Edward M. Mikhail
P= PH
2b-
P = am: Aus] dz
3x4 3x4 3x4
Since Equation (12a) implies equality up to a scale factor, we write three condition equations by dividing each of the
first three equations in (12a) by the fourth equation. Once solved for, the H matrix may be used as in Equation (12a) to
compute absolute ground coordinates, or as in (12b) to compute the absolute camera transformation matrices.
The photogrammetric camera parameters can be extracted from the camera transformation matrix, P, in Equation (12c)
in which the matrix A is a function of x,, yo, and f, for the standard case of three interior orientation parameters. The
matrix M is an orthogonal rotation matrix, i.e. a function of three independent rotation angles, &,P,K. S is a vector of
the three ground coordinates of the camera perspective center, Xz, Yz, Z,. The details of an algorithm for extracting
camera parameters can be found in [Barakat and Mikhail, 1998] and [Faugeras, 1993].
3.2 Multiple Frame Simultaneous Ground Point Intersection
Following are the two condition equations for each image point on each image, i, that are used to solve for the ground
coordinates of an object point (X, Y, Z):
F. x, (p'auX + p'aY + p'3Z + p'ai)- (p'nX 4 ploY 9 p'aZ p'ia )=0 sat)
3a-
F, = y,(paX +p'2Y+ p'nZ + p )- (pn X + p'»Y + p'aZ + p' )=0
where Pk is the (jk) element of the absolute camera transformation matrix for image i. An approximate linear
P elements as constants and the object point coordinates as
solution is obtained by treating the observations and
mizing the sum of the squared errors to the equations.
parameters, and using the least squares criterion of mini
Although some published object reconstruction techniques stop here with the linear solution, we perform a rigorous
refinement by linearizing F, and F, with respect to parameters and observations. Therefore, we use the general least
squares model, Av + BA = f ; see [Mikhail, 1976].
3.3 Experiments
For each of the data sets in Section 2.4, object reconstruction experiments are run and the results are evaluated by check
points. The object reconstruction steps are: 1) establish the relationship between the image coordinates only by solving
for the T elements or the a;; and bj, which is also the first step for image transfer; 2) Use a minimum of 5 ground control
points and the relationship shown in Equation (12a) to solve for the 15 elements of the 3D projective transformation
matrix, H; and 3) Compute the check point ground positions using the image coordinates and the absolute camera
transformation matrices and compare to their known values. For each of the data sets, results are shown for both the
linear solution and the nonlinear refinement. Two-frame versus three-frame ground point intersections are also
considered.
Table 4 shows the results from object reconstruction with simulated data, using image coordinates on the two oblique
frames to compute the check points. The non-linear refinement does not improve the results for this data set. Models 2-
4 show some improvement compared to Model 1.
10 wm perturbation 15 wm perturbation
25 um perturbation
|
XRMS | YRMS | ZRMS | XRMS | YRMS | ZRMS | XRMS | YRMS Z RMS
(meters) | (meters) (meters) | (meters) (meters) | (meters) (meters) | | (meters) (meters)
| | 0.04 0.05 0.11 0.15 | 0:20 013. | 015 | 023
| 0.14 0.08 0.17 o1 | 005 | 012.014 T 013 | oi
| Non- 004 | 0.07 0.12 oF ] Oi$ 1.920 [013 1.01 3 | 922
| Linear | 2-4 0.14 0.08 | 0.17 0.13 0.07 0.12 0.14 | 013 | 020
Table 4. Object Reconstruction with Simulated Data, 8 control points, 7 check points
590 International Archives of Photogrammetry and Remote Sensing. Vol.. XXXIII, Part B3. Amsterdam 2000.
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