Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 1 W.D. Philpot, Cornell University, Fall 2012

Transcription

1 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 1 7. PHOTOGRAMMETRY Photogrammetry is the art, science and technology of obtaining reliable information about physical objects and the environment through processes of recording, measuring, and interpreting photographic images and patterns of recorded radiant electromagnetic energy and other phenomena. Metric photogrammetry is the process of making precise measurements from photos, while interpretive photogrammetry is the process of recognizing and identifying objects and judging their significance through careful and systematic analysis (from P. Wolf, Elements of Photogrammetry). 7.1 Basic Optics A simple, converging lens will produce an image at a distance, i, from the lens (along the optic axis) of an object at a distance, o, on the opposite side of the lens (Figure 7.1a). The object distance and image distance are related by: (7.1) o i f where f is the focal length of the lens. The size of the image is v tan and the size of the object is u tan. When discussing lenses, the relative size of the image and object defines the magnification of the system, and the magnification, m, is given by: i image size m (7.2) o object size Figure 7.1: Formation of an image by a simple converging lens: a) the object is relatively close to the lens, b) the object distance is much greater than the image distance. In remote sensing, the distance to the object is normally the altitude, h, of the aircraft or satellite which is much greater than the image distance (o >> i) in which case the image distance approaches the focal length (i f). In such systems we refer to the scale of the image rather than its magnification, and define scale in terms of the focal length of the lens system: s f (7.3) h Since scale is expressed as a ratio of the image size to the object size objects in a "large scale" image (or map) objects are larger than in a "small scale" image. Conversely, the total area covered by a large scale image is less than that of a small scale image.

2 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 2 Figure 7.2: Illustration of imaging for a nadir-viewing camera. For remote sensing systems, it is common to represent the imaging system as in Figure 7.2 where the camera is nadir viewing, the ground is flat, and the object distance is the altitude of the lens of the camera. In Figure 7.2, the ground length imaged is labeled the FOV (field of view) and the image size is labeled w. The image scale may then be represented as: w object size s f h FOV image size (7.4) Another fundamental characteristic of an imaging system is its spatial resolution. Spatial resolution defines the ability of an optical system to distinguish between two distinct points. For photographic systems, resolution is normally defined in terms of line pairs (lp) per unit length. When a bar pattern such as that shown in Figure 7.3. The resolution of the system (including the lens system and film or sample spacing of the detector ) is then the smallest separation, d, for which a pair of bars can be distinguished on the film. Figure 7.3: A bar pattern with spacing, d, between consecutive bars. The spatial resolution of a digital frame camera will depend on the spacing of the elements in the detector array. The closer the spacing, the higher the resolution will be for a given lens system and focal length. For film the resolution depends on the size of the grains of the silver salts that form the image. The smaller the grain size, the higher the spatial resolution of the film and the slower the film speed. The highest spatial resolutions available for aerial photographic films are typically 200 lp mm -1. This corresponds to a typical grain size of ~1 m. The detector spacing for a comparable digital frame camera would be ~5-10 m. The line pair definition of spatial resolution is reasonable for film, where the image formation depends on grains of silver salts whose orientation is random. It is less appropriate for

