Raymond H. Chan

Xue-Cheng Tai, Hao Liu, Raymond Chan · gatech

For problems in image processing and many other fields, a large class of effective neural networks has encoder-decoder-based architectures. Although these networks have made impressive performances, mathematical explanations of their architectures are still underdeveloped. In this paper, we study the encoder-decoder-based network architecture from the algorithmic perspective and provide a mathematical explanation. We use the two-phase Potts model for image segmentation as an example for our explanations. We associate the segmentation problem with a control problem in the continuous setting. Then, multigrid method and operator splitting scheme, the PottsMGNet, are used to discretize the continuous control model. We show that the resulting discrete PottsMGNet is equivalent to an encoder-decoder-based network. With minor modifications, it is shown that a number of the popular encoder-decoder-based neural networks are just instances of the proposed PottsMGNet. By incorporating the Soft-Threshold-Dynamics into the PottsMGNet as a regularizer, the PottsMGNet has shown to be robust with the network parameters such as network width and depth and achieved remarkable performance on datasets with very large noise. In nearly all our experiments, the new network always performs better or as good on accuracy and dice score than existing networks for image segmentation.

Hao Liu, Xue-Cheng Tai, Raymond Chan · gatech

Deep neural network is a powerful tool for many tasks. Understanding why it is so successful and providing a mathematical explanation is an important problem and has been one popular research direction in past years. In the literature of mathematical analysis of deep neural networks, a lot of works is dedicated to establishing representation theories. How to make connections between deep neural networks and mathematical algorithms is still under development. In this paper, we give an algorithmic explanation for deep neural networks, especially in their connections with operator splitting. We show that with certain splitting strategies, operator-splitting methods have the same structure as networks. Utilizing this connection and the Potts model for image segmentation, two networks inspired by operator-splitting methods are proposed. The two networks are essentially two operator-splitting algorithms solving the Potts model. Numerical experiments are presented to demonstrate the effectiveness of the proposed networks.

Chun-Wun Cheng, Christina Runkel, Lihao Liu et al.

Image segmentation is a fundamental task in image analysis and clinical practice. The current state-of-the-art techniques are based on U-shape type encoder-decoder networks with skip connections, called U-Net. Despite the powerful performance reported by existing U-Net type networks, they suffer from several major limitations. Issues include the hard coding of the receptive field size, compromising the performance and computational cost, as well as the fact that they do not account for inherent noise in the data. They have problems associated with discrete layers, and do not offer any theoretical underpinning. In this work we introduce continuous U-Net, a novel family of networks for image segmentation. Firstly, continuous U-Net is a continuous deep neural network that introduces new dynamic blocks modelled by second order ordinary differential equations. Secondly, we provide theoretical guarantees for our network demonstrating faster convergence, higher robustness and less sensitivity to noise. Thirdly, we derive qualitative measures to tailor-made segmentation tasks. We demonstrate, through extensive numerical and visual results, that our model outperforms existing U-Net blocks for several medical image segmentation benchmarking datasets.

Ruoning Li, Kangning Cui, Raymond H. Chan et al.

In this work, a novel algorithm called SVM with Shape-adaptive Reconstruction and Smoothed Total Variation (SaR-SVM-STV) is introduced to classify hyperspectral images, which makes full use of spatial and spectral information. The Shape-adaptive Reconstruction (SaR) is introduced to preprocess each pixel based on the Pearson Correlation between pixels in its shape-adaptive (SA) region. Support Vector Machines (SVMs) are trained to estimate the pixel-wise probability maps of each class. Then the Smoothed Total Variation (STV) model is applied to denoise and generate the final classification map. Experiments show that SaR-SVM-STV outperforms the SVM-STV method with a few training labels, demonstrating the significance of reconstructing hyperspectral images before classification.

Dong Li, Shuisheng Zhou, Tieyong Zeng et al.

