A Proposed Fully Constrained Least Squares for Solving Sparse Endmember Fractions with Linear Spectral Mixture Model

Linear spectral mixture analysis (LSMA) is a widely used technique in remote sensing to estimate abundance fractions of materials present in an image pixel. In order for an LSMA-based estimator to produce accurate amounts of material abundance, it generally requires two constraints imposed on the linear mixture model used in LSMA, which are the abundance sum-to-one constraint and the abundance nonnegativity constraint. The first constraint requires the sum of the abundance fractions of materials present in an image pixel to be one and the second imposes a constraint that these abundance fractions be nonnegative. While the first constraint is easy to deal with, the second constraint is difficult to implement since it results in a set of inequalities and can only be solved by numerical methods. Consequently, most LSMA-based methods are unconstrained and produce solutions that do not necessarily reflect the true abundance fractions of materials. In this case, they can only be used for the purposes of material detection, discrimination, and classification, but not for material quantification. The authors present a fully constrained least squares (FCLS) linear spectral mixture analysis method for material quantification. Since no closed form can be derived for this method, an efficient algorithm is developed to yield optimal solutions. In order to further apply the designed algorithm to unknown image scenes, an unsupervised least squares error (LSE)-based method is also proposed to extend the FCLS method in an unsupervised manner.

Fully constrained linear spectral mixture analysis (FCLSMA) has been used for material quantification in remotely sensed imagery. In order to implement FCLSMA, two constraints are imposed on abundance fractions, referred to as Abundance Sum-to-one Constraint (ASC) and Abundance Nonnegativity Constraint (ANC). While the ASC is linear equality constraint, the ANC is a linear inequality constraint. A direct approach to imposing the ASC and ANC has been recently investigated and is called fully constrained least-squares (FCLS) method. Since there is no analytical solution resulting from the ANC, a modified fully constrained least-squares method (MFCLS) which replaces the ANC with an Absolute Abundance Sum-to-one Constraint (AASC) was proposed to convert a set of inequality constraints to a quality constraint. The results produced by these two approaches have been shown to be very close. In this paper, we take an oopposite approach to the MFCLS method, called least-squares with linear inequality constraints (LSLIC) method which also solves FCLSMA, but replaces the ASC with two linear inequalities. The proposed LSLIC transforms the FCLSMA to a linear distance programming problem which can be solved easily by a numerical algorithm. In order to demonstrate its utility in solving FCLSMA, the LSLIC method is compared to the FCLS and MFCLS methods. The experimental results show that these three methods perform very similarly with only subtle differences resulting from their problem formations.

In recent years, sedimentation conditions in Dongting Lake have varied greatly because of signifi cant changes in runoff and sediment load in the Changjiang (Yangtze) River following the construction of Three Gorges Dam. The topography of the lake bottom has changed rapidly because of the intense exchange of water and sediment between the lake and the Changjiang River. However, time series information on lake-bottom topographic change is lacking. In this study, we introduced a method that combines remote sensing data and in situ water level data to extract a record of Dongting Lake bottom topography from 2003 to 2011. Multi-temporal lake land/water boundaries were extracted from MODIS images using the linear spectral mixture model method. The elevation of water/land boundary points were calculated using water level data and spatial interpolation techniques. Digital elevation models of Dongting Lake bottom topography in different periods were then constructed with the multiple heighted waterlines. The mean root-mean-square error of the linear spectral mixture model was 0.036, and the mean predicted error for elevation interpolation was −0.19 m. Compared with fi eld measurement data and sediment load data, the method has proven to be most applicable. The results show that the topography of the bottom of Dongting Lake has exhibited uneven erosion and deposition in terms of time and space over the last nine years. Moreover, lake-bottom topography has undergone a slight erosion trend within this period, with 58.2% and 41.8% of the lake-bottom area being eroded and deposited, respectively.

