3. THE SOLUTION OF TRANSFORMATION PARAMETERS

Transcription

1 Deartment of Geosatial Siene. HE SOLUION OF RANSFORMAION PARAMEERS Coordinate transformations, as used in ratie, are models desribing the assumed mathematial relationshis between oints in two retangular oordinate sstems; in these notes, the u, and the, sstems. o determine the arameters of an transformation, oordinates of oints ommon to both sstems must be nown. hese oints are nown as "ontrol oints" or "ommon oints". he number of ommon oints required for the solution of transformation arameters deends on the number of arameters in the transformation. In D transformations, eah ommon oint gies rise to two equations, thus ommon oints will gie n equations. herefore, if the four arameters of a Linear Conformal transformation are to be determined, then a minimum of two ommon oints are required to sole for the arameters. For Affine transformations, (si arameters) and rd-order Polnomial transformations (u to twent arameters), minimums of three and ten ommon oints resetiel are required for a solution of the arameters. It is good measurement ratie to determine oordinate transformation arameters b using more than the minimum number of ommon oints. his introdues redundant equations into the solution for the arameters and the theor of least squares is emloed to alulate the best estimates. Parameters alulated in this manner are usuall more reliable and the least squares roess allows reision estimation of the arameters as well as an assessment (ia residuals) of how well the transformation model fits the ommon oints. B using least squares, seeral tes of transformations an be "tested" on the ommon oints to assess their suitabilit... he Solution Proess he solution for the arameters for an transformation inoles the following stes (i) Choose the aroriate transformation model assumed to lin the, and u, oordinate sstems. (ii) Selet the ommon oints ensuring that there are suffiient to allow a redundant set of equations. (iii) Selet the aroriate least squares adjustment model to be used to estimate the arameters. (i) Selet the aroriate weight matri for the model. () Sole for the arameters and residuals., R.E. Deain Coordinate ransformations ()

2 Deartment of Geosatial Siene (i) Assess the suitabilit of the model b analsis of the arameters and residuals... Solution of D Linear Conformal ransformation Parameters he D Linear Conformal transformation, onsisting of rotation, saling and translation is set out in Setions. and.6. he transformation model for the b equations (.),,,, ommon oints is gien a b u t + t b a (.) Elements of the solution roess defined aboe is set in detail below.... Mathematial Models wo least squares models an be onstruted to sole for the arameters of a D Linear Conformal transformation. Whih one is seleted will deend on how the u, and, oordinates are to be treated. he two hoies of models are. (a) Onl the, oordinates are treated as obserations with residuals and. his leads to an adjustment model of the form + B f. he solution for the arameters is diret (requiring no iteration) and relatiel simle; it is b far the most oular model. his tehnique of siml adding residuals to aount for the inonsisten in the model is similar to that used in statistial regression roblems. his adjustment model suffers in omarison to model (b) due to its inabilit to roerl treat both sets of oordinates as obserations. hat is, one annot roerl assign (or onstrut) oariane matries for both the, and u, oordinates (the obserations) of the ommon oints. Hene, solutions from this model ma not be the best estimates. On the other hand, in man aliations, nothing is nown of the reision of the oordinates and this adjustment model ma be the most aroriate. (b) Both the, and u, oordinates are treated as obserations with residuals,, u and. his leads to an adjustment model of the form A + Bδ f requiring an iteratie solution roess. his model has an adantage oer model (a) as the reision of the oordinates (the obserations) an be roerl taen into aount in the solution., R.E. Deain Coordinate ransformations ()

3 Deartment of Geosatial Siene Model (a) + B f Equations (.) an be eressed in the form of obseration equations where and are small unnown orretions or residuals siml added to the equations to aount for the assumed inonsisten in the model. We ould thin of these residuals as onsisting of two arts; one art assoiated with the u, sstem and the other assoiated with the transformed, sstem; the subsrits and attahed to the residuals siml reflet the fat that the hae been added to the "transformed" side of the model. a b u t + t + b a (.) Re-arranging (.) so that all the "unnowns" are on to the left of the equals sign and the obserations are to the right gies au b t a + bu t (.) For ommon oints and u 4 unnown arameters, the artitioned matri reresentation of the n equations (.) is u a u b u t t u + u u u u (.4) hese equations are reresented b the matri equation + B f (.5) where B is an (n,) olumn etor of residuals is an (n,u) matri of oeffiients, R.E. Deain Coordinate ransformations ()