3 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 3 digital imaging systems which have arrays of (usually) square detection elements. Since the spacing of the elements is different along the rows and columns of the array than along a diagonal, the spatial resolution actually varies with the orientation of the array. Spacing of the detectors in a high resolution CCD array is roughly an order of magnitude greater than that of high resolution film. At present it is also difficult to create large arrays. For the time being, large format (e.g., 9-inch) aerographic film is inherently capable of both higher resolution and greater area coverage than a CCD array for an equivalent lens system. The digital systems are able to compensate, however, with lens design and adjustments to the focal distance. 7.2 Photogrammetry definitions "Photogrammetry is the art, science, and technology of obtaining reliable information about physical objects and the environment through processes of recording, measuring, and interpreting photographic images and patterns of recorded radiant electromagnetic energy and other phenomena." Paul Wolfe, Elements of Photogrammetry. The above definition, a classic, incorporates all aspects of qualitative interpretation as well as quantitative measurements made from photos. More precisely, the definition includes: 1) metric photogrammetry: making precise measurements from photos. 2) interpretative photogrammetry: recognizing and identifying objects and judging their significance through careful and systematic analysis. We will consider only metric photogrammetry here. There is also a distinction between "hardcopy" or "analog" photogrammetry work done using hardcopy photographic products and generally involving the use of optical and mechanical systems and "digital" or "softcopy" photogrammetry in which the image is in digital form and all manipulations are performed by computer in a virtual domain. The same basic geometric principles apply to all forms of photogrammetry. 7.3 Definitions The discussion of photogrammetry is reduced here to a series of definitions and figures describing the main concepts. In general, the idea is to relate the geometry of the image to the geometry of the ground, removing (or accounting for) distortions due to the motion and attitude of the aircraft, and the fact that the ground is not flat General terms Camera: A lightproof chamber or box in which the image of an exterior object is projected upon a sensitized plate or film through an opening usually equipped with a "lens," a "shutter" and a variable "aperture." Mapping Camera or Surveying Camera: An aerial or terrestrial camera which is equipped with means for maintaining and indicating the interior orientation of the photographs with sufficient accuracy for surveying purposes. (Interior orientation: calibrated focal length, location of calibrated principal point, and calibrated lens distortion). Shutter: Camera mechanism which controls the length of time that the sensitized plate or film is exposed to light; (types: focal-plane shutter, between-the-lens shutter, louver shutter).

4 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 4 Lens: One or more pieces (lens "elements") of glass, or other transparent material, shaped to form an image of an object by refraction Terms Associated with Camera Lens Aperture Stop: Camera mechanism which limits the size of the pencil of light rays passing through the lens system. Adjusting the size of the aperture stop regulates the brightness of the image. (The aperture stop may be considered as the effective lens diameter; it is sometimes called the "diaphragm"). Relative Aperture: The ratio of the equivalent focal length to the effective lens diameter; (also called the "f number" or "speed of lens"). (e.g., if the focal length = 6 in. and the lens diameter = 1.5 in., then the relative aperture = 4; this would be expressed as f/4). Aberration: Failure of an optical system to bring all light rays received from a point object to a single image point; (types: astigmatism, lateral chromatic aberration, longitudinal chromatic aberration, spherical aberration, coma, curvature of field, distortion). Nodal Point: One of two points on the lens (optical) axis such that, when all "object distances" ("o") are measured from one nodal point and all "image distances" ("i") are measured from the other nodal point, they satisfy the lens relation: (1/f) = (1/i) + (1/o)... where f = focal length. The total distance between an object and its image equals o + i plus or minus a small distance called the "nodal-point separation." Focal Plane: The plane perpendicular to the lens axis, in which images of exterior objects are focused by the lens. Equivalent Focal Length: The distance measured along the lens axis from the rear nodal point to the plane of best average definition over the entire field used in the camera (i.e., the distance between the real nodal point and the focal plane). Calibrated Focal Length: An adjusted value of the equivalent focal length which is determined during camera calibration; the C.F.L. is computed such that the effect of lens distortion is distributed over the entire field used in the camera. Resolving Power (of lens): An expression of lens definition, usually stated as the maximum number of lines per mm (line-space pairs) that can be seen as separate lines in the image. Angle of Coverage: The apex angle of the cone of light rays passing through the front nodal point of the lens. Lenses are generally classified according to their angle of coverage: "narrowangle," less than 60 degrees; "normal-angle," 60 to 75 degrees; "wide-angle," 75 to 100 degrees; "super-wide-angle," greater than 100 degrees Terms Associated with Vertical Aerial Photographs Vertical Photograph: An aerial photograph made with the camera axis vertically downward; generally refers to photographs with less than 2 to 3 degrees tilt. Perspective Center: The point of origin or termination of bundles of perspective light rays. (In a perfectly adjusted lens-camera system, the "exterior" and "interior" perspective centers correspond, respectively, to the front and rear nodal points). Ground Nadir: The point on the ground that is vertically beneath (directly below) the perspective center of the camera lens.

