arXiv subjects

Gene Cheung

Publications and source records attributed to Gene Cheung.

At least 55 records · Page 3Linked to original sources

arXiv2022

Pre-demosaic Graph-based Light Field Image Compression

An unfocused plenoptic light field (LF) camera places an array of microlenses in front of an image sensor in order to separately capture different directional rays arriving at an image pixel. Using a conventional Bayer pattern, data captured at each pixel is a single color component (R, G or B).The sensed data then undergoes demosaicking (interpolation of RGB components per pixel) and conversion to an array of sub-aperture images (SAIs). In this paper, we propose a new LF image coding scheme based on graph lifting transform (GLT), where the acquired sensor data are coded in the original captured form without pre-processing. Specifically, we directly map raw sensed color data to the SAIs, resulting in sparsely distributed color pixels on 2D grids, and perform demosaicking at the receiver after decoding. To exploit spatial correlation among the sparse pixels, we propose a novel intra-prediction scheme, where the prediction kernel is determined according to the local gradient estimated from already coded neighboring pixel blocks. We then connect the pixels by forming a graph, modeling the prediction residuals statistically as a Gaussian Markov Random Field (GMRF). The optimal edge weights are computed via a graph learning method using a set of training SAIs. The residual data is encoded via low-complexity GLT. Experiments show that at high PSNRs -- important for archiving and instant storage scenarios -- our method outperformed significantly a conventional light field image coding scheme with demosaicking followed by High Efficiency Video Coding (HEVC).

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arXiv2021

Fast Graph Sampling for Short Video Summarization using Gershgorin Disc Alignment

We study the problem of efficiently summarizing a short video into several keyframes, leveraging recent progress in fast graph sampling. Specifically, we first construct a similarity path graph (SPG) $\mathcal{G}$, represented by graph Laplacian matrix $\mathbf{L}$, where the similarities between adjacent frames are encoded as positive edge weights. We show that maximizing the smallest eigenvalue $λ_{\min}(\mathbf{B})$ of a coefficient matrix $\mathbf{B} = \text{diag}(\mathbf{a}) + μ\mathbf{L}$, where $\mathbf{a}$ is the binary keyframe selection vector, is equivalent to minimizing a worst-case signal reconstruction error. We prove that, after partitioning $\mathcal{G}$ into $Q$ sub-graphs $\{\mathcal{G}^q\}^Q_{q=1}$, the smallest Gershgorin circle theorem (GCT) lower bound of $Q$ corresponding coefficient matrices -- $\min_q λ^-_{\min}(\mathbf{B}^q)$ -- is a lower bound for $λ_{\min}(\mathbf{B})$. This inspires a fast graph sampling algorithm to iteratively partition $\mathcal{G}$ into $Q$ sub-graphs using $Q$ samples (keyframes), while maximizing $λ^-_{\min}(\mathbf{B}^q)$ for each sub-graph $\mathcal{G}^q$. Experimental results show that our algorithm achieves comparable video summarization performance as state-of-the-art methods, at a substantially reduced complexity.

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arXiv2021

Fast sensor placement by enlarging principle submatrix for large-scale linear inverse problems

Sensor placement for linear inverse problems is the selection of locations to assign sensors so that the entire physical signal can be well recovered from partial observations. In this paper, we propose a fast sampling algorithm to place sensors. Specifically, assuming that the field signal $\mathbf{f}$ is represented by a linear model $\mathbf{f}=\pmbϕ\mathbf{g}$, it can be estimated from partial noisy samples via an unbiased least-squares (LS) method, whose expected mean square error (MSE) depends on chosen samples. First, we formulate an approximate MSE problem, and then prove it is equivalent to a problem related to a principle submatrix of $\pmbϕ\pmbϕ^\top$ indexed by sample set. To solve the formulated problem, we devise a fast greedy algorithm with simple matrix-vector multiplications, leveraging a matrix inverse formula. To further reduce complexity, we reuse results in the previous greedy step for warm start, so that candidates can be evaluated via lightweight vector-vector multiplications. Extensive experiments show that our proposed sensor placement method achieved the lowest sensor sampling time and the best performance compared to state-of-the-art schemes.

