COORDINATE TRANSFORMATION. Lecture 6

Transcription

1 COORDINATE TRANSFORMATION Lecture 6 SGU 1053 SURVEY COMPUTATION 1

2 Introduction Geomatic professional are mostly confronted in their work with transformations from one two/three-dimensional coordinate system to another. This includes the transformation of geographic coordinates delivered by the surveyor into Cartesian map coordinates or the transformation from one 2D Cartesian (x,y) system of a specific map projection into another 2D Cartesian (x,y) system of a defined map projection. Integration of spatial data into one common coordinate system. 2

3 Coordinate Systems 2D Geographic Coordinates Latitude and Longitude The most widely used global coordinate system consists of lines of geographic latitude (phi) and longitude (lambda). 3

4 3D geographic Coordinates Latitude, Longitude and Height h 3D geographic coordinates are obtained by introducing the ellipsoidal height h to the system. The ellipsoidal height (h) of a point is the vertical distance of the point in question above the ellipsoid. It is measured in distance units along the ellipsoidal normal from the point to the ellipsoid surface. 3D geographic coordinates can be used to define a position on the surface of the Earth (point P) 4

5 Geocentric Coordinates (X,Y,Z) or 3D Cartesian The system has its origin at the mass-centre of the Earth with the X- and Y-axes in the plane of the equator. The X-axis passes through the meridian of Greenwich, and the Z-axis coincides with the Earth's axis of rotation. The three axes are mutually orthogonal and form a right-handed system. Geocentric coordinates can be used to define a position on the surface of the Earth (point P) 5

6 2D Cartesian Coordinates (X,Y) The 2D Cartesian coordinate system is a system of intersecting perpendicular lines, which contains two principal axes, called the X- and Y-axis. The horizontal axis is usually referred to as the X-axis and the vertical the Y-axis (note that the X-axis is also sometimes called Easting and the Y-axis the Northing). The intersection of the X- and Y-axis forms the origin. 6

7 2D Polar Coordinates,d Another possibility of defining a point in a plane is by polar coordinates,d. This is the distance d from the origin to the point concerned and the angle a between a fixed (or zero) direction and the direction to the point. The angle a is called azimuth or bearing and is measured in a clockwise direction. It is given in angular units while the distance d is expressed in length units. 7

8 Transformation: Geographic to Geocentric to Geographic Coordinates 8

9 Transformation: Geographic to Geocentric Coordinates The conversion from the latitude and longitude coordinates into the geocentric coordinates is rather straightforward and turns ellipsoidal latitude (ϕ) and longitude (λ), and possibly also the ellipsoidal height (h), into X,Y and Z, using 3 direct equations. If the ellipsoidal semi-major axis is a, semi-minor axis b, and inverse flattening 1/f, then: X = (υ+ h) cos ϕ cos λ Y = (υ + h) cos ϕ sin λ Z = [(1- e 2 ) υ + h] sin ϕ where υ is the prime vertical radius of curvature at latitude ϕ and is equal to υ = a /(1 e 2 sin 2 ϕ ) 0.5 e is the eccentricity of the ellipsoid where e 2 = (a 2 b 2 ) / a 2 = 2f f 2 9

10 Transformation: Geocentric to Geographic Coordinates The inverse equations for the reverse conversion are more complicated and require either an iterative calculation of the latitude and ellipsoidal height, or it makes use of approximating equations like those of Bowring. 10

11 2D Cartesian Coordinate Transformations 2D Cartesian coordinate transformations can be used to transform 2D Cartesian coordinates (x,y) from one 2D Cartesian coordinate system to another 2D Cartesian coordinate system. The three primary transformation methods are: the Conformal transformation, the Affine transformation, and the Polynomial transformation. 11

