Photogrammetry Fundamentals

Overview#

Photogrammetry is the science and technology of obtaining reliable measurements and spatial information about physical objects and the environment through the processes of recording, measuring, and interpreting photographic images. The word itself derives from the Greek roots photos (light), gramma (something drawn or written), and metron (to measure) -- literally, "measuring from light-drawings."

"Photogrammetry is the art, science, and technology of obtaining reliable information about physical objects and the environment through the process of recording, measuring, and interpreting photographic images and patterns of electromagnetic radiant energy and other phenomena." -- ASPRS, Manual of Photogrammetry (5th Ed.), Ch. 1, p. 1

The discipline emerged in the mid-19th century when Aime Laussedat first demonstrated the use of terrestrial photographs for topographic mapping in 1849. Aerial photogrammetry became practical after World War I, when surplus aircraft and cameras were repurposed for civilian mapping. By the mid-20th century, photogrammetry had become the primary means of producing topographic maps worldwide.

Metric vs. Interpretive Photogrammetry

Photogrammetry divides broadly into two branches:

Surveying professionals are primarily concerned with metric photogrammetry -- the extraction of accurate three-dimensional coordinates and spatial relationships from imagery. Every formula, coordinate system, and adjustment procedure discussed in this article serves that quantitative goal.

"Metric photogrammetry involves making precise measurements from photos to determine the relative locations of points -- it is quantitative photogrammetry." -- Ghilani & Wolf, Elementary Surveying: An Introduction to Geomatics (15th Ed.), Ch. 27, p. 791

Types of Photogrammetry#

Terrestrial and Close-Range Photogrammetry

Terrestrial photogrammetry uses ground-based cameras to photograph objects from fixed, known positions. When the camera-to-object distance is short (typically less than 300 meters), the practice is specifically called close-range photogrammetry. Applications include:

• Architectural documentation and historic preservation
• Industrial measurement (ship hulls, machinery, structural deformation)
• Accident scene reconstruction
• Archaeological site recording
• Structural monitoring of dams, bridges, and tunnels

The mathematical principles are identical to aerial photogrammetry, but the camera geometry differs -- the optical axis is typically horizontal or oblique rather than near-vertical.

Aerial Photogrammetry

Aerial photogrammetry captures images from airborne platforms -- historically manned aircraft, and increasingly unmanned aerial systems (UAS). The camera is typically oriented with its optical axis near-vertical (within 3 degrees of the plumb line for vertical photography). This is the most widely practiced form of photogrammetry for mapping and is the primary focus of surveying exam preparation.

Satellite Photogrammetry

Satellite-based sensors (e.g., WorldView, Pleiades, SPOT) capture imagery from orbital altitudes. The geometry differs from frame cameras -- many satellite sensors use pushbroom (linear array) acquisition rather than frame-based imaging, which introduces different mathematical models. Ground sample distances (GSD) for commercial satellites now reach 30 cm or finer.

The Collinearity Condition#

The collinearity condition is the fundamental mathematical model of photogrammetry. It states that, at the instant of exposure, an object point, the corresponding image point, and the perspective center of the camera lens are all collinear -- they lie on a single straight line.

"The collinearity equations express the condition that an object point, its image, and the exposure station all lie along a straight line in three-dimensional space." -- Wolf, Dewitt & Wilkinson, Elements of Photogrammetry with Applications in GIS (4th Ed.), Ch. 11, p. 293

Mathematical Formulation

Let the object point have ground coordinates $(X, Y, Z)$, the exposure station (perspective center) have coordinates $(X_L, Y_L, Z_L)$, and the corresponding image point have coordinates $(x, y)$ in the image coordinate system with principal point at $(x_0, y_0)$ and focal length $f$. The collinearity equations are:

$x - x_0 = -f \cdot \frac{m_{11}(X - X_L) + m_{12}(Y - Y_L) + m_{13}(Z - Z_L)}{m_{31}(X - X_L) + m_{32}(Y - Y_L) + m_{33}(Z - Z_L)}$

$y - y_0 = -f \cdot \frac{m_{21}(X - X_L) + m_{22}(Y - Y_L) + m_{23}(Z - Z_L)}{m_{31}(X - X_L) + m_{32}(Y - Y_L) + m_{33}(Z - Z_L)}$

where $m_{ij}$ are the elements of the $3 \times 3$ rotation matrix $\mathbf{M}$ that relates the image coordinate system to the ground coordinate system. The rotation matrix is defined by three sequential rotations about the axes, parameterized by the angles $\omega$ (omega), $\phi$ (phi), and $\kappa$ (kappa):

• $\omega$ -- rotation about the $X$-axis (tip/roll)
• $\phi$ -- rotation about the $Y$-axis (tilt/pitch)
• $\kappa$ -- rotation about the $Z$-axis (swing/yaw)

Interior and Exterior Orientation

Each photograph is characterized by two sets of orientation parameters:

Interior orientation is established through camera calibration. Exterior orientation is determined through photogrammetric processes such as space resection, relative orientation, and aerotriangulation (bundle adjustment).

