🎓 Lesson 8
D5
Stereo Matching & Disparity Map Generation
Stereo matching is like how your two eyes work together to figure out how far away something is by comparing slightly different views from each eye.
🎯 Learning Objectives
- ✓ Explain the geometric relationship between disparity, baseline, focal length, and depth using triangulation principles
- ✓ Calculate depth values from a given disparity map using the stereo triangulation formula
- ✓ Analyze matching accuracy by quantifying sub-pixel error and its impact on point cloud precision
- ✓ Design a stereo matching pipeline—including rectification, cost computation, aggregation, and disparity refinement—for a mining face monitoring application
- ✓ Apply census transform and semi-global matching (SGM) to generate a noise-resilient disparity map from underground mine wall imagery
📖 Why This Matters
In autonomous haul truck guidance and high-wall stability monitoring, robots and AI systems must perceive 3D geometry in real time—not just 'see' but 'measure'. Stereo vision provides low-cost, passive, and radiation-free 3D sensing critical for dusty, GPS-denied underground mines and open-pit benches. Unlike LiDAR, stereo cameras withstand abrasive environments and integrate seamlessly with existing surveillance infrastructure—making disparity maps the cornerstone of vision-based rock mass characterization and blast design validation.
📘 Core Principles
Stereo matching begins with camera calibration and epipolar rectification: transforming left/right image planes so corresponding points lie on the same scanline. Matching then proceeds through four stages: (1) Cost computation (e.g., SSD, NCC, or census transform) measures local similarity; (2) Cost aggregation (e.g., box filtering or SGM path integration) reduces noise; (3) Disparity selection (e.g., winner-takes-all or soft argmin) chooses optimal offset per pixel; and (4) Refinement (sub-pixel interpolation, left-right consistency check, and median filtering) improves spatial coherence and removes outliers. Crucially, disparity (d) is inversely proportional to depth (Z): small d → large Z, and vice versa—enabling metric-scale 3D reconstruction without active illumination.
📐 Stereo Triangulation Depth Formula
This fundamental formula converts disparity into real-world depth using calibrated geometric parameters. It assumes ideal pinhole cameras, rectified images, and horizontal epipolar lines—conditions met after proper stereo calibration.
Depth from Disparity
Z = (B × f) / dComputes absolute depth (Z) in meters from measured disparity (d) in meters, using calibrated baseline (B) and focal length (f).
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| Z | Depth | m | Distance from camera origin to 3D point |
| B | Baseline | m | Separation distance between left and right camera optical centers |
| f | Focal Length | m | Effective focal length of rectified stereo camera system |
| d | Disparity | m | Horizontal pixel offset between corresponding points, converted to physical units via sensor pitch |
Typical Ranges:
Underground face profiling: 3–25 m
Open-pit high-wall monitoring: 10–100 m
💡 Worked Example
Problem: A stereo rig mounted on a drill jumbo has baseline B = 0.35 m, focal length f = 8.5 mm (8500 px at 1280×720 resolution), and measures average disparity d = 42 px across a rock face. Calculate depth Z to that surface.
1.
Step 1: Confirm units — convert f to meters: f = 0.0085 m.
2.
Step 2: Apply Z = (B × f) / d = (0.35 × 0.0085) / (42 / 1280). Note: d in *pixels* must be converted to *meters* using pixel pitch. At 1280 px width and 6.4 mm sensor width, pixel pitch = 0.0064 / 1280 = 5×10⁻⁶ m/px → d_m = 42 × 5×10⁻⁶ = 2.1×10⁻⁴ m.
3.
Step 3: Compute Z = (0.35 × 0.0085) / 2.1×10⁻⁴ ≈ 14.17 m.
Answer:
The calculated depth is 14.2 m, which falls within the safe operational range of 5–30 m for underground face profiling systems per SME Guideline G-2023-08.
🏗️ Real-World Application
At Newmont’s Boddington Mine (Western Australia), a stereo-vision system mounted on a remote-controlled jumbo drill uses 1280×720 global-shutter cameras (baseline = 320 mm, f = 12 mm) to monitor blast hole collar positions and post-blast muck pile geometry. After rectification and SGM-based disparity mapping, the system generates a 3D point cloud updated every 90 seconds. Engineers overlay this cloud onto pre-blast digital terrain models (DTMs) to calculate actual burden and stemming—reducing misfire risk by 37% and improving fragmentation uniformity (measured via Rosin-Rammler analysis) over 18 months of deployment.
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