K-Means algorithm is a popular clustering method. However, it has two limitations: 1) it gets stuck easily in spurious local minima, and 2) the number of clusters k has to be given a priori. To solve these two issues, a multi-prototypes convex merging based K-Means clustering algorithm (MCKM) is presented. First, based on the structure of the spurious local minima of the K-Means problem, a multi-prototypes sampling (MPS) is designed to select the appropriate number of multi-prototypes for data with arbitrary shapes. A theoretical proof is given to guarantee that the multi-prototypes selected by MPS can achieve a constant factor approximation to the optimal cost of the K-Means problem. Then, a merging technique, called convex merging (CM), merges the multi-prototypes to get a better local minima without k being given a priori. Specifically, CM can obtain the optimal merging and estimate the correct k. By integrating these two techniques with K-Means algorithm, the proposed MCKM is an efficient and explainable clustering algorithm for escaping the undesirable local minima of K-Means problem without given k first. Experimental results performed on synthetic and real-world data sets have verified the effectiveness of the proposed algorithm.

Raymond H. Chan, Ruoning Li

Hyperspectral images often have hundreds of spectral bands of different wavelengths captured by aircraft or satellites that record land coverage. Identifying detailed classes of pixels becomes feasible due to the enhancement in spectral and spatial resolution of hyperspectral images. In this work, we propose a novel framework that utilizes both spatial and spectral information for classifying pixels in hyperspectral images. The method consists of three stages. In the first stage, the pre-processing stage, Nested Sliding Window algorithm is used to reconstruct the original data by {enhancing the consistency of neighboring pixels} and then Principal Component Analysis is used to reduce the dimension of data. In the second stage, Support Vector Machines are trained to estimate the pixel-wise probability map of each class using the spectral information from the images. Finally, a smoothed total variation model is applied to smooth the class probability vectors by {ensuring spatial connectivity} in the images. We demonstrate the superiority of our method against three state-of-the-art algorithms on six benchmark hyperspectral data sets with 10 to 50 training labels for each class. The results show that our method gives the overall best performance in accuracy. Especially, our gain in accuracy increases when the number of labeled pixels decreases and therefore our method is more advantageous to be applied to problems with small training set. Hence it is of great practical significance since expert annotations are often expensive and difficult to collect.

Kangning Cui, Seda Camalan, Ruoning Li et al.

Artisanal and Small-scale Gold Mining (ASGM) is an important source of income for many households, but it can have large social and environmental effects, especially in rainforests of developing countries. The Sentinel-2 satellites collect multispectral images that can be used for the purpose of detecting changes in water extent and quality which indicates the locations of mining sites. This work focuses on the recognition of ASGM activities in Peruvian Amazon rainforests. We tested several semi-supervised classifiers based on Support Vector Machines (SVMs) to detect the changes of water bodies from 2019 to 2021 in the Madre de Dios region, which is one of the global hotspots of ASGM activities. Experiments show that SVM-based models can achieve reasonable performance for both RGB (using Cohen's $κ$ 0.49) and 6-channel images (using Cohen's $κ$ 0.71) with very limited annotations. The efficacy of incorporating Lab color space for change detection is analyzed as well.

Kangning Cui, Ruoning Li, Sam L. Polk et al.

Hyperspectral images, which store a hundred or more spectral bands of reflectance, have become an important data source in natural and social sciences. Hyperspectral images are often generated in large quantities at a relatively coarse spatial resolution. As such, unsupervised machine learning algorithms incorporating known structure in hyperspectral imagery are needed to analyze these images automatically. This work introduces the Spatial-Spectral Image Reconstruction and Clustering with Diffusion Geometry (DSIRC) algorithm for partitioning highly mixed hyperspectral images. DSIRC reduces measurement noise through a shape-adaptive reconstruction procedure. In particular, for each pixel, DSIRC locates spectrally correlated pixels within a data-adaptive spatial neighborhood and reconstructs that pixel's spectral signature using those of its neighbors. DSIRC then locates high-density, high-purity pixels far in diffusion distance (a data-dependent distance metric) from other high-density, high-purity pixels and treats these as cluster exemplars, giving each a unique label. Non-modal pixels are assigned the label of their diffusion distance-nearest neighbor of higher density and purity that is already labeled. Strong numerical results indicate that incorporating spatial information through image reconstruction substantially improves the performance of pixel-wise clustering.

