RotaTouille: Rotation Equivariant Deep Learning for Contours
Odin Hoff Gardaa, Nello Blaser
2025-08-25
Summary
This paper introduces a new deep learning method called RotaTouille designed to work effectively with shapes and curves, like those found in images or data representing orbits.
What's the problem?
When dealing with shapes, a model should ideally recognize a shape even if it's rotated or if you start tracing its outline from a different point. Traditional deep learning models struggle with this because they treat rotated or shifted shapes as completely new, different inputs. The problem is that these models aren't naturally equipped to understand that a rotated or cyclically shifted shape is fundamentally the same object.
What's the solution?
RotaTouille solves this by using a special type of mathematical operation called 'complex-valued circular convolution'. This allows the model to automatically recognize shapes regardless of their rotation or starting point on the outline. The researchers also developed specific building blocks for the model – like layers that handle changes in detail and layers that summarize the overall shape – that maintain this rotational and shift-invariant behavior. Essentially, they built a model that 'understands' shapes in a way that isn't affected by how they're oriented or where you begin describing them.
Why it matters?
This work is important because it makes deep learning models more efficient and accurate when working with shapes and curves. This has applications in areas like computer vision (recognizing objects in images), meteorology (analyzing weather patterns), and engineering (monitoring rotating machinery). By building models that are naturally good at handling rotations and shifts, we can reduce the amount of data needed for training and improve the reliability of these systems.
Abstract
Contours or closed planar curves are common in many domains. For example, they appear as object boundaries in computer vision, isolines in meteorology, and the orbits of rotating machinery. In many cases when learning from contour data, planar rotations of the input will result in correspondingly rotated outputs. It is therefore desirable that deep learning models be rotationally equivariant. In addition, contours are typically represented as an ordered sequence of edge points, where the choice of starting point is arbitrary. It is therefore also desirable for deep learning methods to be equivariant under cyclic shifts. We present RotaTouille, a deep learning framework for learning from contour data that achieves both rotation and cyclic shift equivariance through complex-valued circular convolution. We further introduce and characterize equivariant non-linearities, coarsening layers, and global pooling layers to obtain invariant representations for downstream tasks. Finally, we demonstrate the effectiveness of RotaTouille through experiments in shape classification, reconstruction, and contour regression.