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

Linhao Li; Zhaojie Pan; Langkun Chen

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

Linhao Li, Zhaojie Pan, Langkun Chen

arXiv:2609.02063v1Sep 2, 2026cs.CV

TL;DR

SISR must recover high-frequency detail and long-range structure from an ill-posed low-resolution observation, while Fourier bases are limited for localized aperiodic variations. LaST-SR derives an approximate spatial steady–transient decomposition and implements it with complementary global Fourier and local complex-frequency branches plus collaborative aggregation. Across five benchmarks, it achieves the best compared PSNR/SSIM for ×2 and ×4 SISR.

Problem
SISR is ill-posed because degraded observations discard or alias high-frequency information, while Fourier bases constrain localized aperiodic variation.
Method
LaST-SR uses CFD to combine global full-spectrum Fourier modeling with window-conditioned local complex-frequency modeling, then uses STCA for cross-branch interaction and joint aggregation.
Results
LaST-SR achieves the best PSNR/SSIM among compared methods on five benchmarks for both ×2 and ×4 SISR.
Takeaways & Limitations
The approximate spatial steady–transient formulation provides a theoretical basis for jointly modeling image-wide structure and localized aperiodic variations in SISR.
Takeaways & Limitations
The decomposition is approximate: mixed Fourier–complex-pole families are collected into a residual, and the local formulation relies on normalized coordinates without a negative-real-part constraint.

Abstract

from arXiv · show

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 $\times2$ and $\times4$ SISR. Ablation studies further validate the effectiveness of the proposed architecture and its key modeling mechanisms.

1 Introduction

SISR is an ill-posed reconstruction problem requiring both local detail cues and long-range structural dependencies. LaST-SR addresses Fourier limitations on localized aperiodic variations through approximate steady–transient decomposition, complementary frequency branches, and collaborative aggregation.

Problem: SISR reconstructs a high-resolution image from a degraded low-resolution observation despite discarded or aliased high-frequency information.Multiple high-resolution images can correspond to the same low-resolution observation, making SISR inherently ill-posed.
Motivation: CNNs capture local patterns but struggle with explicit image-wide structure, while global self-attention often incurs quadratic cost.These limitations motivate efficient global feature modeling for structurally consistent reconstruction.
Approach: LaST-SR derives an approximate steady–transient decomposition for static two-dimensional feature maps using a Laplace-domain formulation.The formulation provides a spatial counterpart to the pole–residue interpretation used for time-dependent dynamical systems.
Architecture: CFD combines a global full-spectrum Fourier branch for image-wide dependencies with a window-conditioned local complex-frequency branch for localized aperiodic variations.STCA then enables cross-branch interaction and joint aggregation of the resulting features.
Results: LaST-SR achieves the best PSNR/SSIM among compared methods on five benchmarks for both ×2 and ×4 SISR.Ablations further validate the decomposition, complementary branch roles, and STCA.

2 Related Work

Related work connects LaST-SR to frequency-domain SISR and complex-exponential signal modeling. The paper builds on Fourier-based global interaction and Laplace-inspired pole–residue decompositions while extending them toward spatial feature modeling.

Frequency-domain SISR: Frequency-domain SISR methods use Fourier or transform-domain features to improve global interaction and combine them with local or adaptive representations.Examples include Fourier convolution, wavelet subbands, progressive frequency decomposition, and weighted Fourier operators.
Complex-exponential modeling: Complex-exponential methods extend oscillatory Fourier modes with exponential modulation and estimate poles or modal coefficients for signal decomposition.Classical examples include Prony-type methods, Matrix Pencil, and ESPRIT.
Laplace neural operators: LNO introduces learnable pole–residue parameterization to separate dynamical responses into steady-state and transient components.LaST-SR uses this conceptual background for spatial feature modeling in SISR.

3 Theoretical Analysis

The theoretical analysis represents feature maps as continuous functions, applies Fourier and complex-pole expansions to convolution, and approximates the result with steady and transient response families. Mixed pole families are retained as an approximation residual, motivating global and windowed implementations.

Feature-map representation: An intermediate feature tensor X is represented as samples of a continuous vector-valued function on a finite two-dimensional spatial domain.The analysis then fixes one sample and channel to study a scalar component.
Fourier representation: The global spectral structure is described with a truncated Fourier representation under periodic extension.Fourier coefficients and spatial frequencies define the oscillatory basis used for steady modeling.
Convolution model: A translation-invariant convolution operator with learnable kernel κ acts on the feature function through spatial convolution.The kernel is parameterized using complex exponentials following the complex-exponential modeling paradigm.
Steady–transient decomposition: The two-dimensional Laplace convolution theorem yields four pole families, while the approximation retains Fourier–Fourier and complex-pole–complex-pole families.The mixed Fourier–complex-pole families are collected into an approximation residual rather than assumed to vanish.
Response interpretation: The retained first and second terms represent steady and transient responses with corresponding effective coefficients.Steady responses use globally supported oscillatory modes, whereas transient responses represent content-dependent aperiodic variations.
Windowed realization: The transient response is implemented in normalized local windows because full-map evaluation requires large complex tensors and globally shared modal coefficients limit regional adaptation.Windowing derives window-specific modal coefficients while sharing poles and residues across windows.

