COORDINATE TRANSFORMATIONS

1 PLATE 17-1 COORDINATE TRANSFORMATIONSTWO DIMENSIONAL TRANSFORMATIONSThe two dimensional conformal COORDINATE transformation isalso known as the four parameter similarity transformationsince it maintains scale relationships between the two Scaling2. Rotation3. Translation in X and YxYXy1342 BCA1324 ABC(a)(b)PLATE 17-22D CONFORMAL TRANSFORMATIONSS teps in transforming coordinates measured in thecoordinate system shown in (b) into that shown in (a).SABaABbxSxySyPLATE 17-32D CONFORMAL TRANSFORMATIONSStep 1: SCALING. To make the length between A and Bin (b) equal to the length between A and B in (a).andCall this scaled system (b ).TyY'XX'Yx'Y'4Y44X'4X4 Txy PLATE 17-42D CONFORMAL TRANSFORMATIONSStep 2: ROTATION. Rotate COORDINATE system in scaledsystem (b ) so that points in (b ) coincide with pointsin (X , Y ) ' = x' Cos - y' Sin Y' = x' Sin + y' Cos PLATE 17-52D CONFORMAL TRANSFORMATIONSSTEP 3: TRANSLATIONSUse coordinates of common point in scaled-rotated system(X , Y ) to compute T and = X - X xT = Y - Y yTan1baSaCosPLATE 17-6 OBSERVATION EQUATIONSC ombining equations for scale, rotation, and translationyields:X = (S Cos )x - (S Sin )y + TXY = (S Sin )x + (S Cos )y + TYLet S Cos = a, S Sin = b, T = c, and T = dXYAdd residuals to develop observation - by + c = X + vXay + bx + d = Y + vYNOTE:Axaya10yaxa01xbyb10ybxb01xcyc10yc xc01 XabcdLXAYAXBYBXCYCVvXAvYAvXBvYBvXCvYCPLA TE 17-7 LEAST SQUARES EXAMPLET ransform points in (x, y) system into (E,N).

2 PointENxyA1,049, , ,049, , ,049, , observation equations in form, AX = L + 17-8 LEAST SQUARES EXAMPLES olve system using unweighted least squares = (AA) ALT-1 TSO a = , b = , T = 1,050, , and xT = 50, 17-9 USING OBSERVATION EQUATIONSTRANSFORM REMAINING POINTSTABULATE RESULTST ransformed Control PointsPOINTXYVXVYA1,049, , ,049, , ,049, , Parameters and estimated errors a= b= 1,050, 50, PointsPOINTXY Sx Sy11,049, , ,047, , ,046, , ,045, , = 183 13' " Scale = 's Reference Variance = 17-102D AFFINE TRANSFORMATIONThe Six Parameter TransformationOBSERVATION EQUATIONSax + by + c = X + VXdx + ey + f = Y + VYEach axis has a different scale the most probable values for the 2D affinetransformation parameters for the data above. Transformpoints 306 and 307 into the (X, Y) 17-122D AFFINE TRANSFORMATIONO bservation 17-132D AFFINE TRANSFORMATIONS olution: X = (AA)ALT-1 TPLATE 17-14 TABULATE RESULTST ransformed Control Points POINT X Y VX VY-------------------------------------- ------------------- 1 3 5 7 Parameters.

3 A = b = c = d = e = f = 's Reference Variance = Points POINT X Y x y--------------------------------------- ----------------- 1 3 5 7 306 307 17-152D PROJECTIVETRANSFORMATION(The Eight Parameter transformation )OBSERVATION EQUATIONSNote that these equations are exact solution to compute initial values for (a3xb3y1)2xfb3a1xb1yc1(a3xb3y1)2yfa3a2xb 2yc2(a3xb3y1)2yfa3a2xb2yc2(a3xb3y1)2yPLA TE 17-16 LINEARIZED EQUATIONSFor every point, the matrix for is:wherePLATE 17-17 EXAMPLEC ompute the transformation parameters for the followingdata using a 2D projective parameter values solved by using only first = = = = = = = = 17-18 EXAMPLEITERATION 1 PLATE 17-19 TABULATE RESULTST ransformation Parameters.