Many spectral unmixing approaches ranging from geometry, algebra to statistics have been proposed, in which nonnegative matrix factorization (NMF)-based ones form an important family. The original NMF-based unmixing algorithm loses the spectral and spatial information between mixed pixels when stacking the spectral responses of the pixels into an observed matrix. Therefore, various constrained NMF methods are developed to impose spectral structure, spatial structure, and spectral-spatial joint structure into NMF to enforce the estimated endmembers and abundances preserve these structures. Compared with matrix format, the third-order tensor is more natural to represent a hyperspectral data cube as a whole, by which the intrinsic structure of hyperspectral imagery can be losslessly retained. Extended from NMF-based methods, a matrix-vector nonnegative tensor factorization (NTF) model is proposed in this paper for spectral unmixing. Different from widely used tensor factorization models, such as canonical polyadic decomposition CPD) and Tucker decomposition, the proposed method is derived from block term decomposition, which is a combination of CPD and Tucker decomposition. This leads to a more flexible frame to model various application-dependent problems. The matrix-vector NTF decomposes a third-order tensor into the sum of several component tensors, with each component tensor being the outer product of a vector (endmember) and a matrix (corresponding abundances). From a formal perspective, this tensor decomposition is consistent with linear spectral mixture model. From an informative perspective, the structures within spatial domain, within spectral domain, and cross spectral-spatial domain are retreated interdependently. Experiments demonstrate that the proposed method has outperformed several state-of-the-art NMF-based unmixing methods.

Hyperspectral image provides a way to analyze component and the fraction by spectral unmixing. The classical method is Least Squares algorithm. But it is defined without any constraint on the abundance. The Fully constrained Least Squares (FCLS) algorithm allows resulted abundances satisfying sum-to-one and non-negative, which is practical. It has been widely used till now because it may work quickly with a small unmxing error. But when the number of endmembers is large, it gets much slow. And unmxing error is high if the endmembers are not found out completely. As for improving the speed of the algorithm, there have been many research works. But fewer algorithms are developed for increasing accurateness. Here in this paper, a primal-dual interior-point method is tested for the fully constraint unmixing with the linear unmixing model being modified slightly. A dual problem is established firstly, and then the process of the algorithm is designed to decrease distance between original problem and the dual problem based on result of FCLS. In order to promise high unmxing accurateness, the unmxing error of resulted abundances is added into the optimizing goal. Several algorithms of fully constraint abundance estimation are tested on both simulation and real hyperspectral images. The experimental results show that the prompted algorithm may find the optimal abundance for existing endmembers even though the endmember set is not complete. Accurateness of unmxing is increased, especially on the real image.

The biophysical composition index (BCI)-based linear spectral mixture model (LSMM) is used in this study to extract the impervious surface (IS), vegetation, and soil coverage of the main urban region (MUR) of Hefei City over the 2001–2021 period. In addition, the Soil Conservation Service-Curve Number (SCS-CN) model is first applied to simulate the surface runoff (SR) in the MUR of Hefei City over the past 21 years, then assessed for simulation accuracy using typical waterlogging points in the study area. On this basis, the spatiotemporal evolution of IS and SR and their relationships in the MUR of Hefei City are investigated and discussed in this study. The obtained results showed that (1) the root-mean-square error (RMSE), mean absolute error (MAE), and systematic error (SE) values of the BCI index-based LSMM are smaller than those of the LSMM, demonstrating a higher extraction accuracy of urban IS extraction of the BCI index-based LSMM. (2) The IS area of the MUR of Hefei City exhibits an increasing trend from 107.555 km2 in 2001 to 387.660 km2 in 2021. In addition, the change rate and change intensity values indicate an increasing–decreasing–increasing trend. The highest change rate and change intensity values are 24.839 km2/year and 23.094%, respectively, and were observed in the 2001–2005 period. (3) The simulated SR (165–195 mm) in the MUR of Hefei City demonstrates an increasing trend in the 2001–2021 period at a rainfall intensity value of 200 mm/d. In addition, the simulated SR amount in the central area exhibits slight changes, while that in the surrounding areas shows substantial variations. (4) The distribution of IS and SR in the MUR of Hefei City reveals strong directional variations, which are all affected by geographical conditions. The IS coverage and SR show high positive correlation coefficients in different years. (5) The present study provides primary data for effective urban planning, water resources management and regulation, and disaster prevention and mitigation in Hefei City, as well as a scientific reference for future studies on urban IS, SR, and their quantitative relationships in other regions.

Hyperspectral unmixing is recognized as an important tool to learn the constituent materials and corresponding distribution in a scene. The physical spectral mixture model is always important to tackle this problem because of its highly ill-posed nature. In this article, we introduce a linear spectral mixture model (LMM)-based end-to-end deep neural network named SNMF-Net for hyperspectral unmixing. SNMF-Net shares an alternating architecture and benefits from both model-based methods and learning-based methods. On the one hand, SNMF-Net is of high physical interpretability as it is built by unrolling <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{p}$ </tex-math></inline-formula> sparsity constrained nonnegative matrix factorization ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{p}$ </tex-math></inline-formula> -NMF) model belonging to LMM families. On the other hand, all the parameters and submodules of SNMF-Net can be seamlessly linked with the alternating optimization algorithm of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{p}$ </tex-math></inline-formula> -NMF and unmixing problem. This enables us to reasonably integrate the prior knowledge on unmixing, the optimization algorithm, and the sparse representation theory into the network for robust learning, so as to improve unmixing. Experimental results on the synthetic and real-world data show the advantages of the proposed SNMF-Net over many state-of-the-art methods.

