Whittaker-Henderson smoother for long satellite image time series interpolation

TL;DR

This paper introduces a neural network-based extension of the Whittaker smoother for satellite image time series, enabling adaptive, large-scale, heteroscedastic smoothing with improved efficiency.

Contribution

It reformulates the Whittaker smoother as a differentiable neural layer with a neural network to infer smoothing parameters, and extends it to handle heteroscedastic noise with a scalable GPU implementation.

Findings

The method efficiently processes large satellite datasets on GPU.

It successfully adapts smoothing locally along the time series.

Differences with the baseline are limited, indicating potential for further improvement.

Abstract

Whittaker smoother is a widely adopted solution to pre-process satellite image time series. Yet, two key limitations remain: the smoothing parameter must be tuned individually for each pixel, and the standard formulation assumes homoscedastic noise, imposing uniform smoothing across the temporal dimension. This paper addresses both limitations by casting the Whittaker smoother as a differentiable neural layer, in which the smoothing parameter is inferred by a neural network. The framework is further extended to handle heteroscedastic noise through a time-varying regularization, allowing the degree of smoothing to adapt locally along the time series. To enable large-scale processing, a sparse, memory-efficient, and fully differentiable implementation is proposed, exploiting the symmetric banded structure of the underlying linear system via Cholesky factorization. Benchmarks on GPU demonstrate that this implementation substantially outperforms standard dense linear solvers, both in speed and memory consumption. The approach is validated on SITS acquired over the French metropolitan territory between 2016 and 2024. Results confirm the feasibility of large-scale heteroscedastic Whittaker smoothing, though reconstruction differences with the homoscedastic baseline remain limited, suggesting that the transformer architecture used for smoothing parameter estimation may lack the temporal acuity needed to capture abrupt noise variations such as singleday cloud contamination.