5 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 5 Nadir Point or Photographic Nadir: The point on the photograph which corresponds to the ground nadir. The point at which a vertical line (plumb line) from the perspective center to the ground nadir intersects the photograph. Fiducial Marks: Index marks, usually four, which are rigidly connected with the camera lens through the camera body and which form images on the photographic negative. Fiducial Axes: Straight lines joining opposite fiducial marks on a photograph. X-Axis: The fiducial axis that is more nearly in the direction of flight. Y-Axis: The fiducial axis that is more nearly perpendicular to the direction of flight. Photograph Perpendicular: The perpendicular from the interior perspective center (real nodal point) to the plane of the photograph. Principal Distance: The length of the photograph perpendicular. This distance is equal to the calibrated focal length corrected for both the enlargement or reduction ratio and the film shrinkage or expansion. Principal Point: The point on the photograph located at the foot of the photograph perpendicular. Photograph Center: The point on the photograph that is located at the intersection of the fiducial axes. (The photograph center is sometimes called the "center of collimation.") In a perfectly adjusted camera, the photograph center and the principal point are identical (i.e., unless camera calibration indicates otherwise, the principal point is generally assumed to coincide with the photography center). Scale: The ratio of a distance on a photograph or map to its corresponding distance on the ground. Although photographic scale varies from point to point (due to relief and/or tilt), it is usually taken as f/h'... where f = focal length and H' = height of camera above mean ground elevation. Scale may be expressed as a ratio (1:24,000), a fraction (1/24,000), or an equivalence (1 in. = 2,000 ft.). Relief Displacement: If a ground object is below (above) the elevation of the ground nadir, its image will be displaced radially inward (outward) with respect to the nadir point. Relief displacements may be measured accurately from the photography center if two conditions are met: (1) the photography is truly vertical (i.e., the nadir and principal points coincide), and (2) the camera is in perfect adjustment (i.e., the principal point and photograph center coincide) Terms Associated with Stereoscopic Photographs Stereoscopic Pair or Overlapping Photographs: Two photographs which were taken from two different camera stations and which include all or part of the same scene ("stereopair"). That portion of the scene which is common to both photographs (i.e., the overlap area) is called the "stereoscopic model," "stereomodel" or "model." The stereomodel may also be referred to as the "stereoscopic image," since the mental impression of three dimensions can only be obtained in the overlap portion of the photographs. Overlap: The amount by which one photograph covers the same area as covered by another (customarily expressed as a percentage). The overlap between aerial photographs in the same flight line is called "end lap," and the overlap between photographs in adjacent, parallel flight lines is called "side lap."

6 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 6 Floating Mark: (Associated with stereometers or stereoscopic plotting machines). A mark (dot or cross) seen as occupying a position in the three-dimensional space formed by the stereoscopic fusion of a stereopair and used as a reference mark in examining or measuring the stereo-model. Stereometer or Parallax Bar: A measuring device containing a micrometer movement by means of which the separation of two index marks (i.e., floating marks) can be changed to measure parallax difference on a stereopair of photographs. Air Base: The line joining two adjacent camera stations, or the length of this line. Photobase: The length of the air base as represented on a photograph. Corresponding Image Points: The images of the same object point on two or more photographs (formerly called "conjugate points"). Parallax: The apparent displacement of the position of an object with respect to a reference point or system, when viewed from a different position of observation. Absolute Stereoscopic Parallax: On a stereopair of equal principal distances, the absolute stereoscopic parallax of a point is the algebraic difference between the distances of the two corresponding images from their respective nadir points, the two distances being measured parallel to the air base; (generally denoted "parallax, but also called "x parallax," "linear parallax" and "horizontal parallax"). Parallax Difference: The difference in the absolute stereoscopic parallaxes of two points imaged on a pair of photographs. Customarily used to determine the elevation difference between the two objects. Y Parallax: The difference between the perpendicular distances of the two corresponding images of a point from the vertical plane containing the air base. The existence of y parallax is an indication of tilt in either or both photographs and/or a difference in flight height (camera elevation); its existence will interfere with stereoscopic examination. Angular Parallax, Parallactic Angle or Angle of Convergence: The angle subtended by the eye base of the observer at the object viewed. Base Height Ratio: The ratio (B:H) between the air base and the flight height of a stereopair of photographs Terms Associated with Tilted Aerial Photographs Tilt: The angle at the perspective center between the photograph perpendicular and the plumb line; also, the dihedral angle between the plane of the photograph and the horizontal plane. Tilt may be separated into tilt about the x-axis (representing "roll" of the aircraft, and noted by Greek letter omega, ), and tilt about the y-axis (representing "pitch" of the aircraft, and noted by the Greek letter phi, ). The direction of tilt is expressed by "swing" (when referred to the photograph axes) or by "azimuth" (when referred to the exterior coordinate system). Principal Plane: The vertical plane through the perspective center containing the photograph perpendicular and the nadir point (and the "isocenter"). Principal Line: The trace (intersection) of the principal plane upon the photograph; also, the line on the photograph which passes through the principal point and the nadir point (and the "isocenter").

