Raw sensor data often contain too many distortions to be used as a map. The process of correcting these errors is known as geometric correction. One of the key subjects of this thesis is the fusion between the in situ measurements per-formed at the test sites and the polarimetric dual frequency EMISAR data.
In order to match these remotely sensed data with the in situ data the exact geographical position of a particular pixel in the SAR image has to be known.
However, as explained in Section2.3the EMISAR data to be used are geomet-rical distorted and a geometric correction is therefore required.
3.3.1 Introduction
The image to be geometrically rectified or warped is called the input image and the rectified result is called the output image. In this context the transformation is done by aligning the input image with another image called the reference image. The geometric rectification of the input image is performed using Ground Control Points (GCP’s), which are features identified in both the input and the reference images.
The geometric rectification involves two steps described in the sections below.
In Section 3.3.2 is the use of first order polynomials as deformation models reviewed and in Section3.3.3the bilinear resampling technique is outlined. The techniques of geometric transformations are described in e.g. Sonka (1993) [77]
and Niblack (1985) [52].
3.3.2 Deformation models
The deformation is often described by polynomials of up to third and fourth orders. It is these polynomials that connect the geometrical relationship between the input and the output image, or alternatively the output and the input image.
3.3 Geometric rectification 33
The output-to-input transformation of first order given by x = a0+a1x´+a2y´
y = b0+b1x´+b2y,´
maps the point (´x,y) in the output image to the point (x, y) in the input image,´ see Figure3.1. The coefficients of the polynomialsa0, a1, a2, b0, b1, b2are derived through the GCP’s in the two coordinate systems. Although only three point pairs are necessary as a minimum for estimating the aandb coefficients of the first order polynomial, additional point pairs are often used. This will ensure a better estimation of the coefficients in a least squares sense and a quality measure of how well the transformation fits the points.
Thea-coefficients for a transformation of first order is given by
where N is the number of point pairs, Ais the design matrix for the model,θ is a vector containing the coefficients and ε is the residual vector. The vector containing the coefficients is estimated from
ˆθ= (ATA)−1ATx,
assuming that the measurement error is zero. The estimate of the observed geometric errorσ2o for thexcoordinates is
ˆ
σo2=||x−Aˆθ||2
N−rg(A), (3.8)
where || · || is the Euclidean norm and rg(A) is the rank of matrix A. The b-coefficients and ˆσ2o for they coordinates are derived in a similar manner.
k, l
Figure 3.1: Illustration of the bilinear resampling. The point (´x,y) in the output´ image is mapped to the point (x, y) in the input image. The neigbours to (x, y) are located at (k, l), (k+ 1, l), (k, l+ 1) and (k+ 1, l+ 1).
In practical applications polynomials of up to third and fourth orders are used.
In the case of second order transformation we have x = a0+a1x´+a2y´+a3x´2+a4y´2+a5x´´y y = b0+b1x´+b2y´+b3x´2+b4y´2+b5x´´y, and here at least 6 point pairs are needed.
3.3.3 Resampling
After establishing an appropriate transformation each pixel in the output image is assigned a value. However, the position (x, y) of the pixels in the input image does typically not fit the integer coordinates (´x,y) of the output image. This´ is because the collections of transformed points give the samples of the output image with non-integer coordinates. However, values on the integer output grid are needed and it is therefore necessary to interpolate between the non-integer points in the input image. This process is called resampling.
The methods that most frequently are used for resampling are nearest neighbour (NN), bilinear and cubic convolution. The simplest method for assigning a value for the output image is the nearest neighbour, which chooses the pixel value in the input image that has its centre closest to the position (x, y) determined by the transformation. This method has the disadvantage of introducing a position error of at most half a pixel. On the other hand, the method preserves
3.3 Geometric rectification 35
the original values and the method is therefore preferred in situations where a classification is the intention or the preservation of the radiometric information is needed.
The bilinear resampling makes use of the four pixel values that surround the calculated position (x, y) in the input image. The interpolated pixel value ´p, which is assigned the position (´x,y) in the output image is a weighted average´ between the calculated position and its four neighbours. With reference to Figure3.1 the bilinear interpolation is given by the equation
´
p(´x,y)´ = (1−b)(1−a)p(k, l) +b(1−a)p(k+ 1, l) +(1−b)ap(k, l+ 1) +abp(k+ 1, l+ 1).
Due to the averaging nature of the bilinear interpolation the resampling will cause a small decrease in resolution as well as blurring of the output image.
However, the problem of the step-like boundaries using nearest neighbour re-sampling is reduced.
In the cubic convolution or bicubic interpolation the fifteen neighbouring points are used in the resampling. The interpolation copes with the bilinear blurring as well as the step-like boundary problem of the nearest neighbour interpola-tion. Furthermore it is superior in terms of preserving fine details. The cubic convolution can be performed in many ways, but the result is also a weighted average that is assigned the position (´x,y) in the output image.´
The test areas at Gjern and Mols Bjerge are relatively flat and very small and consequently the local incidence angleϕcan be considered constant within each area. Using (2.1) this again implies that the ground resolution ∆Rg can be considered constant within each area. A polynomial of first order is therefore used in the geometric rectification of the polarimetric EMISAR data covering the test sites.
Due to the varying topography of the areas surrounding the test sites the GCP’s have been manually collected within or as close as possible to the test sites. Here the reference images are UTM rectified ortho-photos from 1995 and as input we use the EMISAR data. Due to the speckled nature of these EMISAR data it has been difficult to identify features in both the reference and the input images.
The number of GCP’s for each test site is therefore only 5 to 8. However, in spite of this small number of GCP’s the affine transformation has proved to be quite accurate in terms of how well the transformation fits the points in the areas of the test sites, see page172and210.
The resampling of the restored EMISAR data is performed using the bilinear
interpolation. This method is suitable because the restored regions to be rec-tified are coherent and homogeneous and without large discontinuities between them. The blurring effect of the bilinear resampling is therefore not significant.
The cubic resampling could be used as well, however, there are no distinct fine details in the restored samples containing the test sites and consequently not much is gained. Concerning the nearest neighbour (NN) interpolation we are not interested in preserving the radiometric content in particular but rather the gap between mean amplitude levels. In addition the NN interpolation will cause a geometrical distortion, which is unwanted given the small size of the test areas.