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 7View Image), 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 (2006Jump To The Next Citation Point).
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Figure 8: Orientation of epipolar planes in space and the respective epipolar lines in the images for two observers (e.g., STEREO spacecraft A and B) looking at the Sun (from Inhester, 2006Jump To The Next Citation Point).

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 [X, Y ] (Figure 8View Image); (2) identification of corresponding point-like or curvi-linear features in each image (with coordinates (XA, YA) and (XB, YB ); and (3) geometric triangulation to retrieve the 3-coordinates (x,y,z) 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 s, yielding two sets of 2D coordinates [XA (s),YA (s)] and [XB (s),YB (s)]. 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 s, as described in the next paragraph for the case of two spacecraft A and B with different distances to the Sun.

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Figure 9: The geometry of triangulating or projecting a point P from spacecraft A and B is shown, where the (epipolar) XZ plane is coincident with the Sun center position O and the two spacecraft positions A and B (left panel), while the vertical YZ plane is perpendicular (right panel). The distances of the spacecraft from the Sun are dA and dB, the observed angles of point P with respect to the Sun center O are αA and αB, intersecting the X-axis at positions xA and xB with the angles γA and γB. The spacecraft separation angle is αsep. The point P has the 3D coordinates (x,y,z) and heliographic longitude l and latitude b (from Aschwanden et al., 2008bJump To The Next Citation Point).

We define a coordinate system (X, Y, Z) that has the origin O in the Sun center, the Z-axis is the line-of-sight from spacecraft A to Sun center, and the XZ plane coincides with the plane of the spacecraft A and B (Figure 9View Image left). The two spacecraft have distances of d A and d B from the Sun center and observe a point P at an angle of α A and α B in X-direction from Sun center (corresponding to the difference of x-pixels in the image) and at an angle δA and δB in Y-direction from Sun center (corresponding to difference in y-pixels in the image). The spacecraft have a separation angle of αsep in the XZ plane. The point P has the 3D coordinates (x, y,z) or heliographic longitude l and latitude difference b with respect to the central meridian defined by the line-of-sight axis of spacecraft A. The projected positions of the point P on the X-axis are xA from spacecraft A and xB from spacecraft B. So, our main problem is to solve for the variables (x,y,z ) using the observables (dA, dB,αA, αB, δA,δB,αsep).

In the triangles (O, A, xA) and (O, B, xB) we can determine the angles γA and γB simply from the geometric rule that the sum of the three angles in a planar triangle amounts to π, i.e.,

γA = π-− αA , (15 ) 2
π γB = 2-− αB − αsep . (16 )
Using the sine relation in a planar triangle [a∕ sin (α ) = b∕sin(β) = c∕ sin(γ)] we obtain the sides xA and x B in the two triangles,
xA = dA tan(αA ), (17 )
xB = dB sin-αB-. (18 ) sin γB
Furthermore we have the relations in the rectangular triangles, tan γ = z∕ (x − x) A A and tan γB = z ∕(xB − x), from which we can determine the coordinates x,
x = xB-tan-γB-−-xA-tan-γA-, (19 ) tan γB − tan γA
and z,
z = (xA − x )tan γA. (20 )
The coordinate y can be obtained from a relation in the Y Z plane (Figure 9View Image right),
y = (d − z)tan δ . (21 ) A A
The distance of point P from the Sun center O is then
∘------------ r = x2 + y2 + z2 , (22 )
and the height of the point at position P is
h = r − R . (23 ) ⊙
The method of stereoscopic triangulation is also called tie-point method, because a second point [XB (si),YB(si)] in image B is tied to a first point [XA (si),YA (si)] 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 αsep = 7.3° (Aschwanden et al., 2008cJump To The Next Citation Point), and for 9 loops observed on 2007 Jun 8 with a spacecraft separation angle of αsep = 12° (Feng et al., 2007aJump To The Next Citation Point). Further stereoscopic triangulations have been applied to oscillating loops (Aschwanden, 2009cJump To The Next Citation Point), to polar plumes (Feng et al., 2009Jump To The Next Citation Point), to an erupting filament (Liewer et al., 2009Jump To The Next Citation Point), or to an erupting prominence (Bemporad, 2009Jump To The Next Citation Point), including a strongly rotating, erupting, quiescent polar crown prominence (Thompson, 2011).

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