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arXiv2021

Unfolding Projection-free SDP Relaxation of Binary Graph Classifier via GDPA Linearization

Algorithm unfolding creates an interpretable and parsimonious neural network architecture by implementing each iteration of a model-based algorithm as a neural layer. However, unfolding a proximal splitting algorithm with a positive semi-definite (PSD) cone projection operator per iteration is expensive, due to the required full matrix eigen-decomposition. In this paper, leveraging a recent linear algebraic theorem called Gershgorin disc perfect alignment (GDPA), we unroll a projection-free algorithm for semi-definite programming relaxation (SDR) of a binary graph classifier, where the PSD cone constraint is replaced by a set of "tightest possible" linear constraints per iteration. As a result, each iteration only requires computing a linear program (LP) and one extreme eigenvector. Inside the unrolled network, we optimize parameters via stochastic gradient descent (SGD) that determine graph edge weights in two ways: i) a metric matrix that computes feature distances, and ii) a sparse weight matrix computed via local linear embedding (LLE). Experimental results show that our unrolled network outperformed pure model-based graph classifiers, and achieved comparable performance to pure data-driven networks but using far fewer parameters.

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arXiv2021

Fast & Robust Image Interpolation using Gradient Graph Laplacian Regularizer

In the graph signal processing (GSP) literature, it has been shown that signal-dependent graph Laplacian regularizer (GLR) can efficiently promote piecewise constant (PWC) signal reconstruction for various image restoration tasks. However, for planar image patches, like total variation (TV), GLR may suffer from the well-known "staircase" effect. To remedy this problem, we generalize GLR to gradient graph Laplacian regularizer (GGLR) that provably promotes piecewise planar (PWP) signal reconstruction for the image interpolation problem -- a 2D grid with random missing pixels that requires completion. Specifically, we first construct two higher-order gradient graphs to connect local horizontal and vertical gradients. Each local gradient is estimated using structure tensor, which is robust using known pixels in a small neighborhood, mitigating the problem of larger noise variance when computing gradient of gradients. Moreover, unlike total generalized variation (TGV), GGLR retains the quadratic form of GLR, leading to an unconstrained quadratic programming (QP) problem per iteration that can be solved quickly using conjugate gradient (CG). We derive the means-square-error minimizing weight parameter for GGLR, trading off bias and variance of the signal estimate. Experiments show that GGLR outperformed competing schemes in interpolation quality for severely damaged images at a reduced complexity.

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arXiv2021

Signed Graph Metric Learning via Gershgorin Disc Perfect Alignment

Given a convex and differentiable objective $Q(\M)$ for a real symmetric matrix $\M$ in the positive definite (PD) cone -- used to compute Mahalanobis distances -- we propose a fast general metric learning framework that is entirely projection-free. We first assume that $\M$ resides in a space $\cS$ of generalized graph Laplacian matrices corresponding to balanced signed graphs. $\M \in \cS$ that is also PD is called a graph metric matrix. Unlike low-rank metric matrices common in the literature, $\cS$ includes the important diagonal-only matrices as a special case. The key theorem to circumvent full eigen-decomposition and enable fast metric matrix optimization is Gershgorin disc perfect alignment (GDPA): given $\M \in \cS$ and diagonal matrix $§$, where $S_{ii} = 1/v_i$ and $\v$ is $\M$'s first eigenvector, we prove that Gershgorin disc left-ends of similarity transform $\B = §\M §^{-1}$ are perfectly aligned at the smallest eigenvalue $λ_{\min}$. Using this theorem, we replace the PD cone constraint in the metric learning problem with tightest possible linear constraints per iteration, so that the alternating optimization of the diagonal / off-diagonal terms in $\M$ can be solved efficiently as linear programs via the Frank-Wolfe method. We update $\v$ using Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) with warm start as entries in $\M$ are optimized successively. Experiments show that our graph metric optimization is significantly faster than cone-projection schemes, and produces competitive binary classification performance.

