---
kind: concept
slug: rotation-equivariant-cnn
title: Rotation-equivariant CNN
summary: >-
  A CNN backbone whose feature maps keep one block per discrete rotation, so a rotated input gives a rotated feature map and the weights are shared across orientations. ReDet uses it for oriented aerial detection, with a 12 Mb backbone against 103 Mb and higher mAP on DOTA-v1.5, plus an orientation-aware RoI pooling step. One paper so far.
tags: [equivariance, group-convolution, oriented-object-detection, aerial-imagery, rotation-equivariance]
papers: ["2103.07733"]
related: [oriented-object-detection, object-detection, equivariance-constraint, euclidean-equivariant-network, convolutional-neural-network]
status: draft
updated_at: 2026-10-19
---

## Overview

A standard CNN is not equivariant to rotation. Rotating the input does not
turn the feature map in any simple way, so the same object at two angles gives
two different region features [arxiv:2103.07733 §1]. A map is equivariant to a
group of transforms when transforming the input transforms the output in a
known way [arxiv:2103.07733 eq. 1].

The rotation-equivariant design builds that property into the layers for a
discrete set of rotations. Orientation becomes an extra feature dimension, and
the weights are shared across it. An orientation-invariant region feature can
then be recovered by undoing the object's rotation, given its angle
[arxiv:2103.07733 eq. 7] [arxiv:2103.07733 §3].

The alternatives the paper names are more parameters, more rotation-augmented
training data, and RRoI warping, which aligns only the spatial dimension
[arxiv:2103.07733 §1] [arxiv:2103.07733 §2.3]. Every variant on this page keeps
a feature map shaped (channels, orientations, height, width) [arxiv:2103.07733 §4.1].

## Variants

### Discrete C_N backbone with orientation-aware RoI pooling

Each layer of a ResNet-50 with FPN is rebuilt as a layer equivariant to
translations and the cyclic rotation group C_N. The main results use N = 8
[arxiv:2103.07733 §4.1] [arxiv:2103.07733 tab. 1]. The authors use the e2cnn
library for these layers [arxiv:2103.07733 §4.1].

The detector is oriented: an RPN proposes horizontal RoIs, and a RoI
Transformer head turns them into rotated RoIs [arxiv:2103.07733 §4]. The
pooling step, RiRoI Align, first samples the rotated RoI in space as earlier
RRoI methods do. It then shifts the orientation channels so the channel
nearest the RoI angle comes first, and linearly blends the two channels around
that angle [arxiv:2103.07733 §4.2] [arxiv:2103.07733 eq. 8]
[arxiv:2103.07733 eq. 9]. Keeping all orientation responses beats max-pooling
over orientations [arxiv:2103.07733 tab. 3].

Card: [ReDet: A Rotation-equivariant Detector for Aerial Object Detection](/papers/2103.07733).

### Other variants

No other rotation-equivariant design has a paper card yet, so none is listed.

## When to use which

Constraints below come from the paper's own tables. The paper reports no
latency, FLOPs or memory measurements for the detector
[arxiv:2103.07733 §4.1].

| Situation | Use | Why |
|---|---|---|
| Oriented objects at arbitrary angles, model size matters | C_8 backbone with RiRoI Align ([ReDet](/papers/2103.07733)) | 12 Mb backbone against 103 Mb, and 66.86 against 65.03 mAP on the DOTA-v1.5 test set [arxiv:2103.07733 tab. 1] |
| Smallest backbone is the priority | C_16: 6 Mb, 64.36 mAP; C_8: 12 Mb, 66.86 mAP | The paper sweeps C_4, C_8 and C_16 with a size and accuracy trade-off [arxiv:2103.07733 tab. 1] |
| Image classification is the main task | Standard CNN | Top-1 on ImageNet without FPN is 76.55 for ResNet-50 against 71.20 for the C_8 version [arxiv:2103.07733 tab. 1] |
| Short training budget | ReDet at its 1x schedule | Rotation-augmented baseline at 1x reaches 64.07 mAP, against 66.66 for the ReDet* row, in 13 h of training [arxiv:2103.07733 tab. 4] |
| Longer training is affordable and no architecture change is wanted | Rotation-augmented baseline at 2x | At 22 h it reaches 67.34 mAP, above the ReDet* row at 1x [arxiv:2103.07733 tab. 4] |

Invariance is approximate. The backbone is equivariant only to the discrete
group, and RiRoI Align blends neighbouring orientations. Expect invariance at
the grid angles and approximate invariance between them
[arxiv:2103.07733 §4.1] [arxiv:2103.07733 §4.2] [arxiv:2103.07733 §5.3].

