Oriented 3D box IoU

Overview

3D detection and pose papers score a predicted box against a ground-truth box with 3D intersection over union: the intersection volume divided by the union volume 2012.09988 §5.1. The metric is invariant to scale and to rigid transforms of both boxes 2012.09988 §5.1. The hard part is the intersection. For boxes that can rotate in all directions, it is the volume of a convex polyhedron, not a simple product of overlaps.

The core idea of the exact method is to clip each box's faces against the other box, collect the points that survive, and take their convex hull as the intersection 2012.09988 §5.1. Every variant on this page either keeps that exact geometry or simplifies it by assuming something about the boxes.

Variants

Exact oriented IoU (polygon clipping and convex hull)

The approach described in the Objectron paper. Clip the faces of one box in the frame of the other, repeat with the roles swapped, then take the convex hull of the points that remain 2012.09988 §5.1. The paper calls this the exact 3D IoU for oriented boxes. Card: Objectron.

Symmetric-object rule

For objects that look the same under some rotation, such as cups and bottles, the paper rotates the predicted box about its symmetry axis and keeps the rotation that maximises IoU 2012.09988 §5.1. This is a rule layered on top of the exact method, not a different metric. It is described in the paper but not implemented in the released evaluator (see Contradictions). Card: Objectron.

Monte Carlo estimate (released code only)

The released repository also contains a sampling-based IoU, with 10,000 samples per box by default. The paper does not describe it, and the released evaluator uses the exact method instead. Card notes: Objectron, Paper vs code. Use it only with your own accuracy check, because the paper gives no error analysis for it.

Simplifications the paper rejects

Two cheaper forms are named in the paper as too simple for oriented boxes 2012.09988 §5.1:

• Axis-aligned IoU, which assumes both boxes are aligned with the axes.
• Ground-plane IoU, which projects the boxes onto the ground plane and assumes a shared height. Because of this, it removes roll and pitch.

Both are correct only when their assumption holds. Do not use them for boxes that can tilt. Card: Objectron.

When to use which

Whatever the variant, report the threshold you used. The paper's headline numbers are AP at a fixed 3D IoU threshold, and the paper reports them per category 2012.09988 tab. 2a. Across nine categories, the MobilePose-style baseline reaches 0.18 to 0.71 and the two-stage model 0.52 to 0.85 2012.09988 tab. 2a. These are model results at 0.5 IoU, not properties of the IoU itself.

Implementation sketch

The exact method, as described in 2012.09988 §5.1. Fill in the helper functions for your box representation. The empty-input guard is an implementation choice and is not in the paper.

# Exact IoU of two oriented 3D boxes  [arxiv:2012.09988 §5.1]
# Input:  boxes a, b, each with eight corners, metric units
# Output: IoU in [0, 1]

def oriented_iou(a, b):
    # Clip the faces of each box against the other, both ways  [§5.1]
    pts = clipped_points(a, b)
    pts += clipped_points(b, a)
    # Implementation guard, not from the paper: no points, no overlap
    if not pts:
        return 0.0
    # Intersection volume is the convex hull of the surviving points  [§5.1]
    inter = hull_volume(pts)
    # Union is the two box volumes minus the intersection  [§5.1]
    union = volume(a) + volume(b) - inter
    return inter / union


def clipped_points(moving, fixed):
    # Express the moving box in the frame of the fixed box  [§5.1]
    # Clip its faces against the fixed box and keep the surviving
    # corners of the moving box that lie inside it  [§5.1]
    # Return the points in world coordinates
    ...

Where variants differ: the symmetric-object rule wraps oriented_iou in a search over rotations about the symmetry axis 2012.09988 §5.1. The Monte Carlo estimate replaces the hull with sampled points inside both boxes (released code only). The axis-aligned and ground-plane forms replace the clipping with interval or plane tests and are not recommended for tilted boxes 2012.09988 §5.1.

Contradictions

Only one paper has a card on this page, so there is no paper-against-paper disagreement. There is a gap between the paper and its released code, reported in the Objectron card:

• Symmetric objects. The paper describes a rotation search for cups and bottles 2012.09988 §5.1. The released evaluator scores the box as given, with no rotation search. Its only symmetry code computes a different metric, ADD-S Objectron, Paper vs code.
• Which IoU is used. The paper describes the exact method. The released repository also has a Monte Carlo estimate, which the paper does not describe, and the evaluator uses the exact one Objectron, Paper vs code.

Papers

• Objectron: proposes the exact 3D IoU for oriented boxes and the symmetric-object rule, and reports 3D-AP baselines on nine categories.