7 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 7 Isocenter: The point on the photograph where the bisector of the angle of tilt strikes the photograph; (located in the principal plane as well as on the principal line). The isocenter is the unique point common to the plane of the tilted photograph, its principal plane, and the plane of the assumed truly vertical photograph taken from the same camera station and having an equal principal distance; (i.e., the isocenter is located at the intersection of 3 planes). Photograph Parallel: The image of any horizontal line which is perpendicular to the principal plane. Thus, all photograph parallels are perpendicular to the principal line. Isometric Parallel: The photograph parallel that passes through the isocenter; also, the line of intersection between the plane of the photograph and a horizontal plane (or a truly vertical photograph) having an equal principal distance from the same perspective center. (The isometric parallel was once called the "axis of tilt"; in present usage, the axis of tilt is defined as a line perpendicular to the principal plane, but passing through the perspective center--not through the isocenter). Tilt Displacement: The displacement of images on a tilted photograph, being radially outward (inward) with respect to the isocenter if the images are on the low (high) side of the isometric parallel. (The "low" side of a tilted photograph is the side closer to ground). Swing: The angle about the principal point of a photograph, measured clockwise from the positive y-axis to the principal line at the nadir point. Swing also refers to a rotation of the photograph (or photo-coordinate system) around the photograph perpendicular (or photo-graph z-axis, or exterior Z-axis). In air navigation, swing represents aircraft rotation about the aircraft's vertical axis and is referred to as "yaw" or "crab." (Swing is often noted by the Greek letter kappa, ). Azimuth: The horizontal angle measured clockwise about the ground nadir from a reference plane (usually the ground-survey north meridian) to the principal plane. (The azimuth of a photograph is the ground-survey direction of tilt, while swing is the direction of tilt with respect to the photograph axes). 7.4 Figures Figure 7.4: Frame Camera. The lens assembly selects and refracts light rays originating from points in object space in such a manner that the rays come to focus as points in the focal plane.

8 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 8 The camera cone supports the lens assembly and prevents stray light from entering the camera. The camera body, either an integral part of the camera cone or detachable from it, houses the drive mechanism and other equipment associated with film transport. The upper surface of the fiducial marks, which are either attached to the camera body or to a separate cone within the camera cone, define the location of the focal plane. The camera magazine houses the film (or plates) and the film-flattening device. b o a focal plane (plane of infinite focus ) focal length nodal points nodal point separation infinite distance A O B Figure 7.5: Nodal Points of a 4-element lens. The lens of virtually all cameras consists of several lens elements. The nodal points are two (imaginary) points on the optical axis such that when a light ray from object space is directed to the front nodal point, it will emerge from the rear nodal point with no apparent directional change. The distance between the rear nodal point and the focal plane is the nominal focal length. The distance along the optical axis to the plane of best average definition is called the equivalent focal length. The calibrated focal length is an adjusted value (calculated, not measured) of the equivalent focal length which accounts for lens distortion over the entire field.

9 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 9 +y y-fiducial axis fiducial marks x-fiducial axis -x + x photographic center fiducial marks Figure 7.6: Aerial Photograph. Although the photograph center (located at the intersection of the fiducial axes) is generally assumed to be identical to the principal point, this must be verified through camera calibration before accurate measurements can be made. On a truly vertical photograph, the principal point and nadir point coincide -y Negat ive f S = f/ H rear nodal point front nodal point reduced pr int cont act print enlarged print H dat um plane Figure 7.7: Relation to Datum. The scale (S) of the photograph based at the datum plane is equal to the focal length (f) divided by the height (H) of the camera above the datum. S = f/h Horizontal distances on the datum are equal to corresponding horizontal distances measured on the photograph, multiplied by the scale factor H/f. Scale = image distance/ground distance 1:24,000 ==> 1 = 2,000 ft. 1 mm = 24,000 mm

