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Analytical Photogrammetry Hande Demirel, PhD, Assistant Prof. Istanbul Technical University (ITU), Faculty of Civil Engineering, Geodesy and Photogrammetry Department Division of Photogrammetry Coordinate Transformations Division of PHOTOGRAMMETRY A problem frequently encountered in photogrammetric works is conversion from one rectangular coordinate system to another. This is because photogrammetrists commonly determine coordinates of unknown points in convenient arbitrary rectangular coordinate system. Spring 2008, Istanbul Technical University, Istanbul-Turkey Coordinate Transformations Division of PHOTOGRAMMETRY The arbibitrary coordinates must be then converted to a final system such as; Photo coordinate system Ground coordinate system The procedure for converting from one coordinate system to another is known as coordinate transformation. Spring 2008, Istanbul Technical University, Istanbul-Turkey Coordinate Transformations Division of PHOTOGRAMMETRY The procedure requires that some points have their coordinates known (or measured) in both the arbitrary and the final coordinate systems. Such points are called control points. Spring 2008, Istanbul Technical University, Istanbul-Turkey Coordinate Transformations Division of PHOTOGRAMMETRY Two-dimensional conformal coordinate transformation: The term two-dimensional means that the coordinate systems lie on plane surfaces. A conformal transformation is one in which true shape is preserved after transformation. To perform: At least two points to be known in both the arbitrary and final coordinate system. If more than two control points are available, an improved solution may be obtained by applying the method of least squares. Spring 2008, Istanbul Technical University, Istanbul-Turkey Two-dimensional conformal coordinate transformation: Division of PHOTOGRAMMETRY Three basic steps: Scale change Rotation Translation Spring 2008, Istanbul Technical University, Istanbul-Turkey Coordinate Transformations with Redundancy Division of PHOTOGRAMMETRY Least squares procedure aX bY TE E vE aY bX TN N vN If n points are available whose coordinates are known in both systems, 2n equations may be formed containning the four unknown transformation parameters. Spring 2008, Istanbul Technical University, Istanbul-Turkey Two-dimensional Affine Coordinate Transformation Division of PHOTOGRAMMETRY Two-dimensional Affine Coordinate Transformation is only a slight modification of the two-dimensional conformal transformation, to include different scale factors in the x and y directions and to compensate for nonorthogonality (nonperpendicularity) of the axis system. The affine transformation achieves these additional features by including two additional unknown parameters for a total of six. Spring 2008, Istanbul Technical University, Istanbul-Turkey Three-Dimensional Conformal Coordinate Transformation Division of PHOTOGRAMMETRY It is essential in photogrammetry for two basic problems: To transform arbitrary stereomodel coordinates to a ground or object space system To from continuous three dimensional “strip models” from independent streomodels. Seven independent parameters: Three rotation angles A scale factor Three translation parameters. Spring 2008, Istanbul Technical University, Istanbul-Turkey More on Coordinate Transformations Division of PHOTOGRAMMETRY Wolf, P., Dewitt, B., 2000, McGraw- Hill, Elements of Photogrammetry Coordinate Transformations pg.518-550 Spring 2008, Istanbul Technical University, Istanbul-Turkey Projective Equations Division of PHOTOGRAMMETRY The numerical resection problem involves the transformation (rotation and translation) of the ground coordinates to photo coordinates for comparison purposes in the least squares adjustment. Before we begin this process, lets derive the rotation matrix that will be used to form the collinearity condition. Spring 2008, Istanbul Technical University, Istanbul-Turkey Projective Equations Division of PHOTOGRAMMETRY In photogrammetry, the coordinates of the points imaged on the photograph are determined through observations. The next procedure is to compare these photo coordinates with the ground coordinates. On the photograph, the positive x-axis is taken in the direction of flight. For any number of reasons, this will most probably never coincide with the ground X- axis. Spring 2008, Istanbul Technical University, Istanbul-Turkey Projective Equations Division of PHOTOGRAMMETRY