8.1 Fundamentals of Aerial Photogrammetry

Fundamentals of Aerial Photogrammetry

Aerial photogrammetry is the science and art of obtaining reliable quantitative measurements and spatial information about physical objects and terrain through the recording, measuring, and interpreting of aerial photographic images. In Nigerian surveying practice, regulated under the Surveyors Council of Nigeria (SURCON) standards, aerial photogrammetry forms the foundation for medium- and large-scale topographic mapping, boundary mapping, and national spatial data infrastructure.

Unlike map projections which are orthogonal (parallel rays projected perpendicularly onto a horizontal plane), aerial photographs are central perspective projections. In a central projection, all light rays pass through a single point—the rear nodal point of the camera lens. Consequently, the scale of an aerial photograph is not uniform across the image; it varies continuously as a function of focal length, flying height, and ground elevation variations.

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Photogrammetric Scale & Geometry

The scale of a vertical aerial photograph is defined as the ratio of an image distance ($d$) to the corresponding ground distance ($D$). From the geometric similarity of perspective triangles formed by the lens camera station and the ground terrain, photo scale $S$ is expressed as:

$S = \frac{f}{H - h}$

Where:

• $f$ is the calibrated focal length of the camera lens (typically $152.4\text{ mm}$ or $6\text{ inches}$ for traditional metric frame cameras).
• $H$ is the flying height of the aircraft above mean sea level (MSL) or reference datum.
• $h$ is the elevation of the ground point above the reference datum.
• $H - h$ represents the flying height above terrain ($H_g$).

When terrain elevation $h$ varies, scale varies accordingly. Points at higher terrain elevations are closer to the camera and appear at a larger scale, while points in lower valleys appear at a smaller scale. Average photo scale $S_{\text{avg}}$ is calculated using the average terrain elevation $h_{\text{avg}}$ of the project site.

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Flight Design & Overlap Geometry

Photogrammetric flight planning requires methodical arrangement of parallel flight lines (strips) to ensure complete stereoscopic coverage without gaps (holidays). Flight planning parameters dictate the number of photos, exposure intervals, and line spacing.

1. Forward Overlap (Endlap)

To view photographs in 3D stereoscopic vision, every point on the ground must appear on at least two consecutive photographs along the flight line. Standard flight specifications require a 60% forward overlap (with acceptable tolerance of $55% - 65%$). This creates a stereoscopic model in the overlap area known as the stereopair. The distance between consecutive exposure stations is called the air base ($B$).

$\text{Endlap} = \left(1 - \frac{B}{W}\right) \times 100\%$

Where $W$ is the ground coverage width of a photograph parallel to the flight line.

2. Side Overlap (Sidelap)

Adjacent parallel flight strips must overlap laterally to bind individual stereo models into a unified block network. Standard specifications dictate a 30% side overlap (ranging between $20% - 40%$). The lateral spacing between flight lines is called strip spacing ($W_{\text{strip}}$).

3. Aircraft Crab and Drift

Wind during flight operations induces rotational and positional distortions:

• Drift Angle: Lateral displacement of the aircraft off the planned flight line caused by crosswinds. Drift creates unphotographed gaps between strips if not corrected.
• Crab Angle: The deliberate angular rotation of the aircraft fuselage heading relative to the actual track line to compensate for crosswinds. If the camera mount is not rotated into the wind vector, the image edges become skewed relative to the flight line, reducing effective stereoscopic overlap.

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Stereoscopy & 3D Parallax

Stereoscopy is the phenomenon of perceiving 3D depth by viewing two overlapping images of the same terrain taken from different exposure stations. The human eyes, separated by an inter-pupillary distance of approximately $63\text{ mm}$, view the object from slightly different angles, creating retinal parallax that the brain interprets as 3D depth.

In photogrammetry, absolute stereoscopic parallax ($p$) is defined as the algebraic difference in the direction of two images of a point on two successive photographs taken from endpoints of a baseline. For a pair of aligned vertical photographs:

$p = x - x'$

Where $x$ is the coordinate of the point on the left photograph and $x'$ is the coordinate of the same point on the right photograph relative to their respective principal points.

The fundamental parallax equation for height determination is:

$p = \frac{B \cdot f}{H - h}$

Rearranging to solve for terrain elevation $h$ above datum or point elevation:

$h = H - \frac{B \cdot f}{p}$

Differencing two points gives the height difference $\Delta h$ between a top point ($T$) and bottom point ($B$):

$\Delta h = \frac{\Delta p \cdot (H - h_B)}{p_T}$

Where $\Delta p = p_T - p_B$ is the differential parallax between the top and bottom of the feature.

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Camera Calibration & Interior Orientation

Photogrammetric metric cameras undergo rigorous calibration to determine the geometric characteristics of the optical system. Interior Orientation (IO) reconstructs the bundle of rays inside the camera at the exact instant of exposure. The calibration certificate provides four key parameters:

1. Calibrated Focal Length ($f$): The precise distance from the rear nodal point of the lens to the plane of the image focal plane, optimized to equalize positive and negative lens distortions across the field of view.
2. Principal Point Coordinates ($x_0, y_0$): The origin point where the optical axis intersects the image plane. In an ideal camera, this coincides with the intersection of fiducial mark lines; calibration provides small offsets $(\Delta x_0, \Delta y_0)$.
3. Radial Lens Distortion ($K_1, K_2, K_3$): Symmetric optical aberration caused by lens refraction variations, pushing image points radially inward (barrel distortion) or outward (pincushion distortion) from the principal point. Radial distortion $\Delta r$ is modeled by: $\Delta r = K_1 r^3 + K_2 r^5 + K_3 r^7$
4. Decentering Lens Distortion ($P_1, P_2$): Asymmetric distortion resulting from physical misalignment or tilt of individual lens elements during camera assembly.

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Relief Displacement Equation

Relief displacement ($d$) is the shift or displacement in the position of an image point caused by the elevation of the feature above or below the selected reference datum. Relief displacement occurs radially outward from the principal point (or nadir point in a strictly vertical photo).

The mathematical formula governing relief displacement is:

$d = \frac{r \cdot h}{H}$

Where:

• $d$ is the magnitude of relief displacement measured on the photograph.
• $r$ is the radial distance from the principal point to the top of the displaced image feature.
• $h$ is the height of the feature above the reference datum.
• $H$ is the flying height of the aircraft above the same reference datum.

Key characteristics of relief displacement:

• Relief displacement is zero at the principal point ($r = 0$).
• It increases linearly with distance $r$ from the principal point toward the image margins.
• It increases directly with feature height $h$.
• It decreases inversely with higher flying altitudes $H$.
• Points above datum displace outward; points below datum displace inward.