Object detection

The task and its vocabulary

Given an image, output a set of $(\text{box}, \text{class}, \text{score})$ triples, with boxes as $(x, y, w, h)$ or corner coordinates. Benchmarks: PASCAL VOC (20 classes) and COCO (80 classes, 118K training images, small/medium/large objects). Two quantities define everything downstream.

Intersection over Union between boxes $A$ and $B$:

Worked: $A$ spans $(0,0)$–$(10,10)$, $B$ spans $(5,5)$–$(15,15)$. Areas 100 each; intersection $5\times5 = 25$; union $200 - 25 = 175$; $\mathrm{IoU} = 25/175 \approx 0.143$. VOC counts a detection as correct if $\mathrm{IoU}\ge 0.5$; COCO averages over thresholds $0.5, 0.55, \dots, 0.95$, rewarding tight boxes.

From sliding windows to region proposals: the two-stage family

The naive approach — classify every window at every scale — is hopeless (millions of crops). R-CNN (2014) used selective search for about 2,000 class-agnostic region proposals, ran a CNN on each warped crop and classified with SVMs: 47 s per image. Fast R-CNN (2015) ran the CNN once on the whole image and pooled features for each proposal with RoI pooling, training classification and box regression jointly with a multi-task loss (softmax + smooth-$L_1$). Faster R-CNN (2015) replaced selective search with a Region Proposal Network: at each position of the feature map, $k$ reference anchors (e.g., 3 scales × 3 aspect ratios) each get an objectness score and four box deltas. Boxes are parametrized relative to their anchor so the regression targets are scale-free:

Proposals from the RPN feed the Fast R-CNN head; the whole thing runs at 5–17 fps and remained the accuracy reference for years. Mask R-CNN adds a mask branch (5.6 Segmentation).

One-stage detectors: predict everything at once

YOLO (2016) divides the image into an $S\times S$ grid; each cell predicts $B$ boxes with confidences and one class distribution, all from a single forward pass — 45 fps, with weaker small-object accuracy. SSD predicts from multiple feature-map resolutions with default boxes (anchors) to handle scale. One-stage detectors face a brutal imbalance: of ~100K anchors per image, perhaps 10 are objects, and the easy background examples swamp the loss. RetinaNet fixed this with focal loss:

where $p_t$ is the predicted probability of the true class. For $\gamma = 2$, an easy background example with $p_t = 0.99$ is down-weighted by $(0.01)^2 = 10^{-4}$ — effectively removed — while a hard example with $p_t = 0.3$ keeps $0.49$ of its weight. With focal loss (and FPN), one-stage detectors matched two-stage accuracy. The YOLO line continued (v3 multi-scale, v4/v5 training tricks, v8–v11 anchor-free with decoupled heads via Ultralytics) and remains the practical default for real-time detection.

Feature Pyramid Networks

Small objects need high-resolution features, large objects need semantic depth. FPN adds a top-down pathway: upsample the deepest map, add the laterally connected (1×1-projected) shallower map, repeat — producing a pyramid $P_3 \dots P_7$ where every level is both high-resolution-ish and semantically strong. Objects are assigned to levels by size. FPN is now a standard component of detectors and segmenters.

Non-maximum suppression

Detectors fire several overlapping boxes per object. NMS: sort detections by score; take the top one; remove all remaining boxes of the same class with $\mathrm{IoU} > \tau$ (typically 0.5) against it; repeat with the next survivor. Mini-example: boxes with scores 0.9, 0.8, 0.7 where the 0.8 box overlaps the 0.9 box at IoU 0.6 and the 0.7 box at IoU 0.2 → keep 0.9, drop 0.8, keep 0.7. Soft-NMS decays scores instead of deleting. DETR removes the need for NMS altogether.

Mean average precision, by hand

For one class, rank all detections across the dataset by score. Walking down the list, mark each as TP if it matches an unmatched ground-truth box with $\mathrm{IoU}\ge 0.5$, else FP; compute cumulative precision and recall after each detection. Average precision is the area under the (interpolated) precision–recall curve; mAP averages AP over classes (and over IoU thresholds for COCO).

Worked example: 3 ground-truth objects, 5 detections ranked by score: TP, FP, TP, TP, FP.

All-point interpolation replaces each precision with the maximum precision at any recall $\ge$ the current one: at recall 1/3 → 1.00; at 2/3 → 0.75; at 1 → 0.75. AP $= \tfrac{1}{3}(1.00 + 0.75 + 0.75) = 0.833$. A detector that produced the same boxes in a different confidence order would get a different AP — ranking quality is part of the score.

DETR: detection as set prediction

DETR (2020) feeds CNN features into a transformer encoder–decoder (7.1 Attention Is All You Need: the big picture); $N$ learned object queries each produce one (class, box) prediction, with a "no object" class filling the rest. Training uses bipartite matching: the Hungarian algorithm finds the one-to-one assignment between predictions and ground truths minimizing a cost of class probability plus box $L_1$ and generalized IoU, and the loss is computed on matched pairs only. No anchors, no NMS, no hand-designed priors. The original converged slowly and struggled on small objects; Deformable DETR, DINO-DETR and RT-DETR fixed both, and transformer detectors now top COCO leaderboards, while YOLO-style models dominate deployment for speed.

In practice

Use torchvision.models.detection (Faster R-CNN, RetinaNet, FCOS with pretrained COCO weights) or ultralytics (YOLO: YOLO("yolo11n.pt").train(data="data.yaml", epochs=50)). Label with CVAT, Label Studio or Roboflow; know the COCO JSON and YOLO txt formats; augment boxes consistently (Albumentations handles box transforms). Typical failures: tiny objects (use higher input resolution or FPN levels), crowded scenes (NMS merges neighbors), domain shift, and class confusion among similar categories — evaluate per class and per object size.

Where this shows up later

• Mask R-CNN extends Faster R-CNN to instance segmentation — 5.6 Segmentation.
• DETR's encoder–decoder and Hungarian matching reappear in panoptic and open-vocabulary detectors built on CLIP-style encoders — 7.1 Attention Is All You Need: the big picture, 8.10 Multimodal models.
• Focal loss and class imbalance connect back to 4.3 Loss functions catalog and 3.5 Evaluating models: metrics, validation and the bias–variance trade-off; IoU-style losses to 5.6 Segmentation.