# LaST-SR: Laplace-Inspired Steady-Transient Complex-Frequency Decomposition for Single Image Super-Resolution

arXiv: https://arxiv.org/abs/2609.02063 · HTML: https://mlsanity.com/paper/2609.02063/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 33 references that are in the map.
- First posted on arXiv 2 Sep 2026.

## Builds on (most cited first, 19 of 19)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [SwinIR: Image Restoration Using Swin Transformer](https://mlsanity.com/paper/2108.10257/) (2021)
- [Enhanced Deep Residual Networks for Single Image Super-Resolution](https://mlsanity.com/paper/1707.02921/) (2017)
- [Image Super-Resolution Using Deep Convolutional Networks](https://mlsanity.com/paper/1501.00092/) (2014)
- [Accurate Image Super-Resolution Using Very Deep Convolutional Networks](https://mlsanity.com/paper/1511.04587/) (2015)
- [Residual Dense Network for Image Super-Resolution](https://mlsanity.com/paper/1802.08797/) (2018)
- [Activating More Pixels in Image Super-Resolution Transformer](https://mlsanity.com/paper/2205.04437/) (2022)
- [Sketch-based Manga Retrieval using Manga109 Dataset](https://mlsanity.com/paper/1510.04389/) (2015)
- [Deeply-Recursive Convolutional Network for Image Super-Resolution](https://mlsanity.com/paper/1511.04491/) (2015)
- [Fast, Accurate, and Lightweight Super-Resolution with Cascading Residual Network](https://mlsanity.com/paper/1803.08664/) (2018)
- [Cross Aggregation Transformer for Image Restoration](https://mlsanity.com/paper/2211.13654/) (2022)
- [LAPAR: Linearly-Assembled Pixel-Adaptive Regression Network for Single Image Super-Resolution and Beyond](https://mlsanity.com/paper/2105.10422/) (2021)
- [SwinFIR: Revisiting the SwinIR with Fast Fourier Convolution and Improved Training for Image Super-Resolution](https://mlsanity.com/paper/2208.11247/) (2022)
- [Omni Aggregation Networks for Lightweight Image Super-Resolution](https://mlsanity.com/paper/2304.10244/) (2023)
- [N-Gram in Swin Transformers for Efficient Lightweight Image Super-Resolution](https://mlsanity.com/paper/2211.11436/) (2022)
- [MaIR: A Locality- and Continuity-Preserving Mamba for Image Restoration](https://mlsanity.com/paper/2412.20066/) (2024)
- [DiffFNO: Diffusion Fourier Neural Operator](https://mlsanity.com/paper/2411.09911/) (2024)
- [Exploring Frequency-Inspired Optimization in Transformer for Efficient Single Image Super-Resolution](https://mlsanity.com/paper/2308.05022/) (2023)
- [Interpretable Detail-Fidelity Attention Network for Single Image Super-Resolution](https://mlsanity.com/paper/2009.13134/) (2020)

## Abstract (arXiv)

Single-image super-resolution (SISR) requires global context modeling for structurally consistent reconstruction. Fourier operators are increasingly adopted for global feature modeling. However, their periodic spectral bases constrain the representation of localized aperiodic variations, limiting the recovery of irregular structures and fine details. In dynamical systems, the Laplace neural operator extends Fourier modes to complex frequencies and decomposes the output signal into complementary steady-state and transient responses to jointly model periodic and aperiodic information. We derive, for the first time, an approximate steady-transient decomposition for two-dimensional feature maps, providing an analytical basis for the proposed complex-frequency decomposition. Accordingly, we propose LaST-SR, centered on a Complex-Frequency Decomposition module that couples a global full-spectrum Fourier branch for image-wide dependencies and long-range structural consistency with a window-conditioned local complex-frequency branch for localized, content-dependent aperiodic variations. To fuse the resulting features, we further design a Steady-Transient Collaborative Aggregation module for cross-branch interaction and joint aggregation. Experiments on five benchmarks show that LaST-SR achieves the best PSNR/SSIM among the compared methods for ×2 and ×4 SISR. Ablation studies further validate the effectiveness of the proposed architecture and its key modeling mechanisms.