Depth Estimation from Monocular Images

# Depth Estimation from Monocular Images

## Introduction & Motivation

Monocular depth: estimate 3D depth from single images. Self-supervised learning, geometric constraints. Applications: 3D reconstruction, autonomous driving.

Motivation: Recover 3D structure from 2D images.

Applications: 3D scene reconstruction, SLAM systems.

---

## Core Concepts & Theory

### Depth Prediction

Estimate depth map from image.

### Self-Supervised Learning

Photometric loss without ground truth.

### Geometric Constraints

Camera models and epipolar geometry.

### Scale Ambiguity

Addressing depth scale uncertainty.

---

## Mathematical Formulation

Photometric Loss:
$$\mathcal{L}_{ ext{photo}} = \sum_p \| I_t(p) - I_s( ext{warp}(p)) \|_1$$

Depth Map:
$$D: (u,v) o d$$

Reprojection:
$$p' = K D(p) K^{-1} p$$

---

## Advanced Theory & Extensions

### Multi-Frame Supervision

Temporal consistency.

### Occlusion Handling

Disocclusion regions.

### Confidence Estimation

Uncertainty quantification.

---

## Computational Considerations

Inference: O(H·W·D).

Warping: O(H·W).

Loss computation: O(H·W).

---

## Practical Implementation Strategies

### Encoder-Decoder Architecture

Multi-scale depth prediction.

### Disparity Prediction

Inverse depth for numerical stability.

### Post-processing

Refinement and smoothing.

---

## Benchmark Datasets & Evaluation

KITTI: Autonomous driving.

NYU Depth: Indoor scenes.

Cityscapes: Urban driving.

---

## Key Challenges & Limitations

### Scale Ambiguity

Single image depth scale.

### Occlusion Regions

Disoccluded areas.

### Texture-less Regions

Weak gradients.

---

## Hyperparameter Tuning

Loss weight: 0.1-1.0.

Smoothness weight: 0.01-0.1.

Learning rate: 1e-4 to 1e-3.

---

## Real-World Applications & Case Studies

3D Reconstruction: Scene structure recovery.

Autonomous Driving: Obstacle distance.

AR/VR: Depth for rendering.

---

## Integration with Other Methods

Depth + optical flow for ego-motion; + semantic segmentation.

---

## Summary & Key Takeaways

Monocular depth estimation recovers 3D structure from single images.

Principles:
1. Self-supervised: Photometric loss.
2. Geometric: Camera model constraints.
3. Multi-scale: Hierarchical prediction.
4. Temporal: Frame consistency.
5. Uncertainty: Confidence estimation.

---

## Appendix: Practical Labs

### Lab 1: Depth Map Generation

import numpy as np

def generate_depth_map(image_shape, min_depth=0.1, max_depth=100):
 """Generate depth map prediction"""
 h, w = image_shape
 depth = np.random.uniform(min_depth, max_depth, (h, w))
 
 # Smooth spatially
 depth = np.convolve(depth.flatten(), np.ones(5)/5, mode='same').reshape((h, w))
 
 return depth

np.random.seed(42)
depth = generate_depth_map((224, 224))
assert depth.shape == (224, 224), "Correct depth shape"
assert 0.1 <= depth.min() and depth.max() <= 100, "Valid depth range"
print("✓ Depth map generation working")

### Lab 2: Photometric Loss

import numpy as np

def photometric_loss(image_t, image_s_warped, mask=None):
 """Compute photometric loss for depth estimation"""
 diff = np.abs(image_t - image_s_warped)
 
 if mask is not None:
 diff = diff * mask
 
 loss = np.mean(diff)
 return loss

np.random.seed(42)
img_t = np.random.rand(224, 224, 3)
img_s = np.random.rand(224, 224, 3)
mask = np.ones((224, 224, 3))
loss = photometric_loss(img_t, img_s, mask)
assert loss >= 0, "Non-negative loss"
print(f"✓ Photometric loss: {loss:.4f}")

### Lab 3: Depth Smoothness

import numpy as np

def depth_smoothness_loss(depth_map):
 """Regularize depth map smoothness"""
 # Compute gradients
 grad_x = np.abs(np.diff(depth_map, axis=1))
 grad_y = np.abs(np.diff(depth_map, axis=0))
 
 # Mean absolute gradients
 smoothness = (np.mean(grad_x) + np.mean(grad_y)) / 2
 
 return smoothness

np.random.seed(42)
depth = np.random.rand(224, 224)
smooth = depth_smoothness_loss(depth)
assert smooth >= 0, "Non-negative smoothness"
print(f"✓ Smoothness loss: {smooth:.4f}")

### Lab 4: Depth Visualization

import numpy as np

def colorize_depth(depth_map):
 """Convert depth to visualization"""
 # Normalize to [0, 1]
 normalized = (depth_map - depth_map.min()) / (depth_map.max() - depth_map.min() + 1e-8)
 
 # Apply colormap (simplified)
 colored = np.stack([normalized, normalized, 1-normalized], axis=-1)
 
 return colored

np.random.seed(42)
depth = np.random.rand(224, 224)
colored = colorize_depth(depth)
assert colored.shape == (224, 224, 3), "Correct color shape"
print("✓ Depth visualization working")

---