🎓 Lesson 2
D2
Optics Fundamentals for Robotic Vision: FOV, Depth of Field, and MTF
Field of view (FOV) is how much a camera can see at once, like the width of your vision when you look straight ahead without moving your eyes.
🎯 Learning Objectives
- ✓ Calculate horizontal and vertical FOV for a given lens–sensor combination using sensor dimensions and focal length
- ✓ Analyze depth of field limits for robotic vision setups under varying working distances and f-numbers
- ✓ Interpret MTF curves to select lenses that meet minimum resolution requirements for sub-millimeter feature detection in mining inspection tasks
- ✓ Design a vision subsystem for underground robotic drill guidance by balancing FOV, DoF, and MTF constraints
📖 Why This Matters
In autonomous mining robots—such as drill rig navigators or muck pile analyzers—vision systems must reliably detect drill hole positions, rock fractures, or conveyor belt anomalies under low-light, dusty, and vibration-heavy conditions. A poorly chosen lens may miss critical edge features (due to narrow FOV), blur targets at varying standoff distances (shallow DoF), or fail to resolve 2-mm cracks in blast-hole collars (low MTF). Understanding these optics fundamentals prevents costly misalignment, false rejections, and unsafe automation decisions.
📘 Core Principles
FOV is governed by geometry: wider sensors and shorter focal lengths increase FOV, but reduce magnification and working distance flexibility. Depth of field depends on aperture (f-number), focal length, sensor format, and object distance—narrower apertures (higher f-numbers) increase DoF but reduce light throughput, demanding careful trade-offs in low-illumination mine environments. MTF is not a single number—it’s a curve plotting contrast preservation (%) vs. spatial frequency (lp/mm); high-frequency performance (>50 lp/mm) is essential for detecting fine structural defects in rock surfaces or drill bit wear patterns. Real-world optical performance is degraded by diffraction, aberrations, and sensor sampling (Nyquist limit), making MTF-based lens selection non-negotiable for metrology-grade robotic vision.
📐 FOV Calculation
Horizontal FOV (HFOV) determines the maximum lateral coverage at a given working distance—critical for ensuring full drill collar visibility in borehole guidance systems. It is derived from basic trigonometry using sensor width and focal length.
Horizontal Field of View (HFOV)
HFOV (°) = 2 × arctan(w / (2 × f))Calculates angular field of view based on sensor width and focal length.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| w | Sensor width | mm | Active width of image sensor |
| f | Focal length | mm | Effective focal length of lens |
Typical Ranges:
Underground drill guidance: 16° – 35°
Conveyor belt anomaly detection: 45° – 75°
💡 Worked Example
Problem: A robotic drill guidance camera uses a 1/1.8″ CMOS sensor (width = 7.21 mm) and a 12 mm fixed-focal-length lens. The robot operates at a working distance of 1.5 m from the drill collar surface. Calculate the HFOV in degrees and linear width at 1.5 m.
1.
Step 1: Compute half-angle θ = arctan((sensor_width / 2) / focal_length) = arctan(3.605 / 12) = arctan(0.3004) ≈ 16.7°
2.
Step 2: Full HFOV = 2 × θ = 33.4°
3.
Step 3: Linear HFOV at 1.5 m = 2 × 1.5 × tan(16.7°) = 3 × 0.3004 ≈ 0.901 m (901 mm)
Answer:
The HFOV is 33.4°, yielding a linear coverage of 901 mm at 1.5 m—sufficient to image a standard 762 mm drill collar with margin for robot positioning error.
🏗️ Real-World Application
Rio Tinto’s AutoHaul® drill guidance retrofit used a 2/3″ global-shutter sensor (8.8 mm width) with a 25 mm lens (f/2.8) mounted on a boom-arm robot operating 2.0–2.5 m from blast holes. Engineers selected this configuration after MTF analysis confirmed ≥45% contrast retention at 60 lp/mm (required to resolve 0.3 mm drill-bit flank wear), while maintaining 18° HFOV—tight enough to avoid tunnel wall clutter but wide enough to capture collar centering fiducials. Depth of field was validated at ±120 mm around 2.25 m using f/5.6 to tolerate thermal expansion-induced focus drift.
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