Geometry of the Conventional Aerial Photograph: Simplifying Assumptions, Scale, and Component Relations (Especially for GATE-Geospatial 2022)

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A conventional aerial photograph is generally regarded as a central projection, the properties of which provide the basis for the mathematical treatment of photogrammetry.

Geometrical of Conventional Aerial Photograph: Assumptions and Reality

But such consideration is only an idealization and is conditional upon the following assumptions:

  • The projection centre is in or near the lens. No such centre exists, and the bundle of rays passes through different parts of the lens as shown in Figure below.
Function of a Lens in Focusing a Point as an Image
  • A pencil of rays is represented geometrically as only a straight line as shown in Figure below. These rays deviate in some way in the lens because of lens distortion.
Aerial Camera Lens System Showing Nodal Points
  • A point in the object space is imaged by a plane in the negative space. In fact, there is no image plane but an image zone with a definite thickness due to the emulsion; and a collection of points rather than one point is imaged.
  • The emulsion base (either film or glass plate) is taken to be a perfectly flat surface and highly stable. In fact, deformations occur.

Scale

  • By making these assumptions, the aerial photograph is a graphic record of the light rays or, in mathematical terms, a presentation of the relationship between the aerial camera and the ground.
  • Thus, for a truly vertical photograph (Figure below) , the geometry of the relationship can be expressed as

In order that a sharp image can be obtained, Newton՚s Lens Equation must be satisfied, i.e.. :

Where f is the focal length of the camera. As the air camera has a fixed focus for the object distance at infinity, the value for b becomes infinity, thus:

Which means that the image distance is equal to the focal length when the object distance is at infinity. From this the scale (S) of the aerial photograph can be found to be, by substitution in 2.1:

Relation between the Photograph Negative and the Terrain

Photogrammetry Photogrammetry is the Science and Technology …
  • From the above Figure, it is noteworthy that the optical axis of the camera is vertical to the terrain plane which is flat (i.e.. no relief) . The perspective centre of the lens which gives the geometric centre of the aerial photograph is, therefore, point O.
  • This is generally known as the principal point, which can be fixed on the photograph by joining the opposite pairs of fiducial marks registered on the sides of the aerial photograph by the aerial camera during exposure.

The Relation between the Aircraft and the Aerial Photograph Produced

The Relation between the Aircraft and the Aerial Photograph …
  • The principal point is normally taken as the origin of a coordinate system (known as the photo or plate coordinate system, as distinct from the ground or terrestrial coordinate system) . The X-axis of the photograph is the line between opposite fiducial marks which are parallel to the direction of the aircraft flight, whilst the Y-axis is that line normal to the X-axis (Fig 2.18) .
  • To consider the other direction of movement of the aircraft, i.e.. up and down, a third axis - the Z-axis - needs to be introduced. This also passes through the principal point and is perpendicular to the plane of the photograph. This coincides with the optical axis of the camera. Thus, any movement of the aircraft can be defined with reference to this three-axis coordinate system.

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