Efficient denoising of hyperspectral imagery (HSI) relies on an representational model that is capable of capturing the spatial and spectral correlation in HSI. Recently, an intrinsic representation (IR) approach based on the linear spectral mixture model (LSMM) was proposed for unsupervised feature extraction. The IR model constitutes a sound representational model due to its ability to account for the physical data generation process of HSI, the spatial correlation effect, and the noise variance heterogeneity effect. In this paper, we explore the potential of IR for the denoising of HSI. A noisy pixel in HSI is expressed as a nonnegative linear combination of several endmembers, plus some Gaussian noise with heterogeneous noise variances. In order to perform denoising, the IR approach is used to adaptively estimate both the endmembers and the nonnegative coefficients (i.e., the abundances), which are finally used to reconstruct the clean image. The experiments on both simulated and real hyperspectral images demonstrate that the IR approach not only can resist the influence of noise, but also can preserve the image details.

Water bodies are a fundamental element of urban ecosystems, and water mapping is critical for urban and landscape planning and management. Remote sensing has increasingly been used for water mapping in rural areas; however, when applied to urban areas, this spatially- explicit approach is a challenging task due to the fact that the water bodies are often of a small size and spectral confusion is common between water and the complex features in the urban environment. Water indexes are the most common method of water extraction at the pixel level. More recently, spectral mixture analysis (SMA) has been widely employed in analyzing the urban environment at the subpixel level. The objective of this study is to develop an automatic subpixel water mapping method (ASWM) which can achieve a high accuracy in urban areas. Specifically, we first apply a water index for the automatic extraction of mixed land-water pixels, and the pure water pixels that are generated in this process are exported as the final result. Secondly, the SMA technique is applied to the mixed land-water pixels for water abundance estimation. As for obtaining the most representative endmembers, we propose an adaptive iterative endmember selection method based on the spatial similarity of adjacent ground surfaces. One classical water index method (the modified normalized difference water index (MNDWI)), a pixel-level target detection method (constrained energy minimization (CEM)), and two widely used SMA methods (fully constrained least squares (FCLS) and multiple endmember spectral mixture analysis (MESMA)) were chosen for the water mapping comparison in the experiments. The results indicate that the proposed ASWM was able to detect water pixels more efficiency than other unsupervised water extraction methods, and the water fractions estimated by the proposed ASWM method correspond closely to the reference fractions with the slopes of 0.97, 1.02, 1.04, and 0.98 and the R-squared values of 0.9454, 0.9486, 0.9665, and 0.9607 in regression analysis corresponding to different test regions. In the quantitative accuracy assessment, the ASWM method shows the best performance in water mapping with the mean kappa coefficient of 0.862, mean producer’s accuracy of 82.8%, and mean user’s accuracy of 91.8% for test regions.

It is often difficult for pedologists to “see” topsoils indicating differences in properties such as soil particle size. Satellite images are important for obtaining quick information for large areas. However, mapping extensive areas of bare soil using a single image is difficult since most areas are usually covered by vegetation. Thus, the aim of this study was to develop a strategy to determine bare soil areas by fusing multi-temporal satellite images and classifying them according to soil textures. Three different areas located in two states in Brazil, with a total of 65,000 ha, were evaluated. Landsat images of a specific dry month (September) over five consecutive years were collected, processed, and subjected to atmospheric correction (values in surface reflectance). Non-vegetated areas were discriminated from vegetated ones using the Linear Spectral Mixture Model (LSMM) and Normalized Difference Vegetation Index (NDVI). Thus, we were able to fuse images with only bare soil. Field samples were taken from bare soil pixel areas. Pixels of soils with different textures (soil texture classifications) were used for supervised classification in which all areas with exposed soil were classified. Single images reached an average of 36 % bare soil, where the mapper could only “see” these points. After using the proposed methodology, we reached a maximum of 85 % in bare areas; therefore, a pedologist would have proper conditions for generating a continuous map of spatial variations in soil properties. In addition, we mapped soil textural classes with accuracy up to 86.7 % for clayey soils. Overall accuracy was 63.8 %. The method was tested in an unknown area to validate the accuracy of our classification method. Our strategy allowed us to discriminate and categorize different soil textures in the field with 90 % accuracy using images. This method can assist several professionals in soil science, from pedologists to mappers of soil properties, in soil management activities.

