Circular Image Deturbulence using Quasi-conformal Geometry

TL;DR

The paper presents Circular Quasi-Conformal Deturbulence (CQCD), an unsupervised method for removing geometric distortions from images affected by inhomogeneous media. It leverages a circular architecture with forward and inverse deformation mappings regularized by quasi-conformal theory via Beltrami coefficients to ensure bijective, invertible transformations, while tight-frame encoding provides multi-scale, direction-aware features for precise deformation estimation. The approach enforces cycle-consistency by reconstructing distorted inputs from restored images, leading to robust restoration without paired labels. Experiments on synthetic and real turbulence demonstrate superior restoration quality and accurate deformation-field estimation, with ablations highlighting the importance of QC regularization and tight-frame encoding for stability and performance.

Abstract

The presence of inhomogeneous media between optical sensors and objects leads to distorted imaging outputs, significantly complicating downstream image-processing tasks. A key challenge in image restoration is the lack of high-quality, paired-label images required for training supervised models. In this paper, we introduce the Circular Quasi-Conformal Deturbulence (CQCD) framework, an unsupervised approach for removing image distortions through a circular architecture. This design ensures that the restored image remains both geometrically accurate and visually faithful while preventing the accumulation of incorrect estimations. The circular restoration process involves both forward and inverse mapping. To ensure the bijectivity of the estimated non-rigid deformations, computational quasi-conformal geometry theories are leveraged to regularize the mapping, enforcing its homeomorphic properties. This guarantees a well-defined transformation that preserves structural integrity and prevents unwanted artifacts. Furthermore, tight-frame blocks are integrated to encode distortion-sensitive features for precise recovery. To validate the performance of our approach, we conduct evaluations on various synthetic and real-world captured images. Experimental results demonstrate that CQCD not only outperforms existing state-of-the-art deturbulence methods in terms of image restoration quality but also provides highly accurate deformation field estimations.

Figures (8)

• Figure 1: Illustration of how the Beltrami coefficient measures the conformality distortion of a quasi-conformal map
• Figure 2: Illustration of the proposed Circular Quasi-Conformal Deturbulence (CQCD) framework. First, a deformation mapping is estimated to geometrically correct the distorted input. A pixel-wise network then removes both existing blurs and those introduced during resampling. The restored image is then mapped back using the inverse deformation to reconstruct a distorted image, which is compared to the original via a fidelity loss, completing the cycle.
• Figure 3: Schematic illustration of the Tight-Frame (TF) Block.
• Figure 4: Illustration of Deformation Estimator.
• Figure 5: Comparison of unsupervised deturbulence based on Jellyfish (mild), Pterois (medium) and Tiles (severe). From top to bottom, restoration results from 5 distorted images, while the middle from 10 and the right from 15 are presented for different methods.

Theorems & Definitions (2)

• Definition 1
• Theorem 1