Object Detection: R-CNN, YOLO, and SSD

🎯 Learning Objectives

• Distinguish object detection from classification
• Explain the R-CNN family evolution (R-CNN → Fast R-CNN → Faster R-CNN)
• Implement YOLO's grid-based detection approach
• Understand anchor boxes, IoU, NMS, and mAP

📋 Prerequisites

• CNN fundamentals
• Bounding box representation
• Classification vs regression

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1. 📖 Core Content

1.1 The Detection Problem

1. Localization: Predict bounding box coordinates (x, y, w, h)
2. Classification: Predict class for each box
3. Multiple objects: Handle variable number of objects

1.2 Intersection over Union (IoU)

$$\text{IoU} = \frac{\text{Area of Overlap}}{\text{Area of Union}}$$

• IoU > 0.5: Typically considered "good" detection
• IoU > 0.7: Tight/accurate detection
• IoU = 0: No overlap

1.3 Non-Maximum Suppression (NMS)

1. Sort detections by confidence score
2. Pick highest confidence detection
3. Remove all other detections with IoU > threshold (e.g., 0.5)
4. Repeat until no detections left

1.4 R-CNN Family

1.5 YOLO (You Only Look Once)

python
Run Inline
# runnable
import numpy as np
def yolo_encoding(image_grid_size, num_boxes, num_classes):
    """
    YOLO output encoding
    Each grid cell predicts:
    - For each box: (x, y, w, h, confidence)
    - Class probabilities: (num_classes,)
    Total output: grid × grid × (num_boxes × 5 + num_classes)
    """
    S = image_grid_size  # e.g., 7 for YOLOv1
    B = num_boxes        # e.g., 2
    C = num_classes      # e.g., 20 (PASCAL VOC)
    output_shape = (S, S, B * 5 + C)
    print(f"YOLO output shape: {output_shape}")
    print(f"Total predictions: {S * S * B} boxes")
    print(f"Output vector per cell: {B * 5 + C} values")
    return output_shape
# Example
shape = yolo_encoding(7, 2, 20)

$$\mathcal{L} = \lambda_{coord} \sum_{i=0}^{S^2} \sum_{j=0}^B \mathbb{1}_{ij}^{obj} [(x_i - \hat{x}_i)^2 + (y_i - \hat{y}_i)^2 + (\sqrt{w_i} - \sqrt{\hat{w}_i})^2 + (\sqrt{h_i} - \sqrt{\hat{h}_i})^2]$$ $$+ \sum_{i=0}^{S^2} \sum_{j=0}^B \mathbb{1}_{ij}^{obj} (C_i - \hat{C}_i)^2 + \lambda_{noobj} \sum_{i=0}^{S^2} \sum_{j=0}^B \mathbb{1}_{ij}^{noobj} (C_i - \hat{C}_i)^2$$ $$+ \sum_{i=0}^{S^2} \mathbb{1}_{i}^{obj} \sum_{c \in classes} (p_i(c) - \hat{p}_i(c))^2$$

1.6 Anchor Boxes

1.7 Mean Average Precision (mAP)

1. Sort all detections by confidence
2. Compute precision-recall curve
3. Average Precision (AP) = Area under PR curve
4. mAP = Mean of AP across all classes

$$AP = \sum_{k=1}^n P(k) \cdot \Delta R(k)$$

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📝 Practice Questions

Q1

<strong>Q1</strong>: For two boxes A=(10,10,20,20) and B=(15,15,25,25), compute IoU.

A: (10,10) to (30,30), area = 20×20 = 400 B: (15,15) to (40,40), area = 25×25 = 625

Overlap: x_min=15, y_min=15, x_max=30, y_max=30 Overlap area = 15×15 = 225

Union = 400 + 625 - 225 = 800

IoU = 225/800 = 0.281

This is a fairly low IoU — the boxes overlap but one is much larger. Q2

<strong>Q2

<strong>Q2</strong>: YOLOv1 uses 7×7 grid, 2 boxes, 20 classes. What's the output tensor size per image?

Output = 7 × 7 × (2 × 5 + 20) = 49 × 30 = 1470 values

This is the total number of values in the output tensor. For each of the 49 grid cells, the model predicts 30 values (10 for box coordinates/confidence + 20 class probabilities). Q3

<strong>Q3

<strong>Q3</strong>: Why does YOLO use √w and √h in its loss instead of w and h?

Using w and h directly would penalize large boxes and small boxes unequally. The same absolute error in width means more for a small box than a large one.

By using √w and √h, the error is proportional to the box size:

• Small box (w=10): Δw=2 → √12 - √10 = 3.46 - 3.16 = 0.30
• Large box (w=100): Δw=2 → √102 - √100 = 10.10 - 10.00 = 0.10

The square root transformation makes the loss more sensitive to small box errors. Q4

<strong>Q4</strong>: NMS threshold is 0.5. Detection A (score 0.9), B (0.8), C (0.7). IoU(A,B)=0.4, IoU(A,C)=0.7. Which are kept?

1. Pick A (highest score: 0.9)
2. Remove C (IoU with A = 0.7 > 0.5 threshold)
3. B remains (IoU with A = 0.4 < 0.5)
4. Next iteration: Pick B (next highest: 0.8)
5. No more detections to compare

Kept: A and B. C is suppressed because it overlaps too much with A. Q5

<strong>Q5

<strong>Q5

<strong>Q5</strong>: Compare one-stage (YOLO, SSD) vs two-stage (Faster R-CNN) detectors.

Both approaches have converged in recent years — modern detectors like YOLOv8, EfficientDet, and DETR blur the boundaries.

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