Landscape changes in avulsive river systems: Case study of Taquari River on Brazilian Pantanal wetlands

Based on Linear Spectral Mixture Model (LSMM), Zonal Statistic, Transect analyses, and Transformation analyses of urban land cover, we extracted out the Impervious Surface Coverage (ISC) of Changsha metropolitan area in 2005, 2009, and 2013, and analyzed its spatiotemporal patterns in 2005-2013. The results were: (1) we reported that increasing and improving proportion of dark vegetation can reduce the misclassification of low albedo and shadow. (2) The main increase of ISC in Changsha metropolitan area was in 20052009. Impervious surface expanded from the center to all around. The construction of urban greenspace was paid closely attention in 2009-2013. (3) The transformation between impervious surface and vegetation appeared mainly in downtown and counties. The transform from soil to impervious surface mainly arose in 20052009. (4) The Impervious Surface Coverage (ISC) and Land Surface Temperature (LST) had linear relationship in Changsha Metropolitan Area. The primary reason for the rising of urban land surface temperature (LST) was high density of impervious surface, not low density and scattered ones.

Understanding spatial patterns of land use and land cover is essential for studies addressing biodiversity, climate change and environmental modeling as well as for the design and monitoring of land use policies. The aim of this study was to create a detailed map of land use land cover of the deforested areas of the Brazilian Legal Amazon up to 2008. Deforestation data from and uses were mapped with Landsat-5/TM images analysed with techniques, such as linear spectral mixture model, threshold slicing and visual interpretation, aided by temporal information extracted from NDVI MODIS time series. The result is a high spatial resolution of land use and land cover map of the entire Brazilian Legal Amazon for the year 2008 and corresponding calculation of area occupied by different land use classes. The results showed that the four classes of Pasture covered 62% of the deforested areas of the Brazilian Legal Amazon, followed by Secondary Vegetation with 21%. The area occupied by Annual Agriculture covered less than 5% of deforested areas; the remaining areas were distributed among six other land use classes. The maps generated from this project - called TerraClass - are available at INPE's web site (http://www.inpe.br/cra/projetos_pesquisas/terraclass2008.php).

The urban impervious surface, an important part in the city system, has a great influence on theecologicalenvironment in urban areas. The coverage of it is an important indicator for the evaluation ofurbanization. TheRemotesensing data has prominent features such as information-rich and accurate and it can provide data basis for large area extraction of impervious surface. GF-1 satellite is the first satellite of high-resolution earth observation system in China. With the homemade GF-1 satellite remote sensing image date as a resolution, this research, by the combination of V-I-S model and linear spectral mixture model, has first made estimation on the impervious surface of Tianjin City and then employed the remote sensingimage date with high resolution to test the precision of the estimated results. The results not only show that this method will make high precision available, but also reveal that Tianjin City has a wide coverage of impervious surface in general level, especially a high coverage rate both in the center and the coastal areas. The average coverage of impervious surface of the Tianjin city is very high and the coverage of impervious surface in the center and the coastal areas of Tianjin city reach seventy percent.City managers can use these data to guide city management and city planning.

The Earth Observation database is almost on the scale of Zettabyte (1021 Bytes). Produced at a rapid rate, those data also present great diversity, due to the range of sensor types. In such a manner, this kind of data is also classified as Big Data, and present opportunities, such as the possibility to analyze and integrate different data, as well as challenges, mainly regarding storing and processing steps in order to be available to users. The distribution of this database is normally through catalogues, which searching criteria are limited to traditional metadata, as acquisition date, sensor characteristics and geographical localization. Thus, there is a demand for a tool that enables users to search for images based on phenomena in lieu of date or location in a data fusion perspective. In this manner, this work resulted in a Remote Sensing Image Information Mining (ReSIIM) prototype able to make smart searches in big databases based on well-known and basic targets found in Remote Sensing imagery: cloud, cloud shadow, clear land (land area), water, forest, bare soil, built-up and burned area. For that, the aforementioned targets metadata are extract and stored in databases, enabling to refine and boost searches. Besides the spatialization and discussion of the aforementioned targets along Brazil, ReSIIM was assessed according to its accuracy to identify adequate images in databases for Linear Spectral Mixture Model (LSMM) application, which is one of the main methods available to detect burned areas in the Brazilian Amazon. With the adequate search criteria, ReSIIM was able to retrieve all adequate images, indicating its high potential for this and further applications.