Continuous Representation Methods, Theories, and Applications: An Overview and Perspectives

TL;DR

This paper reviews recent advances in continuous representation methods, highlighting their theoretical foundations and diverse applications across fields like vision, graphics, bioinformatics, and remote sensing.

Contribution

It provides a systematic overview of design approaches, theoretical insights, and practical applications of continuous representations, and outlines future research directions.

Findings

Continuous methods outperform discrete frameworks in resolution and adaptability.

Various design strategies like basis functions and neural implicit representations are analyzed.

Theoretical analysis includes error bounds, convergence, and regularization insights.

Abstract

Recently, continuous representation methods emerge as novel paradigms that characterize the intrinsic structures of real-world data through function representations that map positional coordinates to their corresponding values in the continuous space. As compared with the traditional discrete framework, 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 representation method designs such as basis function representation, statistical modeling, tensor function decomposition, and implicit neural representation; (ii) Theoretical foundations of continuous representations such as approximation error analysis, convergence property, and implicit regularization; (iii) Real-world applications of continuous representations derived from computer vision, graphics, bioinformatics, and remote sensing. Furthermore, we outline future directions and perspectives to inspire exploration and deepen insights to facilitate continuous representation methods, theories, and applications. All referenced works are summarized in our open-source repository: https://github.com/YisiLuo/Continuous-Representation-Zoo