Usually, one of the pair of photographs used for stereographic measurements is an "orthophoto" and has an emission angle of zero degrees -- in other words, the camera is looking straight down at the surface. The second photograph should be taken with an "off-nadir" line of sight (have a large emission angle). Figure 1 will be used here to illustrate the basis for computing the height of an object from the parallax displacement between two such images. This discussion requires some knowledge of trigonometric functions. For readers who are a bit rusty on their trig, see Definition I of the sine and cosine on this web page.

The object in this case is simply a vertical structure whose base is at position A0 and whose top is at position A1. In "Photo1," the orthophoto looking straight down at the surface, the points A0 and A1 coincide. But in Photo2, the off-nadir image, the points are separated by a distance, h2, that is the equal to the sine of the emission angle, e, time the true height, h0, of the object as viewed from the planet surface.

For simplicity, the direction of the "X" axis of Photo2 is the same as the direction of the line of site in the plane of the image, although in general this is not the case.
 
 

Figure 1. Diagram illustrating the computation of elevation of a point, A1, relative to a ground reference point, B. The arcs drawn between lines indicate repetitions of the emission angle of the off-nadir image, Photo2.
 
 

Given the knowledge that point A1 is directly above point A0, the height of the object bounded by those two points can be determined by simply dividing the measured distance, h2, between A0 and A1 in Photo2 by the sine of the emission angle. For cases in which the 3-dimensional object is known to be a narrow vertical structure such as the object whose base is A0 and whose peak is at A1, the orthophoto is unneeded. The height of, for example, an office building could be determined from a single off-nadir image simply by measuring the length of the vertical wall visible in the image and dividing it by the sine of the emission angle.

However, if the point A1 were the peak of a mountain rather than a visible vertical object and we intend to determine the mountain's height, the base point A0 would be buried under the mountain. There is, of course, no way to guess from the photograph where that buried point might be projected on the image plane if the mountain wasn't there. The displacement relative to some point visible on the plains beyond the perimeter of the mountain is what would have to be measured. In Figure 1, the point chosen as the reference is point B, on the surface of the plains and visible to the camera.

As shown in the series of equations in the upper left corner of Figure 1, the height h0 above the ground reference point B can be derived from the measured distance between points A1 and B in the direction of the line of sight of the off-nadir photograph. In the orthophoto, Photo1, that distance is d0, and in the off-nadir Photo2 the distance is Dx2. The height can be expressed in terms of measurable distances in the two photos by Equation [1]:

[Equation 1]                    h0 = (Dx2 - d0 cos(e)) / sin(e)

Notice that the distance between the reference point B and A0 in the direction perpendicular to the line of sight (the X-axis) in Photo2 is c0, the same as in the orthophoto, Photo1 and in the side view from the surface. The parallax displacement must be zero in the direction perpendicular to the line of sight in an off-nadir image and must have its maximum value in directions parallel to the line of sight. This fact figures into error-checking of the results, as explained subsequently.
 
 

Preparation of MGS Images for Stereographic Measurements

Equation 1 can be applied directly to a pair of photographs taken when the motion of the camera relative to the imaged scene is negligible and the only distortions are those produced by parallax. However, for spacecraft photography the motion of the camera is a significant factor. The image smear can often be great enough in MGS images that circular craters appear noticeably elliptical even though the spacecraft was almost directly above the imaged area, giving the impression that some MGS orthophotos were taken at large emission angles.

Fortunately, the MGS ancillary data contains the information necessary to correct for image smear. The skew angle and aspect ratio, explained more fully in the PDS web site documents, can be used to create an image that would be orthographically correct if the imaged surface were completely flat. Such a partially orthorectified image may be created by a two-step process. First the image is resized (resampled) in the vertical direction by the specified aspect ratio divided by the sine of the specified skew angle. The image is then skewed so that the angle between the vertical and horizontal edges of the image is the skew angle rather than the original 90 degrees in the raw image.

The distance between point A0  and  point  B in Figure 1, as viewed from both the surface and in the orthophoto, Photo1, is d0, while in the off-nadir Photo2 the distance between the two points is d0cos(e).The cosine of any angle is always less than 1, so d2 is always less than d0. Photo2 must therefore be stretched by the factor 1/cos(e) in the line-of-sight direction of the off-nadir image in order to show the correct distances between A0 and B as viewed from directly overhead. If the surface being photographed were completely flat, the orthophotograph, Photo1, could be completely recovered from the off-nadir Photo2, by this stretching operation. This is exactly the effect of the resampling and skewing procedure.

