of number of bands to be selected is provided by Virtual Dimensionality (VD). Informative bands preserving maximum information are selected based on the statistical feature, the variance using K-means Clustering technique. Further, our proposed work involves the utilization of the effectiveness of Simplex Growing Algorithm (SGA) on endmember extraction in association with clustering based band selection. Using Fully Constrained Least Squares (FCLS) method, abundance fraction is estimated based on endmember signatures, which are derived using Endmember Extraction Algorithm (EEA). The proposed work is investigated and compared with that of N-FINDR and Vertex Component Analysis (VCA) algorithms. The performance of the proposed algorithm is evaluated using Root Mean Square Error (RMSE), Spectral Angle Distance (SAD) and computation time. Experimental results show that the proposed clustering based band selection with SGA endmember extraction algorithm reduces the average SAD by 8 to 10 % and the average RMSE by nearly 1 %, compared to that of N-FINDR and VCA algorithms. In terms of computation time, the proposed band selection based DR with SGA algorithm is seven times faster than conventional transform based DR with SGA algorithm.

This paper presents a new simplex-based method for unsupervised endmember extraction, called maximum abundance sum-to-one constraint (ASC) fraction (MAF). The ASC fractions refer to the spectral unmixing results with the abundance sum-to-one constraint unmixing only. The algorithm assumes the existence of the pure pixels in the input data for every endmember in the scene, and exploits the fact that pixels with maximum ASC fractions are corresponding to the vertices of a simplex. In order to demonstrate the performance of the proposed MAF, the N-findr algorithm (N-FINDR) and vertex component analysis (VCA) based merely on PCA dimensional reduction are used for comparison. Experiments using both simulated and real hyperspectral data show that MAF is effective in searching optimal results, with a low computational complexity.

The simplex growing algorithm (SGA) developed by Chang et al. (A growing method for simplex-based endmember extraction algorithms. IEEE Transactions on Geoscience and Remote Sensing 44(10): 2804–2819, 2006b) has been used for finding endmembers and was studied in Chap. 10 of Chang (Real time progressive hyperspectral image processing: endmember finding and anomaly detection, Springer, New York, 2016). It can be considered a sequential version of the well-known N-finder endmember finding algorithm (N-FINDR) developed by Winter (Proceedings of 13th international conference on applied geologic remote sensing, Vancouver, BC, Canada, pp. 337–344, 1999a; Image spectrometry V, Proceedings of SPIE 3753, pp. 266–277, 1999b) to find endmembers one after another by growing simplexes one vertex at a time. However, one of the major hurdles for N-FINDR and SGA is the calculation of a simplex volume (SV), as discussed in Chap. 2, which poses a great challenge in designing any algorithms using a SV to find endmembers. This chapter develops an orthogonal projection (OP)-based approach to SGA, called OPSGA, which essentially resolves this computational issue. The idea is based on a geometric SV (GSV) from structures of simplexes. If we consider a j-vertex simplex S j specified by j previously found endmembers as a base and the next endmember mj+1 to be found as a new vertex to be added to S j to form a new (j + 1)-vertex simplex, Sj+1, then calculating the GSV of this new (j + 1)-vertex simplex, Sj+1, is equivalent to multiplying the GSV of S j , which is considered a base with the OP on the base S j from mj+1, which is considered its height. As a result, finding mj+1 to yield a Sj+1 with the maximal GSV is equivalent to finding mj+1 with the maximal OP on S j . On this interpretation, OPSGA converts the issue of calculating a determinant-based SV (DSV) to finding OP without actually computing matrix determinants. Accordingly, OPSGA can be considered a technique for calculating a GSV by OP (GSV-OP). To further reduce the computational complexity, OPSGA is also extended to a recursive Kalman filtering-like recursive hyperspectral sample processing of OPSGA (RHSP-OPSGA), which has several advantages and benefits in terms of computational savings and hardware implementation over N-FINDR and SGA.

Simplex growing algorithm (SGA) is an endmember finding algorithm which grows simplexes with maximal volumes one vertex at a time to find new endmembers where simplex volumes (SV) are calculated by determinants of endmember matrix. However, three issues arise in SV calculation by matrix determinants. One is excessive computation. Another is the matrix determinant-calculated SV does not generally yield true SV if the matrix is not a square matrix in which case dimensionality reduction is required. A third one is numerical instability. To resolve these issues this paper presents a new geometric approach to SGA, called geometric simplex growing algorithm (GSGA) which takes advantage of simplex geometric structures to calculate simplex volume (SV) by considering a simplex to be formed by its base and height so that SV can be calculated simply by multiplying its height with its base where the height is the magnitude of the newly generated endmember and the base is the volume of the simplex formed by previous endmembers. With such a simple geometric structure GSGA has been shown to be the best among all variants currently derived from SGA in the literature.