Evaluation covers DOTA-v1.0, DOTA-v1.5 and HRSC2016 only
[arxiv:2103.07733 tab. 6] [arxiv:2103.07733 tab. 7] [arxiv:2103.07733 tab. 8].
Results on other domains are not stated.

## Implementation sketch

Pseudocode for the RiRoI Align pooling step. Shapes are noted on each line.

```pseudocode
# RiRoI Align for one rotated RoI  [arxiv:2103.07733 §4.2]
# Input:  feature maps (K, N, H, W); RRoI (x, y, w, h, theta)
# Output: region features (K, N, S, S), orientation-aligned

def riroi_align(feature_maps, roi):
    # Spatial part: sample the RRoI grid, as in RRoI Align  [§4.2 (a)]
    region = rroi_sample(feature_maps, roi)          # (K, N, S, S)

    # Index r of the orientation bin below theta, and the fraction  [eq. 8]
    position = roi.theta * N / (2 * pi)
    r = floor(position)
    alpha = position - r

    # Output channel o blends input channels (o - r) and (o - r + 1)  [eq. 9]
    # The sign convention is a known difference; see Pitfalls
    aligned = zeros_like(region)
    for o in range(N):
        first = (o - r) % N
        second = (o - r + 1) % N
        aligned[:, o] = ((1 - alpha) * region[:, first]
                         + alpha * region[:, second])
    return aligned
```

## Contradictions

Only one paper is in the wiki, so no disagreement between papers is recorded.
Inside the ReDet paper, the abstract's gains and the table values do not agree.
See Pitfalls.

## Pitfalls

- **Abstract gains do not match the tables.** The card records the abstract's
  gains as 1.2, 3.5 and 2.6 mAP over the previous best. Against the strongest
  other row in the tables, the gains are 0.68 on DOTA-v1.0 multi-scale
  [arxiv:2103.07733 tab. 6], 1.90 on DOTA-v1.5 OBB multi-scale
  [arxiv:2103.07733 tab. 7] and 2.62 on HRSC2016 under the VOC2012 metric
  [arxiv:2103.07733 tab. 8]. Implement from the tables.
- **Interpolation count.** The paper ablates 1, 2 and 4 blended orientation
  channels [arxiv:2103.07733 tab. 3]. The card's review of the released code
  found that the kernel always blends two channels, so the 1 and 4 variants
  cannot be reproduced from that code. The card's Paper vs code section has
  the details.
- **Sign convention.** The card reports that the paper's equation and the
  released kernel index the channels in opposite directions. The sketch above
  follows the kernel. Check both before reimplementing.
- **Extra RoI scaling.** The released code enlarges each RRoI, with width times
  1.2 and height times 1.4, before RiRoI Align. The paper does not describe this
  step, so results may differ without it [see card, Paper vs code].
- **Group size is a setting, not a constant.** The RiRoI module defaults to 8
  orientations. Other group sizes need an explicit setting, and the two-channel
  blend still applies [see card, Pitfalls].
- **Rotation augmentation in the top result.** The DOTA-v1.0 80.10 result uses
  rotation augmentation and test-time rotation, so it is not a pure equivariance
  result [arxiv:2103.07733 §5.4].
- **Pretraining is outside the repository.** The ImageNet pretraining setup is
  given in the paper only [arxiv:2103.07733 §5.2].

## Papers

- [ReDet: A Rotation-equivariant Detector for Aerial Object Detection](/papers/2103.07733):
  a C_8-equivariant ResNet-50 with FPN backbone and RiRoI Align pooling for
  oriented aerial object detection.