l6 – transformations in the euclidean plane ngen06(tek230) – algorithms in geographical...
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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)
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.
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.
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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).
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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
L6- Transformations
Transformations in the Euclidean Plane
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
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Transformations in the Euclidean Plane
Original grid Congruence transformation Similarity transformation
Affine transformation Projektive transformation Polynomial transformation
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 -
Type of transformation Invariant (maintain) Does not maintain No. of unknown parameters
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 pointsHow?
How many common points?
L6- Transformations
Congruence (Euclidean) transformation
It models translation and rotation.
L6- Transformations
Congruence (Euclidean) transformation model
Note: to determine the 3 transformation parameters , and α) at least 2 common points are required.
or in matrix form:
10
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Similarity transformation
It models translation, rotation, and uniform scaling.
L6- Transformations
Similarity transformation model
Note: to determine the 4 transformation parameters , , m, and α ) at least 2 common points are required.
Or in matrix form:
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Affine transformation
It models translation, rotation, non-uniform scaling in different directions, and shear.
non-uniform scaling shear
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Affine transformation model
Note: to determine the 6 transformation parameters , , , , α , and ) at least 3 common points are required.
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Projective transformation
Figure: Projective transformation. The planes do not have to be parallel. 15
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Projective transformation modelThe 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
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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 = 6n = 2 no. of parameters = 12n = 3 no. of parameters = 20n = 4 no. of parameters = ?n = 5 no. of parameters = ?
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑢𝑛𝑘𝑛𝑜𝑤𝑛𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠=(𝑛+1)×(𝑛+2)
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Applying empirical transformations
Example: how to estimate affine parameters?
Original affine transformation
Rewritten affine transformation
where
Or
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Step 1
Equations system for solving the unknown parameters using 3 common points
𝑎0 ,𝑎1 ,𝑎2
𝑏0 ,𝑏1 ,𝑏2
, , , , α ,
Note: In case of having more than 3 common points, Least Squares technique is used.
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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.
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Standard error of the empirical transformation
The estimated standard error (transformation quality at the common points)
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Choice of empirical transformation
In general, it depends on type of
geometric distortions between the two coordinate
systems
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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).
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)
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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).
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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.
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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.
L6- Transformations