Simplex growing algorithm (SGA) is an endmember finding algorithm which grows simplexes one vertex at a time by finding vertexes which yield maximal simplex volumes via determinant calculation. This paper presents a new version of SGA, called orthogonal projection based simplex growing algorithm (OPSGA) which takes advantage of a simplex's geometric structure to calculate simplex volume by multiplying its height with its base where the height is the magnitude of the newly generated endmember and the base is the volume of the simplex formed by previous endmembers. With such a simple structure OPSGA can be very easily implemented with significant saving of computing time. Most importantly, OPSGA bridges gaps between fully constrained SGA and most commonly used orthogonal projection-based unconstrained technique, Automatic target generation process (ATGP) to find endmembers.

The simplex growing algorithm (SGA) was recently developed as an alternative to the N-finder algorithm (N-FINDR) and shown to be a promising endmember extraction technique. This paper further extends the SGA to a versatile real-time (RT) processing algorithm, referred to as RT SGA, which can effectively address the following four major issues arising in the practical implementation for N-FINDR: (1) use of random initial endmembers which causes inconsistent final results; (2) high computational complexity which results from an exhaustive search for finding all endmembers simultaneously; (3) requirement of dimensionality reduction because of large data volumes; and (4) lack of RT capability. In addition to the aforementioned advantages, the proposed RT SGA can also be implemented by various criteria in endmember extraction other than the maximum simplex volume.

The simplex growing algorithm (SGA) has been widely used for finding endmembers. It can be considered as a sequential version of the well-known endmember finding algorithm, N-finder algorithm (N-FINDR), which finds endmembers one at a time by growing simplexes. However, one of the major hurdles for N-FINDR and SGA is the calculation of simplex volume (SV) which poses a great challenge in designing any algorithm using SV as a criterion for finding endmembers. This paper develops an orthogonal projection (OP)-based SGA (OP-SGA) which essentially resolves this computational issue. It converts the issue of calculating SV to calculating the OP on previously found simplexes without computing matrix determinants. Most importantly, a recursive Kalman filter-like OP-SGA, to be called recursive OP-SGA (ROP-SGA), can be also derived to ease computation. By virtue of ROP-SGA, several advantages and benefits in computational savings and hardware implementation can be gained for which N-FINDR and SGA do not have.

Endmember extraction (EE) is a fundamental and crucial task in hyperspectral unmixing. Among other methods vertex component analysis (VCA) has become a very popular and useful tool to unmix hyperspectral data. VCA is a geometrical based method that extracts endmember signatures from large hyperspectral datasets without the use of any a priori knowledge about the constituent spectra. Many Hyperspectral imagery applications require a response in real time or near-real time. Thus, to met this requirement this paper proposes a parallel implementation of VCA developed for graphics processing units. The impact on the complexity and on the accuracy of the proposed parallel implementation of VCA is examined using both simulated and real hyperspectral datasets.

Spectral unmixing is one of the most important techniques for analyzing hyperspectral images. Many of the hyperspectral unmixing algorithms developed in recent years have been developed under an assumption that pure pixels exist, and some algorithms, such as N-finder algorithm (N-FINDR) and vertex component analysis (VCA), can only search data points in this case. However, the pure-pixel assumption may be seriously violated for highly mixed data. Whether a pure pixel exists or not, the endmember extraction can be regarded as an optimization problem. In this paper, we incorporate the linear mixture model (LMM) and particle swarm optimization (PSO) to develop LMM constrained PSO (LMMC-PSO) for endmember extraction from highly mixed data. The main contribution of the proposed method is that we redefine the particle motion rules. Each particle in LMMC-PSO moves in the search space according to the LMM, rather than with a velocity. The LMM is one kind of relationship between endmembers and mixed pixels, which can help guide efforts to build a path from mixed pixels to endmembers in PSO. The proposed algorithm was tested and evaluated with both synthetic and real hyperspectral data sets. The experimental results indicated that the proposed method obtains better results with highly mixed data than the algorithms of VCA, minimum volume constrained nonnegative matrix factorization, minimum volume simplex analysis (MVSA), robust MVSA, the convex analysis-based minimum volume enclosing simplex, and simplex identification via variable splitting and augmented Lagrangian.

Endmember extraction plays an important role in spectral unmixing. Traditional endmember extraction algorithms, such as EEAs, only use spectral information to get the endmember, but ignore the spatial characteristic of the remote sensing image. Because of this, the algorithms are susceptible to the noise and anomaly image, which reduces the accuracy of endmember extraction. Focusing on this problem, a new EEA (OSP-LSC) combining subspace projection and local spatial information is proposed. Based on the theory of convex simplex, the algorithm sequentially extracts the endmembers by combining the subspace projection and simplex volume analysis. During the extracting process, the local spectral similarity constraint is used to improve the robustness to the noise and anomaly pixel, which also avoids to the huge computational cost caused by global spatial information. Furthermore, the simplex volume calculation is free of dimensionality which may cause the possible loss of original information, The experimental results on synthetic and real hyperspectral image show that the experimental results on synthetic and real hyperspectral image show that the proposed algorithm can improve the accuracy of the endmember extraction, and is more robust to the noise and anomaly pixels compared to the spectral based EEAs.