Zhenda Shen, Yanqi Cheng, Raymond H. Chan et al.

Implicit neural representations (INRs) have garnered significant interest recently for their ability to model complex, high-dimensional data without explicit parameterisation. In this work, we introduce TRIDENT, a novel function for implicit neural representations characterised by a trilogy of nonlinearities. Firstly, it is designed to represent high-order features through order compactness. Secondly, TRIDENT efficiently captures frequency information, a feature called frequency compactness. Thirdly, it has the capability to represent signals or images such that most of its energy is concentrated in a limited spatial region, denoting spatial compactness. We demonstrated through extensive experiments on various inverse problems that our proposed function outperforms existing implicit neural representation functions.

Yanqi Cheng, Lipei Zhang, Zhenda Shen et al.

The utilisation of Plug-and-Play (PnP) priors in inverse problems has become increasingly prominent in recent years. This preference is based on the mathematical equivalence between the general proximal operator and the regularised denoiser, facilitating the adaptation of various off-the-shelf denoiser priors to a wide range of inverse problems. However, existing PnP models predominantly rely on pre-trained denoisers using large datasets. In this work, we introduce Single-Shot PnP methods (SS-PnP), shifting the focus to solving inverse problems with minimal data. First, we integrate Single-Shot proximal denoisers into iterative methods, enabling training with single instances. Second, we propose implicit neural priors based on a novel function that preserves relevant frequencies to capture fine details while avoiding the issue of vanishing gradients. We demonstrate, through extensive numerical and visual experiments, that our method leads to better approximations.

Chu Chen, Yang Liu, Se Weon Park et al.

Chemical Exchange Saturation Transfer (CEST) MRI demonstrates its capability in significantly enhancing the detection of proteins and metabolites with low concentrations through exchangeable protons. The clinical application of CEST, however, is constrained by its low contrast and low signal-to-noise ratio (SNR) in the acquired data. Denoising, as one of the post-processing stages for CEST data, can effectively improve the accuracy of CEST quantification. In this work, by modeling spatial variant z-spectrums into low-dimensional subspace, we introduce Implicit Regression in Subspace (IRIS), which is an unsupervised denoising algorithm utilizing the excellent property of implicit neural representation for continuous mapping. Experiments conducted on both synthetic and in-vivo data demonstrate that our proposed method surpasses other CEST denoising methods regarding both qualitative and quantitative performance.

Yaofang Liu, Yumeng Ren, Aitor Artola et al.

The rapid advancement of video diffusion models has been hindered by fundamental limitations in temporal modeling, particularly the rigid synchronization of frame evolution imposed by conventional scalar timestep variables. While task-specific adaptations and autoregressive models have sought to address these challenges, they remain constrained by computational inefficiency, catastrophic forgetting, or narrow applicability. In this work, we present Pusa, a groundbreaking paradigm that leverages vectorized timestep adaptation (VTA) to enable fine-grained temporal control within a unified video diffusion framework. Besides, VTA is a non-destructive adaptation, which means it fully preserves the capabilities of the base model. By finetuning the SOTA Wan2.1-T2V-14B model with VTA, we achieve unprecedented efficiency -- surpassing the performance of Wan-I2V-14B with $\leq$ 1/200 of the training cost (\$500 vs. $\geq$ \$100,000) and $\leq$ 1/2500 of the dataset size (4K vs. $\geq$ 10M samples). Pusa not only sets a new standard for image-to-video (I2V) generation, achieving a VBench-I2V total score of 87.32\% (vs. 86.86\% of Wan-I2V-14B), but also unlocks many zero-shot multi-task capabilities such as start-end frames and video extension -- all without task-specific training. Meanwhile, Pusa can still perform text-to-video generation. Mechanistic analyses reveal that our approach preserves the foundation model's generative priors while surgically injecting temporal dynamics, avoiding the combinatorial explosion inherent to vectorized timesteps. This work establishes a scalable, efficient, and versatile paradigm for next-generation video synthesis, democratizing high-fidelity video generation for research and industry alike. Code is open-sourced at https://github.com/Yaofang-Liu/Pusa-VidGen

Yang Liu, Yaofang Liu, Jinshan Pan et al.