4 A1 = b1 = = = = c2 = = b3 = 's Reference Variance = Number of Iterations = 2 Transformed Control Points POINT X Y VX VY-------------------------------------- -------------------- 1 1, 2 3 4 -1, 5 1, -2, 6 -3, 3, Points POINT X Y x y ---------------------------------------- ------------------ 1 1, 2 3 4 -1, 5 1, -2, 6 -3, 3, 7 -2, 1, 8 -6, -4, ()Sin()0 Sin()Cos()andXxyzPLATE 17-20 THREE DIMENSIONAL CONFORMALCOORDINATE TRANSFORMATIONS imilar to two-dimensional conformal transformation with3 rotational about x axis, = MX'11wherezzxx2121X2x2y2z2,andM2 Cos()0 Sin()010 Sin()0 Cos()PLATE 17-21 THREE DIMENSIONAL CONFORMALCOORDINATE TRANSFORMATIONR otation about Y axis, = MX22 1whereYyXx11 XXXYZ,andM3 Cos()Sin()0 Sin()Cos()0001 PLATE 17-22 THREE DIMENSIONAL CONFORMALCOORDINATE TRANSFORMATIONR otation about Z axis, = MX3 2whereXm11m12m13m21m22m23m31m32m33 PLATE 17-23 THREE DIMENSIONAL CONFORMALCOORDINATE TRANSFORMATIONF inal combined expression: = M M M X' = M X'321where M iswhere.

5 M = Cos() Cos()11m = Sin() Sin() Cos() + Cos() Sin()12m = -Cos() Sin() Cos() + Sin() Sin()13m = -Cos() Sin()21m = -Sin() Sin() Sin() + Cos() Cos()22m = Cos() Sin() Sin() + Sin() Cos()23m = Sin()31m = -Sin() Cos()32m = Cos() Cos()33 XSo0 XoXo100 YSoYoYoYo010 ZSoZoZoZo001dSddddTxdTydTzXXoYYoZZoPLATE 17-24 THREE DIMENSIONAL CONFORMALCOORDINATE TRANSFORMATIONOBSERVATION EQUATIONS:X = S( m x + m y + m z ) + Tx111213Y = S( m x + m y + m z ) + Ty212223Z = S( m x + m y + m z ) + Tz313233 Linearized Observation Equations for a single [m13xm23ym33z]ZS[m12xm22ym32z]XS[Sin()Co s()xSin()Sin()yCos()z]YS[Sin()Cos()Cos() xSin()Cos()Sin()ySin()Sin()z]ZS[m12xm22y m32z]XS[m21xm11y]YS[m22xm12y]ZS[m23xm13y ]PLATE 17-25 THREE DIMENSIONAL CONFORMALCOORDINATE transformation wherePLATE 17-26 EXAMPLEPtXYZx Sy Sz are the most probable values for the 3D transformationparameters? J matrix K matrix~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ ~~~~~~~~~~ matrix~~~~~~~~ Points---------------------------------- ---------------------------------------- -- NAME x

6 Y z Sx Sy Sz-------------------------------------- -------------------------------------- 1 2 3 4 POINTS---------------------------------- ---------------------------------------- -- NAME X VX Y VY Z VZ-------------------------------------- -------------------------------------- 1 2 3 4 17-27 EXAMPLET ransformation Coefficients---------------------------- Scale = = 2 17' " 0 00' " Phi = -0 33' " 0 00' "Kappa = 224 32' " 0 00' " Tx = Ty = Tz = Standard Deviation: of Freedom: 5 Iterations: 2 Transformed 17-28 STATISTICALLY VALID PARAMETERSThe adjusted parameters divided by their standarddeviation represents a t statistic.

7 Thus the parameterchecked for statistical is:PLATE 17-29 EXAMPLEA ssume results of two dimensional projectivetransformation with 2 degrees of freedom are:ParameterSt-valuea = = = = = = these parameters statistically different from 0 at a 5%level of significance?H: parameter = 0oH: parameter 0aRejection Region is when t-value > t/2, vt-value > = t = t /2, , 2 Yes, all parameters are statistically different from 0 sincetheir t-values are greater than