Fourier neural operators (FNOs) are effective and efficient surrogates for approximating solutions of PDEs and generalize across discretizations. However, owing to the reliance on frequency truncation to maintain learning efficiency of FNOs, empirical studies suggest that FNOs exhibit spectral bias toward low-frequency information, which may hinder the learning capability especially for certain PDEs with strong high-frequency oscillations. To address this limitation, we propose SirenFNO, a novel framework that leverages sinusoidal representation networks (SIRENs) to learn implicit neural representations and performs mode-wise kernel parameterization. Our SIREN parameterization learns a full-grid spectrum with a constant and discretization-independent parameter count, thereby eliminating the need for frequency truncation. We further extend SirenFNO with functional tensor decompositions to enhance parameter and learning efficiency. Empirical results show that our SirenFNO consistently outperforms FNO with approximately 4 to 15 times parameter reductions with preserved discretization invariance, and our functional decomposition variants obtain performance improvements with a maximum of 73 times fewer parameters across multiple PDE benchmarks.

Historically, image formation parameters and hyperparameters have been tuned by hand for image restoration due to the lack of a broadly applicable selection framework. This has substantially limited the application of state-of-the-art image processing methods. We show that recasting the problem of image restoration to include the additional task of inpainting pixels permits an automatic hyperparameter selection procedure for image restoration through a pixel-wise hold-out cross-validation. This procedure is algorithm-agnostic and works on single images without any prior information. We show that the estimator in the procedure is affinely unbiased under very general settings and can be used for model selection. We provide a reference implementation to demonstrate how to extend existing methods to support inpainting and present empirical results. We hope that this eliminates manual tuning in image restoration pipelines.

In machine learning, it is common to interpret each data sample as a multivariate vector disregarding the correlations among covariates. However, the data may actually be functional, i.e., each data point is a function of some variable, such as time, and the function is discretely sampled. The naive treatment of functional data as traditional multivariate data can lead to poor performance due to the correlations. In this article, we focus on subspace clustering for functional data or curves and propose a new method robust to shift and rotation. The idea is to define a function or curve and all its versions generated by shift and rotation as an equivalent class and then to find the subspace structure among all equivalent classes as the surrogate for all curves. Experimental evaluation on synthetic and real data reveals that this method massively outperforms prior clustering methods in both speed and accuracy when clustering functional data.

In this paper we present Collaborative Low-Rank Subspace Clustering. Given multiple observations of a phenomenon we learn a unified representation matrix. This unified matrix incorporates the features from all the observations, thus increasing the discriminative power compared with learning the representation matrix on each observation separately. Experimental evaluation shows that our method outperforms subspace clustering on separate observations and the state of the art collaborative learning algorithm.

In machine learning it is common to interpret each data point as a vector in Euclidean space. However the data may actually be functional i.e.\ each data point is a function of some variable such as time and the function is discretely sampled. The naive treatment of functional data as traditional multivariate data can lead to poor performance since the algorithms are ignoring the correlation in the curvature of each function. In this paper we propose a method to analyse subspace structure of the functional data by using the state of the art Low-Rank Representation (LRR). Experimental evaluation on synthetic and real data reveals that this method massively outperforms conventional LRR in tasks concerning functional data.

We propose Ordered Subspace Clustering (OSC) to segment data drawn from a sequentially ordered union of subspaces. Similar to Sparse Subspace Clustering (SSC) we formulate the problem as one of finding a sparse representation but include an additional penalty term to take care of sequential data. We test our method on data drawn from infrared hyper spectral, video and motion capture data. Experiments show that our method, OSC, outperforms the state of the art methods: Spatial Subspace Clustering (SpatSC), Low-Rank Representation (LRR) and SSC.

Alpha matting is an ill-posed problem, as such the user must supply dense partial labels for an acceptable solution to be reached. Unfortunately this labelling can be time consuming. In this paper we introduce the w-penalty function, which when incorporated into existing matting techniques allows users to supply extremely sparse input. The formulated objective function encourages driving matte values to 0 and 1. The experiments demonstrate the proposed model outperforms the state-of-the-art KNN matting algorithm. MATLAB code for our proposed method is freely available in the MatteKit package.

A novel framework for enhancing the resolution of a low-resolution multispectral or hyperspectral image using a high resolution panchromatic image or multispectral image is proposed in this paper. This framework can be further used to perform more general types of image fusion. To create the enhanced image, a convex objective function is minimised, which preserves both the pixel affinity learnt from the high resolution image and spectral information from the low resolution image. A fast approximation method is discussed. Quantitive and qualitative analysis against existing methods shows that our method is comparable to state of the art with faster running time and greater flexibility. MATLAB code for our proposed method and the compared methods are freely available in the FuseBox package.

Image matting is the process of extracting the foreground component from an image. Since matting is an under constrained problem most techniques address the case where users supply some dense labelling to indicate known foreground and background regions. In contrast to other techniques our proposed technique is unique in that focuses on achieving satisfactory results with extremely sparse input, e.g. a handful of individual pixel labels. We propose an iterative extension to the class of affinity matting techniques. Analysis of results from affinity matting with sparse labels reveals that the low quality alpha mattes can be processed and re-used for the next iteration. We demonstrate this extension using the recent KNN matting and show that this technique can greatly improve matting results.

Image mating is the process of isolating the foreground in images and video. This task is challenging as it is severely under constrained. At each pixel we must estimate the foreground and background colour and the blending between them (alpha value). Most approaches calculate an affinity matrix and then minimise a system of linear equations to find the alpha matte. In this work we propose an extension to this class of affinity based matting techniques by introducing a Total Variation constraint over the alpha matte. We show that our Total Variation Regularisation method improves results in the presence of hard boundaries, gaps and holes.