Most existing super-resolution methods and datasets have been developed to improve the image quality in well-lighted conditions. However, these methods do not work well in real-world low-light conditions as the images captured in such conditions lose most important information and contain significant unknown noises. To solve this problem, we propose a SRRIIE dataset with an efficient conditional diffusion probabilistic models-based method. The proposed dataset contains 4800 paired low-high quality images. To ensure that the dataset are able to model the real-world image degradation in low-illumination environments, we capture images using an ILDC camera and an optical zoom lens with exposure levels ranging from -6 EV to 0 EV and ISO levels ranging from 50 to 12800. We comprehensively evaluate with various reconstruction and perceptual metrics and demonstrate the practicabilities of the SRRIIE dataset for deep learning-based methods. We show that most existing methods are less effective in preserving the structures and sharpness of restored images from complicated noises. To overcome this problem, we revise the condition for Raw sensor data and propose a novel time-melding condition for diffusion probabilistic model. Comprehensive quantitative and qualitative experimental results on the real-world benchmark datasets demonstrate the feasibility and effectivenesses of the proposed conditional diffusion probabilistic model on Raw sensor data. Code and dataset will be available at https://github.com/Yaofang-Liu/Super-Resolving

Kangning Cui, Ruoning Li, Sam L. Polk et al.

Hyperspectral images (HSIs) provide exceptional spatial and spectral resolution of a scene, crucial for various remote sensing applications. However, the high dimensionality, presence of noise and outliers, and the need for precise labels of HSIs present significant challenges to HSIs analysis, motivating the development of performant HSI clustering algorithms. This paper introduces a novel unsupervised HSI clustering algorithm, Superpixel-based and Spatially-regularized Diffusion Learning (S2DL), which addresses these challenges by incorporating rich spatial information encoded in HSIs into diffusion geometry-based clustering. S2DL employs the Entropy Rate Superpixel (ERS) segmentation technique to partition an image into superpixels, then constructs a spatially-regularized diffusion graph using the most representative high-density pixels. This approach reduces computational burden while preserving accuracy. Cluster modes, serving as exemplars for underlying cluster structure, are identified as the highest-density pixels farthest in diffusion distance from other highest-density pixels. These modes guide the labeling of the remaining representative pixels from ERS superpixels. Finally, majority voting is applied to the labels assigned within each superpixel to propagate labels to the rest of the image. This spatial-spectral approach simultaneously simplifies graph construction, reduces computational cost, and improves clustering performance. S2DL's performance is illustrated with extensive experiments on three publicly available, real-world HSIs: Indian Pines, Salinas, and Salinas A. Additionally, we apply S2DL to landscape-scale, unsupervised mangrove species mapping in the Mai Po Nature Reserve, Hong Kong, using a Gaofen-5 HSI. The success of S2DL in these diverse numerical experiments indicates its efficacy on a wide range of important unsupervised remote sensing analysis tasks.

Chao Wang, Robert Plemmons, Sudhakar Prasad et al. · mila

An optical imager that exploits off-center image rotation to encode both the lateral and depth coordinates of point sources in a single snapshot can perform 3D localization and tracking of space debris. When actively illuminated, unresolved space debris, which can be regarded as a swarm of point sources, can scatter a fraction of laser irradiance back into the imaging sensor. Determining the source locations and fluxes is a large-scale sparse 3D inverse problem, for which we have developed efficient and effective algorithms based on sparse recovery using non-convex optimization. Numerical simulations illustrate the efficiency and stability of the algorithms.