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arXiv2021

Unrolling of Deep Graph Total Variation for Image Denoising

While deep learning (DL) architectures like convolutional neural networks (CNNs) have enabled effective solutions in image denoising, in general their implementations overly rely on training data, lack interpretability, and require tuning of a large parameter set. In this paper, we combine classical graph signal filtering with deep feature learning into a competitive hybrid design -- one that utilizes interpretable analytical low-pass graph filters and employs 80% fewer network parameters than state-of-the-art DL denoising scheme DnCNN. Specifically, to construct a suitable similarity graph for graph spectral filtering, we first adopt a CNN to learn feature representations per pixel, and then compute feature distances to establish edge weights. Given a constructed graph, we next formulate a convex optimization problem for denoising using a graph total variation (GTV) prior. Via a $l_1$ graph Laplacian reformulation, we interpret its solution in an iterative procedure as a graph low-pass filter and derive its frequency response. For fast filter implementation, we realize this response using a Lanczos approximation. Experimental results show that in the case of statistical mistmatch, our algorithm outperformed DnCNN by up to 3dB in PSNR.

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arXiv2021

Point Cloud Sampling via Graph Balancing and Gershgorin Disc Alignment

3D point cloud (PC) -- a collection of discrete geometric samples of a physical object's surface -- is typically large in size, which entails expensive subsequent operations like viewpoint image rendering and object recognition. Leveraging on recent advances in graph sampling, we propose a fast PC sub-sampling algorithm that reduces its size while preserving the overall object shape. Specifically, to articulate a sampling objective, we first assume a super-resolution (SR) method based on feature graph Laplacian regularization (FGLR) that reconstructs the original high-resolution PC, given 3D points chosen by a sampling matrix $\H$. We prove that minimizing a worst-case SR reconstruction error is equivalent to maximizing the smallest eigenvalue $λ_{\min}$ of a matrix $\H^{\top} \H + μ\cL$, where $\cL$ is a symmetric, positive semi-definite matrix computed from the neighborhood graph connecting the 3D points. Instead, for fast computation we maximize a lower bound $λ^-_{\min}(\H^{\top} \H + μ\cL)$ via selection of $\H$ in three steps. Interpreting $\cL$ as a generalized graph Laplacian matrix corresponding to an unbalanced signed graph $\cG$, we first approximate $\cG$ with a balanced graph $\cG_B$ with the corresponding generalized graph Laplacian matrix $\cL_B$. Second, leveraging on a recent theorem called Gershgorin disc perfect alignment (GDPA), we perform a similarity transform $\cL_p = §\cL_B §^{-1}$ so that Gershgorin disc left-ends of $\cL_p$ are all aligned at the same value $λ_{\min}(\cL_B)$. Finally, we perform PC sub-sampling on $\cG_B$ using a graph sampling algorithm to maximize $λ^-_{\min}(\H^{\top} \H + μ\cL_p)$ in roughly linear time. Experimental results show that 3D points chosen by our algorithm outperformed competing schemes both numerically and visually in SR reconstruction quality.

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arXiv2021

Learning Sparse Graph Laplacian with K Eigenvector Prior via Iterative GLASSO and Projection

Learning a suitable graph is an important precursor to many graph signal processing (GSP) pipelines, such as graph spectral signal compression and denoising. Previous graph learning algorithms either i) make some assumptions on connectivity (e.g., graph sparsity), or ii) make simple graph edge assumptions such as positive edges only. In this paper, given an empirical covariance matrix $\bar{C}$ computed from data as input, we consider a structural assumption on the graph Laplacian matrix $L$: the first $K$ eigenvectors of $L$ are pre-selected, e.g., based on domain-specific criteria, such as computation requirement, and the remaining eigenvectors are then learned from data. One example use case is image coding, where the first eigenvector is pre-chosen to be constant, regardless of available observed data. We first prove that the subspace of symmetric positive semi-definite (PSD) matrices $H_{u}^+$ with the first $K$ eigenvectors being $\{u_k\}$ in a defined Hilbert space is a convex cone. We then construct an operator to project a given positive definite (PD) matrix $L$ to $H_{u}^+$, inspired by the Gram-Schmidt procedure. Finally, we design an efficient hybrid graphical lasso/projection algorithm to compute the most suitable graph Laplacian matrix $L^* \in H_{u}^+$ given $\bar{C}$. Experimental results show that given the first $K$ eigenvectors as a prior, our algorithm outperforms competing graph learning schemes using a variety of graph comparison metrics.

