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B. SHMUTTERIsrael Institute of Technology
Haifa, Israel
Connecting Adjacent Models
The distribution of residual errors in the connection ofadjacent models in a strip is discussed.
STRIP TRIANGULATION with independentmodels is used widely in photogrammet
ric practice. One of the problems inherent invarious triangulation procedures is the connection of two consecutive models in thestrip to each other. Whatever procedure isemployed to perform the connection, it isusually based on an adjustment whichutilizes redundant observations, these beingthe observed coordinates of points commonto the two adjacent models. Because of theredundant observations, the adjustmentyields residual errors, so that after the connection of the models has been performed,
in which:
X = coordinates of a point observed in the previous model,x = coordinates of the same point observed inthe new model,Xv = a translation,A = a scale factor, andU = an orthogonal rotation matrix whoseelements are computed from three angles.
Each common point (including the projection centers) contributes one equation of type(1), i.e., three equations in coordinates, forthe solution of the unknowns. Since Equation 1 is non-linear with respect to the sought
ABSTRACT: The paper discusses the problem of distributing residualerrors that remain after connecting two consecutive models to eachother in a strip. The strip coordinates ofpoints located in the commonzone of adjacent models are determined uniquely.
each point common to both models is associated with two sets of strip coordinates:one set that was determined while observingcoordinates in the previous model, andanother that resulted from the transformationof the observed coordinates of the "new"model. The differences between the valuesof these two sets of coordinates are the residual errors. Since each point should be determined by a unique set ofstrip coordinates,the question arises of how to handle the residuals in order to assign a single set of coordinates to each point. An analysis ofthis problem is presented.
It is assumed that the connection of a newmodel in the strip to the previous one is carried out by a coordinate transformation, thetransformation elements being derived fromthe equation:
scale factor and rotation angles, the unknowns have to be solved for by an iterationprocess which makes use of linear expressions obtained from expanding the aboveequations around an initial solution. Secondly, in order to fulfill Equation 1, all themeasured quantities X and x have to be corrected. Hence, the demand for a unique determination of the coordinates of the common points, while solving the unknown elements for the connection procedure, leads toan adjustment which involves condition equations with unknowns
X o + AU (x+vn) - (X +vp ) = O. (2)
The subscript n denotes the new model,and p - the previous one.
After linearizing Equation 2 the following expression is obtained:
x = Xo + Aax (1) (3)
617
618 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1975
(11)
(13)
(12)
the ob-
1K = - ,..".....-;-, WA· + 1
where the W represent the final discrepancies between the coordinates andare defined as follows: .
W = X (transformed) -X.The corrections to be added toserved coordinates are given by
(5)
BV + AX + W = O.
V =BTK
K = - (BBT)-I (AY+W)
Y = - (AT (BBT)-I A)··I AT (BBT)-I W
where A is a coefficient matrix and Y is avector which represents the sought corrections to the elements assumed in the initial solution (denoted in the equation by abar).
Each common point yields three equations for solving the unknowns and theentire system of the condition equationshas the form
that which would be obtained if the problem were stated in terms of usual observation equations.
The solution of the orientation elementsis based on an iteration process; for eachstep the matrix A and the discrepancies Ware computed anew utilizing the corrections Y found in the previous step. Afterthe last iteration step (convergence of theprocess) the corrections Y will be very
(4) small, practically equal to zero. Thus, theThe V are the corrections to the observed correlates K may be computed fromcoordinates and W the discrepancies (thebracketed term is Equation 3).
If the correlations between the observedcoordinates are disregarded, which iscommon practice in many triangulationprocedures, and the observations are assumed equally weighted, the conditionEquation 4 provides the following solutions:
(14)
Taking into account the form of the matrixB, the required correction to the observedcoordinates are computed as follows: Thecorrections for the coordinates of the previous model equal
(6)1]The matrix B has a quasi-diagonal formand can be presented by
[
BI 0o B 2
B = . .
o 0
and the unknown corrections to the orientation elements are solved from
According to Equation 10 the solution ofthe required quantities Y is identical with
(15)
and, for the corrections of the new model,one has
The corrections vn ought to be added tothe observed coordinates of the newmodel, and the corrected coordinates inserted into the transformation equations.Instead, regarding the fact that the transformation is linear with respect to thecoordinates, the values V n can be transformed separately, and their transformedvalues then added to the transformedcoordinates which were obtained duringthe solution of the orientation and utilizedfor the determination of the discrepanciesW.
v - >..av - - 1..2
W (16)n (transformed) - n - 1..2 +1 .
Equation 16 holds because a is an orthogonal matrix, and aaT is a unit matrix.
Now we arrive at the following conclusions: The discrepancies between the coor-
(8)
(10)
BBT = (1..2 + 1) I
(BBT)-I = 1..2 ~ 1 I.
Each matrix B i is associated with the corrections to one common point. All matrices B i are equal to each other and havethe following form:
B i = [Xii -I]. (7)
Since ii is an orthogonal matrix and I aunit matrix, we have
It follows therefore that the correlates forcomputing corrections to the observedcoordinates are given by
1K = - 1..2 +1 I (AY + W) (9)
CONNECTING ADJACENT MODELS 619
dinates which remain after performing theconnection between two models are distributed according to the ratios stated by Equation 14 and 16.
Usually the scale variations between twoconsecutive models are small and the valueof A is very close to unity. Hence,
It follows, therefore, that when the scalealterations between consecutive models
are small, the common practice of averaging the two sets of coordi nates for eachpoint is fully justified.
The author believes that the justificationfor averaging the coordinates as presentedabove will satisfy all who have intuitivelyapplied that rule, even when they haddoubts about its theoretical basis.
REFERENCES
1. B. Shmutter, Triangulation with independent models, Photogrammetric Engineering,June 1969.
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