Supervised nonlinear unmixing with vector-valued kernel functions 

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Motivations and organisation

Motivations

Hyperspectral imaging, also known as imaging spectroscopy, has emerged as one of the most important technologies in the realm of remote sensing. The premise behind hyperspectral imag-ing is that it provides images with thorough spectral information for each pixel, enabling the identification of various materials and the extraction of various physical parameters in remotely sensed scenes. It is therefore not surprising that hyperspectral imaging has vastly contributed to increasing our understanding of the world.
The earliest successes of hyperspectral imaging began due to its potential and unprecedented applications in remote sensing. Naturally, the development of hyperspectral sensors with hun-dreds of contiguous spectral channels was a significant step beyond multispectral sensors with only tens of broad spectral channels. Originally, hyperspectral images were valued due to their ability to identify a wide range of surface cover materials that could not be identified using images with lower spectral resolution such as multispectral images. For example, hyperspectral images have been successfully used to identify mineral components and map vegetation species [Goetz et al., 1985, Pieters and Mustard, 1988]. Furthermore, they have been used to study plant canopy [Peterson et al., 1988], measure water vapor in the atmosphere [Gao and Goetz, 1990], es-timate chlorophyl content in water [Hamilton et al., 1993], analyze vegetation species [Clark and Swayze, 1995], and perform geologic mapping [Kruse, 1998]. The utility of hyperspectral imaging has been also proven for target detection in the military field, and for studying the composition of stars and planets in astronomy. Although hyperspectral imaging was originally developed for remote sensing applications, it has also spread into non-remote sensing applications such as food quality monitoring, forensic inspections, and disease diagnosis to cite a few. Nowadays, the amount of hyperspectral data is continually increasing and becoming more available to the public. As a result, it can be expected that its applications to new fields will be soon explored.
Nevertheless, effective applications and future advancements in the hyperspectral imaging domain heavily rely on the development of appropriate signal processing techniques. Indeed, the various hyperspectral imaging applications require the development and adaptation of new or existing signal processing techniques to fully exploit the specificities of this data. Some of the well-known and established signal processing techniques in the area of hyperspectral imaging are dimensionality reduction, feature extraction, target detection, anomaly detection, classification, and unmixing to cite a few [Bioucas-Dias et al., 2012]. Among the previously cited techniques, unmixing has particularly witnessed growing attention and contributions over the past years. This is mainly due to the fact that unmixing allows to analyze the hyperspectral data with very high accuracy. Research dedicated to the unmixing topic has led to the development of blind unmixing techniques, and to the development of new paradigms for jointly exploiting the spatial and spectral dimensions of the data. Moreover, recent research trends in this topic have proposed nonlinear unmixing schemes in order to increase the interpretation and analysis accuracy.
This thesis focuses on unmixing and contributes to the aforementioned research trends in this topic. In particular, we propose a new technique for blind unmixing, we incorporate spatial information in (linear and nonlinear) unmixing, and we finally propose a new nonlinear mixing model. The thesis organization is detailed in the following section.