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arXiv2020

Sampling Signals on Graphs: From Theory to Applications

The study of sampling signals on graphs, with the goal of building an analog of sampling for standard signals in the time and spatial domains, has attracted considerable attention recently. Beyond adding to the growing theory on graph signal processing (GSP), sampling on graphs has various promising applications. In this article, we review current progress on sampling over graphs focusing on theory and potential applications. Although most methodologies used in graph signal sampling are designed to parallel those used in sampling for standard signals, sampling theory for graph signals significantly differs from the theory of Shannon--Nyquist and shift-invariant sampling. This is due in part to the fact that the definitions of several important properties, such as shift invariance and bandlimitedness, are different in GSP systems. Throughout this review, we discuss similarities and differences between standard and graph signal sampling and highlight open problems and challenges.

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arXiv2020

Joint Demosaicking / Rectification of Fisheye Camera Images using Multi-color Graph Laplacian Regularization

To compose a 360 image from a rig with multiple fisheye cameras, a conventional processing pipeline first performs demosaicking on each fisheye camera's Bayer-patterned grid, then translates demosaicked pixels from the camera grid to a rectified image grid---thus performing two image interpolation steps in sequence. Hence interpolation errors can accumulate, and acquisition noise in the captured pixels can pollute neighbors in two consecutive processing stages. In this paper, we propose a joint processing framework that performs demosaicking and grid-to-grid mapping simultaneously---thus limiting noise pollution to one interpolation. Specifically, we first obtain a reverse mapping function from a regular on-grid location in the rectified image to an irregular off-grid location in the camera's Bayer-patterned image. For each pair of adjacent pixels in the rectified grid, we estimate its gradient using the pair's neighboring pixel gradients in three colors in the Bayer-patterned grid. We construct a similarity graph based on the estimated gradients, and interpolate pixels in the rectified grid directly via graph Laplacian regularization (GLR). Experiments show that our joint method outperforms several competing local methods that execute demosaicking and rectification in sequence, by up to 0.52 dB in PSNR and 0.086 in SSIM on the publicly available dataset, and by up to 5.53dB in PSNR and 0.411 in SSIM on the in-house constructed dataset.

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arXiv2020

Graph Metric Learning via Gershgorin Disc Alignment

We propose a fast general projection-free metric learning framework, where the minimization objective $\min_{\textbf{M} \in \mathcal{S}} Q(\textbf{M})$ is a convex differentiable function of the metric matrix $\textbf{M}$, and $\textbf{M}$ resides in the set $\mathcal{S}$ of generalized graph Laplacian matrices for connected graphs with positive edge weights and node degrees. Unlike low-rank metric matrices common in the literature, $\mathcal{S}$ includes the important positive-diagonal-only matrices as a special case in the limit. The key idea for fast optimization is to rewrite the positive definite cone constraint in $\mathcal{S}$ as signal-adaptive linear constraints via Gershgorin disc alignment, so that the alternating optimization of the diagonal and off-diagonal terms in $\textbf{M}$ can be solved efficiently as linear programs via Frank-Wolfe iterations. We prove that the Gershgorin discs can be aligned perfectly using the first eigenvector $\textbf{v}$ of $\textbf{M}$, which we update iteratively using Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) with warm start as diagonal / off-diagonal terms are optimized. Experiments show that our efficiently computed graph metric matrices outperform metrics learned using competing methods in terms of classification tasks.