Xiaohao Cai, Raymond Chan, Mila Nikolova et al. · mila

In this paper, we propose a SLaT (Smoothing, Lifting and Thresholding) method with three stages for multiphase segmentation of color images corrupted by different degradations: noise, information loss, and blur. At the first stage, a convex variant of the Mumford-Shah model is applied to each channel to obtain a smooth image. We show that the model has unique solution under the different degradations. In order to properly handle the color information, the second stage is dimension lifting where we consider a new vector-valued image composed of the restored image and its transform in the secondary color space with additional information. This ensures that even if the first color space has highly correlated channels, we can still have enough information to give good segmentation results. In the last stage, we apply multichannel thresholding to the combined vector-valued image to find the segmentation. The number of phases is only required in the last stage, so users can choose or change it all without the need of solving the previous stages again. Experiments demonstrate that our SLaT method gives excellent results in terms of segmentation quality and CPU time in comparison with other state-of-the-art segmentation methods.

Kangning Cui, Wei Tang, Rongkun Zhu et al.

Understanding the spatial distribution of palms within tropical forests is essential for effective ecological monitoring, conservation strategies, and the sustainable integration of natural forest products into local and global supply chains. However, the analysis of remotely sensed data in these environments faces significant challenges, such as overlapping palm and tree crowns, uneven shading across the canopy surface, and the heterogeneous nature of the forest landscapes, which often affect the performance of palm detection and segmentation algorithms. To overcome these issues, we introduce PalmDSNet, a deep learning framework for real-time detection, segmentation, and counting of canopy palms. Additionally, we employ a bimodal reproduction algorithm that simulates palm spatial propagation to further enhance the understanding of these point patterns using PalmDSNet's results. We used UAV-captured imagery to create orthomosaics from 21 sites across western Ecuadorian tropical forests, covering a gradient from the everwet Chocó forests near Colombia to the drier forests of southwestern Ecuador. Expert annotations were used to create a comprehensive dataset, including 7,356 bounding boxes on image patches and 7,603 palm centers across five orthomosaics, encompassing a total area of 449 hectares. By combining PalmDSNet with the bimodal reproduction algorithm, which optimizes parameters for both local and global spatial variability, we effectively simulate the spatial distribution of palms in diverse and dense tropical environments, validating its utility for advanced applications in tropical forest monitoring and remote sensing analysis.

Xue-Cheng Tai, Hao Liu, Lingfeng Li et al.

The Transformer architecture has revolutionized the field of sequence modeling and underpins the recent breakthroughs in large language models (LLMs). However, a comprehensive mathematical theory that explains its structure and operations remains elusive. In this work, we propose a novel continuous framework that rigorously interprets the Transformer as a discretization of a structured integro-differential equation. Within this formulation, the self-attention mechanism emerges naturally as a non-local integral operator, and layer normalization is characterized as a projection to a time-dependent constraint. This operator-theoretic and variational perspective offers a unified and interpretable foundation for understanding the architecture's core components, including attention, feedforward layers, and normalization. Our approach extends beyond previous theoretical analyses by embedding the entire Transformer operation in continuous domains for both token indices and feature dimensions. This leads to a principled and flexible framework that not only deepens theoretical insight but also offers new directions for architecture design, analysis, and control-based interpretations. This new interpretation provides a step toward bridging the gap between deep learning architectures and continuous mathematical modeling, and contributes a foundational perspective to the ongoing development of interpretable and theoretically grounded neural network models.

Chu Chen, Aitor Artola, Yang Liu et al.

