HOPE: Enhanced Position Image Priors via High-Order Implicit Representations

Deep Image Prior (DIP) has shown that networks with stochastic initialization and custom architectures can effectively address inverse imaging challenges. Despite its potential, DIP requires significant computational resources, whereas the lighter Implicit Neural Positional Image Prior (PIP) often yields overly smooth solutions due to exacerbated spectral bias. Research on lightweight, high-performance solutions for inverse imaging remains limited. This paper proposes a novel framework, Enhanced Positional Image Priors through High-Order Implicit Representations (HOPE), incorporating high-order interactions between layers within a conventional cascade structure. This approach reduces the spectral bias commonly seen in PIP, enhancing the model's ability to capture both low- and high-frequency components for optimal inverse problem performance. We theoretically demonstrate that HOPE's expanded representational space, narrower convergence range, and improved Neural Tangent Kernel (NTK) diagonal properties enable more precise frequency representations than PIP. Comprehensive experiments across tasks such as signal representation (audio, image, volume) and inverse image processing (denoising, super-resolution, CT reconstruction, inpainting) confirm that HOPE establishes new benchmarks for recovery quality and training efficiency.