It is strongly preferred, however, that there be as many background peaks as possible. With only two background peaks, there is no redundancy in the calibration of the photo pixel coordinate system to the angles at which an object would appear in the field; hence no error analysis is possible. It is possible to guess the error based on the pixel resolution, but there is no way to know other sources of error such as atmospheric refraction and projection distortion. Ideally, one should have at least six background peaks.
Click here for a more complete description of the techniques one should employ for taking a digital photograph to get the best possible results for pixel coordinate analysis.
Next, the software tool feeds the pixel coordinates of each rival point into the calibrated transformation to calculate the field angles of each rival point. Finally, given the field angles and distance away from the camera, the software tool calculates the elevation of each rival point relative to the camera.
A user guide is available.
Source code in C++, organized in three files, is available:
You can examine the data file and photo used in the calculation.
Even if it is known that the projection is gnomonic, it is not known exactly where the optical axis of the camera is. However, the result is usually not very sensitive to the presumed optical axis. Users can sometimes manually adjust the optical axis to minimize the residual error.
Projection error is small near the optical axis but can increase rapidly at angles farther from the optical axis. For this reason, it is suggested that background and rival points be not too far from the axis. The more background points are used, the better, but if background points that are too far from the axis cause the residual error to increase, then this is almost certainly due to projection error, and these points should not be used. Users should experiment with the inclusion and exclusion of background peaks that might involve significant projection error.
Panoramic photos are difficult to use because they must necessarily be highly distorted and involve large projection error. A photo of this nature will be evident in the large residual error no matter what background points are used.
Another way of dealing with unknown atmospheric refraction is to minimize the residual error with respect to the degree of refraction; with sufficient redundancy, this technique actually measures the degree of refraction!
Seven peaks in the photograph for Buckner were used as background peaks: Katsuk Pk, Monument Pk, Robinson Mtn, Tower Mtn, Fisher Pk, Black Pk, and Silver Star Mtn. The following table shows Geopix's calculated height for each peak as if it were a rival peak, using the other six peaks as background peaks.
|Silver Star Mtn||31.1||8876||8821±61||-55||43.7|
In the table above, the calculated error is √3 times the standard deviation reported by Geopix. √3 was chosen as the number of sigmas because that is the ratio of the half-width to the standard deviation of a uniform distribution.
Bob Bolton provided a photograph looking north from South Sister, which was also good enough to provide a similar type of validation test. The results of this test are as follows:
|Three Fingered Jack||42.2||7841||7832±24||-9||41.6|
Paul Klenke took pictures looking both directions from the two rival summits of American Ridge on the east side of Mt Rainier, and these images were both good for that type validation test. The results of the test on the shot looking from the E summit to the W summit are as follows:
|Cowlitz Chimneys S pk||18.4||7605||7575±67||-30||17.8|
and from the W summit to the E summit:
|Nelson Ridge N pk||9.1||7049||7056±25||7||9.8|
Paul's images also provide an opportunity to calculate the relative heights of the rival summits looking in both directions, to check the consistency of the results. According to the shot looking from the E summit to the W summit, the W summit is 3.4 feet higher, plus or minus 1.0 feet. According to the shot looking from W to E, the E summit is 4.1 feet lower, plus or minus 0.5 feet. The consistency between these two results is encouraging.
In every test case, the calculated height is within the calculated error of the (known) actual height, and the error is comparable to or less than the pixel resolution. Hence Geopix is a very credible tool.