## Measuring and Mapping from Single Photographs

Individual measurements of lengths and sizes—the width of a brook, the diameter of a tree crown, the distance between vegetation patches—can be taken easily in analog or digital photographs using Equation 3-1, if the image shows an object or distance with known length. This could, for example, be a scale bar as in Figure 2-3 or the (calculable) distance between two ground control points as in Figure 9-8. Adapting Equation 3-1, the required ground measure D can be calculated by comparing its image length d with that of the scale bar.

D = d x Dscale bar/dscale bar (Equation 3-7)

Other possibilities exist for measuring object heights in single photographs, but they are restricted to objects rising vertically from the ground. Here, the relief displacement between the top and bottom point of the object, e.g. a building or tree, can be measured and related to the object's distance from the image principal point in order to calculate the object's height (Jensen, 2007; Lillesand et al., 2008). If the exact time, date and place of image acquisition is known, another possibility is to use the shadow cast by the object for calculating its height (Jensen, 2007). Neither method can be used for objects where the top and bottom points are not perpendicularly aligned and both visible. The height of a mountain, sloping river bank, or rock face of uncertain verticality cannot be determined with these methods, making them rather useless for geoscientific applications.

It has to be kept in mind that the accuracy of such measurements from single images depends not only on the measurement precision but also more importantly on the validity of the equation, i.e. closeness of the image acquisition situation to perfect central projection over flat terrain. Even for flat terrain, SFAP images rarely meet these requirements, as the optical axis is rarely exactly vertical and SFAP camera lenses often show considerable distortions (see Chapter 6).

Usually, continuous spatial mapping is more of interest than single measurements when analyzing aerial photographs. In order to be suitable as a base for mapping or monitoring the photographs have to be geometrically corrected and georeferenced. This can be done either by polynomial rectification using a set of ground control points or by orthorectification, as follows.

### 3.4.1.1. Polynomial Rectification by GCPs

Polynomial equations formed by ground control point coordinates and their corresponding image-point coordinates are used in order to scale, offset, rotate and warp images and fit them into the ground coordinate system (Mather, 2004). The highest exponent of the polynomial equation (the polynomial order) determines the degree of complexity used in this transformation—1st order transformations are linear and can take account of scale, skew, offset and rotation, while 2nd and 3rd order transformations (quadratic and cubic polynomials) also may correct nonlinear distortions. Because the polynomials are computed from the GCP points only and then applied to the entire image, they only produce good results if the GCP locations and distribution adequately represent the geometric distortions of an image. For aerial photographs, which are subject to radial displacement, this is usually difficult to achieve.

Vertical or oblique images of flat terrain (Figs. 3-1 and 3-3) can be quite successfully rectified with 1st or 2nd order polynomials, but the relief distortions present in images of variable terrain are much more difficult to correct (consider the impossibility of fitting a 2nd or 3rd order polynomial to the surface shown in Fig. 3-2). Although higher orders of transformation can be used to correct more complicated terrain distortions, the risk of unwanted edge effects by extrapolation beyond the GCP-covered area increases.

Further similar methods of image rectification may utilize ground control points, i.e. spline transformation or rubber sheeting, which optimize local accuracy at the control points, but all of these methods have in common that the rectification of the image areas between the GCPs is interpolated by the polynomial equation and not a direct function of radial relief displacement. Thus, the rectified image would not be truly and completely distortion-free. while polynomial rectification by GCPs may well be sufficient for low-distortion images or applications with limited demand for accuracy, seriously distorted images and more precise applications require full modelling of the distortion parameters (relief displacement, lens distortion, and image obliqueness) for geocorrection.

3.4.1.2. Orthorectification or Orthophoto Correction

The continuously changing distortions caused by relief displacement for images of variable terrain cannot be removed sufficiently with polynomial rectifications, because the surface is too complex to be described by simple mathematical algorithms. A complete and differential rectification of the distorted photograph into a plani-metrically correct image with orthographic (= map) projection can be achieved only if the exact amount and direction of displacement for each pixel can be calculated and removed. Following the principles shown in Figures 3-2 and 3-12, this is possible if the interior and exterior orientation of the camera and the terrain heights are known. Orthorectification procedures make use of digital elevation models (DEMs) in relation to which the photographs are oriented in space so that the relief displacement (with the added effect of image obliqueness and lens distortion) of each single pixel can be determined. In the new, ortho-rectified image file, each pixel is then placed in its correct planimetric position.

Depending on the type and source of the DEM, the elevation values either describe the ground surface (digital terrain model or DTM) or the true surface including all objects rising above the terrain (e.g. woodland, buildings; digital surface model or DSM) (Kasser and Egels, 2002; Jensen, 2007). Thus, it can be differentiated between conventional orthophotos and true orthophotos, where the latter present an image truly geocorrected for all terrain and object elevations. For orthorectifying SFAP images, a DEM with appropriately high resolution is normally not available from external sources—thus, the best solution would be to generate a DEM from the SFAP images themselves first and subsequently use this for orthophoto correction. If the DEM is generated by automatic elevation extraction procedures, it would necessarily be a DSM, resulting in a true orthophoto and thus in the most geometrically accurate image product achievable. A brief overview of DEM generation is given below in this chapter.

If a single rectified photograph or orthophoto does not fully cover the study area, an aerial mosaic may be constructed by stitching the georeferenced images together (controlled mosaic, wolf and Dewitt, 2000; see Chapter 11.3.3). With semi-controlled mosaics, even areas devoid of ground control points can be bridged by registering the images of the gap areas visually to the georeferenced images around them. There are several image-processing techniques that help to minimize a jigsaw-puzzle appearance of the resulting mosaic. The seamlines between the individual mosaic pieces can be manually or automatically placed in order to be most inconspicuous, and radiometric matching techniques may be used for taking color and brightness differences into account.

The resulting GCP-rectified or orthorectified images or mosaics may now serve a variety of purposes—as photo-maps annotated with reference grid and coordinates, for measuring distances and areas of individual objects, as a background for compiling thematic map layers in a GIS environment, or as components of a time series for change monitoring. For further details and examples, see Chapters 11 and 13.

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