Introduction

[33] Topographic features in radar images are always distorted to some degree. The amount of distortion depends on the topographic relief and on the incidence angle of the observation. Magellan's extended mission (Cycles 2 and 3)-with a viewing geometry different than that used in Cycle 1-provided the opportunity to derive heights and depths of features at a lateral resolution comparable to that of the SAR images themselves. In this chapter, the mechanics of radar stereoscopic (stereo) imaging are presented, along with a description of the Magellan stereo image data set and examples that apply the techniques to Magellan images.

Radar Image Distortion-Why Stereo Works

The location of a resolution cell in a radar image is determined across track by the range (distance) between the antenna and the feature (measured as a time delay of the echo), and along track by the Doppler shift expected for a given piece of terrain. The range-Doppler coordinates of a resolution cell can be easily transformed into planetary coordinates (i.e., latitude and longitude) under an assumption of the large-scale shape of the surface. In the case of Magellan data, the large-scale shape of the surface (at scales of hundreds of km) is modeled as gently sloping (the "topo model") or as a portion of a perfect planetary sphere.

Use of the range coordinate to locate a resolution cell leads to a distortion of topographic features known as relief displacement. For example, high areas return an echo sooner than the surroundings, and thus are displaced toward the spacecraft in the image plane; the opposite is true for low areas. These effects are known as "foreshortening" and "elongation," respectively. In the extreme case of foreshortening, an echo is received from the top of a mountain before an echo from the near-range base of the mountain, leading to "layover," where, on the image, the top is superimposed on the base. Foreshortening and elongation can often complicate analyses of radar images, but they also allow the use of stereometric techniques to determine feature heights.

The amount of relief displacement is a simple function of the height of the feature and the incidence angle of the observation (Figure 4-1). Smaller incidence angles produce greater distortion for a given amount of topographic relief. Thus, by comparing the distortions measured on two images taken at different incidence angles, the heights of features can be determined. Terrain relief can also be perceived visually, using same-side (illuminated from the same direction) stereo image pairs and a stereoscope.

Estimating Heights From Stereo Data

Feature heights may be obtained from same-side stereo image pairs (Figure 4-1), such as those from Cycles 1 and 3, and from opposite-side stereo image pairs (Figure 4-2), such as those from Cycles I and 2, or 2 and 3. In either case, an accurate measurement can be made only if the two points between which a height difference is to be found can be identified as unambiguously identical in both images. Errors in identification of the points will propagate as errors in the height determination of the feature. The appearance of features is often more similar in same-side image pairs than in opposite-side image pairs, making the selection of points....

Figure 4-1. Solving for heights using same-side stereo images. Measurements are taken from the separate images of a same-side stereo image pair to determine the height of point P above the reference plane, R. As wavefront W₁ intercepts point P, the projection in range space (i.e., time delay) places the echo from P at point p₁ on plane R. Similarly in the second image, the echo from point P is placed at point p₂ Identify a point r_1,2 in a low-relief area on the plane R, and measure the distance from r_1,2 to p₁ on image 1, and from r_1,2 to p₂ on image 2. The difference between these distances (d₁ - d₂) is s, the parallax of point P. Using the incidence angles of the two observations ( and ), the height h of the feature is obtained by dividing s by the parallax-to-height ratio (cot - cot ).

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....easier and more accurate on the two same-side images. On the other hand, because the displacements on the opposite side image pairs are opposite in direction, a larger parallax than that from a same-side pair is obtained, making height determinations more accurate, provided identical points are identified on both images.

Stereo measurements are best made from high-resolution digital data (e.g., Magellan F-MIDRs), in which the precise pixel location of features can be made on a video monitor. Hard-copy prints may also be used, but the image pair must be enlarged to a single scale. Once the separation of the two features has been measured on the images and converted to a ground distance, the height difference is easily calculated, using the parallax-to-height ratio of Figure 4-1 or 4-2. The incidence angles of the two observations can be obtained from Table 4-1 (represented graphically in Figure 4-3), or from the ancillary files on Magellan CD-ROMS. Note that the parallax-to-height ratio is obtained with slightly different formulas for same- and opposite-side stereo pairs (Figures and 4-2).

Magellan Stereo Data

Magellan Cycle I data were obtained in a left-looking variable incidence angle mode. To maximize image quality incidence angles were varied as a function of spacecraft altitude. In the Cycle 2 right-looking mode, the incidence angle was constant at about 25 deg for most of each orbit. Incidence angles in Cycle 3 were selected to provide images suitable to produce a stereo pair with corresponding Cycle 1 images. Figures 2-4 through 2-6 show the coverage for each of the three imaging cycles, and Table 4-1 lists the associated incidence angles.

Figure 4-2. Solving for heights using opposite-side stereo images. The procedure is similar to that used for same-side images. Here the parallax, s, is equal to d₂ - d₁. Using the incidence angles of the two observations ( and ), height h of the feature is obtained by dividing s by the parallax-to-height ratio (cot + cot ).