4 Deartment of Geosatial Siene f is a (u,) etor of unnown arameters is an (n,) olumn etor of numeri terms (oordinates) he equations for the solution of arameters and residuals is set out in setion.5. noting that in this ase and f hae relaed δ and f resetiel. he general form of the normal equations BWB h BWf(or N t ) assuming that W Q I are ( u + ) u a b ( u + ) ( u) ( u + ) u t n t smmetri n (.6) Centroidal oordinates Comutational saings an be made b reduing oordinates to a entroid. For the ommon oints, the oordinates of the entroid in the, sstem are and the entroidal oordinates of these same oints are then Similar relationshis an be written for entroidal oordinates in the u, sstem. A useful roert of the entroidal oordinates of the ommon oints is that their sums equal zero, ie,, R.E. Deain Coordinate ransformations () 4

5 Deartment of Geosatial Siene u hus, relaing, and u, oordinates with their entroidal ounterarts, and u, redues the transformation (.) to a b u b a (.7) It should be noted here that translations t and t are both zero when entroidal oordinates are used indiating that the entroids, and u, are the same oint. For ommon oints and u unnown arameters, the artitioned matri reresentation of the n equations resulting from the entroidal model (.7) is u u a u b u + u u u u (.8) hese equations are reresented b the matri equation (.5). With W Q I, the normal equations hae the following simle form ontaining onl three different numbers u + u + n ( ) he solutions for the arameters a and b are ( ) a n b ( u + ) ( u ) a b u u + + g h (.9) (.), R.E. Deain Coordinate ransformations () 5

6 Deartment of Geosatial Siene b ( u) ( u + ) (.) he translations t and t are obtained b re-arranging (.) as t a b u t b a or t au b t + bu a (.) After alulation of the arameters, a b t t the residuals are alulated using (.8). Model (b) A + Bδ f Equations (.) an be written in funtional form f au + b + t f bu + a + t (.) where the etor of arameters is a b t t and the etor of obserations is l u with ofator matries as estimates of reision. Aling the riniles of setion. gies a set of equations of the form A + Bδ f where, for a single oint, the oeffiient matries A and B are f f f f F u a b A ˆ l, f f f f l b a u f f f f F a b t t u B ˆ l, f f f f u a b t t he etor of numeri terms f is, R.E. Deain Coordinate ransformations () 6

7 Deartment of Geosatial Siene au b t + f F ( l, ) bu a t + Note that in the matries aboe a b t t is a etor of aroimate alues of the arameters and the matries A, B and the etor f are ealuated with these aroimate alues. If the are unnown, then the are set to zero for the first iteration. Centroidal oordinates Comutational saings an be made b using gentroidal oordinates. he number of arameters is redued to two, sine translations t and t are eliminated. Equations (.) an be written in funtional form using entroidal oordinates f au + b f bu + a (.4) where the etor of arameters is a b and the etor of obserations is l u with ofator matries as estimates of reision. Aling the riniles of setion. gies a set of equations of the form A + Bδ f where, for a single oint, the oeffiient matries and etor of numeri terms beome f f f f F u a b A ˆ l, f f f f l b a u f f F a b u B ˆ l, f f u a b he etor of numeri terms f is au b + f F ( l, ) bu a + For a single oint, the matri equation A + Bδ f has the following form, R.E. Deain Coordinate ransformations () 7

8 Deartment of Geosatial Siene a b u δ a au b + b a + u u δ b bu a + For the 4 ommon oints, the artitioned matri reresentation of the equation A + Bδ f resulting from the entroidal model gien b the funtional equations (.4) is gien as u a b u au b + b a u a b u bu a + u au b + b a a b + u δ a bu a + u δ b au b + b a u bu a + a b u u4 4 au4 b4 + 4 b a 4 u4 bu4 a u4 4, R.E. Deain Coordinate ransformations () 8