Low-dimensional Representations of Hyperspectral Data for Use in CRF-based Classification

Probabilistic graphical models have strong potential for use in hyperspectral image classification. One important class of probabilisitic graphical models is the Conditional Random Field (CRF), which has distinct advantages over traditional Markov Random Fields (MRF), including: no independence assumption is made over the observation, and local and pairwise potential features can be defined with flexibility. Conventional methods for hyperspectral image classification utilize all spectral bands and assign the corresponding raw intensity values into the feature functions in CRFs. These methods, however, require significant computational efforts and yield an ambiguous summary from the data. To mitigate these problems, we propose a novel processing method for hyperspectral image classification by incorporating a lower dimensional representation into the CRFs. In this paper, we use representations based on three types of graph-based dimensionality reduction algorithms: Laplacian Eigemaps (LE), Spatial-Spectral Schroedinger Eigenmaps (SSSE), and Local Linear Embedding (LLE), and we investigate the impact of choice of representation on the subsequent CRF-based classifications

Schroedinger Eigenmaps for Manifold Alignment of Multimodal Hyperspectral Images

Multimodal remote sensing is an upcoming field as it allows for many views of the same region of interest. Domain adaption attempts to fuse these multimodal remotely sensed images by utilizing the concept of transfer learning to understand data from different sources to learn a fused outcome. Semisupervised Manifold Alignment (SSMA) maps multiple Hyperspectral images (HSIs) from high dimensional source spaces to a low dimensional latent space where similar elements reside closely together. SSMA preserves the original geometric structure of respective HSIs whilst pulling similar data points together and pushing dissimilar data points apart. The SSMA algorithm is comprised of a geometric component, a similarity component and dissimilarity component. The geometric component of the SSMA method has roots in the original Laplacian Eigenmaps (LE) dimension reduction algorithm and the projection functions have roots in the original Locality Preserving Projections (LPP) dimensionality reduction framework. The similarity and dissimilarity component is a semisupervised component that allows expert labeled information to improve the image fusion process. Spatial-Spectral Schroedinger Eigenmaps (SSSE) was designed as a semisupervised enhancement to the LE algorithm by augmenting the Laplacian matrix with a user-defined potential function. However, the user-defined enhancement has yet to be explored in the LPP framework. The first part of this thesis proposes to use the Spatial-Spectral potential within the LPP algorithm, creating a new algorithm we call the Schroedinger Eigenmap Projections (SEP). Through experiments on publicly available data with expert-labeled ground truth, we perform experiments to compare the performance of the SEP algorithm with respect to the LPP algorithm. The second part of this thesis proposes incorporating the Spatial Spectral potential from SSSE into the SSMA framework. Using two multi-angled HSI’s, we explore the impact of incorporating this potential into SSMA

Data Representation for Learning and Information Fusion in Bioinformatics

This thesis deals with the rigorous application of nonlinear dimension reduction and data organization techniques to biomedical data analysis. The Laplacian Eigenmaps algorithm is representative of these methods and has been widely applied in manifold learning and related areas. While their asymptotic manifold recovery behavior has been well-characterized, the clustering properties of Laplacian embeddings with finite data are largely motivated by heuristic arguments. We develop a precise bound, characterizing cluster structure preservation under Laplacian embeddings. From this foundation, we introduce flexible and mathematically well-founded approaches for information fusion and feature representation. These methods are applied to three substantial case studies in bioinformatics, illustrating their capacity to extract scientifically valuable information from complex data

Semi-Supervised Normalized Embeddings for Fusion and Land-Use Classification of Multiple View Data

Land-use classification from multiple data sources is an important problem in remote sensing. Data fusion algorithms like Semi-Supervised Manifold Alignment (SSMA) and Manifold Alignment with Schroedinger Eigenmaps (SEMA) use spectral and/or spatial features from multispectral, multimodal imagery to project each data source into a common latent space in which classification can be performed. However, in order for these algorithms to be well-posed, they require an expert user to either directly identify pairwise dissimilarities in the data or to identify class labels for a subset of points from which pairwise dissimilarities can be derived. In this paper, we propose a related data fusion technique, which we refer to as Semi-Supervised Normalized Embeddings (SSNE). SSNE is defined by modifying the SSMA/SEMA objective functions to incorporate an extra normalization term that enables a latent space to be well-defined even when no pairwise-dissimilarities are provided. Using publicly available data from the 2017 IEEE GRSS Data Fusion Contest, we show that SSNE enables similar land-use classification performance to SSMA/SEMA in scenarios where pairwise dissimilarities are available, but that unlike SSMA/SEMA, it also enables land-use classification in other scenarios. We compare the effect of applying different classification algorithms including a support vector machine (SVM), a linear discriminant analysis classifier (LDA), and a random forest classifier (RF); we show that SSMA/SEMA and SSNE robust to the use of different classifiers. In addition to comparing the classification performance of SSNE to SSMA/SEMA and comparing classification algorithm, we utilize manifold alignment to classify unknown views

Efficient Nonlinear Dimensionality Reduction for Pixel-wise Classification of Hyperspectral Imagery