Thesis organisation

The thesis is composed of seven chapters as shown in Figure 1.1. The present chapter, chapter 1, gives an overview of the overall work presented in the thesis and its organization. Chapter 2 introduces the hyperspectral unmixing framework. The following four chapters, namely chapters 3 to 6, are the core of the thesis in the sense that they explain the proposed unmixing techniques. The organization of these chapters is detailed in what follows:
In the third Chapter, we introduce an original approach for unsupervised unmixing based on collaborative sparse regularization. This chapter addresses the problem of blind and fully constrained unmixing of hyperspectral images. Unmixing is performed without the use of any dictionary, and assumes that the number of constituent materials in the scene and their spectral signatures are unknown. Two models with increasing complexity are developed to achieve this challenging task, depending on how noise interacts with hyperspectral data. The first one leads to a convex optimization problem, and is solved with the Alternating Direction Method of Mul-tipliers (ADMM). The second one accounts for signal-dependent noise, and is addressed with a Reweighted Least Squares algorithm.
In the fourth chapter, we propose to incorporate spatial regularization within the unmixing formulation using a graph-based framework. More precisely, this chapter introduces two graph based regularizations in the linear and unsupervised unmixing framework. The proposed regular-izations rely upon the construction of a graph representation of the hyperspectral image. In the first case, a quadratic Laplacian regularization is used to promote smoothness in the estimated abundance maps and collaborative estimation between homogeneous areas of the image. In the second case, a graph-based Total Variation regularization is used to promote piece-wise constant reconstructed spectra with respect to the graph structure. The resulting optimization problems are convex and solved using the Alternating Direction Method of Multipliers (ADMM). The per-formance of the proposed algorithms is demonstrated by comparing them to other well-known algorithms using both simulated and real hyperspectral data.
In the fifth chapter, we propose a new nonlinear mixing model that allows to incorporate spectral prior regarding the nonlinearities at different bands. This chapter presents a kernel based nonlinear mixing model for hyperspectral data where the nonlinear function belongs to a Hilbert space of vector valued functions and the endmembers are assumed to be known. The proposed model extends existing ones by accounting for band dependent and neighboring non-linear contributions. The key idea is to work under the assumption that nonlinear contributions are dominant in some parts of the spectrum while they are less pronounced in other parts. In addition to this, we motivate the need for taking into account nonlinear contributions origi-nating from the ground covers of neighboring pixels by practical considerations, precisely the adjacency effect. The proposed nonlinear function is associated to a matrix valued kernel that allows to jointly model a wide range of nonlinearities and include prior information regarding band dependencies. Furthermore, the choice of the nonlinear function input allows to incorporate neighboring effects. A particular class of kernels is investigated where the kernel’s design relies upon the construction of a graph where each band represents a pixel. The optimization problem is strictly convex and the corresponding iterative algorithm is based on the alternating direction method of multipliers (ADMM). Finally, experiments conducted using synthetic and real data demonstrate the effectiveness of the proposed approach.
In the sixth chapter, we propose to incorporate a spatial regularization within the supervised kernel-based nonlinear unmixing formulation. Using tools from vector-valued functions in a RKHS, we incorporate a regularization that promotes smooth spatial variations of the nonlinear part. The spatial regularizer and the nonlinear contributions are jointly modelled by a vector-valued function that lies in a reproducing kernel Hilbert space (RKHS). The design of the kernel relies upon the construction of a graph representation of the hyperspectral image where each node represents a pixel. The unmixing problem is strictly convex and reduces to a quadratic programming (QP) problem. Simulations on synthetic data illustrate the effectiveness of the proposed approach.
Finally, the seventh chapter concludes the thesis by summarizing the advantages and main contributions of the methods developed in the previous chapters. It also provides concluding remarks and an outlook on future research directions.
In this chapter, we begin with an introduction on hyperspectral imaging. Afterwards, the concept of mixed and pure spectra often encountered in remote sensing applications is explained. Linear and nonlinear spectral mixing models and unmixing techniques are reviewed. Finally, the importance of exploiting spatial and spectral prior information in the unmixing procedure is discussed.

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Hyperspectral imaging concept

Hyperspectral imaging is defined as the simultaneous acquisition of images in hundreds of narrow and adjacent wavelength bands such that for each pixel in the image a reflectance spectrum can be derived [Goetz et al., 1985]. Hyperspectral images are produced using airborne or spaceborne hyperspectral sensors also known as imaging spectrometers.

Hyperspectral sensors

Hyperspectral sensors measure the solar radiation scattered from the surface with relatively high spectral and spatial resolutions. To provide perspective on the spectral and spatial characteristics of these sensors, consider the following sensor examples. The airborne visible/infrared imaging spectrometer (AVIRIS)1 delivers hyperspectral images in 224 contiguous wavelength bands in the wavelength range 400 − 2500 nm, and a spatial resolution of 20 m when flown onboard an airborne at approximately 20 km above ground level. Other examples of hyperspectral sensors are Hyperion2 and compact high resolution imaging spectrometer (CHRIS)3. In contrast with AVIRIS, these sensors are flown onboard spacebornes. Table 2.1 gives the spectral and spatial characteristics of the three previously cited sensors. It can be noted that CHRIS covers the visible (VIS) and near infrared (NIR) portions of the electromagnetic spectrum, whereas AVIRIS and Hyperion also cover the shortwave infrared (SWIR) portion of the electromagnetic spectrum. The spectral resolution for the three sensors is around 10 nm, and the spatial resolution is between 20 and 36 m. In addition to these three currently operational sensors, several sensors are under construction and will be part of planned missions. The reader is referred to [Staenz et al., 2012] for a summary of current and future spaceborne hyperspectral sensors and missions initiatives planned in different countries around the world.
Historically, a great part of the developments that helped establish and demonstrate the hy-perspectral technology emerged at the national aeronautics and space administration (NASA) jet propulsion laboratory (JPL) [Wendisch and Brenguier, 2013, chap. 2]. This was mainly due to the pioneering work of Alexander Goetz and his colleagues who developed imaging spectrometers as well as important image processing software and atmospheric correction algorithms [MacDonald et al., 2009]. In 1983, the driving force for this technology was the deployment of the Airborne imaging spectrometer (AIS) which provided the first hyperspectral images in 128 spectral bands covering the spectral range from 1200 to 2400 nm with a spectral resolution of 9.6 nm. The development and deployment of AIS mainly served as a technology demonstrator and a starting point for hyperspectral sensors design. Soon thereafter, in 1987, AIS was followed by the well-known hyperspectral sensor AVIRIS [Goetz, 1991, Kruse et al., 1999]. The considerations that influenced the design of AVIRIS with the characteristics as described in table 2.1 are reported in [Porter and Enmark, 1987]. As mentioned previously, AVIRIS is currently providing data for scientific use. Nevertheless, AVIRIS will be soon replaced by an AVIRIS – Next Generation (AVIRIS-NG)4 providing data with higher quality compared to the classic AVIRIS.

