The Grid transformation - Basic Concepts

Why the Affine transformation is not suitable for precision tasks?

At the end of section Transformations | Affine (see Fig. TR.A15) we came to the conclusion that the Affine transformation is not suitable for tasks in which high precision is required, because it does not remove the deformation of the map. The reason is due to the fact that the Affine algorithm performs a simple geo-referencing technique, not a rectification of the raster map. It simply scales the map by a factor, but it doesn’t modify the raster at all. Also, the calculated scale factor is unique for the whole raster, thus assuming that the map has been subjected to a regular deformation, i.e. the entire map is deformed by the same amount in all directions. But obviously this is not the case, because in reality the physical agents deforming the original paper map (heat, humidity, paper degradation, etc.) might have acted with different levels in single localized areas of the map. So, if we need to obtain geometric  information (coordinates, distances) of a specific localized area of the map, we are interested in correcting the deformation of that single localized area, regardless of the average deformation of the whole map.

The algorithm

Having said that the Affine transformation is not suitable for some type of tasks, such as boundary disputes, what is the right approach for these tasks? Well, let’s go back to the situation of paper maps at the time when they were not yet available as raster images. How should we act in that case? For example, on the paper map shown in Fig. BC.G1 we want to retrieve the Easting of the point indicated by the red spot. To do so, we measure with an engineer’s scale rule the distance from the grid line on the right hand side having an Easting of -24200. As we’ve already seen in section Basic Concepts | The map deformation, this area of the map was deformed (enlarged), the distance measured between the two grid lines was 201.10 instead of 200.00.

 

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Fig. BC.G1To understand the Grid transformation, let’s go back on how should we act on paper maps.

 

Here are the steps to perform in order to correct this:

 

1.Measure the distance between the 2 grid lines containing the desired point:

 

-201.10

 

2.Compare the distance retrieved in step 1 with the nominal value of the interval (200 meters in this scale) and, if the two values differ, calculate the difference by dividing the measured value by the nominal value:

 

-200 / -201.10 = 0.9945

 

3.Measure the point East from the reference grid line and multiply it by the adjusting coefficient found in step 2, hence determining the adjusted distance (deformation removed):

 

-139.10 ● 0.9945 = -138.34

 

4.Add the adjusted distance found in step 3 to the absolute East coordinate of the reference grid line:

 

-24200 + (-138.34) = -24338.34

 

This method is particularly appropriate for maps in which the points have been originally inserted by just measuring their coordinates from the grid lines, because it exactly reflects the creation of the map. Well, this concept of map calibration is also valid for raster maps as well. But of course, being available as image files, and using appropriate CAD software we can conveniently add much more precision, rather than just estimating decimals of a millimeter with an engineer’s scale rule.

So, following the manual approach described above, the Grid transformation algorithm is based on the assumption that each grid square formed by the grid lines has been subjected to a deformation in both East and North direction. In Fig. BC.G2 and Fig. BC.G3 below we have manually exaggerated the deformation of a square so that the calibration procedure would be obvious, whereas the real much smaller deformation would be much less obvious. With such a magnified deformation we can see that the grid square is no longer a square, but it is a quadrilateral because its sides on the grid lines, are no longer parallel.

Therefore, if we need to retrieve the coordinates of a point P inside it (shown by a blue spot in Fig. BC.G2), we should calculate the deformation of the quadrilateral compared to the original square. Fig. BC.G3 shows the geometrical schema of the algorithm. Because the horizontal/vertical grid lines forming the quadrilateral have lost their parallelism, we simply extend them until they intersect in both East and North directions (but obviously we also consider the special case in which the grid lines have maintained parallel). Then we conjunct these two intersection points with point P that we are interested in, so determining the lines AC and DB which exactly reflect the quadrilateral deformation referred to point P. We then calculate the length of these two segments. If there was no deformation, this length should be exactly 200 meters (for the scale of this map) for both AC and DB. Otherwise different values will tell us how much the original square has deformed in the corresponding direction. In Fig. BC.G3 for example, let’s assume that we found:

 

AC = 199

 

DB = 201

 

 

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Fig. BC.G2The Grid transformation algorithm is based on the on the assumption that each grid square has been subjected to a deformation.

 

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Fig. BC.G3The geometrical schema of the Grid transformation algorithm.

 

So what we need to do is to calculate at what coordinates should point P be when the quadrilateral was still a perfect square, with both AC and DB = 200. The mathematical demonstration of this algorithm goes beyond the purpose of this guide but it is quite easy to understand from a geometric point of view. In fact, what this transformation does is a reverse-deforming of the quadrilateral until AC and DB both go back to being 200 meters long.

The sequence of images in Fig. BC.G4 shows this reverse-deforming. The original position of point P is indicated by the blue circle which remains fixed, whereas the red circle shows point P moving until it reaches its final position (box 4). In other words, we have returned point P back to the position it was at before the map had been subjected to any deformation. The distance between the blue and red circles shown in Fig. BC.G4 box 4 is graphic evidence of the map deformation.

Different from the Affine transformation then, the Grid technique is not only a simple geo-referencing, but it also applies a real map rectification, as it modifies the raster image by rectifying its grid squares. Therefore this solution calculates and corrects the deformation of each single square of the map in accordance to the assumption that the map had been subjected to an unequal deformation, i.e. grid squares might have different deformations from each other. This is the reason why for maps provided with grid lines this method is much more accurate compared to the Affine transformation explained in the previous chapter.

 

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Fig. BC.G4The Grid transformation performs a reverse-deforming of the grid squares (quadrilaterals).

 

In the Transformations | Grid section you'll find a practical example on how to perform this transformation with CorrMap.