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

TL;DR

The paper addresses the need for a rotation-invariant yet invertible 2D image representation suitable for learning and alignment tasks. It introduces the selective disk bispectrum, a minimal subset of disk bispectrum coefficients that enables exact image reconstruction up to rotation and significantly reduces computational cost. It provides a complete inversion procedure, a fast numerical approximation with accuracy guarantees, and a noise-bias correction for multi-reference alignment on rotated data, demonstrated on rotated MNIST. The resulting method is scalable, interpretable, and practical for rotation-invariant shape analysis in large-scale imaging scenarios.

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.

Figures (4)

• Figure 1: We propose the selective disk bispectrum, which reduces the time complexity of the disk bispectrum from $\mathcal{O}(m^3/N_m)$ to $\mathcal{O}(m)$, where $m$ is the number of disk harmonic frequencies, and $N_m$ the maximum frequency. We provide the first inversion of the disk bispectrum from its coefficients back to image space.
• Figure 2: Graphical illustration of the selective disk bispectrum inversion. The arrows indicate in which order the DH coefficients are reconstructed.
• Figure 3: (a) The selective disk bispectrum (SDB) map is significantly faster than the full disk bispectrum map, especially for large images. (b) Bispectrum invariance error decreases with higher image resolution. (c-d) The SDB coefficient inversion sucessfully reconstructs images for resolution $28\times28$ (c) and $112\times112$ (d), and achieves a similar reconstruction error as the DH Transform fast_dhcs_2023 inversion.
• Figure 4: (a) Dataset samples and original image used in MRA experiments for MNIST 6. (b-e) Relative error compares the similarity between the MRA reconstructed image and the original image used to generate the noisy rotated dataset for (b) $\sigma^2 = 0$ (c) $\sigma^2 = 0.01$, (d) $\sigma^2 = 0.05$, (e) $\sigma^2 = 0.1$. Sample reconstructions are shown below each result plot.

Theorems & Definitions (11)

• Proposition 3.1: Equivariance
• Definition 4.1: Disk Bispectrum zhao2014rotationally
• Proposition 4.2: Complexity of the disk bispectrum
• Definition 4.3: Selective Disk Bispectrum
• Proposition 4.4: Complexity of the selective disk bispectrum
• Theorem 4.5: Inversion
• Theorem 4.6: Complete Invariance
• Proposition 4.7: Complexity of the selective disk bispectrum
• Theorem 4.8
• Corollary 4.9