1. Distortions in imagery:
attitude variation
roll pitch yaw

2. Distortions in imagery:
Earth rotation

3. Distortions in imagery:
panoramic distortion
Earth curvature
- adds to panoramic distortion

4. Distortions in imagery:
relief distortion

5. Correction based on Image Navigation
It requires:
accurate knowledge of the satellite's orbital parameters, to locate its position along the orbit as a function of time;
one to five ground reference points (points of known topographic coordinates identified by the operator on the image), depending on the type of model adopted, to correct errors in computing the orbit's height.
This method is very fast, and is adequate to correct well-characterized distortions, such as the effect of Earth's curvature.
It is used to correct data obtained by sensors with a wide field of view (AVHRR, Meteosat) because:
they are used in applications that require fast processing of many images; for example, meteorological applications;
cloud cover is always present at their acquisition scale, and the images frequently cover large water surfaces. In both these cases the other methods cannot be applied;
It produces greater positioning errors than the other methods, but these errors can be tolerated at the typical restitution scale of these images. Also, the coarse geometric resolution makes it difficult to position control points on the image. Under these circumstances, the other methods cannot give good results.

6. Image correction by means of bivariate polynomials
Requires a large set of Ground Gontrol Points (GCPs) whose coordinates are known both over the image matrix and on the reference cartography.
A Ground Control Point is an object or a landscape element precisely identifiable both on the image and on the reference cartography. Topographic maps of the imaged surface must be available at an adequate scale. An operator manually identifies a number of GCPs large enough to obtain a reliable statistical model.
If a georeferenced image is already available, it can be used instead of the topographic map. An image-to-image correction is faster, as GCPs can be identified more easily.
A statistical procedure is used to find the two bivariate polynomial functions that best approximate the relationships among image and cartographic coordinates. These functions are then used to create a new, corrected image.
Also, this procedure does not require the knowledge of the distortions which affect the image, and can be applied to every type of image, as long as they are not acquired over mountainous terrain. If relief is not negligible, the orthorectification procedure must be used instead.

8. The best GCPs are usually artificial features, such as:
intersections of roads and channels, railways, bridges, small buildings or isolated structures, coastal defenses
The GCPs should be homogeneously distributed on the image.
only in this case the mapping functions are truly valid for the entire image
yes no no

9. the relationships between image and map coordinates of the GCPs can be described by two bivariate polynomial functions:
where r, c are the GCPs' image coordinates (known); x, y are the GCPs' map coordinates (known); ajk, bjk are the polynomials' coefficients (unknown); m is the transformation degree (user-defined)
Example: 1st order transformation
dstX = a0 + a1srcX + a2srcY
dstY = b0 + b1srcX + b2srcY

10. The inverse mapping functions is used to calculate the expected GCPs' map coordinates from their image coordinates;
the results are compared with the measured GCPs' map coordinates;
the root mean square error (RMS) of the difference between calculated and measured coordinates for a single GCP is named residual:
- xi, yi are the cartographic coordinates of the GCP, obtained by the operator on the reference cartography;
- xr, yr are the coordinates of the same GCP obtained from the mapping functions.
The RMS is calculated in cartographic units (meters, degrees) or in pixels. A sub-pixel RMS is considered acceptable.

11. First order polynomial Second order and +
Linear, larger residuals, the image is rotated and translated
Second order and +
Smaller residuals, the image is warped (“rubber sheeting”)
Large errors should be expected if GCP dataset is enevenly distributed or unaccurate

12. Once the mapping functions have been properly obtained, and the inverse functions have been calculated, it is possible to build the new, corrected image calculating:
the coordinates of the corners of the corrected image;
the number of cells in the corrected image's rows and columns;
the map coordinates of the cells in the corrected image;
DN = 0 or NoData is assigned to the outer triangles.

13. In the new image, the pixels do not spatially correspond with the original ones. So, the value of each pixel in the corrected image must be obtained by interpolating the values of the pixels that surround it in the original one.

14. Nearest neighbor interpolation: the original image's pixel nearest to the corrected image's cell is located, and its value is assigned to the cell itself.
It should be used when:
the corrected image must be quantitatively analyzed; for example, prior to image classification;
the geometric correction procedure is applied on thematic data.
From an aesthetic viewpoint, this method produces the worst results, thus it is not used when the images are prepared for visual interpretation.

15. Bilinear interpolation: this procedure interpolates the values of the four pixels surrounding the corrected image's cell.
This procedure modifies the original image values; it is not advisable to use it whenever the corrected image must be classified or analyzed quantitatively.
The interpolation softens the intensity differences among pixels, thus the corrected image looks more pleasant than the one obtained using the nearest neighbor procedure; this procedure is often used to prepare data for visual interpretation.

16. Cubic convolution: this procedure interpolates the sixteen pixels surrounding the corrected image's cell using third-degree polynomials.
The original image's values are strongly altered by this procedure; furthermore, the smoothing effect reduces the perception of small features. Thus, this method must be avoided whenever the corrected image must be quantitatively analyzed.
The visual results obtained are even more pleasant than those obtained using the bilinear interpolation method. This procedure should then be used to prepare the images for visual interpretation.
Calculation is much slower.

17. Interpolation nearest neighbor bilinear interpolation
cubic convolution