The collinearity equations contain six unknowns per photograph (the exterior orientation parameters). Each measured image point provides two equations. Therefore, a minimum of three well-distributed ground control points visible on a single photograph are required to solve for its exterior orientation -- a process known as space resection.

"The minimum number of ground control points needed for space resection is three, but additional points are always desirable for redundancy and reliability." -- Mikhail, Bethel & McGlone, Introduction to Modern Photogrammetry, Ch. 5, p. 163

The collinearity equations also form the basis of bundle adjustment, the simultaneous least-squares solution of all exterior orientation parameters, ground point coordinates, and (optionally) interior orientation parameters from all measured image points across an entire block of photographs. Bundle adjustment is the standard method for aerotriangulation in modern photogrammetry.

Stereo Geometry and Parallax#

Stereoscopic Viewing

Human depth perception relies on binocular vision -- the brain fuses two slightly different views of the same scene (one from each eye) into a three-dimensional perception. Photogrammetry replicates this principle using overlapping photographs taken from different positions along a flight line. The lateral shift in position between two successive exposure stations is called the air base ($B$).

When two overlapping photographs of the same terrain are viewed through a stereoscope (or digitally with 3D display systems), the observer perceives a three-dimensional stereomodel of the terrain.

"The three-dimensional impression created by viewing overlapping photographs stereoscopically forms the basis for all stereoscopic measurement in photogrammetry." -- Wolf, Dewitt & Wilkinson, Elements of Photogrammetry with Applications in GIS (4th Ed.), Ch. 8, p. 195

Parallax

Parallax is the apparent shift in position of an object when viewed from two different vantage points. In photogrammetry, parallax is the key to extracting elevation information from stereo pairs.

• x-parallax (absolute stereoscopic parallax): The difference in the $x$-coordinates of the same point as imaged on the left and right photographs of a stereo pair. This parallax is directly related to the elevation of the point. Points at higher elevations exhibit greater x-parallax.

• y-parallax: The difference in $y$-coordinates of the same point on the two photographs. Ideally, y-parallax is zero for a perfectly aligned stereo pair. Non-zero y-parallax indicates tilt, unequal flying heights, or other orientation errors.

Parallax Equations for Height

For a vertical photograph stereo pair with flying height $H$ above datum, air base $B$, and focal length $f$, the elevation $h$ of a point above datum can be determined from its parallax. Let $p$ be the absolute stereoscopic parallax of the point. Then:

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

The difference in elevation between two points, $A$ and $B$, can be computed from their parallax difference $\Delta p = p_A - p_B$:

$\Delta h = h_A - h_B = \frac{H - h_B}{p_B} \cdot \Delta p$

This relationship is the foundation of stereoscopic height measurement. The precision of height determination depends on the magnitude of the parallax difference, which in turn depends on the base-to-height ratio ($B/H$). Larger base-to-height ratios provide better height accuracy.

Scale of a Photograph#

The scale of a vertical aerial photograph depends on two quantities: the camera focal length $f$ and the flying height above the terrain $H'$:

$S = \frac{f}{H'}$

where $H' = H - h$ is the flying height above the terrain, $H$ is the flying height above the datum, and $h$ is the terrain elevation above datum. Because terrain elevation varies across the photograph, the scale is only truly uniform for flat terrain. For undulating terrain, the scale varies point by point.

"The scale of a vertical photograph varies from point to point, depending on the elevation of the terrain. A single-valued scale can only be given for a specific elevation." -- Ghilani & Wolf, Elementary Surveying: An Introduction to Geomatics (13th Ed.), Ch. 27, p. 806

Worked Example

A vertical aerial photograph is taken with a camera having a focal length of $f = 152.4 \text{ mm}$ from a flying height of $H = 3{,}048 \text{ m}$ above mean sea level. What is the photo scale at a point with ground elevation $h = 305 \text{ m}$?

$S = \frac{f}{H - h} = \frac{152.4 \text{ mm}}{3{,}048 - 305 \text{ m}} = \frac{0.1524 \text{ m}}{2{,}743 \text{ m}} = \frac{1}{18{,}000}$

The photo scale at that point is approximately $1:18{,}000$.

If the terrain elevation were instead $h = 610 \text{ m}$:

$S = \frac{0.1524}{3{,}048 - 610} = \frac{0.1524}{2{,}438} = \frac{1}{16{,}000}$

The higher terrain yields a larger scale (less ground area covered per unit photo distance), illustrating how scale varies with elevation.

Relief Displacement#

Relief displacement is the radial shift in the image position of a point caused by its elevation above (or below) a reference datum. Objects with height (buildings, trees, terrain above datum) appear to lean radially outward from the center of the photograph (the nadir point or principal point for a truly vertical photograph).