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arXiv2020

Fast Graph Sampling Set Selection Using Gershgorin Disc Alignment

Graph sampling set selection, where a subset of nodes are chosen to collect samples to reconstruct a smooth graph signal, is a fundamental problem in graph signal processing (GSP). Previous works employ an unbiased least-squares (LS) signal reconstruction scheme and select samples via expensive extreme eigenvector computation. Instead, we assume a biased graph Laplacian regularization (GLR) based scheme that solves a system of linear equations for reconstruction. We then choose samples to minimize the condition number of the coefficient matrix---specifically, maximize the smallest eigenvalue $λ_{\min}$. Circumventing explicit eigenvalue computation, we maximize instead the lower bound of $λ_{\min}$, designated by the smallest left-end of all Gershgorin discs of the matrix. To achieve this efficiently, we first convert the optimization to a dual problem, where we minimize the number of samples needed to align all Gershgorin disc left-ends at a chosen lower-bound target $T$. Algebraically, the dual problem amounts to optimizing two disc operations: i) shifting of disc centers due to sampling, and ii) scaling of disc radii due to a similarity transformation of the matrix. We further reinterpret the dual as an intuitive disc coverage problem bearing strong resemblance to the famous NP-hard set cover (SC) problem. The reinterpretation enables us to derive a fast approximation scheme from a known SC error-bounded approximation algorithm. We find an appropriate target $T$ efficiently via binary search. Extensive simulation experiments show that our disc-based sampling algorithm runs substantially faster than existing sampling schemes and outperforms other eigen-decomposition-free sampling schemes in reconstruction error.

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arXiv2020

Graph Neural Net using Analytical Graph Filters and Topology Optimization for Image Denoising

While convolutional neural nets (CNNs) have achieved remarkable performance for a wide range of inverse imaging applications, the filter coefficients are computed in a purely data-driven manner and are not explainable. Inspired by an analytically derived CNN by Hadji et al., in this paper we construct a new layered graph neural net (GNN) using GraphBio as our graph filter. Unlike convolutional filters in previous GNNs, our employed GraphBio is analytically defined and requires no training, and we optimize the end-to-end system only via learning of appropriate graph topology at each layer. In signal filtering terms, it means that our linear graph filter at each layer is always intrepretable as low-pass with known biorthogonal conditions, while the graph spectrum itself is optimized via data training. As an example application, we show that our analytical GNN achieves image denoising performance comparable to a state-of-the-art CNN-based scheme when the training and testing data share the same statistics, and when they differ, our analytical GNN outperforms it by more than 1dB in PSNR.

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arXiv2020

3D Point Cloud Enhancement using Graph-Modelled Multiview Depth Measurements

A 3D point cloud is often synthesized from depth measurements collected by sensors at different viewpoints. The acquired measurements are typically both coarse in precision and corrupted by noise. To improve quality, previous works denoise a synthesized 3D point cloud a posteriori after projecting the imperfect depth data onto 3D space. Instead, we enhance depth measurements on the sensed images a priori, exploiting inherent 3D geometric correlation across views, before synthesizing a 3D point cloud from the improved measurements. By enhancing closer to the actual sensing process, we benefit from optimization targeting specifically the depth image formation model, before subsequent processing steps that can further obscure measurement errors. Mathematically, for each pixel row in a pair of rectified viewpoint depth images, we first construct a graph reflecting inter-pixel similarities via metric learning using data in previous enhanced rows. To optimize left and right viewpoint images simultaneously, we write a non-linear mapping function from left pixel row to the right based on 3D geometry relations. We formulate a MAP optimization problem, which, after suitable linear approximations, results in an unconstrained convex and differentiable objective, solvable using fast gradient method (FGM). Experimental results show that our method noticeably outperforms recent denoising algorithms that enhance after 3D point clouds are synthesized.

