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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