geometric correction

CHAPTER 4A. DERMANIS: REMOTE SENSINGnT2 2v v = ∑(vx+ vy) = min(15)k kk = 1is well known to be given by the unique solution of the normal equationsTT( A A)ˆz = A b , which explicitly is−− ⎡aˆTTT 1 TT 1 T ⎤ ⎡QQ 0 ⎤ ⎡Qi⎤⎡(Q Q)Q i⎤zˆ = ( A A)A b = ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ = ⎢ − ⎥⎣bˆ. (16)TTT 1 T⎦ ⎣ 0 Q Q⎦⎣Qj⎦⎣(Q Q)Q j⎦The corresponding individual residualsvˆ= b − Azˆ, (17)as well as the single collective parametervˆT vˆσ ˆ 2 = , (18)fgive a measure of the “accuracy” with which the model has been fitted tothe known points. The integer f = 2 n − s , called the degrees of freedom,is the difference between the number of equations and the number of parameters.A value of σ ˆ significantly larger than 1, designates bad fit-2ting, in which case one or more points with large residuals should be removedand the solution should be repeated with the remaining points. Amore rigorous and more efficient treatment of the problem is givenwithin the framework of statistical estimation theory for the linear model(Gauss-Markov model), see e.g. Sansó (1996).As an example we give the solution for the linear transformation−1i i(x,y)= a + a x + a y = ti + a x + b y= 00 10 01j j(x,y)= b + b x + b y = t j + c x + d y= 00 10 01(19)The solution is obtained after performing all the above matrix calculationsand has the form2ysxi2 2sxsys − s s s ρi xi − ρ ρ yiaˆ= =, (20)− s 1−ρxy yi2sxyxxy2xy2ysxj2 2sxsys − s s s j ρ xj − ρ ρ yjcˆ= =, (21)− s 1−ρxy yj2sxyxxy2xy6