Graph Spectral Image Processing

Recent advent of graph signal processing (GSP) has spurred intensive studies of signals that live naturally on irregular data kernels described by graphs (e.g., social networks, wireless sensor networks). Though a digital image contains pixels that reside on a regularly sampled 2D grid, if one can design an appropriate underlying graph connecting pixels with weights that reflect the image structure, then one can interpret the image (or image patch) as a signal on a graph, and apply GSP tools for processing and analysis of the signal in graph spectral domain. In this article, we overview recent graph spectral techniques in GSP specifically for image / video processing. The topics covered include image compression, image restoration, image filtering and image segmentation

Image processing and understanding based on graph similarity testing: algorithm design and software development

Image processing and understanding is a key task in the human visual system. Among all related topics, content based image retrieval and classification is the most typical and important problem. Successful image retrieval/classification models require an effective fundamental step of image representation and feature extraction. While traditional methods are not capable of capturing all structural information on the image, using graph to represent the image is not only biologically plausible but also has certain advantages. Graphs have been widely used in image related applications. Traditional graph-based image analysis models include pixel-based graph-cut techniques for image segmentation, low-level and high-level image feature extraction based on graph statistics and other related approaches which utilize the idea of graph similarity testing. To compare the images through their graph representations, a graph similarity testing algorithm is essential. Most of the existing graph similarity measurement tools are not designed for generic tasks such as image classification and retrieval, and some other models are either not scalable or not always effective. Graph spectral theory is a powerful analytical tool for capturing and representing structural information of the graph, but to use it on image understanding remains a challenge. In this dissertation, we focus on developing fast and effective image analysis models based on the spectral graph theory and other graph related mathematical tools. We first propose a fast graph similarity testing method based on the idea of the heat content and the mathematical theory of diffusion over manifolds. We then demonstrate the ability of our similarity testing model by comparing random graphs and power law graphs. Based on our graph analysis model, we develop a graph-based image representation and understanding framework. We propose the image heat content feature at first and then discuss several approaches to further improve the model. The first component in our improved framework is a novel graph generation model. The proposed model greatly reduces the size of the traditional pixel-based image graph representation and is shown to still be effective in representing an image. Meanwhile, we propose and discuss several low-level and high-level image features based on spectral graph information, including oscillatory image heat content, weighted eigenvalues and weighted heat content spectrum. Experiments show that the proposed models are invariant to non-structural changes on images and perform well in standard image classification benchmarks. Furthermore, our image features are robust to small distortions and changes of viewpoint. The model is also capable of capturing important image structural information on the image and performs well alone or in combination with other traditional techniques. We then introduce two real world software development projects using graph-based image processing techniques in this dissertation. Finally, we discuss the pros, cons and the intuition of our proposed model by demonstrating the properties of the proposed image feature and the correlation between different image features

Spectral Image Segmentation with Global Appearance Modeling

We introduce a new spectral method for image segmentation that incorporates long range relationships for global appearance modeling. The approach combines two different graphs, one is a sparse graph that captures spatial relationships between nearby pixels and another is a dense graph that captures pairwise similarity between all pairs of pixels. We extend the spectral method for Normalized Cuts to this setting by combining the transition matrices of Markov chains associated with each graph. We also derive an efficient method that uses importance sampling for sparsifying the dense graph of appearance relationships. This leads to a practical algorithm for segmenting high-resolution images. The resulting method can segment challenging images without any filtering or pre-processing

Spectral Segmentation with Multiscale Graph Decomposition

We present a multiscale spectral image segmentation algorithm. In contrast to most multiscale image processing, this algorithm works on multiple scales of the image in parallel, without iteration, to capture both coarse and fine level details. The algorithm is computationally efficient, allowing to segment large images. We use the Normalized Cut graph partitioning framework of image segmentation. We construct a graph encoding pairwise pixel affinity, and partition the graph for image segmentation.We demonstrate that large image graphs can be compressed into multiple scales capturing image structure at increasingly large neighborhood. We show that the decomposition of the image segmentation graph into different scales can be determined by ecological statistics on the image grouping cues. Our segmentation algorithm works simultaneously across the graph scales, with an inter-scale constraint to ensure communication and consistency between the segmentations at each scale. As the results show, we incorporate long-range connections with linear-time complexity, providing high-quality segmentations efficiently. Images that previously could not be processed because of their size have been accurately segmented thanks to this method

Stochastic spectral-spatial permutation ordering combination for nonlocal morphological processing

International audienceThe extension of mathematical morphology to mul-tivariate data has been an active research topic in recent years. In this paper we propose an approach that relies on the consensus combination of several stochastic permutation orderings. The latter are obtained by searching for a smooth shortest path on a graph representing an image. The construction of the graph can be based on both spatial and spectral information and naturally enables patch-based nonlocal processing

HIGH-PERFORMANCE SPECTRAL METHODS FOR COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS

Recent research shows that by leveraging the key spectral properties of eigenvalues and eigenvectors of graph Laplacians, more efficient algorithms can be developed for tackling many graph-related computing tasks. In this dissertation, spectral methods are utilized for achieving faster algorithms in the applications of very-large-scale integration (VLSI) computer-aided design (CAD) First, a scalable algorithmic framework is proposed for effective-resistance preserving spectral reduction of large undirected graphs. The proposed method allows computing much smaller graphs while preserving the key spectral (structural) properties of the original graph. Our framework is built upon the following three key components: a spectrum-preserving node aggregation and reduction scheme, a spectral graph sparsification framework with iterative edge weight scaling, as well as effective-resistance preserving post-scaling and iterative solution refinement schemes. We show that the resultant spectrally-reduced graphs can robustly preserve the first few nontrivial eigenvalues and eigenvectors of the original graph Laplacian and thus allow for developing highly-scalable spectral graph partitioning and circuit simulation algorithms. Based on the framework of the spectral graph reduction, a Sparsified graph-theoretic Algebraic Multigrid (SAMG) is proposed for solving large Symmetric Diagonally Dominant (SDD) matrices. The proposed SAMG framework allows efficient construction of nearly-linear sized graph Laplacians for coarse-level problems while maintaining good spectral approximation during the AMG setup phase by leveraging a scalable spectral graph sparsification engine. Our experimental results show that the proposed method can offer more scalable performance than existing graph-theoretic AMG solvers for solving large SDD matrices in integrated circuit (IC) simulations, 3D-IC thermal analysis, image processing, finite element analysis as well as data mining and machine learning applications. Finally, the spectral methods are applied to power grid and thermal integrity verification applications. This dissertation introduces a vectorless power grid and thermal integrity verification framework that allows computing worst-case voltage drop or thermal profiles across the entire chip under a set of local and global workload (power density) constraints. To address the computational challenges introduced by the large 3D mesh-structured thermal grids, we apply the spectral graph reduction approach for highly-scalable vectorless thermal (or power grids) verification of large chip designs. The effectiveness and efficiency of our approach have been demonstrated through extensive experiments