Vertical Photography

Other than a map and similar to the images we perceive with our eyes, a photograph—either analog or digital—is the result of a central projection, also known as singlepoint perspective. The distances of the central point of convergence—the optical center of the camera lens, or exposure station—to the sensor on one side and the object on the other side determine the most basic property of an image, namely its scale.

Figure 3-1 shows the ideal case of a vertical photograph taken with perfect central perspective over completely flat terrain. The triangles established by a ground distance D—e.g. the distance D2 between points A and P—and the flying height above ground Hg on the terrain side and by the corresponding photo distance d and the focal length fon the camera side are geometrically similar for any given D and d: the scale S or 1/s of the photograph is the same at any point.

(Equation 3-1)

Once the image scale S is known, this equation can be resolved for D in order to calculate the width or length of the ground area covered by the image by using the image width or length on the film or sensor chip as d.

Small-Format Aerial Photography

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Format The Aerial Photograph
FIGURE 3-1 Vertical photograph taken over completely flat terrain. The optical axis, which intersects the image plane at its center o, meets the ground in a right angle at the principal point P. Note that the respective distances between all points are the same on the ground and on the image.

Several other important characteristics of the photograph can be derived from the basic relationships described in Figure 3-1. By transforming Equation (3-1), the commonly used equation for Hg can be obtained:

If the image format is square, the ground area A covered by the photograph can be derived from squaring Equation 3-1. For SFAP cameras, this usually would not be the case, so the rectangular image format (with ¿l as the image length dw as the image width) must be taken into account:

(Equation 3-3)

For digital images, the ground sample distance GSD (see Chapter 2) determines the spatial resolution or smallest visible detail in the photograph. The exact size of the picture element (sensor cell), which is needed for GSD calculation, can be determined from the manufacturer's information on the sensor size in pixels and millimeters (see Eq. 2-3 in Chapter 2).

Because the relationship is a direct linear one, any change in Hg or fwill change the scale and the image distances by the same factor. For example, doubling the flying height results in an image with half the scale S and halved image distances d, tripling the focal length will enlarge the scale and image distances by a factor of three. In reality, most aerial photographs—especially by far the most SFAP—deviate from the situation in Figure 3-1 for three reasons:

• The ground is not completely flat, i.e. the distance between image plane and ground varies within the image.

• The photograph is not completely vertical, i.e. the optical axis is not perpendicular to the ground.

• The central projection is imperfect due to lens distortions, i.e. the paths of rays are bent when passing through the lens.

All three situations spoil the similarity of the triangles and result in scale variations and hence geometric distortions of the objects within the image. The last two problems can be minimized with modern survey and manufacturing techniques for professional high-tech survey cameras and mounts, but may be quite severe for the platforms and cameras often used in small-format aerial photography. The first problem, scale variations and geometric distortions caused by varied terrain, does not depend on camera specifications and occurs with any remote sensing images. However, it also normally would be more severe for SFAP than for conventional aerial photography because of the lower flying heights and thus relatively higher relief variations.

Figure 3-2 illustrates the effects of different elevations on the geometry of a vertical photograph. All points lying on the same elevation as the image center have the same scale—points above this horizontal plane are closer to the camera and therefore have a larger scale, points below this horizontal plane are farther away and have a smaller scale. At the same time, the positions of the points in the image are shifted radially away from (for higher points) or toward (for lower points) the image center (o)—compare the different positions of a and a', and b and b', respectively in Figure 3-2. This happens because they appear under another angle than they would if they were in the same horizontal plane as the ground principal point, seemingly increasing or reducing their distance to the point at the image center.

This so-called relief displacement increases with the distance to the image center (see Fig. 2-7). It is inversely proportional to the flying height and the focal length: the displacement is less severe with larger heights and longer focal lengths because in both cases the rays of light are comparatively less inclined. This effect of varying relief displacement can be exploited according to the requirements of image analysis. Images for monoscopic mapping or image mosaicking—demanding minimum distortion by relief displacement—are best taken using a telephoto lens from a greater flying height. However, images utilized for stereoscopic viewing and analysis should have higher relief displacement and stereo-parallaxes and are best attained

Relief Displacement Principal Point

FIGURE 3-2 Vertical photograph taken over variable terrain. The elevation of the principal point P determines the horizontal plane of local datum. Points lying on this plane remain undistorted, whereas points above or below are shifted radially with respect to the image center. Note that the horizontal distances Di-D4 are the same in the object space but not in the image.

FIGURE 3-2 Vertical photograph taken over variable terrain. The elevation of the principal point P determines the horizontal plane of local datum. Points lying on this plane remain undistorted, whereas points above or below are shifted radially with respect to the image center. Note that the horizontal distances Di-D4 are the same in the object space but not in the image.

with wide-angle lenses and lower heights (see section on base-height ratio and Fig. 3-11 below).

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Responses

  • MATHIAS
    How to calculate ground sample distance from scale and focal length?
    8 years ago

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