Chemical Exchange Saturation Transfer (CEST) MRI enables molecular-level visualization of low-concentration metabolites by leveraging proton exchange dynamics. However, its clinical translation is hindered by inherent challenges: spatially varying noise arising from hardware limitations, and complex imaging protocols introduce heteroscedasticity in CEST data, perturbing the accuracy of quantitative contrast mapping such as amide proton transfer (APT) imaging. Traditional denoising methods are not designed for this complex noise and often alter the underlying information that is critical for biomedical analysis. To overcome these limitations, we propose a new Blind Adaptive Local Denoising (BALD) method. BALD exploits the self-similar nature of CEST data to derive an adaptive variance-stabilizing transform that equalizes the noise distributions across CEST pixels without prior knowledge of noise characteristics. Then, BALD performs two-stage denoising on a linear transformation of data to disentangle molecular signals from noise. A local SVD decomposition is used as a linear transform to prevent spatial and spectral denoising artifacts. We conducted extensive validation experiments on multiple phantoms and \textit{in vivo} CEST scans. In these experiments, BALD consistently outperformed state-of-the-art CEST denoisers in both denoising metrics and downstream tasks such as molecular concentration maps estimation and cancer detection.

Chu Chen, Ander Biguri, Jean-Michel Morel et al.

X-ray Computed Laminography (CL) is essential for non-destructive inspection of plate-like structures in applications such as microchips and composite battery materials, where traditional computed tomography (CT) struggles due to geometric constraints. However, reconstructing high-quality volumes from laminographic projections remains challenging, particularly under highly sparse-view acquisition conditions. In this paper, we propose a reconstruction algorithm, namely LamiGauss, that combines Gaussian Splatting radiative rasterization with a dedicated detector-to-world transformation model incorporating the laminographic tilt angle. LamiGauss leverages an initialization strategy that explicitly filters out common laminographic artifacts from the preliminary reconstruction, preventing redundant Gaussians from being allocated to false structures and thereby concentrating model capacity on representing the genuine object. Our approach effectively optimizes directly from sparse projections, enabling accurate and efficient reconstruction with limited data. Extensive experiments on both synthetic and real datasets demonstrate the effectiveness and superiority of the proposed method over existing techniques. LamiGauss uses only 3$\%$ of full views to achieve superior performance over the iterative method optimized on a full dataset.

Yumeng Ren, Yaofang Liu, Aitor Artola et al.

Diffusion denoising models have become a popular approach for image generation, but they often suffer from slow convergence during training. In this paper, we identify that this slow convergence is partly due to the complexity of the Brownian motion driving the forward-time process. To address this, we represent the Brownian motion using the Karhunen-Loève expansion, truncating it to a limited number of eigenfunctions. We propose a novel ordinary differential equation with augmented random initials, termed KL diffusion, as a new forward-time process for training and sampling. By developing an appropriate denoising loss function, we facilitate the integration of our KL-diffusion into existing denoising-based models. Using the widely adopted DDIM framework as our baseline ensures a fair comparison, as our modifications focus solely on the forward process and loss function, leaving the network architecture and sampling methods unchanged. Our method significantly outperforms baseline diffusion models, achieving convergence speeds that are twice faster to reach the best FID score of the baseline and ultimately yielding much lower FID scores. Notably, our approach allows for highly parallelized computation, requires no additional learnable parameters, and can be flexibly integrated into existing diffusion methods. The code will be made publicly available.

Chao Wang, Huiwen Zheng, Raymond Chan et al.

Tensor Robust Principal Component Analysis (TRPCA) holds a crucial position in machine learning and computer vision. It aims to recover underlying low-rank structures and characterizing the sparse structures of noise. Current approaches often encounter difficulties in accurately capturing the low-rank properties of tensors and balancing the trade-off between low-rank and sparse components, especially in a mixed-noise scenario. To address these challenges, we introduce a Bayesian framework for TRPCA, which integrates a low-rank tensor nuclear norm prior and a generalized sparsity-inducing prior. By embedding the proposed priors within the Bayesian framework, our method can automatically determine the optimal tensor nuclear norm and achieve a balance between the nuclear norm and sparse components. Furthermore, our method can be efficiently extended to the weighted tensor nuclear norm model. Experiments conducted on synthetic and real-world datasets demonstrate the effectiveness and superiority of our method compared to state-of-the-art approaches.