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Several complications arise in using Magellan stereo data for height determinations. First, in most regions, the SAR imaging data are projected onto a low-resolution topographic model, derived from pre-Magellan observations. This was necessary to minimize errors in locating features in regions where the elevation differed greatly from the Venus average. At a local scale, the topographic model does not affect stereometric measurements, but at a regional scale, particularly in areas of large relief, height differences in the topographic model must be added to any stereo-derived height differences. A second complication is related to the geometry of the SAR observations, and to the fact that all relief displacements occur in the cross-track direction. The crosstrack direction is generally to the east in Cycles 1 and 3 and to the west in Cycle 2, but it is not precisely east or west, particularly at high latitudes. The most accurate measurements of relief displacement are made in the crosstrack direction, which can usually be determined by plotting a line perpendicular to the edge of an orbital swath. In most Magellan MIDRs, swath edges are visible either along data gaps or as "shading" artifacts. A third complication occurs when stereo measurements are to be made between orbital swaths that were processed using different spacecraft orbital navigation solutions. These "navigation boundaries" can be identified by viewing stereo image pairs with a stereoscope. The boundary will appear as a linear discontinuity in relief, parallel to the orbital tracks. Height determinations obtained across such boundaries are likely to be unreliable.

Applications of Stereo Analysis

The simplest use of Magellan stereo image pairs is visual examination with a stereoscope. A stereoscope directs one image of the stereo pair to one eye and the other image to the other eye. For most Magellan same-side stereo data, the Cycle 1 image (large incidence angle) goes to the left eye, and the Cycle 3 image (small incidence angle) goes to the right eye (for the Maxwell Montes area, the image positions should be reversed; see Table 4-1). Opposite-side stereo image pairs....

....are often difficult to fuse with a stereoscope. When the images are perceived to merge, variations in relief become apparent as a three-dimensional image. Figures 4-4 and 4-5 are Magellan stereo image pairs that can be viewed with a stereoscope. The perceived relief relative to the horizontal scale is usually about five to ten times the actual relief, depending on the incidence angles used [Leberl et al., 1992]. For precise determinations of relief, measurements of parallax should be made on a computer monitor or on photographic enlargements (see below).

Stereo pairs can also be viewed with color anaglyphs and 3-D color filter glasses. Figure 4-6 is a Magellan stereo anaglyph in which the Cycle 1 image is tinted red and the Cycle 2 image (from the left-looking stereo test) is tinted blue. Standard 3-D glasses, with the red filter on the left eye and the blue filter on the right eye, accomplish the same effect as a stereoscope by providing each image of the pair exclusively to the appropriate eye. Anaglyphs can be generated on color video monitors with Magellan digital image data. Cycle 1 data should be sent to the red channel, and Cycle 3 data should be sent to both the blue and green channels. The amount of lateral offset between the left and right images is not important, as long as the eye is able to fuse the images without too much strain.

Examples of parallax differences used to measure relief are shown in Figures 4-7 and 4-8. Figure 4-7 is an opposite side full-resolution stereo pair (Cycles 1 and 2) of a scalloped dome. The left image, from Cycle 1, was acquired in the left-looking mode at an incidence angle of 40 deg. The right image, from Cycle 2, was acquired in the right-looking mode at an incidence angle of 25 deg. The parallax difference between the left edge of the westernmost pit and the scarp of the dome is 243 - 118 = 125 pixels. Using the formula in Figure 4-2, the parallax-to-height ratio for these images is 3.34. The relief can then be calculated:

This measurement is consistent with the altimeter data, which indicate that the dome summit lies 2.2 to 3.0 km above the surroundings.

Figure 4-8 is a same-side stereo pair (Cycles 1 and 3) of a caldera (volcanic depression). The incidence angles used were 42.5 deg (Cycle 1, left image) and 22.7 deg (Cycle 3, right image). The parallax difference between features on the rim and floor of the caldera is 23 pixels. Using the formula in Figure 4-1 and a parallax-to-height ratio of 1.30, the relief is

This is again consistent with altimeter measurements of the feature. Further examples of stereometric techniques applied to Magellan data may be found in Leberl et al., 1991 and 1992, and Moore et al., 1992.

Digital Elevation Models

Stereo images are also used to derive topographic relief through the automated generation of digital elevation models (DEMs). In this technique, computer algorithms match features between the images of a stereo pair, calculate the....

.....parallax of every pixel, and generate a topographic map for the entire scene. The calculation is identical to that for the manual derivations of heights and depths of features, but the automated DEM procedure has the advantage of producing an elevation measurement for every pixel. Topographic maps produced in this way have a lateral resolution close to the image resolution (~100 m), which is 100 times better than the typical resolution of the altimeter. Geologists studying the morphology (shapes) of Venusian surface features will find the DEMs extremely valuable in their analyses. Where DEMs are not available, geologists can estimate the relief of features using the manual techniques outlined in this chapter.

[43] References

- Leberl, F. W., K. Maurice, J. Thomas, and W. Kober, 1991, "Radargrammetric measurements from the initial Magellan coverage of planet Venus," Photogramm. Eng. Rem. Sens., v. 57, p. 1561-1570.

- Leberl, F. W., J. K. Thomas, and K. E. Maurice, 1992, "Initial results from the Magellan stereo experiment," J. Geophys. Res., v. 97, p. 13,675-13,689.

- Moore, H. J., J. J. Plaut, P. M. Schenk, and J. W. Head, 1992, "An unusual volcano on Venus," J. Geophys. Res., v. 97, p. 13,479-13,493.

 

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