### 3.3 Stereoscopic triangulation or tie-point method

We are turning now to “true stereoscopy”, where an object is simultaneously observed from multiple aspect angles, rather than using the solar rotation to vary the aspect angle over time. Further, we distinguish between stereoscopic triangulation, which can only be applied to point-like or curvi-linear structures (Figure 7), and stereoscopic tomography, where a full 3D density distribution of a voluminous structure is obtained from simultaneous multiple aspect angles. An introductory primer for solar stereoscopy is given in Inhester (2006).

Regarding solar stereoscopy, there exist indeed point-like objects in the solar corona (such as bright points, flare kernels, or centroids of near-spherical CME bubbles) as well as curvi-linear structures (coronal loops, fans, jets, filaments, prominences, plumes), which are suitable for stereoscopic triangulation. The process of stereoscopic triangulation generally requires three steps: (1) coalignment of a stereoscopic image pair into an epipolar coordinate system (Figure 8); (2) identification of corresponding point-like or curvi-linear features in each image (with coordinates and ; and (3) geometric triangulation to retrieve the 3-coordinates of the structure. The first step involves rectification of the stereoscopic pair of images to coalign the X-axis of each one with the epipolar plane (defined by the surface that intersects the two stereoscopic observers and the Sun center), plus rescaling of the pixel size in case the two observers have different distances from the Sun. The second step involves a parameterization of the coordinates of a curvi-linear feature, say as a function of the length coordinate , yielding two sets of 2D coordinates and . The identification of the correct structure B in the second image that corresponds to the feature A in the first image is easy for small stereoscopic angles, because the two images A and B look very similar, but becomes increasingly difficult and ambiguous with larger stereoscopic angles. In practice, approximate 3D magnetic field models can be used to identify the correct correspondence of coronal features. The third step of triangulating the 3D coordinates is straightforward in an epipolar coordinate system, which can be calculated separately for each epipolar plane, corresponding to a particular loop position , as described in the next paragraph for the case of two spacecraft A and B with different distances to the Sun.

We define a coordinate system that has the origin in the Sun center, the -axis is the line-of-sight from spacecraft A to Sun center, and the plane coincides with the plane of the spacecraft A and B (Figure 9 left). The two spacecraft have distances of and from the Sun center and observe a point at an angle of and in -direction from Sun center (corresponding to the difference of x-pixels in the image) and at an angle and in -direction from Sun center (corresponding to difference in y-pixels in the image). The spacecraft have a separation angle of in the plane. The point has the 3D coordinates or heliographic longitude and latitude difference with respect to the central meridian defined by the line-of-sight axis of spacecraft A. The projected positions of the point on the X-axis are from spacecraft A and from spacecraft B. So, our main problem is to solve for the variables using the observables .

In the triangles and we can determine the angles and simply from the geometric rule that the sum of the three angles in a planar triangle amounts to , i.e.,

and
Using the sine relation in a planar triangle [] we obtain the sides and in the two triangles,
and
Furthermore we have the relations in the rectangular triangles, and , from which we can determine the coordinates ,
and ,
The coordinate can be obtained from a relation in the plane (Figure 9 right),
The distance of point from the Sun center is then
and the height of the point at position is
The method of stereoscopic triangulation is also called tie-point method, because a second point in image B is tied to a first point in image A, which enables triangulation of each point according to its epipolar plane. Further details on stereoscopic triangulation can be found in the tutorial of Inhester (2006), which discusses also the identification and matching problem, the tie-point reconstruction, reconstruction errors, ambiguities in identifying corresponding structures, and examples applied to SOHO/EIT data.

First stereoscopic triangulations of coronal loops using STEREO/A and B have been performed for 30 coronal loops in an active region observed on 2007 May 9 with a spacecraft separation angle of  = 7.3° (Aschwanden et al., 2008c), and for 9 loops observed on 2007 Jun 8 with a spacecraft separation angle of  = 12° (Feng et al., 2007a). Further stereoscopic triangulations have been applied to oscillating loops (Aschwanden, 2009c), to polar plumes (Feng et al., 2009), to an erupting filament (Liewer et al., 2009), or to an erupting prominence (Bemporad, 2009), including a strongly rotating, erupting, quiescent polar crown prominence (Thompson, 2011).