# Optical Flow Odometry with Panoramic Image Based on Spherical Congruence Projection
## 1 Introduction  
Panoramic images offer significant advantages in robotics, autonomous driving, and virtual reality due to their extensive field of view (FoV) [1, 2, 3]. In visual odometry, their panoramic nature allows for capturing comprehensive environmental information in a single frame, increasing the feature residence time within the sensor's FoV. This higher information density enhances the accuracy and robustness of visual odometry systems [4]. However, utilizing panoramic images for feature-based odometry poses challenges, particularly in achieving continuous and undistorted feature extraction and tracking across a wide FoV. Original panoramic images suffer from serious image distortions, especially in the polar areas. Existing methods often rely on projecting panoramic images onto specific structures for feature processing, which are categorized into single-plane and multi-plane projection approaches. Single-plane projection methods map panoramic images onto a planar structure. For instance, [PERSON] et al. [5] developed a model for catadioptric and fisheye cameras, which was later extended by [PERSON] et al. [6] for panoramic annular lens (PAL) cameras. [PERSON] et al. [7] employed equirectangular projection (ERP) to process panoramic images. Multi-plane projection methods divide the image into multiple planes with different viewpoints. [PERSON] et al. [8], for example, proposed a cube map structure forfisheye cameras accompanied by a SLAM system. These methods significantly influence feature tracking accuracy, as shown in Figure 1.  
Single-plane projections introduce region-dependent distortions, which lead to tracking inconsistencies, especially under fast motion where feature points experience significant displacements. While multi-plane projections reduce some distortion-related issues, they introduce data discontinuities and retain localized distortions, limiting their robustness under large pixel shifts. To address these challenges, this paper proposes an optical flow odometry method based on a novel spherical congruence projection (SCP). Unlike the equirectangular projection, which introduces significant pixel stretching near the polar regions and exhibits nonlinear geometric distortion with increasing latitude, the SCP method ensures strict scale consistency across the spherical domain. Compared to multi-plane projection approaches, which often suffer from seam discontinuities and localized distortions at plane boundaries, SCP maintains topological and geometric continuity over the entire field of view. This distortion-free mapping enables smooth and consistent pixel tracking, even under fast motion and large displacements. Figure 1: Due to the inherent imaging characteristics of panoramic cameras, the original panoramic images often suffer from significant feature distortion (**top row**). Different projection approaches are adopted to transform the panoramic image into planner structures, such as ERP and multi-plane projections (**bottom row**). However, these methods fail to eliminate distortion. The proposed spherical congruence projection method (**right side in the bottom row**) leverages the spherical structure of panoramic images, effectively reducing feature distortion while preserving the structural integrity of features in the 3D space. In addition, the proposed method integrates a spherical pixel structure with an optical flow tracking framework, allowing for fast and accurate feature association between frames. This dual capability--global distortion elimination and dense, robust tracking--ensures that the proposed system meets both the accuracy and efficiency requirements of panoramic image odometry in real-world scenarios. The main contributions of this paper are summarized as follows:  
* We propose a spherical congruence projection method that provides a globally consistent, distortion-free representation of panoramic images.