The use of projective models such as Direct Linear Transformation (DLT) is well known in photogrammetry and very famous in computer vision. The fitting of random or stochastic observations to these models typically yield very minimum residuals, which may not reflect the actual reality when checked with external information. Now the question is: What is the nature of the residuals that we get from the fitting of stochatic observation to projective models?
I think they can be modeled as random noise, white noise or colored/correlated according to the soundness of the fitting model, of intensity related to the accuracy of computation algorithm.
i hope to understand the question in a right way:
Maybe it can be explained like this: The whole image block will be triangulated to get the needed image orientation. This means, that at the and you will have position and rotation informations for every image by triangulation. This trinagulation is like iterating small differences in image positions and rotations all the time while testing whow acurate the main residual at the end will be. The residulas of the single points show how acurate these single tie points fit to this actual configuration with this minimized average residuals. This means:
If the average of all these residuals would the minimum of all the trials, this configuration would be the best. So for this configuration the position and the rotation configuration of every image in the whole block will be fixed as result of image orientation.
Because triangulation works with all these homologous image points (or also feature points) as tie points beetween different images and because these homologous points are on different positions with different contrasts at the images so differnet image quality and so on, every image will be orientated with different acuracy, even every homologous point will have different acuracy.
Working with analogue photogrammetry in former times the residuals gave hints, which points would not be suitable or should be positioned again. So knowing these residuals for tiepoints or image oprientation states like relative and absolute orientation will give helpful information about the accuracy and about the distrubution of the acuracy over the whole image block.
Is this what you asked for?
With best regards from germany, Deggendorf near bavarian forest
Patrick
Dear Gamal,
That is a good question. I give it a try to answer. But I think your question is also related to a more general issue, that is: the difference between precision versus accuracy.
First of all, in order to fit the data you certainly used some kind of adjustment techniques. The adjustment tries to minimize the errors such, that the dataset you used fits best to the underlying model, in your case to the projective model. That means each image point observation gets a small improvement (residual), in order to fit them to the projective mapping of the known control points. I am sure, you are aware of this adjustment theory. Now, the question is, and I think that is what your question is about, how to interpret the results, and in particular the residuals of the image point observations?
Now, the resulting residuals only tell you about the precision, strictly speaking by the resulting standard deviations which are obtained from the residuals after adjustment. This is a measure of the quality of the observations and of the proper choice of the model, respectively. But it does not tell you about the true value, which is usually meant by the term accuracy (see: https://en.wikipedia.org/wiki/Accuracy_and_precision). The accuracy can be checked by the use of a second independent measurement. You check the accuracy by using another control point, which was not used during the adjustment. That check tells you about the deviation from the averaged results of the fitting. That means it is a measure of the deviation from the true value.
Regarding to your question “what is the nature of the residuals” one can in general not say this, because it depends on the observations, and on each single situation, due to the change of environmental conditions like temperature or illumination etc. So, if your experience is, that the residuals are small then it just means, that the chosen projective model reflects the reality pretty good and the mapped control points are precisely identified in the image, and are free from outliers. But, since the projective model is linear, it can not handle a high level of distortion. So, if you use a different camera with heavy distortion, then your residuals will certainly grow. In addition, if the control points are difficult to identify, hence imprecisely measured, the residuals will grow as well.
Finally, one has to consider the problem of extrapolation. That means, the observations which are used for computing the mapping parameters actually hold only at the selected image points. If one uses other image points, that are not used during the fitting, in order to check the results, they might have a bigger error, just because the model is not fitted at those positions. Of course, one usually overcomes this problem by using an appropriate number of control points, which are uniformly distributed over the image.
Kind regards,
Olaf
https://en.wikipedia.org/wiki/Accuracy_and_precision
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