Mingda Lu, Zitian Ao, Chao Wang et al.

For the 3D localization problem using point spread function (PSF) engineering, we propose a novel enhancement of our previously introduced localization neural network, LocNet. The improved network is a physics-informed neural network (PINN) that we call PiLocNet. Previous works on the localization problem may be categorized separately into model-based optimization and neural network approaches. Our PiLocNet combines the unique strengths of both approaches by incorporating forward-model-based information into the network via a data-fitting loss term that constrains the neural network to yield results that are physically sensible. We additionally incorporate certain regularization terms from the variational method, which further improves the robustness of the network in the presence of image noise, as we show for the Poisson and Gaussian noise models. This framework accords interpretability to the neural network, and the results we obtain show its superiority. Although the paper focuses on the use of single-lobe rotating PSF to encode the full 3D source location, we expect the method to be widely applicable to other PSFs and imaging problems that are constrained by known forward processes.

Yuxiang Hui, Yang Liu, Yaofang Liu et al.

Video restoration task aims to recover high-quality videos from low-quality observations. This contains various important sub-tasks, such as video denoising, deblurring and low-light enhancement, since video often faces different types of degradation, such as blur, low light, and noise. Even worse, these kinds of degradation could happen simultaneously when taking videos in extreme environments. This poses significant challenges if one wants to remove these artifacts at the same time. In this paper, to the best of our knowledge, we are the first to propose an efficient end-to-end video transformer approach for the joint task of video deblurring, low-light enhancement, and denoising. This work builds a novel multi-tier transformer where each tier uses a different level of degraded video as a target to learn the features of video effectively. Moreover, we carefully design a new tier-to-tier feature fusion scheme to learn video features incrementally and accelerate the training process with a suitable adaptive weighting scheme. We also provide a new Multiscene-Lowlight-Blur-Noise (MLBN) dataset, which is generated according to the characteristics of the joint task based on the RealBlur dataset and YouTube videos to simulate realistic scenes as far as possible. We have conducted extensive experiments, compared with many previous state-of-the-art methods, to show the effectiveness of our approach clearly.

Hok Shing Wong, Li Wang, Raymond Chan et al.

We present Deep Tensor Canonical Correlation Analysis (DTCCA), a method to learn complex nonlinear transformations of multiple views (more than two) of data such that the resulting representations are linearly correlated in high order. The high-order correlation of given multiple views is modeled by covariance tensor, which is different from most CCA formulations relying solely on the pairwise correlations. Parameters of transformations of each view are jointly learned by maximizing the high-order canonical correlation. To solve the resulting problem, we reformulate it as the best sum of rank-1 approximation, which can be efficiently solved by existing tensor decomposition method. DTCCA is a nonlinear extension of tensor CCA (TCCA) via deep networks. The transformations of DTCCA are parametric functions, which are very different from implicit mapping in the form of kernel function. Comparing with kernel TCCA, DTCCA not only can deal with arbitrary dimensions of the input data, but also does not need to maintain the training data for computing representations of any given data point. Hence, DTCCA as a unified model can efficiently overcome the scalable issue of TCCA for either high-dimensional multi-view data or a large amount of views, and it also naturally extends TCCA for learning nonlinear representation. Extensive experiments on three multi-view data sets demonstrate the effectiveness of the proposed method.

Zitong Wang, Li Wang, Raymond Chan et al.

We focus on developing a novel scalable graph-based semi-supervised learning (SSL) method for a small number of labeled data and a large amount of unlabeled data. Due to the lack of labeled data and the availability of large-scale unlabeled data, existing SSL methods usually encounter either suboptimal performance because of an improper graph or the high computational complexity of the large-scale optimization problem. In this paper, we propose to address both challenging problems by constructing a proper graph for graph-based SSL methods. Different from existing approaches, we simultaneously learn a small set of vertexes to characterize the high-dense regions of the input data and a graph to depict the relationships among these vertexes. A novel approach is then proposed to construct the graph of the input data from the learned graph of a small number of vertexes with some preferred properties. Without explicitly calculating the constructed graph of inputs, two transductive graph-based SSL approaches are presented with the computational complexity in linear with the number of input data. Extensive experiments on synthetic data and real datasets of varied sizes demonstrate that the proposed method is not only scalable for large-scale data, but also achieve good classification performance, especially for extremely small number of labels.