The problem with this technique, however, is that for two points not in the same horizontal plane such as A1 and B, the distance between them is also stretched by a factor  of 1/cos(e), which merely exaggerates the parallax displacement rather than correcting for it. I refer to enhancements created by this resample-and-skew process as "partially orthorectified." While the process cannot produce a correct orthographic view of a 3-dimensional object, this exaggerated parallax effect can be used to compute the height of those objects as follows:

If Photo2 in Figure 1 were to be processed by the resample-and-skew technique to produce a partially orthorectified image, say Photo3, the displacement along the line of sight on the Photo3, would be measured as:

[Equation 2]                       h0 = (Dx3 - d0) / tan(e)

Equation[2] is the form that was actually used to compute the elevations of points on and around the Face from partially-orthorectified versions of the 1998 and 2001 MGS images relative to the true orthophotograph of June, 2000. It must be emphasized that the height is only relative to whatever ground reference point, such as point B, is chosen. Choosing another point that was at a different elevation would give a different result, although the difference in elevation measured would be equal to the difference in elevation between the two alternative reference points.

The graphical coordinates can be measured by the pointer tool on most commercial image software such as Photoshop. The program that computes the height then transforms the coordinates of the point measured on the off-nadir program into the coordinates it should have in the coordinate system of the orthophoto, with the ground reference point being the origin (0, 0) point in both systems. The X and Y axes directions of the off-nadir photo are generally at a significant angle from the X and Y axes of the second image, so the transformation requires a rotation. To do the coordinate rotation, the direction of true north from the MGS ancillary data relative to the image coordinates is used.

The ancillary data does not explicitly state the direction of the camera line of sight relative to the image axes, but it does give the areographic coordinates of the spacecraft at the time an image is taken. These can be used to compute the azimuth from the spacecraft to the imaged scene, which is the line of sight angle. The azimuth computations assume an areocentric coordinate system, so translations have to be made from the areographic spacecraft coordinates to the areocentric coordinates and then the azimuth had to be projected back onto the areographic plane of the image. The difference between the areographic and areocentric systems and the equation for converting from one system to the other is explained here.
 
 

Stereographic Measurements with Two Off-Nadir Images

Unfortunately, the 2001 image, the true orthophoto of the Mars Face, captured only the western part of the landform and did not include the landform's peak. The 1998 and 2001 images that did capture the entire landform were both off-nadir. Neither photograph shows any point at the position it would have in a true orthophoto, but that position can be computed as illustrated in Figure 2.

Figure 2. Illustration of how the apparent position of a point elevated relative to a ground reference point changes as two spacecraft move. Both spacecraft would be directly overhead when they view the elevated point at position A0. Each spacecraft would be travelling in the direction opposite to the direction of the arrowheads, which indicate the parallax displacements relative to the ground reference. When the spacecraft have passed the overhead position, they would view the scene from an off-nadir position and see the elevated point at positions A1 and A2 respectively.

Suppose that a spacecraft begins filming the surface below while flying directly overhead so that the true position of an elevated point, A, relative to the ground reference point B is seen. As the ground distance between the spacecraft and point A increases, the camera must be tilted by an ever-increasing emission angle to keep the point in the field of view. Playing back the film, after partially orthorectifying each frame, point A would be seen to move from A0, the position it had when viewed directly overhead, along the line of sight, finally reaching position A1 at the end of the movie, where the camera emission angle was the greatest. The arrow between A0 and A1 is the line of sight as well as the orbital path of the spacecraft projected onto the ground.

Suppose that a second spacecraft also took a movie film starting when it was directly over the elevated feature A, but traveling in a different direction so that its partially-orthorectified movie film showed the feature moving from point A0 to point A2. If all but the last two frames of the two movie sequences were lost, the position of the feature as viewed directly overhead could still be inferred given the two spacecrafts' line-of-sight directions. Overlaying the two photographs so that their two images of the ground reference point coincide, the true position of feature A as it would be viewed directly overhead must be at the intersection of the lines passing through A1 and A2 drawn along the lines of sight of the respective photographs.

The height of feature A above the ground reference point can then be computed by measuring the distance along either line of sight to the point of intersection and applying Equation 2 to that distance, as if that intersection point had been seen on a true orthophoto.

A "reality check" on the validity of the measurements is provided by the fact that the elevation can be computed by using the distance measured along either line of sight. The two computed elevations should be the same. If they are not the same, then something is wrong with the measurements. The use of such discrepancies in correcting for measurement errors is described more fully in the next section.

Sources of Error

In order of increasing severity, the sources of significant error in stereographic estimations of height I have identified are: the limits of camera resolution, mismatch of the points selected between images of a pair, and uncertainty in the value of camera resolution. These error sources are described below.

