Photogrammetry and Metric Analysis from Images

The pinhole camera model treats the camera as a central projection: every scene point projects along a ray through the camera centre (the optical centre) to land on the image plane. The transformation from a 3D scene point to a 2D image coordinate involves the camera's intrinsic matrix (focal length and sensor centre) and the extrinsic parameters (camera position and orientation in world coordinates). Together these form the camera projection matrix P, and once P is known, the forward and inverse mappings are fully determined.

Real lenses deviate from the ideal pinhole. Radial distortion causes straight lines to bow outward (barrel distortion, common in wide-angle lenses) or inward (pincushion distortion, more common in telephoto lenses). Tangential distortion arises from lens elements not being perfectly aligned with the optical axis. Correcting these distortions before making measurements is essential: an uncorrected wide-angle image can introduce centimetre-scale errors into a measurement even at short range.

When neither camera calibration data nor a physical survey of the scene is available, metric information can still be extracted from a single image using projective constraints. The two main approaches are vanishing-point analysis and planar homography.

1. Identify vanishing points
   Find at least two sets of parallel horizontal lines in the scene (floor tiles, road markings, building edges) and two sets of vertical lines. Their image intersections give the vanishing points, from which the camera tilt, pan, and (with additional constraints) effective focal length can be estimated.
2. Derive the ground-plane homography
   With two horizontal vanishing points and one vertical vanishing point, plus at least one known reference distance on the ground plane (tile size, road marking width), a homography can be computed that maps image pixels on the ground plane to real-world metric coordinates. Objects on or near the ground can then be measured directly.
3. Measure with uncertainty
   Pixel localisation (selecting the exact pixel corresponding to a point) has an uncertainty of roughly plus or minus one pixel. This propagates through the homography to produce an uncertainty in the scene measurement. For a ground plane at 10 metres with a 4-megapixel camera covering a 10-metre wide field, locating a person's feet to within one pixel gives a ground position uncertainty of approximately 2-5 cm.

A critical limitation: homography applies only to objects lying exactly on the reference plane. A person's head is roughly 1.7-2 metres above the ground. Using a ground-plane homography to measure head position directly introduces a perspective error proportional to the height divided by the camera distance. This is why height estimation from CCTV is a separate, more involved procedure treated in the following topic.

When the case allows access to the original camera at the original location, calibration from a physical reference target is more accurate than reconstruction from scene geometry alone. The standard workflow uses a checkerboard or dot-grid calibration target photographed at multiple positions and orientations.

Zhang's checkerboard calibration method (1999) is the standard. Photographing a flat checkerboard from at least five different poses allows simultaneous estimation of focal length, principal point, and radial and tangential distortion coefficients using a closed-form initial estimate followed by non-linear refinement. The result is a reprojection error measured in pixels; values below 0.5 pixels are considered good for forensic work.

When a scene can be photographed systematically at the time of investigation, Structure-from-Motion produces a far more complete and accurate metric record than any single-image method. The pipeline begins by extracting feature keypoints (typically SIFT or ORB) from each photograph, matching them across image pairs, estimating relative camera poses, and then jointly optimising all camera positions and 3D point coordinates in a process called bundle adjustment.

The model is floating until ground control points (GCPs) with known real-world coordinates are included. Surveying at least three non-collinear GCPs with a total station or RTK-GPS pins the model to a coordinate system and provides the scale. After GCP registration, the dense point cloud can be measured to the accuracy of the GCP survey, typically within a few millimetres for a scene surveyed with a total station at close range.

The 2011 paper by Bramble, Comber, and Tidy in the Journal of Forensic Sciences described a systematic single-camera measurement approach applied to CCTV footage with explicit uncertainty quantification. The paper established procedural norms that UK and international practitioners have since adopted.

• The method must be described in enough detail that a second analyst can replicate the measurement from the same source data.
• The pixel resolution at the measurement location must be estimated, not assumed. A CCTV frame covering a 12-metre frontage with 640 horizontal pixels gives about 53 pixels per metre; at 10 metres, a door of 2 metres width occupies roughly 106 pixels.
• The uncertainty must be propagated from each contributing source: pixel localisation (typically plus or minus 1-2 pixels), reference dimension accuracy, and any camera calibration error.
• The conclusion must be expressed as a range with a stated confidence level, not as a point estimate with implicit certainty.