Jianchao Zhang, Angelica I. Aviles-Rivero, Daniel Heydecker et al.

We consider the problem of segmenting an image into superpixels in the context of $k$-means clustering, in which we wish to decompose an image into local, homogeneous regions corresponding to the underlying objects. Our novel approach builds upon the widely used Simple Linear Iterative Clustering (SLIC), and incorporate a measure of objects' structure based on the spectral residual of an image. Based on this combination, we propose a modified initialisation scheme and search metric, which helps keeps fine-details. This combination leads to better adherence to object boundaries, while preventing unnecessary segmentation of large, uniform areas, while remaining computationally tractable in comparison to other methods. We demonstrate through numerical and visual experiments that our approach outperforms the state-of-the-art techniques.

Xiaohao Cai, Raymond Chan, Xiaoyu Xie et al.

High-dimensional data classification is a fundamental task in machine learning and imaging science. In this paper, we propose a two-stage multiphase semi-supervised classification method for classifying high-dimensional data and unstructured point clouds. To begin with, a fuzzy classification method such as the standard support vector machine is used to generate a warm initialization. We then apply a two-stage approach named SaT (smoothing and thresholding) to improve the classification. In the first stage, an unconstraint convex variational model is implemented to purify and smooth the initialization, followed by the second stage which is to project the smoothed partition obtained at stage one to a binary partition. These two stages can be repeated, with the latest result as a new initialization, to keep improving the classification quality. We show that the convex model of the smoothing stage has a unique solution and can be solved by a specifically designed primal-dual algorithm whose convergence is guaranteed. We test our method and compare it with the state-of-the-art methods on several benchmark data sets. The experimental results demonstrate clearly that our method is superior in both the classification accuracy and computation speed for high-dimensional data and point clouds.

Xiaohao Cai, Raymond Chan, Carola-Bibiane Schonlieb et al.

The piecewise constant Mumford-Shah (PCMS) model and the Rudin-Osher-Fatemi (ROF) model are two important variational models in image segmentation and image restoration, respectively. In this paper, we explore a linkage between these models. We prove that for the two-phase segmentation problem a partial minimizer of the PCMS model can be obtained by thresholding the minimizer of the ROF model. A similar linkage is still valid for multiphase segmentation under specific assumptions. Thus it opens a new segmentation paradigm: image segmentation can be done via image restoration plus thresholding. This new paradigm, which circumvents the innate non-convex property of the PCMS model, therefore improves the segmentation performance in both efficiency (much faster than state-of-the-art methods based on PCMS model, particularly when the phase number is high) and effectiveness (producing segmentation results with better quality) due to the flexibility of the ROF model in tackling degraded images, such as noisy images, blurry images or images with information loss. As a by-product of the new paradigm, we derive a novel segmentation method, called thresholded-ROF (T-ROF) method, to illustrate the virtue of managing image segmentation through image restoration techniques. The convergence of the T-ROF method is proved, and elaborate experimental results and comparisons are presented.

Rui Zhao, Raymond H. Chan

We propose a variational approach to obtain super-resolution images from multiple low-resolution frames extracted from video clips. First the displacement between the low-resolution frames and the reference frame are computed by an optical flow algorithm. Then a low-rank model is used to construct the reference frame in high-resolution by incorporating the information of the low-resolution frames. The model has two terms: a 2-norm data fidelity term and a nuclear-norm regularization term. Alternating direction method of multipliers is used to solve the model. Comparison of our methods with other models on synthetic and real video clips show that our resulting images are more accurate with less artifacts. It also provides much finer and discernable details.