Beyond Instant-NGP: A Unified Variable-Order B-spline Framework for Sparse-View CT Reconstruction

Achieving high-quality CT reconstruction from sparse measurements is essential for reducing radiation exposure without compromising diagnostic image quality. However, the limited number of projections makes image reconstruction severely underdetermined. In this context, Instant-NGP has emerged as an efficient implicit neural representation for sparse-view CT reconstruction, yet its representational properties underlying its effectiveness remain poorly understood. In this work, we provide a systematic mathematical analysis of the original methodology and specific implementation of Instant-NGP, and show that its multi-resolution grid encoding is equivalent to a multi-resolution linear B-spline representation, in which the CT image is approximated by multivariate polynomials constructed from locally supported linear B-spline basis functions. We further show that the approximation capability of this linear B-spline representation is fundamentally limited for piecewise-smooth signals, potentially leading to reconstruction errors and artifacts. Motivated by this observation, we generalize the linear representation to a unified variable-order B-spline framework, allowing higher-order basis functions to provide more accurate local approximation while retaining the efficiency of hash encoding. We theoretically establish that, under appropriate smoothness conditions, increasing the spline order tightens the bound on the optimal approximation error. Extensive quantitative and qualitative experiments on sparse-view CT reconstruction validate our analysis, showing that higher-order B-splines consistently improve reconstruction quality over the linear formulation, with quartic B-splines achieving the strongest overall performance. The code will be open-sourced upon acceptance of the paper.