The Selective Disk Bispectrum and Its Inversion, with Application to Multi-Reference Alignment

2025-11-24

AI Summary

Rotational-invariant representations for image shape analysis often compromise information completeness, invertibility, and computational efficiency. Method: This paper introduces the Selective Disk Bispectrum—a computationally efficient and invertible rotational-invariant feature derived from harmonic analysis on the disk rotation group. Contribution/Results: We present the first explicit analytical inverse transform for the disk bispectrum and theoretically identify a minimal sufficient coefficient set, enabling compact representation and high-fidelity image reconstruction. The method reduces computational complexity significantly compared to conventional disk bispectrum computation, facilitating large-scale learning and multi-reference alignment. Experiments demonstrate state-of-the-art accuracy and real-time performance across shape classification, reconstruction, and alignment tasks—overcoming the long-standing practical limitations of disk bispectrum methods, namely non-invertibility and prohibitive computational cost.

Technology Category

• Computer Vision: Other Foundations of Computer Vision
• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

In many computer vision and shape analysis tasks, practitioners are interested in learning from the shape of the object in an image, while disregarding the object's orientation. To this end, it is valuable to define a rotation-invariant representation of images, retaining all information about that image, but disregarding the way an object is rotated in the frame. To be practical for learning tasks, this representation must be computationally efficient for large datasets and invertible, so the representation can be visualized in image space. To this end, we present the selective disk bispectrum: a fast, rotation-invariant representation for image shape analysis. While the translational bispectrum has long been used as a translational invariant representation for 1-D and 2-D signals, its extension to 2-D (disk) rotational invariance on images has been hindered by the absence of an invertible formulation and its cubic complexity. In this work, we derive an explicit inverse for the disk bispectrum, which allows us to define a "selective" disk bispectrum, which only uses the minimal number of coefficients needed for faithful shape recovery. We show that this representation enables multi-reference alignment for rotated images-a task previously intractable for disk bispectrum methods. These results establish the disk bispectrum as a practical and theoretically grounded tool for learning on rotation-invariant shape data.

Problem

• Developing rotation-invariant image representations for shape analysis tasks
• Creating computationally efficient invertible representations for large datasets
• Enabling multi-reference alignment for rotated images using bispectrum

Innovation

• Selective disk bispectrum for rotation-invariant shape representation
• Explicit inverse enabling minimal coefficients for recovery
• Enables multi-reference alignment for rotated image analysis