Classification, target detection, and compression are all important tasks in analyzing hyperspectral imagery (HSI). Because of the high dimensionality of HSI, it is often useful to identify low-dimensional representations of HSI data that can be used to make analysis tasks tractable. Traditional linear dimensionality reduction (DR) methods are not adequate due to the nonlinear distribution of HSI data. Many nonlinear DR methods, which are successful in the general data processing domain, such as Local Linear Embedding (LLE) [1], Isometric Feature Mapping (ISOMAP) [2] and Kernel Principal Components Analysis (KPCA) [3], run very slowly and require large amounts of memory when applied to HSI. For example, applying KPCA to the 512×217 pixel, 204-band Salinas image using a modern desktop computer (AMD FX-6300 Six-Core Processor, 32 GB memory) requires more than 5 days of computing time and 28GB memory! In this thesis, we propose two different algorithms for significantly improving the computational efficiency of nonlinear DR without adversely affecting the performance of classification task: Simple Linear Iterative Clustering (SLIC) superpixels and semi-supervised deep autoencoder networks (SSDAN). SLIC is a very popular algorithm developed for computing superpixels in RGB images that can easily be extended to HSI. Each superpixel includes hundreds or thousands of pixels based on spatial and spectral similarities and is represented by the mean spectrum and spatial position of all of its component pixels. Since the number of superpixels is much smaller than the number of pixels in the image, they can be used as input for nonlinearDR, which significantly reduces the required computation time and memory versus providing all of the original pixels as input. After nonlinear DR is performed using superpixels as input, an interpolation step can be used to obtain the embedding of each original image pixel in the low dimensional space. To illustrate the power of using superpixels in an HSI classification pipeline,we conduct experiments on three widely used and publicly available hyperspectral images: Indian Pines, Salinas and Pavia. The experimental results for all three images demonstrate that for moderately sized superpixels, the overall accuracy of classification using superpixel-based nonlinear DR matches and sometimes exceeds the overall accuracy of classification using pixel-based nonlinear DR, with a computational speed that is two-three orders of magnitude faster. Even though superpixel-based nonlinear DR shows promise for HSI classification, it does have disadvantages. First, it is costly to perform out-of-sample extensions. Second, it does not generalize to handle other types of data that might not have spatial information. Third, the original input pixels cannot approximately be recovered, as is possible in many DR algorithms.In order to overcome these difficulties, a new autoencoder network - SSDAN is proposed. It is a fully-connected semi-supervised autoencoder network that performs nonlinear DR in a manner that enables class information to be integrated. Features learned from SSDAN will be similar to those computed via traditional nonlinear DR, and features from the same class will be close to each other. Once the network is trained well with training data, test data can be easily mapped to the low dimensional embedding. Any kind of data can be used to train a SSDAN,and the decoder portion of the SSDAN can easily recover the initial input with reasonable loss.Experimental results on pixel-based classification in the Indian Pines, Salinas and Pavia images show that SSDANs can approximate the overall accuracy of nonlinear DR while significantly improving computational efficiency. We also show that transfer learning can be use to finetune features of a trained SSDAN for a new HSI dataset. Finally, experimental results on HSI compression show a trade-off between Overall Accuracy (OA) of extracted features and PeakSignal to Noise Ratio (PSNR) of the reconstructed image

COMPUTATIONAL METHODS IN MACHINE LEARNING: TRANSPORT MODEL, HAAR WAVELET, DNA CLASSIFICATION, AND MRI

With the increasing amount of raw data generation produced every day, it has become pertinent to develop new techniques for data representation, analyses, and interpretation. Motivated by real-world applications, there is a trending interest in techniques such as dimensionality reduction, wavelet decomposition, and classication methods that allow for better understanding of data. This thesis details the development of a new non-linear dimension reduction technique based on transport model by advection. We provide a series of computational experiments, and practical applications in hyperspectral images to illustrate the strength of our algorithm. In wavelet decomposition, we construct a novel Haar approximation technique for functions f in the Lp-space, 0 < p < 1, such that the approximants have support contained in the support of f. Furthermore, a classification algorithm to study tissue-specific deoxyribonucleic acids (DNA) is constructed using the support vector machine. In magnetic resonance imaging, we provide an extension of the T2-store-T2 magnetic resonance relaxometry experiment used in the analysis of magnetization signal from 2 to N exchanging sites, where N >= 2

Extensions of Laplacian Eigenmaps for Manifold Learning

This thesis deals with the theory and practice of manifold learning, especially as they relate to the problem of classification. We begin with a well known algorithm, Laplacian Eigenmaps, and then proceed to extend it in two independent directions. First, we generalize this algorithm to allow for the use of partially labeled data, and establish the theoretical foundation of the resulting semi-supervised learning method. Second, we consider two ways of accelerating the most computationally intensive step of Laplacian Eigenmaps, the construction of an adjacency graph. Both of them produce high quality approximations, and we conclude by showing that they work well together to achieve a dramatic reduction in computational time

An interactive analysis of harmonic and diffusion equations on discrete 3D shapes

AbstractRecent results in geometry processing have shown that shape segmentation, comparison, and analysis can be successfully addressed through the spectral properties of the Laplace–Beltrami operator, which is involved in the harmonic equation, the Laplacian eigenproblem, the heat diffusion equation, and the definition of spectral distances, such as the bi-harmonic, commute time, and diffusion distances. In this paper, we study the discretization and the main properties of the solutions to these equations on 3D surfaces and their applications to shape analysis. Among the main factors that influence their computation, as well as the corresponding distances, we focus our attention on the choice of different Laplacian matrices, initial boundary conditions, and input shapes. These degrees of freedom motivate our choice to address this study through the executable paper, which allows the user to perform a large set of experiments and select his/her own parameters. Finally, we represent these distances in a unified way and provide a simple procedure to generate new distances on 3D shapes