10 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 10 small scale 1:250,000 large scale 1:12,000 ELEVAT ION Above datum = H Above A: H' A= H - h Above B: H' B = H - h A B f PHOT O a' b' a b S = f /(H - h) B h B A B' GROUND h A A' DAT UM Figure 7.8: Relation to Ground Space. The scale of the photograph based at the datum is equal to f/h, but the scale of the photo-graph at any point above (or below) the datum will not be equal to f/h. In Figure 5, the photographic scale at point "A" is equal to f/(h-h A ), and the scale at point "B" is equal to f/(h-h B ). The scale of a vertical photograph at any point on the photograph is equal to the focal length divided by the elevation difference between the camera and corresponding ground point.

11 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 11 Figure 7.9: Relief Displacement. On a map, where the scale is based on the datum, point A would be identical to point A' because A' is the position of A on the datum. On a photograph, however, the image of point A (a) will be displaced from the image of point A' (a') by an amount related to the elevation difference between A and A' Relief displacement on an aerial photograph is always in a radial direction from the nadir point, and it is measured as such. On a truly vertical photograph, relief displacement is commonly measured with respect to the photograph center, since the principal point is generally assumed to be identical to the photograph center. Letting r equal the radial distance between the nadir point and point a, and letting r' equal the radial distance between the nadir point and point a', then relief displacement of A would be r-r', or d A. By similar triangles: Then: f/h = r'/r and f/(h-h) = r/r r' = Rf/H and r = Rf/(H-h) As defined, d a = r a r a ' then: d a = (Rfh a ) / H(H-h a ) But since: r a = Rf/(H h a ) Then substituting:d a = (rh a )/H or h a = (Hd/ a )/r Similarly, the height of a tree could be determined as being: h T h B = d T (H-h B )/(r) where: H = camera height above datum r = radial distance between nadir point and t. h T = height of tree top above datum r b ' = radial distance between nadir point and b. h B = height of tree base above datum d T = r b -r b '

12 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 12 Figure 7.10: Parallax measurements f = focal length of photos 1 and 2 H = elevation above datum of photos 1 and 2 A, B = ground points a, b = images of A and B on photo 1 a', b' = " " " " " " " 2 N 1, N 2 = ground nadir points of photo 1 and photo 2, respectively n1, n2 = images on N1 and N2, respectively, on photo 1; n1 is the nadir point of photo 1. n1', n2' = images of N1 and N2, respectively, on photo 2; n2' is the nadir point of photo 2. x1, y1 and x2, y2 = fiducial axes of photo 1 and photo 2, respectively X1, Y1 and X2, Y2 = rotated fiducial axes of photo 1 and photo 2, respectively, whereby the X1 and X2 axes are colinear with the flight line, and the Y1 and Y2 axes are perpendicular to the flight line. da, db, dn2 = distances measured on photo 1 with respect to X1, Y1 axes; (all positive values). da', db', dn1' = distances measured on photo 2 with respect to X2, Y2 axes; (da' and dn1' are negative). D12 = distance between nadir points (between origins of rotated axes), as established by construction of model. DA, DB, DN1, D 2 = distances between corresponding image points, as established by construction of model.

13 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 13 Parallax measurements are made on stereopairs of vertical aerial photographs which have equal focal lengths and equal camera heights above the datum. The absolute stereoscopic parallax (or simply, "parallax") of a point imaged on both photos of the pair is determined as the algebraic difference between two distances. One distance is measured on one photo, the second distance is measured on the second photo, and both distances are measured parallel to the direction of flight as it exists between the two photos. Each distance is measured from a nadir point to one of the two corresponding image points. Similar to relief displacement measurement on one photo, when certain conditions are met (or assumed met), the photographs centers can be used in place of the nadir points. Y1 DB Y2 photo 1 y1 y2 ph oto2 db b b' n1 dn2 n2 a DN2 X1 x1 DN1 n1' dn1' a' n2' x2 X2 da DA da' D12 Figure 7.11: Since both distances must be measured parallel to the flight direction, it is convenient to establish a new set of photo axes for each photo, such that each photo's new X axis will be parallel to and colinear with the flight direction. In Figure 7, the new (rotated) axes are noted X1, Y1 for photo 1 and X2, Y2 for photo 2. The X1 and X2 axes are colinear with each other as well as with the flight line, which is defined by the corresponding images of the two nadir points (n1, N2, N1', N2'); the Y1 and Y2 axes are perpendicular to the flight line, each passing through their respective nadir point (n1 or N2'). All X-coordinate measurements made with respect to these new photo axes will be parallel to the flight line. On photo 1, the X-coordinate of a point imaged to the right of the Y1 axis will be positive, and the X-coordinate of a point imaged to the left of the Y1 axis will be negative. Similarly, on photo 2, distances measured from Y2 axis toward the direction of flight are positive, while distances measured from the Y2 axis opposite to the direction of flight are negative.