Hyperspectral imaging concept 

Hyperspectral reflectance data

Hyperspectral imaging is sometimes referred to as “imaging spectroscopy”, or “imaging spec-trometry” as in [Goetz et al., 1985], and described as “spatial spectroscopy from afar” as in [Wendisch and Brenguier, 2013, chap. 2]. These nomenclatures highlight the fact that hyper-spectral imaging endows optical imaging with spectroscopic measurements. Spectroscopy is the science concerned with the identification of materials based on their scattering properties [Hapke, 2012]. This science exploits the fact that light is scattered differently by materials depending on their molecular composition, scale and shape. Examining the reflectance values at sufficiently sampled portions of the electromagnetic spectrum or equivalently the overall shape of the re-flectance spectrum allows to identify the composition of the object and infer various properties [Shaw and Burke, 2003]. Spectroscopy is concerned with field or laboratory measurements esti-mated using a spectrometer on a single pixel basis, whereas hyperspectral imaging is concerned with spectral measurements from afar and on an image basis. To sum up, hyperspectral imaging provides the possibility to combine the potential of spectroscopic analysis along with a spatial and contextual analysis in a single system.
The fundamental quantity used to describe hyperspectral images is the spectral reflectance corresponding to the viewed surface. The information related to the composition of the surface being viewed is revealed by analyzing this quantity across the covered spectral range [see Chang, 2007, chap. 2]. By definition, the reflectance is the ratio of incident light reflected by a surface at a specific wavelength band after a part of that light has been absorbed or emitted by the surface [Hapke, 2012]. Being a ratio, the reflectance is a number with no unit ranging from 0 to 1, i.e. when there is no reflection and when there is perfect reflection respectively. In hyperspec-tral imaging, the reflectance is simultaneously estimated for each pixel at hundreds of narrow and adjacent wavelength bands. The ordered set of reflectance values across the wavelength range is referred to as the reflectance spectrum or simply spectrum. The estimated reflectance values depend on the corresponding wavelength band and usually have smooth variations for a sufficiently small spectral resolution. As a result, the overall hyperspectral image reduces to a stack of spectrally contiguous images often represented as a cube. Figure 2.1 gives a schematic example of such a cube having two spatial dimensions and one spectral dimension. On the left is the hyperspectral data cube which consists of a stack of images at different wavelength bands, in the middle is the spectrum from one pixel in the image providing the reflectance values across the range of wavelength bands, and on the right is a plot of the pixel’s spectrum showing the variation of the reflectance values as a function of the wavelength bands.

Table of contents :

1 Motivations and organisation 
1.1 Motivations
1.2 Thesis organisation
2 Introduction on hyperspectral imaging 
2.1 Hyperspectral imaging concept
2.1.1 Hyperspectral sensors
2.1.2 Hyperspectral reflectance data
2.2 Spectral mixing models
2.2.1 Linear mixing model
2.2.2 Nonlinear mixing models
2.3 Spectral unmixing
2.3.1 Linear unmixing
2.3.2 Nonlinear unmixing
2.4 Incorporating spatial information into unmixing
3 Blind & fully constrained linear unmixing 
3.1 Introduction
3.2 Group lasso, unit sum, and positivity (GLUP)
3.2.1 Optimization problem
3.2.2 ADMM algorithm
3.3 Reduced noise GLUP (NGLUP)
3.3.1 Optimization problem
3.3.2 ADMM algorithm
3.4 Experimental results
3.4.1 Synthetic Data
3.4.2 Real data: Cuprite
3.5 Conclusion
4 Graph-based regularized linear unmixing 
4.1 Introduction
4.2 Image to graph mapping
4.2.1 Defining the edge set
4.2.2 Defining the weights
4.2.3 Graph operators
4.3 Graph based regularization
4.3.1 Quadratic Laplacian regularization
4.3.2 Nonlocal TV regularization
4.4 ADMM algorithm
4.4.1 Quadratic Laplacian regularization
4.4.2 Nonlocal TV regularization
4.5 A note on complexity
4.6 Experiments
4.6.1 Experiments with the Quadratic Laplacian regularization
4.6.2 Experiments with nonlocal TV regularization
4.7 Conclusion
5 Supervised nonlinear unmixing with vector-valued kernel functions 
5.1 Introduction
5.2 Nonlinear mixing model
5.2.1 Model Description
5.2.2 RKHS of vector-valued functions
5.2.3 Kernel design
5.3 Estimation algorithm
5.3.1 Optimization problem
5.3.2 Iterative algorithm
5.3.3 Implementation details
5.4 Experiments
5.4.1 Synthetic data
5.4.2 Real data: Gulf of Lion
5.5 Conclusion
6 Spatial regularization for nonlinear unmixing 
6.1 Introduction
6.2 Vector-valued formulation
6.3 Kernel design and regularization
6.4 Estimation algorithm
6.5 Experiments on synthetic data
6.5.1 Illustrative example
6.5.2 Spatial data set
6.6 Conclusion
7 Concluding remarks 
7.1 Summary of objectives and contributions
7.2 Future research directions .

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