4 LaST-SR Architecture

LaST-SR models steady and transient spatial responses through complementary global Fourier and local complex-frequency branches, then jointly refines them with STCA. Its architecture combines low-dimensional branch projections, windowed transient modeling, full-spectrum steady modeling, and cross-branch aggregation.

LaST-SR Architecture: The architecture is organized into sequential STAR blocks containing PDPM, CFD, and STCA, followed by convolutional PixelShuffle reconstruction.The fused feature is projected back to the original channel dimension, then processed with overlapping cross-attention and channel-attention recalibration.
PDPM: Pre-Decomposition Projection Module: PDPM independently projects each branch into lower-dimensional features before steady and transient modeling.The two branch projections do not share parameters and produce Cmid channels, with Cmid < C.
Complex-Frequency Decomposition: LaST-SR uses CFD to model global steady responses with a full-spectrum Fourier branch and localized transient responses with a windowed complex-frequency branch.The steady branch targets image-wide dependencies and long-range structural consistency, while the transient branch represents content-dependent aperiodic variations.
Global Steady Branch: The steady branch retains and modulates the complete two-dimensional Fourier spectrum to preserve global organization and fine-scale structural components.It first computes the discrete Fourier transform of Xss and applies frequency-dependent modulation with channel mixing.
Local Transient Branch: The transient branch shares complex poles and residues across local windows but dynamically generates modal coefficients from each window’s local spectrum.Normalized local coordinates are used during reconstruction; pole real and imaginary parts govern spatial growth or decay and local oscillation, respectively.
Steady–Transient Collaborative Aggregation: STCA enhances both CFD outputs with window attention, forms shared guidance, performs cross-attention, and jointly aggregates the refined branches.The resulting shared guidance jointly refines image-wide structural information and localized aperiodic variations.

5 Experiments

LaST-SR is evaluated under bicubic degradation across five benchmarks against diverse SISR methods. It achieves strong benchmark performance, while ablations show that global and local frequency modeling, cross-branch aggregation, spectral supervision, and pole flexibility each contribute.

Experimental Setup: Evaluation uses bicubic degradation, five standard benchmarks, and PSNR/SSIM measured on the Y channel of YCbCr.Training uses DIV2K, while evaluation uses Set5, Set14, BSD100, Urban100, and Manga109.
Experimental Setup: LaST-SR is compared with representative CNN-, Transformer-, and frequency-domain methods spanning comparable and larger parameter budgets.The comparison includes methods such as CARN, LAPAR-A, SwinIR, SwinFIR-T, Omni-SR, FDSR, CRAFT, LAMNet-large, and MaIR-Small.
Comparison with State-of-the-Art Methods: LaST-SR achieves the best PSNR/SSIM on five benchmarks for both ×2 and ×4 SISR.On Urban100 and Manga109, improvements over the best competing results range from 0.04 dB to 0.09 dB for ×2 and from 0.05 dB to 0.09 dB for ×4.
Ablation Studies: Replacing global Fourier modeling drops PSNR by 5.13 dB on Urban100 and 6.18 dB on Manga109, whereas removing local complex-frequency modeling drops it by 0.11 dB and 0.08 dB.The results support complementary roles for image-wide structural modeling and localized aperiodic variation modeling.
Ablation Studies: Replacing cross-branch attention decreases PSNR by 0.06 dB and 0.02 dB, confirming the contribution of collaborative aggregation.All ablation variants use the same training and evaluation settings.
Ablation Studies: Removing Fourier-magnitude loss lowers PSNR by 0.12 dB, while constraining transient poles to Re(µ) < 0 lowers Urban100 PSNR by 0.10 dB.Unconstrained real parts allow both increasing and decaying spatial envelopes within finite windows.
Feature Responses: The steady response preserves image-wide structure, the transient response emphasizes localized aperiodic variations, and STCA enhances local edges while retaining overall organization.Figure 3 visualizes the branch projections, corresponding CFD responses, and the STCA-fused representation.