14 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 14 For point b (photo 1) and its corresponding image point, b' (photo 2), the parallax of ground point B is the algebraic difference between the X-coordinate of b on photo 1 and the X- coordinate of b' on photo 2; or, pb = Xb - Xb' pb = (db) - (db') = db - db1 With regards to points a and a', it will be seen that the X-coordinate of a' is a negative value. The parallax of ground point A would be: or, pa = Xa - Xa' pa = (da) - (-da') = da + da' The parallax difference between two (non-corresponding) image points is the value used to determine the elevation difference between the imaged ground points. For example, the parallax difference between points a and b is simply the difference between their absolute stereoscopic parallaxes: or pab = pb pa pab = (db - db') - (da + da') In practice, it is more convenient to measure DA and DB instead of db, db', da, and da', since: (db - db') - (da + da') = DA - DB pab = DA - DB Note that pab is a positive quantity, meaning that the ground elevation increases from point A to point B. If, however, pab had been negative--indicating a decrease in elevation--pba would have been used; where: pba = pa - pb = (da + da') - (db - db') = DB - DA The following relations are derived from the figure: Photo X1 Coordinate Photo X2 Coordinate Point Coordinate Distance Point Coordinate Distance n1xn1 dn1 = 0 n1' Xn1' -nd1' n2xn2 dn2 n2' Xn2' dn2' = 0 a Xa da a' Xa' -da' b Xb db b' Xb' db' Ground Point Absolute Stereoscopic Parallax (Parallax) N1pn1 = Xn1 - Xn1' = (dn1) - (-dn1') = dn1' N2pn2 = Xn2 - Xn2' = (dn2) - (dn2') = dn2 Apa = Xa - Xa' = (da) - (-da') = da + da' = D12 - DA Bpb = Xb - Xb' = (db) - (db') = db - db' = D12 - DB

15 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 15 Ground Points Parallax Difference N1 to N2 pnln2 = pn2 - pn1 = dn2 - dn1' = DN1 - DN2 N1 to A pn1a = pa - pn1 = da + da' - dn1' = DN1 - DA N1 to B pn1b = pb - pn1 = db - db' - db1' = DN1 - DB A to N2 pan2 = pn2 - pa = dn2 - da - da' = DA - DN2 N2 to B pn2b = pb - pn2 = db - db' - dn2 = DN2 - DB A to Bpab = pb - pa = db - db' - da - da' = DA - DB Of the several formulae used for computing ground point elevation differences from parallax measurements the following are generally more accurate: where: hwz = (Hw) (pwz) (pw + pwz) w and z =image points of ground points W and Z, with point W at a lower elevation than point Z. hwz =elevation difference from W to Z. Hw =elevation difference between camera and W. H = camera height above average ground surface elevation. pw = absolute stereoscopic parallax of W. pwz = parallax difference between W and Z. b = average photo base; average distance between nadir points and corresponding nadir points; (dn2 + dn1 )/2 or (pn1 + pn2)/ 2 or (H') (pwz) ( b + pwz)

16 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 16 In practice, however, where pwz is relatively small, the following approximation is often employed for slide-rule computations: hwz = (H') (pwz)/b (pwz)/(b) pers pec tive center (rear nod al point) photograph perpendic ular principal plane t f photo principa l line -x t 2 n t 2 i p S +x p - princ ipal point i - isocenter n - nadir point Figure 7.12: Tilted aerial photograph f - focal length s - swing t - angle of tilt n - nadir point p - principal point i - isocenter tan (t/2) = (pi)/f ; (pi) = f tan (t/2) tan (t) -y ground nadir = (pn)/f ; (pn) = f tan (t) isometric paral lel for angles of tilt, t, less than 5 degrees, (pi) is approximately equal to (pn)/2; that is, for t less than 5 degrees, the distance between the principal point and the isocenter is approximately equal to the distance between the isocenter and the nadir point.

