L6 Transformations in the Euclidean Plane

Transcription

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

2 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 9. BIM -> GIS transformations 2

3 Background In geographic analysis, it is common that you have to transform coordinates between coordinate systems, for example when digitizing a paper map or georeferencing remote sensing data. 3

4 Transform map or digitize paper map L6- Transformations Map in a different coordinate system. Old map with unknown reference system Source: Lantmäteriet Image coordinates SWEREF99 TM 4

5 Satellite image to SWEREF99 TM L6- Transformations Only image coordinates, no reference system Satellite image SWEREF99 TM 5

6 L5-Transformations in the L6- Euclidean Transformations Plane Satellite image to SWEREF99 TM Satellite image SWEREF99 TM 6

7 Background 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. 7

8 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. Do not use empirical transformation when there are known analytical relationships between the coordinate systems e.g. never use empirical transformations for map projections. 8

9 A note of warning! Do not use empirical transformation when there are known analytical relationships between the coordinate systems e.g. for map projections. SWEREF SWEREF99 TM Changing map projection: N,E (old projection) -> latitude,longitude -> N,E (new projection) 9

10 A note of warning! Do not use empirical transformation when there are known analytical relationships between the coordinate systems e.g. for map projections. SWEREF SWEREF99 TM SWEREF99 Changing map projection: N,E (old projection) -> latitude,longitude -> N,E (new projection) 10

11 Transformations in the Euclidean Plane L6- Transformations Original grid Congruence transformation Similarity transformation Affine transformation Projektive transformation Polynomial transformation How to classify the transformations? Invariant Invariant: a property that is maintained in the transformation 11

12 Transformations in the Euclidean Plane L6- Transformations Original grid Congruence transformation Similarity transformation Affine transformation Projektive transformation Polynomial transformation Type of transformation Invariant (maintain) Does not maintain No. of unknown parameters Congruence (Euclidean) Shape, size Position 3 Similarity (Helmert) Shape Size, position 4 Affine Parallelism Shape, size, position 6 Projective Double ratio property Parallelism, shape, size, position 8 Polynomial Topological relationships Geometrical properties - 12

13 Coordinate systems in the presentation For example transforming from image to a map projection Large (new) coordinate system L6- Transformations Y A Small (original) coordinate system y A x X 13

14 Coordinate systems in the presentation L6- Transformations Y y A A Source: Lantmäteriet Image coordinates x SWEREF99 TM X 14

15 Y Coordinate systems in the presentation Image: sometimes origin upper left x A A Source: Lantmäteriet y SWEREF99 TM X 15

16 Coordinate systems in the presentation L6- Transformations Y New coordinate system rotate A Original coordinate system To transform the point A from the original to the new coordinate system we need to move (translate) and rotate the original coordinate system. (Other properties may also change) translate X 16

17 Congruence (Euclidean) transformation L6- Transformations Models translation and rotation. 17

18 Congruence (Euclidean) transformation L6- Transformations We know x p, y p and we want to find X p, Y p Y Y p (X p, Y p ) X p X 18

19 Congruence (Euclidean) transformation model First along the X-axis L6- Transformations We need to move the origin of the small coordinate system. X 0 - translation along the x-axis 19

20 Congruence (Euclidean) transformation model We also need to find the distance X 0 to X p 20

21 Congruence (Euclidean) transformation model x p cos(α) Note, the old coordinates 21

22 Congruence (Euclidean) transformation model y p sin(α) Note, the old coordinates 22

23 Congruence (Euclidean) transformation model We also need to find the distance X 0 to X p x p cos(α) y p sin(α) 23

24 Congruence (Euclidean) transformation model Transformation of the x-coordinate 24

25 Congruence (Euclidean) transformation model Then along the Y-axis y p cos(α) Y 0 x p sin(α) 25

26 Congruence (Euclidean) transformation model Then along the Y-axis y p cos(α) x p sin(α) x p sin(α) 26

27 Congruence (Euclidean) transformation model Then along the Y-axis y p cos(α) x p sin(α) y p cos(α) 27

28 Congruence (Euclidean) transformation model Then along the Y-axis y p cos(α) x p sin(α) Y 0 x p sin(α) y p cos(α) 28

29 Congruence (Euclidean) transformation model X, Y, x, y without subscript since valid for all points 29

30 Congruence (Euclidean) transformation model or in matrix form: Note: to determine the 3 transformation parameters (X o, Y o and α) at least 2 common points are required. 30

31 Similarity transformation Models translation, rotation, and uniform scaling. 31

32 Similarity transformation model Derivation of equations similar to congruence but scale change (same in x and y) is included. L6- Transformations y*m*sin(α) x*m*sin(α) 32

33 Similarity transformation model NB! y*m*sin(α) Or in matrix form: Note: to determine the 4 transformation parameters (X o, Y o, m, and α) at least 2 common points are required. 33

34 Affine transformation It models translation, rotation, non-uniform scaling in different directions, and shear. non-uniform scaling shear 34

35 Affine transformation model Note that the figure is not correct in the Lecture notes! 35

36 Affine transformation model m x, m y 36

37 Affine transformation model α, β 37

38 Affine transformation model Move parallel to the y-axis, so it is still x*cosα 38

39 Affine transformation model Here we need both α and β to get the correct distance x*cos(α+β) 39

40 Affine transformation model Note: to determine the 6 transformation parameters (X o, Y o, m x, m y, α, and β) at least 3 common points are required. 40

