L6 – Transformations in the Euclidean Plane NGEN06(TEK230) – Algorithms in Geographical Information Systems by: Irene Rangel, updated by Sadegh Jamali (source: Lecture notes in GIS, Lars Harrie) 1 L6- Transformations

Background In geographic analysis, it is common that you have to transform coordinates between coordinate systems. The most GIS programs provide a set of transformations (congruent, similarity, affine, projective and polynomial). To choose the right transformation it’s important to know their differences from a geometrical viewpoint. 2 L6- Transformations

A note of warning! All transformations in this lecture are of type Empirical Transformations. Empirical transformations are used when the true relationships between the coordinate systems are unknown. 3 L6- Transformations

Aim  … to understand the most common empirical transformations in plane to be able to choose a suitable one.  … to understand the concept of common point and be able to decide which common points to use in an application (how many and the distribution). 4 L6- Transformations

Content 1.Transformations in the Euclidean Plane 2.Congruence (Euclidean) transformation 3.Similarity transformation 4.Affine transformation 5.Projective transformation 6.Polynomial transformation 7.Applying empirical transformations 8.Choice of empirical transformations 5 L6- Transformations

Transformations in the Euclidean Plane How to classify the transformations?Invariant Invariant: a property that is maintained in the transformation 6 L6- Transformations

Transformations in the Euclidean Plane Congruence (Euclidean)Shape, sizePosition3 Similarity (Helmert)ShapeSize, position4 AffineParallelismShape, size, position6 ProjectiveDouble ratio propertyParallelism, shape, size, position8 PolynomialTopological relationshipsGeometrical properties- Type of transformation Invariant (maintain)Does not maintain No. of unknown parameters 7 L6- Transformations

Transformations in the Euclidean Plane To be able to use the transformations, unknown parameters have to be determined. Common points: known points in both coordinate systems Each common point provides 2 relationships (x- and y- directions). In theory: no. of common points = (½). no. of unknowns. In practice: no. of common points > (½). no. of unknowns. Common points How? How many common points? 8 L6- Transformations

Congruence (Euclidean) transformation It models translation and rotation. 9 L6- Transformations

Congruence (Euclidean) transformation model or in matrix form: 10 L6- Transformations

Similarity transformation It models translation, rotation, and uniform scaling. 11 L6- Transformations

Similarity transformation model Or in matrix form: 12 L6- Transformations

Affine transformation It models translation, rotation, non-uniform scaling in different directions, and shear. non-uniform scaling shear 13 L6- Transformations

Affine transformation model 14 L6- Transformations

Projective transformation Figure: Projective transformation. The planes do not have to be parallel. 15 L6- Transformations

Projective transformation model The double ratio property Note: to determine the 8 transformation parameters (a1, a2, a3, b1, b2, b3, d1, d2) at least 4 common points are required. 16 L6- Transformations

Polynomial transformation model Polynomial transformation (degree = n) The total number of unknown parameters are 6, 12, and 20 for transformation of degree 1, 2, and 3 respectively. 17 L6- Transformations

Polynomial transformation model n = 1 no. of parameters = 6 n = 2 no. of parameters = 12 n = 3 no. of parameters = 20 n = 4 no. of parameters = ? n = 5 no. of parameters = ? 18 L6- Transformations

Applying empirical transformations Example: how to estimate affine parameters? Original affine transformation Rewritten affine transformation where 19 Or L6- Transformations Step 1

Equations system for solving the unknown parameters using 3 common points Note: In case of having more than 3 common points, Least Squares technique is used. 20 L6- Transformations Step 2

Common points Each common point gives 2 relationships. Theoretically, it is enough with 3 common points to determine 6 unknowns in affine transformation. But in practice you should always use twice as many points as is theoretically required. The common points should always be evenly-distributed and circumvent the area of interest. 21 L6- Transformations

Standard error of the empirical transformation The estimated standard error (transformation quality at the common points) 22 L6- Transformations

Choice of empirical transformation In general, it depends on type of geometric distortions between the two coordinate systems 23 L6- Transformations

Choice of empirical transformation Transformation between two uniform scale coordinate systems where the scale is the same in both systems ---> Congruence Transformation. Application: transformation between two geodetic reference systems expressed in the same map projection and with equal scale between the systems (e.g. the same measuring techniques and instruments). 24 L6- Transformations

Choice of empirical transformation Transformation between two uniform scale coordinate systems where the scale might differ between the systems ---> Similarity Transformation. Application: transformation between two geodetic reference systems expressed in the same map projection but with different scales between the systems (e.g. NOT the same measuring techniques and instruments) 25 L6- Transformations

Choice of empirical transformation Transformation between two coordinate systems where at least one of the systems might not have a uniform scale ----> Affine transformation. Application: transformation used for digitizing a paper map or a photograph that might have different scales in the two main directions (due to e.g. non-uniform shrinkage of paper). 26 L6- Transformations

Choice of empirical transformation Transformation between two coordinate systems where one is close to a projection of the other ---> Projective Transformation. Application: suitable for rectification of aerial images that are not taken along the plumb line. Rectification is a technique used in photogrammetry to obtain distortion-free images. 27 L6- Transformations

Choice of empirical transformation Transformation between two coordinate systems where one has a bad (or completely unknown) geometry ----> Polynomial transformation. Application: - geocoding of remote sensing images that are normally a mosaic of several minor parts. - digitizing historical maps or other maps with an unknown map projection. 28 L6- Transformations