6 Conclusion

LaST-SR combines a Laplace-inspired steady–transient decomposition with complementary global and local complex-frequency modeling for SISR. Experiments and ablations demonstrate the framework’s effectiveness, while future work targets more complex degradations.

Conclusion: LaST-SR jointly models image-wide structural information and localized aperiodic variations through an approximate steady–transient decomposition for static two-dimensional feature maps.The decomposition is derived under certain assumptions and extends the pole–residue interpretation from dynamical systems to spatial feature modeling.
Conclusion: CFD uses global full-spectrum Fourier and window-conditioned local complex-frequency branches, while STCA collaboratively aggregates their responses.The two branches target complementary spatial information.
Conclusion: Extensive experiments and ablation studies demonstrate the effectiveness of LaST-SR.
Conclusion: Future work will explore super-resolution under more complex degradations.

A Steady–Transient Decomposition for Static Two-Dimensional Image Features

The appendix introduces the modeling conventions and derivation used for the approximate steady–transient decomposition of static two-dimensional feature maps.

Steady–Transient Decomposition for Static Two-Dimensional Image Features: The derivation first states modeling conventions and then obtains a corresponding two-dimensional pole–residue representation.

A.1 Continuous Feature Representation and Periodic-Extension Convention

The analysis represents discrete feature maps with continuous interpolants and uses a periodic representative to define a global Fourier formulation. Truncation then yields the finite representation used in the subsequent derivation.

Continuous Feature Representation: An intermediate feature tensor X is represented for each sample by a continuous interpolant in C(Ω; R^Cmid) ⊂ L2(Ω; R^Cmid).The multi-channel construction assigns complex-valued parameters to input–output channel pairs and aggregates responses over input channels.
Continuous Feature Representation: The interpolant is introduced solely for continuous-domain analysis and is not part of the network implementation.
Periodic-Extension Convention: The domain Ω = [0, Tx) × [0, Ty) is treated as a fundamental domain of the two-dimensional torus, producing a periodic representative for global Fourier analysis.This convention does not assume that the original image feature is physically periodic.
Periodic-Extension Convention: Boundary mismatches may introduce jump discontinuities at periodic seams, which remain admissible in the L2 setting.
Fourier Representation: The subsequent derivation restricts a rectangular partial Fourier sum to Ω and uses the resulting truncated representation.Fourier completeness provides approximation as the truncation indices increase.
Fourier Representation: Because the feature channel is real-valued, its Fourier coefficients satisfy the corresponding real-signal symmetry.

A.2 Derivation of the Steady–Transient Decomposition

The derivation uses a two-dimensional one-sided Laplace formulation to express convolution responses through Fourier and learnable complex-pole families. It retains same-type pole families as complementary steady and transient responses while collecting mixed families into an explicit residual before adopting a structured approximation.

Laplace-domain construction: A positive-quadrant extension and one-sided-support convention establish the two-dimensional one-sided Laplace representation for truncated feature maps.The construction is spatial and does not assign temporal causality to image coordinates.
Pole–residue expansion: The convolution response is expanded through Fourier poles and learnable complex poles under common-convergence and noncoincident-pole assumptions.Noncoincident poles ensure nonzero denominators in the partial-fraction expansion.
Response decomposition: The exact response decomposes into steady, transient, and mixed-family residual terms: u(x, y) = uss(x, y) + utr(x, y) + rmix(x, y).The steady term is the Fourier–Fourier family, the transient term is the complex-pole–complex-pole family, and mixed families form rmix.
Response roles: The retained steady and transient families represent globally supported oscillatory structure and spatially varying aperiodic variations, respectively.The transient branch’s local adaptivity is further enhanced by the subsequent windowed implementation.
Approximation: Mixed pole families generally do not vanish, so the main-paper dual-branch model is a structured approximation that omits the explicit residual.The approximation retains same-type pole families for structural symmetry and parameter efficiency.

A.3 Fourier-Domain Form of the Convolution Response

Under the periodic-extension convention, convolution preserves global Fourier basis functions and modifies only their complex-valued coefficients. This yields a Fourier spectral-multiplier form that analytically supports implementing the steady response with a global Fourier-domain operator.

Periodic convolution setup: The periodic-convolution analysis examines a truncated periodic input and periodic kernel on the two-dimensional torus.The convolution is well defined when the input is in L2(T2) and the kernel is in L1(T2).
Fourier mode preservation: Periodic convolution preserves spatial Fourier basis functions while modulating their coefficients through complex-valued mode multipliers.The retained steady response is spanned by the same global Fourier-mode family.
Steady-response implementation: The steady response therefore admits a Fourier spectral-multiplier representation obtained by transforming, modulating, and inverse-transforming retained frequency modes.This form matches the general structure of spectral convolution in Fourier neural operators.