Continuous representation methods, theories, and applications: an overview and perspective

Recently, continuous representation methods have emerged as novel paradigms that characterize the intrinsic structures of real-world data throughfunction representations that map positional coordinates to their corresponding values in the continuous space. As compared with the traditional discreteframework, the continuous framework demonstrates inherent superiority for data representation and reconstruction (e.g., image restoration, novel view synthesis,and waveform inversion) by offering inherent advantages including resolution flexibility, cross-modal adaptability, inherent smoothness, and parameter efficiency.In this review, we systematically examine recent advancements in continuous representation frameworks, focusing on three aspects: (i) continuous representationmethod designs, such as basis function representation, statistical modeling, tensor function decomposition, and implicit neural representation; (ii) theoreticalfoundations of continuous representations, such as approximation error analysis, convergence property, and implicit regularization; (iii) real-world applicationsof continuous representations derived from computer vision, graphics, bioinformatics, and remote sensing. Furthermore, we outline future directions andperspectives to inspire exploration and deepen insights to facilitate continuous representation methods, theories, and applications.