41 Projective transformation Typically from camera. One plane is the film in the camera and the other plane the real World. Imagine two papers with a line passing both papers. The points will have different coordinates on the two paper planes. Figure: Projective transformation. The planes do not have to be parallel. 41

42 Projective transformation Common in image analysis. Less common in GIS rare with flat surfaces 42

43 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. 43

44 Polynomial transformation model Polynomial transformation (degree = n) No unique equation depends on the order of the polynomial. 44

45 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. 45

46 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 =? number of unknown parameters = (n + 1) (n + 2) 46

47 To be able to use the transformations the unknown parameters have to be determined. L6- Transformations Finding the unknown parameters - Common points For this we need common points = points that are known in both coordinate systems. Large (new) coordinate system Y A Small (original) coordinate system C B y a c x b X 47

48 Common points SWEREF99 TM Satellite image 48

49 How many common points? Y Large (new) coordinate system A Small (original) coordinate system C B y a c x b X Each common point provides 2 relationships: x- and y-directions Congruence transformation, three unknown. 49

50 How many common points? Y Large (new) coordinate system A Small (original) coordinate system C B y a c x b X Theoretically, 3 common points is sufficient to determine 6 unknowns in affine transformation. But in practice you should always use twice as many points as is theoretically required. A bad common point may ruin the transformation. 50

51 Distribution of common points Cover all area of interest. Area outside point extrapolation. Well distributed. Satellite image SWEREF99 TM 51

52 Applying empirical transformations Example: how to estimate affine parameters? L6- Transformations Step 1 Here only 3 points are used to simplify. Use more in reality. Original affine transformation Rewritten affine transformation Where: To make it linear. Note that it is the same as a polynomial transformation of degree 1. 52

53 Step 2 Equation system for solving the unknown parameters using 3 common points Matrix form: X = A*a where: X 1 X = X 2 X 3 1 x 1 y 1 A = 1 x 2 y 2 1 x 3 y 3 a 0 a = a 1 a 2 53

54 Step 2 (same for Y) Equation system for solving the unknown parameters using 3 common points Matrix form: Y = A*b where: Y 1 Y = Y 2 Y 3 1 x 1 y 1 A = 1 x 2 y 2 1 x 3 y 3 b 0 b = b 1 b 2 54

55 Step 3 (same for both X and Y) Solving the linear equation system Y 1 Y = Y 2 Y 3 1 x 1 y 1 A = 1 x 2 y 2 1 x 3 y 3 b 0 b = b 1 b 2 In Matlab: >> Y = A*b >> b = A\Y 55

56 More than 3 common points L6- Transformations Remember that we should have more than 3 common points. Then we get an overdetermined equation system: X 1 = a 0 + a 1 x 1 + a 2 y 1 X 2 = a 0 + a 2 x 2 + a 2 y 2 X 3 = a 0 + a 3 x 3 + a 2 y 3 X 4 = a 0 + a 4 x 4 + a 2 y 4 X 5 = a 0 + a 5 x 5 + a 2 y 5 X 6 = a 0 + a 6 x 6 + a 2 y 6 Vector a does not change X 1 X = X 2 X 3 X 4 X 5 X 6 1 x 1 y 1 A = 1 x 2 y 2 1 x 3 y 3 1 x 4 y 4 1 x 5 y 5 1 x 6 y 6.. a 0 a = a 1 a 2 56

57 More than 3 common points L6- Transformations Remember that we should have more than 3 common points. Then we get an overdetermined equation system: More than 3 common points means that there is no unique solution. Use least squares technique. In Matlab: >> a = A\X X 1 X = X 2 X 3 X 4 X 5 X 6 1 x 1 y 1 A = 1 x 2 y 2 1 x 3 y 3 1 x 4 y 4 1 x 5 y 5 1 x 6 y 6.. a 0 a = a 1 a 2 57

58 Finding the original transformation parameters Original affine transformation Rewritten affine transformation X 0 = a 0 Y 0 = b 0 Where: α β α 58

59 Finding the original transformation parameters Original affine transformation Rewritten affine transformation Where: m x 1 m y 59

60 Finding the original transformation parameters Original affine transformation Rewritten affine transformation Where: Note that above only holds when α < 90 degrees 60

61 Evaluating the accuracy of a transformation Standard error of the empirical transformation 61

62 Evaluating the accuracy of a transformation The estimated standard error is based on the common points ONLY! 62

63 Choice of empirical transformation In general, it depends on the type of geometric distortions that may exist between the two coordinate systems. For example if translation, rotation and uniform scale changes are required, Use similarity transformation. 63

64 Choice of empirical transformation Transformation between two uniform scale coordinate systems where the scale is the same in both systems ---> Congruence Transformation. Application (rare in GIS): 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). Note: Same map projection 64

65 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). Common in geodesy and often mapping agencies provide transformation parameters. Note: Same map projection 65

66 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). 66

67 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. Only for non-accurate applications. For high accuracy use photogrammetric methods. 67

68 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. 68

69 Choice of empirical transformation Why not use a transformation that models many geometric properties? For example similarity models only size and position, but affine models also differences in shape. We only know what happen at the common points, nothing about what happens between the points. The more geometric properties a transformation models the more can happen between the points. 69