(1) Camera Resolution

Given the knowledge that an infinitesimally small point is located within a specific pixel in an image, the uncertainty in its true position is 1/2 the size of the pixel in meters, or 1/2 the resolution of the image. The uncertainty in the true distance between that point and some other point is the full resolution, r, because the measured distance is the computed as the difference in measured positions of the two points. For the case of two points on the X axis of the image, the true distance between two points Pa and Pb is xb- xa, but the measured distance could be anywhere between

The parallax displacement is computed by subtracting the measured distance in one image from the distance measured in the second image. When one of the pair of images is an orthophoto, the height is computed from Equation [2} by dividing that distance by the tangent of the emission angle, so the uncertainty in the estimated height is:

(2) Mismatch of Selected Points

The three images were taken at different resolutions and under different lighting conditions. In addition, the 1998 Face image was taken under hazy conditions, making the boundaries of features indistinct, some more so than others. These factors make it nearly impossible to determine exactly which pixel in one image of a pair matches a previously selected pixel in the other image. From experience, even when the greatest care is exercised, it is hard to say that one pixel matches any better than its nearest neighbors.
 
 

(3) Uncertainty in the Value of Camera Resolution

This is the least obvious of the different types of uncertainty, but is probably the largest when the distance from the ground reference point are large for the feature whose elevation is being measured. The resolution for all MGS images appears to be given to the nearest 1/100 of a meter, so the exact resolution is +/- .005 meters. This translates to an uncertainty of +/- 10 meters if a distance measured between the ground reference point and the feature of interest is 2000 meters -- a significant uncertainty. An uncertainty in the distance measurement in turn translates into an even larger uncertainty in height measurement of about +/- 25 meters.

Such large distances are a factor in the comparison of the 1998 and 2001 image but not for the comparisons of either to the 2000 image, which was confined to a much smaller surface area.
 
 

Control of Measurement Uncertainty

The total of the three sources of uncertainty is somewhere between 20 and 40 meters and is at the upper end of the range when large distances are involved. Yet as stated in the previous article, the measured differences in elevations between the three stereo pairs implies an uncertainty of about +/- 7.5 meters, which is about the same as the limiting uncertainty of camera resolution, excluding the more severe sources of error. The smaller than expected error margins are apparently due to the "reality checks" previously mentioned.

When the pair of X/Y graphical coordinates for a point for which the elevation is to be measured are given to the program that computes the elevation, the program returns a pair of values. For the pair of images that were both off-nadir, (98:01), the value computed using distances measured along both lines of sight of the two respective images are returned. If the difference in the two values was less than 10 meters, the average was taken. If the difference in the two values was greater than 10 meters apart, the original X and Y coordinates were incremented or decremented and the program was run again until the difference in values was less than 10 meters. Often, the difference in values on the first try can be large as 50 meters, and sometimes more. However, in most cases, an adjusted coordinate value can be found within a radius of 3 or 4 pixels of the original value for which the difference in computed elevation along the two lines of sight is within 10 meters.

It appears that this trial-and-error process produces results that are much more accurate than was anticipated for the following reason:

When the correct pixel coordinates are chosen (i.e., the coordinates of the point in one image that truly matches the point selected in the other image of the pair), the elevation values computed along both lines of sight must be exactly the same. For a large radius around the correct pixel, elevation values computed for the pair diverge from each other with increasing distance from the correct point. At even larger radii from the correct pixel the elevation values may begin to converge again on a spurious value for a point that does not really match the feature for which the elevation is being measured. But typically, such incorrect points will be much farther away from the correct pixel than was the originally chosen pixel -- much more than the 3 or 4 pixels typically separating the correct position from the original position chosen.

For the two image pairs that include the 2000 orthophoto, (98:00 and 01:00), the differences in distances can be measured only along one line of sight: the line of sight of the off-nadir image. However, a "reality check" of sorts is still possible. For these two image pairs, elevations were rejected when the measured displacement perpendicular to the line of sight of the off-nadir image was too large. Recall that in Figure 1, the distance c0 perpendicular to this line of sight should be exactly the same in the off-nadir image as for the orthophoto, and the differences in distance should be zero when the correct pixels are matched in the image pair. This is a reasonably good check for pixel mismatches but not for the uncertainty in the value of the resolution. However, due to the small area in the 2000 orthophoto of the Face, the distances are not large enough for this type of uncertainty to make a significant contribution in the measured elevations.

For further verification of the accuracy of the elevation numbers, recomputed elevations using the 98:00 ground reference point to the west of the Face, which was measured to be 95 meters higher than the original 98:01 ground reference to the northeast. The measured peak height of the Face from this reference point was 290 meters, 90 meters lower than the height measured from the original reference point -- close to the expected difference assuming the second reference point is 95 meters higher than the first. Also, a ground point fairly close to the second reference point was measured to be at an elevation of 70 meters relative to the original reference point but was found to have an elevation of -25 meters relative to the second reference point. This again is very close to the expected 95-meter difference in measured height assuming 100% accuracy. The elevation of the original eastern reference point relative to the western point was found to be -95 meters as would be expected for highly accurate measurements. None of these measurements required further trial-and-error adjustments of pixel coordinates; the values previously adjusted for the original measurements provided pairs of values that were within 10 meters of each other.

While it would be useful to have resolutions given with another decimal place of precision, I do not expect that the results would differ much from those already computed. The methods I use appear to produce results within +/- 10 meters of the true values.