17 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 17 The scale of a tilted photograph changes uniformly throughout the photograph. If the scale near the center is correct, then the scale is too small on the high side and too large on the low side. Although the scale changes along the principal line (i.e., in the direction of tilt), the scale does not change along any line that is perpendicular to the principal line. As such, points at different distances from the isometric parallel will be at different scales (in accordance with their distance from the isometric parallel), but points equidistant from and on the same side of the isometric parallel (i.e., points along any line which is parallel to the isometric parallel) will be at the same scale. The scale of any image on a tilted photograph can be expressed as: S = (f - d sin t)/(h-h) Where "d" is the perpendicular distance between the isometric parallel and the image, being positive on the upward side of the photograph.

18 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 18 o pers pec tive center t B A equivalent vertic al photo d' d d'' Figure 7.13: Tilt displacement f n' i n f P p - principal point of tilted photo n - nadir point of tilted photo i - isocenter of tilted photo a, b, c, d, e - points on tilted photo; (a and d are on principal line) a', d', n' - corresponding points on an equivalent vertical photo; (Note that n' is both the nadir point and the principal point of the vertical photo) a", b", c", d", e" - points derived by rotating the equivalent vertical photo into the plane of the tilted photo f - focal length of tilted and equivalent vertical photo t - angle of tilt A, B - angles, as shown I - angle, about isocenter, from principal line to point a a'' a' tilted photo

19 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 19 pers pec tive center equivalent vertical photo d' d d'' i a a'' a' tilted photo principal line t t radial displac ement measured from is ocenter Figure 7.14: Tilt displacement in 3-dimensions isometric parallel

20 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 20 e e'' isometric paral lel principa l line d d'' n i p I a a'' b b'' c, c'' Figure 7.15: Tilt displacement The tilted photo and its equivalent vertical photo are identical along the isometric parallel, where they intersect. At any other position on the tilted photo, the image of a point will be displaced either outward or inward with respect to its equivalent position on the vertical photo. All tilt displacments take place in a radial direction from the isocenter. In Figure 8.9, the principal line is colinear with the x axis (cf., Figure 8.10). Ground points A and D appear at a and d on the tilted photo and at a' and d' on the equivalent vertical photo. If the vertical photo is rotated about the isometric parallel until is is in the plane of the tilted photo, point a' would fall at a", and point d', at d". The displacements caused by tilt are a" to a (toward the isocenter) and d" to d (away from the isocenter). In the general case, the tilt displacement of any point on the upward side of the isometric parallel (e.g., point b) is directed toward the isocenter (b" to b) and is given by the following: d u = f(tan(t+a) - 2 tan (t/2) - tan (A)) cos (I) For points lying on the downward side of the isometric parallel (e.g., point e), the tilt displacements are directed away from the isocenter (e" to e) and are given by the following: d ḋ = f(tan(b) - 2 tan (B-t) - tan (t/2)) cos (I) Note that since points a and d are on the principal line, cos (I) = 1.0

21 Philipson & Philpot: Remote Sensing Fundamentals Photogrammetry 21 isometric parallel 2, Figure 7.16: Relief and Tilt displacement p - principal point; n - nadir point; i - isocenter n p i princ ipal line The position of each point marked 1 represents the image of a ground point at the datum; that marked 2 represents the position after the image has undergone relief displacement (radial, with respect to the nadir point); and that marked 3 represents the position after the image has undergone tilt displacement (radial, with respect to the isocenter). Notes compiled primarily from: 1. American Society of Photogrammetry Manual of photogrammetry. 3rd Edition. 2 vols. George Banta Co., Menasha, Wisc pp. (see Chap. 24). 2. Moffitt, F.H Photogrammetry. International Textbook Co., Scranton, Pa. (see later edition of other photogrammetry texts).