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arXiv2020

Feature Graph Learning for 3D Point Cloud Denoising

Identifying an appropriate underlying graph kernel that reflects pairwise similarities is critical in many recent graph spectral signal restoration schemes, including image denoising, dequantization, and contrast enhancement. Existing graph learning algorithms compute the most likely entries of a properly defined graph Laplacian matrix $\mathbf{L}$, but require a large number of signal observations $\mathbf{z}$'s for a stable estimate. In this work, we assume instead the availability of a relevant feature vector $\mathbf{f}_i$ per node $i$, from which we compute an optimal feature graph via optimization of a feature metric. Specifically, we alternately optimize the diagonal and off-diagonal entries of a Mahalanobis distance matrix $\mathbf{M}$ by minimizing the graph Laplacian regularizer (GLR) $\mathbf{z}^{\top} \mathbf{L} \mathbf{z}$, where edge weight is $w_{i,j} = \exp\{-(\mathbf{f}_i - \mathbf{f}_j)^{\top} \mathbf{M} (\mathbf{f}_i - \mathbf{f}_j) \}$, given a single observation $\mathbf{z}$. We optimize diagonal entries via proximal gradient (PG), where we constrain $\mathbf{M}$ to be positive definite (PD) via linear inequalities derived from the Gershgorin circle theorem. To optimize off-diagonal entries, we design a block descent algorithm that iteratively optimizes one row and column of $\mathbf{M}$. To keep $\mathbf{M}$ PD, we constrain the Schur complement of sub-matrix $\mathbf{M}_{2,2}$ of $\mathbf{M}$ to be PD when optimizing via PG. Our algorithm mitigates full eigen-decomposition of $\mathbf{M}$, thus ensuring fast computation speed even when feature vector $\mathbf{f}_i$ has high dimension. To validate its usefulness, we apply our feature graph learning algorithm to the problem of 3D point cloud denoising, resulting in state-of-the-art performance compared to competing schemes in extensive experiments.

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arXiv2019

Robust Deep Graph Based Learning for Binary Classification

Convolutional neural network (CNN)-based feature learning has become state of the art, since given sufficient training data, CNN can significantly outperform traditional methods for various classification tasks. However, feature learning becomes more difficult if some training labels are noisy. With traditional regularization techniques, CNN often overfits to the noisy training labels, resulting in sub-par classification performance. In this paper, we propose a robust binary classifier, based on CNNs, to learn deep metric functions, which are then used to construct an optimal underlying graph structure used to clean noisy labels via graph Laplacian regularization (GLR). GLR is posed as a convex maximum a posteriori (MAP) problem solved via convex quadratic programming (QP). To penalize samples around the decision boundary, we propose two regularized loss functions for semi-supervised learning. The binary classification experiments on three datasets, varying in number and type of features, demonstrate that given a noisy training dataset, our proposed networks outperform several state-of-the-art classifiers, including label-noise robust support vector machine, CNNs with three different robust loss functions, model-based GLR, and dynamic graph CNN classifiers.

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arXiv2019

Graph Sampling for Matrix Completion Using Recurrent Gershgorin Disc Shift

Matrix completion algorithms fill missing entries in a large matrix given a subset of observed samples. However, how to best pre-select informative matrix entries given a sampling budget is largely unaddressed. In this paper, we propose a fast sample selection strategy for matrix completion from a graph signal processing perspective. Specifically, we first regularize the matrix reconstruction objective using a dual graph signal smoothness prior, resulting in a system of linear equations for solution. We then select appropriate samples to maximize the smallest eigenvalue $λ_{\min}$ of the coefficient matrix, thus maximizing the stability of the linear system. To efficiently solve this combinatorial problem, we derive a greedy sampling strategy, leveraging on Gershgorin circle theorem, that iteratively selects one sample (equivalent to shifting one Gershgorin disc) at a time corresponding to the largest magnitude entry in the first eigenvector of a modified graph Laplacian matrix. Our algorithm benefits computationally from warm start as the first eigenvectors of incremented Laplacian matrices are computed recurrently for more samples. To achieve computation scalability when sampling large matrices, we further rewrite the coefficient matrix as a sum of two separate components, each of which exhibits block-diagonal structure that we exploit for alternating block-wise sampling. Extensive experiments on both synthetic and real-world datasets show that our graph sampling algorithm substantially outperforms existing sampling schemes for matrix completion and reduces the completion error, when combined with a range of modern matrix completion algorithms.

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