Intersection over Union (IoU)

Intuition

How well does a predicted box match the true box? Divide the overlapping area by the total area covered by either box. 1 = perfect match, 0 = no overlap.

Precise explanation

Boxes as [x1, y1, x2, y2] with $x2 > x1$, $y2 > y1$ (continuous coordinates):

• area $= (x2-x1)(y2-y1)$;
• intersection corners: $\max(x1)$, $\max(y1)$, $\min(x2)$, $\min(y2)$; width/height clipped at 0;
• $\text{IoU} = \frac{|A\cap B|}{|A| + |B| - |A\cap B|}$. Pairwise for $A$ (m, 4), $B$ (n, 4): broadcast A[:, None, :] against B[None, :, :] to get (m, n) matrices. Detection benchmarks count a prediction as correct if IoU ≥ 0.5 (or several thresholds).

Worked example

A = [0, 0, 2, 2], B = [1, 1, 3, 3]: intersection 1×1 = 1; union 4 + 4 − 1 = 7 → IoU 1/7 ≈ 0.143.

Common mistakes

• Forgetting to clip negative widths (non-overlapping boxes give negative "intersection").
• Mixing [x, y, w, h] and [x1, y1, x2, y2] formats.
• +1 pixel conventions from integer-pixel datasets (we use continuous coordinates).

Why this matters

IoU drives detection evaluation (mAP), anchor matching during training and non-maximum suppression (CV-24).

References

• Dive into Deep Learning — Aston Zhang, Zachary C. Lipton, Mu Li, Alexander J. Smola (S013)