12 Conformal Transformation A conformal transformation is a linear (or first-order) transformation and relates two 2D Cartesian coordinate systems through a rotation, a uniform scale change, followed by a translation. The rotation is defined by one rotation angle ( ), and the scale change by one scale factor (s). The translation is defined by two origin shift parameters (x o,y o ). The equation is: The simplified equation is: X' = s X cos( ) - s Y sin( ) + x o Y' = s X sin( ) + s Y cos( ) + y o X' = ax - by + x o Y' = bx + ay + y o where a=s cos( ) and b=s sin( ). The transformation parameters (or coefficients) are a,b,x o,y o. 12

13 Affine Transformation An affine transformation is a linear (or first-order) transformation and relates two 2D Cartesian coordinate systems through a rotation, a scale change in x- and y-direction, followed by a translation. The transformation function is expressed with 6 parameters:one rotation angle (a), two scale factors, a scale factor in the x-direction (s x ) and a scale factor in the y-direction (s y ), and two origin shifts (x o, y o ). The equation is: X' = s x X cos(α) - s y Y sin(α) + x o Y' = s x X sin(α) + s y Y cos(α) + y o 13

14 The simplified equation is: X' = ax - by + x o Y' = cx + dy + y o where the transformation parameters (or coefficients) are a,b,c,d,x o,y o. 14

15 The diffference between a conformal and an affine transformation is illustrated in the figure below. Both are linear transformations which means that the lines of the grid remain straight after the transformation. The uniform scale change of the conformal transformation retains the shape of the original rectangular grid. The different scale in x and y-direction of the affine transformation changes the shape of the original rectangular grid, but the lines of the grid remain straight. 15

16 Polynomial Transformation A polynomial transformation is a non-linear transformation and relates two 2D Cartesian coordinate systems through a translation, a rotation and a variable scale change. The transformation function can have an infinite number of terms. The equation is: X' = x o + a 1 X+ a 2 Y+ a 3 XY + a 4 X 2 + a 5 Y 2 + a 6 X 2 Y+ a 7 XY 2 + a 8 X Y' = y o + b 1 X+ b 2 Y+ b 3 XY + b 4 X 2 + b 5 Y 2 + b 6 X 2 Y+ b 7 XY 2 + b 8 X

17 Polynomial transformations are sometimes used to georeference uncorrected satellite imagery or aerial photographs or to match vector data layers that don't fit exactly by stretching or rubber sheeting them over the most accurate data layer. The figure below shows a grid with no uniform scale distortions. It may occur in an aerial photograph, caused by the tilting of the camera and the terrain relief (topography). An approximate correction may be derived through a high-order polynomial transformation. The displacements caused by relief differences can be corrected using a Digital Elevation Model (DEM). 17

18 2D Cartesian coordinate transformations are generally used to assign map coordinates (x,y) to an uncorrected image or scanned map. The type of transformation (usually an affine transformation) depends on the geometric errors in the data set. This is illustrated in the below figure (a). After georeferencing, the image can be aligned (rectified) so that the pixels are exactly positioned within the map coordinate system (figure (b)). For each image pixel in the new coordinate system, a new pixel value has to be determined by means of an interpolation from surrounding pixels in the old image. This is called image resampling. 18

19 The parameters (or coefficients or unknowns) of a conformal, affine or polynomial transformation are usually computed with ground control points (GCPs) or common points (also called tie points) such as corners of houses or road intersections, as long as they have known coordinates in both systems. The number of control points required to determine the 4 parameters (a,b,x o,y o ) of a conformal transformation must be at least 2. An affine transformation requires at least 3 control points to determine the 6 parameters (a,b,c,d,x o,y o ), and 6 control points are required to determine the 12 parameters (x o,a1-a5,y o,b1-b5) of a simple second-order polynomial transformation. 19

20 Common Coordinate Transformation in Malaysia GDM2000 to PMGSN94 GDM2000 to EMGSN97 GDM2000 to MRT48 GDM2000 to BT68 (Sabah) GDM2000 to BT68 (Sarawak) GDM2000 to MRSOGDM GDM2000 to BRSOGDM GDM2000 to CassiniGDM PMGSN94 to MRT48 EMGSN97 to BT68 (Sabah) EMGSN97 to BT68 (Sarawak) MRT48 to MRSO BT68 to BRSO MRSO to Cassini 20