Relief Displacement Formula

The magnitude of the relief displacement $d$ is given by:

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

where:

• $d$ = relief displacement on the photograph
• $r$ = radial distance from the principal point to the image of the top of the object
• $h$ = height of the object (or elevation above datum)
• $H$ = flying height above the base of the object (or above datum)

Key properties of relief displacement:

• It increases linearly with the radial distance from the photo center
• It increases with object height
• It decreases with greater flying height
• It is always directed radially outward from the photo center
• At the exact center of the photograph ($r = 0$), there is no relief displacement regardless of object height

"Relief displacement is radial from the principal point on a truly vertical photograph. It causes images of objects that have height above the datum to be displaced outward from the photo center." -- Wolf, Dewitt & Wilkinson, Elements of Photogrammetry with Applications in GIS (4th Ed.), Ch. 6, p. 147

Worked Example

On a vertical photograph taken from $H = 3{,}000 \text{ m}$ above datum, a building's rooftop image falls $r = 85.0 \text{ mm}$ from the principal point. If the building is $h = 45 \text{ m}$ tall, the relief displacement is:

$d = \frac{85.0 \times 45}{3{,}000} = 1.275 \text{ mm}$

The building's roof image is displaced 1.275 mm radially outward from where its base would appear if the building had no height.

Coordinate Systems#

Photogrammetric work requires careful management of multiple coordinate systems. The principal systems are:

Image (Photo) Coordinate System

A two-dimensional coordinate system defined in the plane of the photograph, with its origin at the principal point (where the optical axis intersects the image plane). The $x$-axis is typically aligned with the direction of flight, and the $y$-axis is perpendicular. Units are typically millimeters or micrometers. Fiducial marks (on film cameras) or sensor geometry (on digital cameras) establish this system.

Camera (Sensor) Coordinate System

A three-dimensional coordinate system centered at the perspective center (rear nodal point of the lens). The $x_c$ and $y_c$ axes are parallel to the image coordinate axes, and the $z_c$-axis coincides with the optical axis, pointing toward the ground. The focal length $f$ is the distance from the perspective center to the image plane along $z_c$.

Object (Ground) Coordinate System

The three-dimensional coordinate system in which ground control points and final map coordinates are expressed. This may be a local project coordinate system, a State Plane coordinate system, UTM, or a geodetic (latitude/longitude/height) system. The relationship between camera coordinates and ground coordinates is defined by the exterior orientation parameters through the rotation matrix $\mathbf{M}$ and translation vector.

Transformation Between Systems

The collinearity condition equations embody the transformation from ground coordinates to image coordinates. The inverse problem -- computing ground coordinates from image measurements -- requires at least two photographs (stereo) because a single image point defines a ray in space but not a unique point along that ray. The intersection of two such rays from overlapping photos determines the three-dimensional position of the ground point.

The transformation chain for a full photogrammetric workflow proceeds: image coordinates $\rightarrow$ camera coordinates $\rightarrow$ model coordinates (via relative orientation) $\rightarrow$ ground coordinates (via absolute orientation). In modern bundle adjustment, the intermediate model coordinate step is bypassed -- the collinearity equations relate image coordinates directly to ground coordinates.

Key Takeaways#

• Photogrammetry is the science of obtaining reliable spatial measurements from photographs, with applications spanning topographic mapping, engineering surveys, and cultural documentation.
• The collinearity condition is the mathematical foundation: at the instant of exposure, the object point, image point, and perspective center are collinear. The collinearity equations relate image coordinates to ground coordinates through six exterior orientation parameters per photograph.
• Interior orientation (focal length, principal point, lens distortion) defines the camera's internal geometry; exterior orientation ($X_L$, $Y_L$, $Z_L$, $\omega$, $\phi$, $\kappa$) defines the camera's position and attitude in space.
• Stereoscopic parallax -- the shift in x-coordinates between overlapping photographs -- is the basis for height measurement. The parallax equation $h = H - Bf/p$ enables elevation determination from stereo pairs.
• Photo scale varies with terrain elevation: $S = f/(H - h)$. Higher terrain yields larger scale; lower terrain yields smaller scale on the same photograph.
• Relief displacement causes elevated objects to appear to lean radially outward from the photo center, with magnitude $d = rh/H$. This effect is exploited for height measurement but must be corrected for planimetric accuracy.
• Multiple coordinate systems (image, camera, model, ground) must be carefully managed and related through orientation parameters and transformations.
• The base-to-height ratio ($B/H$) controls the strength of height determination -- larger ratios yield better elevation precision.

References#

1. ASPRS. Manual of Photogrammetry (5th Ed.). American Society for Photogrammetry and Remote Sensing, 2004.
2. ASPRS. Manual of Photogrammetry (6th Ed.). American Society for Photogrammetry and Remote Sensing, 2013.
3. Wolf, P.R., Dewitt, B.A. & Wilkinson, B.E. Elements of Photogrammetry with Applications in GIS (4th Ed.). McGraw-Hill, 2014.
4. Ghilani, C.D. & Wolf, P.R. Elementary Surveying: An Introduction to Geomatics (13th Ed.). Pearson, 2012. Chapter 27.
5. Ghilani, C.D. & Wolf, P.R. Elementary Surveying: An Introduction to Geomatics (15th Ed.). Pearson, 2018. Chapter 27.
6. Mikhail, E.M., Bethel, J.S. & McGlone, J.C. Introduction to Modern Photogrammetry. John Wiley & Sons, 2001.
7. Kraus, K. Photogrammetry: Geometry from Images and Laser Scans (2nd Ed.). Walter de Gruyter, 2007.