The origin of the photographic coordinates is at the principal point which can be expressed as X' x x o Y' y y o Z' f where: x, y are the photo coordinates of the imaged point with reference to the intersection of the fiducial axes xo, yo are the coordinates from the intersection of the fiducial axes to the principal point f is the focal length Spring 2008, Istanbul Technical University, Istanbul-Turkey Projective Equations Division of PHOTOGRAMMETRY Since the origin of the ground coordinates does not coincide with the origin of the photographic coordinate system, a translation is necessary. We can write this as X 1 X X L Y Y Y 1 L Z1 Z Z L where: X, Y, Z are the ground coordinates of the point XL, YL, ZL are the ground coordinates of the ground nadir point Spring 2008, Istanbul Technical University, Istanbul-Turkey Projective Equations Division of PHOTOGRAMMETRY Thus, in the comparison, both ground coordinates and photo coordinates are referenced to the same origin separated only by the flying height. Note that the ground nadir coordinates would correspond to the principal point coordinates in X and Y if the photograph was truly vertical. Spring 2008, Istanbul Technical University, Istanbul-Turkey Direction Cosines Division of PHOTOGRAMMETRY If we look at figure, we can see that point P has coordinates XP, YP, ZP. The length of the vector (distance) can be defined as Vector OP in 3-D space University, Istanbul-Turkey Spring 2008, Istanbul Technical Direction Cosines Division of PHOTOGRAMMETRY 1 OP X Y Z 2 P 2 P 2 2 P The direction of the vector can be written with respect to the 3 axes as: XP cos OP Y cos P OP Z cos P OP Spring 2008, Istanbul Technical University, Istanbul-Turkey Direction Cosines Division of PHOTOGRAMMETRY These cosines are called the direction cosines of the vector from O to P. This concept can be extended to any line in space. For example, figure 2 shows the line PQ. Here we can readily see that the vector can be defined as: XQ XP PQ YQ YP QP Z Z Q P Spring 2008, Istanbul Technical University, Istanbul-Turkey Line vector PQ in space. Direction Cosines Division of PHOTOGRAMMETRY The length of the vector becomes PQ X Q X P YQ YP Z Q Z P 2 2 2 1 2 and the direction cosines are XQ XP cos PQ YQ YP cos PQ ZQ ZP cos Spring 2008, Istanbul Technical University, Istanbul-Turkey PQ Direction Cosines Division of PHOTOGRAMMETRY If we look at the unit vector as shown in figure, one can see that the vector from O to P can be defined as OP xi y j zk Spring 2008, Istanbul Technical University, Istanbul-Turkey Unit vectors. Direction Cosines Division of PHOTOGRAMMETRY and the point P has coordinates (x, y, z)T. Given a second set of coordinates axes (I, J, K), one can write similar relationships for the same point P. Each coordinate axes has an angular relationship to each of the i, j, k coordinate axes. Spring 2008, Istanbul Technical University, Istanbul-Turkey Direction Cosines Division of PHOTOGRAMMETRY For example, figure shows the relationship between J and i .The angle between the axes is defined as (xY). Since i has similar angles to the other two axes, one can write the unit vector in terms of the direction cosines as: i I cosxX i i J cosxY i K cosxZ Rotation between Y and x axes. Spring 2008, Istanbul Technical University, Istanbul-Turkey Direction Cosines Division of PHOTOGRAMMETRY Similarly, we have cosyX coszX j cosyY k coszY cosyZ coszZ Then, the vector from O to P can be written as Spring 2008, Istanbul Technical University, Istanbul-Turkey Direction Cosines Division of PHOTOGRAMMETRY cosxX cosyX coszX OP x cosxY y cosyY z coszY cosxZ cosyZ coszZ cosxX cosyX coszX x X cosxY cosyY coszY y Y cosxZ cosyZ coszZ z Z Spring 2008, Istanbul Technical University, Istanbul-Turkey Direction Cosines Division of PHOTOGRAMMETRY This can be written more generally as X Rx To solve these unknowns using only three angles, 6 orthogonal conditions must be applied to the rotation matrix, R. All vectors must have a length of 1 and any combination of the two must be orthogonal [Novak, 1993]. Thus, designating R as three column vectors [R = (r1 r2 r3)], we have r1T r1 r2t r2 r3T r3 1 r1T r2 r2T r3 r3T r1 0 Spring 2008, Istanbul Technical University, Istanbul-Turkey Direction Cosines Division of PHOTOGRAMMETRY Rotation Combinations Three angles of tilt, swing and azimuth completly define the angular orientation of a tilted photograph in space. If the tilt angle is zero.... Spring 2008, Istanbul Technical University, Istanbul-Turkey