In multispectral and hyperspectral remote sensing, classification of pixels is obtained by means of spectral similarity of known field or library spectra to unknown image spectra. Endmember extraction is the most decisive task in hyperspectral image analysis. Endmember initialization algorithms (EIAs) play a key role and support endmember extraction algorithms (EEAs) in extracting near optimal set of endmembers. Though there are few endmember initialization techniques available, similarity measures are not explored in detail in target generation. Hence, in this paper, it is proposed to explore similarity measures in identifying spectrally distinct signatures to use them as initial endmembers. Individual similarity measures are combined to form hybrid similarity measures to confirm their effectiveness in generating spectrally distinct targets. Initial set of endmembers extracted by proposed algorithm are used for initializing classical EEA, the NFINDR, which is sensitive to endmember initialization, and their performance in final endmembers selection is verified. Experimental results on two hyperspectral data sets show the superior performance of the similarity based endmember initialization algorithm (SMEIA).

The endmember extraction is a challenging problem in spectral unmixing (SU) of a mixed pixel in hyperspectral imagery. There are plenty of attempts to solve the endmember extraction problem. Still, the pure pixel assumption-based algorithms have probably been used most in solving the endmember extraction of SU due to the light computational burden. These pure pixel assumption-based algorithms usually follow one of the criteria: (1) Maximum simplex volume or (2) Extreme projection on a subspace. We propose a novel integrated framework that uses both the criteria mentioned above and the proposed one is referred to as the Convexity-based Pure Index (CPI) algorithm. The CPI generates a fixed number of convex sets based on the number of available bands in the hyperspectral image. The algorithm defines the purity score based on the availability of pixels in the convex sets for the two-band data. The CPI has been compared with contemporary algorithms such as Automatic Target Generation Process (ATGP), Vertex Component Analysis (VCA), Pixel Purity Index (PPI), Successive Volume MAXimization (SVMAX), Alternating Volume MAXimization (AVMAX), TRIple-P: P-norm based Pure Pixel identification (TRIP), Successive Decoupled Volume Max–Min (SDVMM), Negative ABundance-Oriented (NABO), and Entropy-based Convex Set Optimization (ECSO). The metrics, Spectral Angle Distance (SAD) and Spectral Information Divergence (SID) used in the comparison were improved up to 5.9% and 9%, respectively. The CPI outperforms prevailing algorithms on real benchmark data and new AVIRIS-NG data. The robustness of the CPI is also tested for various noisy synthetic data. The efficacy of the proposed algorithm is also tested by using qualitative analysis by visualizing the spectra comparison, and abundance maps for all real data.

Fast implementation of linear and nonlinear simplex growing algorithm for hyperspectral endmember extraction

During the last years, several high-resolution sensors have been developed for hyperspectral remote sensing applications. Some of these sensors are already available on space-borne devices. Space-borne sensors are currently acquiring a continual stream of hyperspectral data, and new efficient unsupervised algorithms are required to analyze the great amount of data produced by these instruments. The identification of image endmembers is a crucial task in hyperspectral data exploitation. Once the individual endmembers have been identified, several methods can be used to map their spatial distribution, associations and abundances. This paper reviews the Pixel Purity Index (PPI), N-FINDR and Automatic Morphological Endmember Extraction (AMEE) algorithms developed to accomplish the task of finding appropriate image endmembers by applying them to real hyperspectral data. In order to compare the performance of these methods a metric based on the Root Mean Square Error (RMSE) between the estimated and reference abundance maps is used.

Spectral unmixing is an important problem for remotely sensed hyperspectral data exploitation. Automatic spectral unmixing can be viewed as a three-stage problem, where the first stage is subspace identification, the next one is endmember extraction, and the final one is abundance estimation. In this sequence, endmember extraction is the most challenging problem. Many researchers have attempted to extract endmembers from hyperspectral images using spectral information only. However, it is well known that the inclusion of spatial information can improve the endmember extraction task. In this article, we introduce a new endmember extraction algorithm that exploits both spectral and spatial information. A main innovation of the proposed algorithm is that spatial information is exploited using entropy, while spectral information is exploited using convex set optimization. In the literature, none of the spatial–spectral algorithms has used entropy as spatial information. The inclusion of this entropy-based spatial information improves the accuracy of the endmember extraction process. The results obtained by the proposed algorithm are compared (using a variety of metrics) with those obtained by other state-of-the-art methods, using both synthetic and real datasets. Our experimental results demonstrate that the proposed algorithm outperforms many available algorithms.