Keng-Shih Lu

Bio: Keng-Shih Lu is an academic researcher from University of Southern California. The author has contributed to research in topics: Discrete cosine transform & Fourier transform. The author has an hindex of 6, co-authored 14 publications receiving 97 citations. Previous affiliations of Keng-Shih Lu include Google & National Taiwan University.

Fast Graph Fourier Transforms Based on Graph Symmetry and Bipartition

Abstract: The graph Fourier transform (GFT) is an important tool for graph signal processing, with applications ranging from graph-based image processing to spectral clustering. However, unlike the discrete Fourier transform, the GFT typically does not have a fast algorithm. In this work, we develop new approaches to accelerate the GFT computation. In particular, we show that Haar units (Givens rotations with angle $\pi /4$ ) can be used to reduce GFT computation cost when the graph is bipartite or satisfies certain symmetry properties based on node pairing. We also propose a graph decomposition method based on graph topological symmetry, which allows us to identify and exploit butterfly structures in stages. This method is particularly useful for graphs that are nearly regular or have some specific structures, e.g., line graphs, cycle graphs, grid graphs, and human skeletal graphs. Though butterfly stages based on graph topological symmetry cannot be used for general graphs, they are useful in applications, including video compression and human action analysis, where symmetric graphs, such as symmetric line graphs and human skeletal graphs, are used. Our proposed fast GFT implementations are shown to reduce computation costs significantly, in terms of both number of operations and empirical runtimes.

Intrinsic Integer-Periodic Functions for Discrete Periodicity Detection

Abstract: In this letter we focus on discrete integer-periodic signals, whose periodicity are different from the periodicity of continuous periodic signals in many aspects. We introduce a class of discrete periodic signals called intrinsic integer-periodic function (IIPF). An IIPF contains only a single period in terms of downsampling, which leads to some interesting properties for analyzing periodic components from a discrete signal. We show that one can use Ramanujan sum to decompose discrete periodic signals into IIPF components. Finally, we also propose an integer periodic spectrum rather than frequency spectrum. Our results show that the proposed integer periodic spectrum outperforms the conventional Ramanujan Fourier transform.

Symmetric line graph transforms for inter predictive video coding

Abstract: In this paper, we study graph-based transforms for inter predictive video coding. We are motivated by the fact that symmetries in the transform basis are very beneficial for computational efficiency. Based on this, we describe the relationship between bisymmetric matrices and the butterfly structure in fast transform algorithms. We introduce a new class of transforms called Symmetric Line Graph Transforms (SLGTs), of which the discrete cosine transform (DCT) is one particular case. As is well known in the case of the DCT, the bisymmetry property allows us to reduce the number of multiplications by half. We show that, beyond the DCT, useful SLGTs can be defined that have efficient implementation. We propose a specific SLGT that is shown to outperform the DCT for classes of inter residual blocks. While the proposed SLGT approaches the performance of a Karhunen Loeve Transform (KLT) at certain distortion levels, it has lower computation cost.

Constrained Null Space Component Analysis for Semiblind Source Separation Problem

Abstract: The blind source separation (BSS) problem extracts unknown sources from observations of their unknown mixtures. A current trend in BSS is the semiblind approach, which incorporates prior information on sources or how the sources are mixed. The constrained independent component analysis (ICA) approach has been studied to impose constraints on the famous ICA framework. We introduced an alternative approach based on the null space component (NCA) framework and referred to the approach as the c-NCA approach. We also presented the c-NCA algorithm that uses signal-dependent semidefinite operators, which is a bilinear mapping, as signatures for operator design in the c-NCA approach. Theoretically, we showed that the source estimation of the c-NCA algorithm converges with a convergence rate dependent on the decay of the sequence, obtained by applying the estimated operators on corresponding sources. The c-NCA can be formulated as a deterministic constrained optimization method, and thus, it can take advantage of solvers developed in optimization society for solving the BSS problem. As examples, we demonstrated electroencephalogram interference rejection problems can be solved by the c-NCA with proximal splitting algorithms by incorporating a sparsity-enforcing separation model and considering the case when reference signals are available.

A graph laplacian matrix learning method for fast implementation of graph fourier transform

Abstract: In this paper, we propose an efficient graph learning approach for fast graph Fourier transform. We consider a maximum likelihood problem with additional constraints based on a matrix factorization of the graph Laplacian matrix, such that its eigenmatrix is a product of a block diagonal matrix and a butterfly-like matrix. We show that a special case of this problem reduces to a learning problem with constraints enforcing certain graph symmetries. Then, we provide an efficient approximation approach for the general problem without enforcing any symmetry constraint. We use this approach to design a fast nonseparable transform for intra predictive residual blocks in video compression. The resulting transform achieves a better rate-distortion performance than the 2D DCT and the hybrid DCT/ADST transform.

Learning graphs from data: A signal representation perspective

Abstract: The construction of a meaningful graph topology plays a crucial role in the effective representation, processing, analysis and visualization of structured data. When a natural choice of the graph is not readily available from the data sets, it is thus desirable to infer or learn a graph topology from the data. In this tutorial overview, we survey solutions to the problem of graph learning, including classical viewpoints from statistics and physics, and more recent approaches that adopt a graph signal processing (GSP) perspective. We further emphasize the conceptual similarities and differences between classical and GSP-based graph inference methods, and highlight the potential advantage of the latter in a number of theoretical and practical scenarios. We conclude with several open issues and challenges that are keys to the design of future signal processing and machine learning algorithms for learning graphs from data.

Learning Graphs From Data: A Signal Representation Perspective

Abstract: The construction of a meaningful graph topology plays a crucial role in the effective representation, processing, analysis, and visualization of structured data. When a natural choice of the graph is not readily available from the data sets, it is thus desirable to infer or learn a graph topology from the data. In this article, we survey solutions to the problem of graph learning, including classical viewpoints from statistics and physics, and more recent approaches that adopt a graph signal processing (GSP) perspective. We further emphasize the conceptual similarities and differences between classical and GSP-based graph-inference methods and highlight the potential advantage of the latter in a number of theoretical and practical scenarios. We conclude with several open issues and challenges that are keys to the design of future signal processing and machine-learning algorithms for learning graphs from data.

Graph Spectral Image Processing

Abstract: 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 2-D 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 paper, 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.

Nested Periodic Matrices and Dictionaries: New Signal Representations for Period Estimation

Abstract: In this paper, we propose a new class of techniques to identify periodicities in data. We target the period estimation directly rather than inferring the period from the signal’s spectrum. By doing so, we obtain several advantages over the traditional spectrum estimation techniques such as DFT and MUSIC. Apart from estimating the unknown period of a signal, we search for finer periodic structure within the given signal. For instance, it might be possible that the given periodic signal was actually a sum of signals with much smaller periods. For example, adding signals with periods 3, 7, and 11 can give rise to a period 231 signal. We propose methods to identify these “hidden periods” 3, 7, and 11. We first propose a new family of square matrices called Nested Periodic Matrices (NPMs), having several useful properties in the context of periodicity. These include the DFT, Walsh–Hadamard, and Ramanujan periodicity transform matrices as examples. Based on these matrices, we develop high dimensional dictionary representations for periodic signals. Various optimization problems can be formulated to identify the periods of signals from such representations. We propose an approach based on finding the least $l_{2}$ norm solution to an under-determined linear system. Alternatively, the period identification problem can also be formulated as a sparse vector recovery problem and we show that by a slight modification to the usual $l_{1}$ norm minimization techniques, we can incorporate a number of new and computationally simple dictionaries.