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\Huge
\begin{center}
Chapter 14 Analytic Photogrammetry        
\end{center}

\LARGE
14.1 Introduction \\
Analytic photogrammetry includes the body of techniques by which, from \\
\hspace*{1cm} measurements of one or more 2D-perspective projections of \\
\hspace*{1cm} a 3D object, one can make inferences about the 3D position, \\
\hspace*{1cm} orientation, and lengths of the observed 3D object parts \\
\hspace*{1cm} in a world reference frame. \\
inference problems: can be construed as nonlinear least-squares problems \\
Photogrammetry provides a collection of methods for determining the position \\
\hspace*{1cm} and orientation of cameras and range sensors in the scene and \\
\hspace*{1cm} relating camera positions and range measurements to 
              scene coordinates. \\
GIS: Geographic Information System \\
GPS: Global Positioning System              

\vspace{1cm}
\noindent
exterior orientation: determines position and orientation of camera in  \\
\hspace*{1cm} absolute coordinate system from projections of calibration \\
\hspace*{1cm} points in scene \\
exterior orientation of camera: specified by all parameters of camera pose \\
pose parameters: perspectivity center position, optical axis direction... \\
exterior orientation specification: requires 3 rotation angles, 3 translations

\vspace{1cm}
\noindent
interior orientation: determines internal geometry of camera \\
interior orientation of camera: parameters determining geometry of 3D rays \\
\hspace*{1cm} from measured image coordinates \\
The parameters of interior orientation relate the geometry of ideal \\
\hspace*{1cm} perspective projection to the physics of a camera. \\
parameters: camera constant, principal point, lens distortion .... \\
complete specification of camera orientation: interior and exterior orientation

\vspace{1cm}
\noindent
relative orientation: determines relative position and orientation between \\
\hspace*{1cm} 2 cameras from projections of calibration points in scene \\
relative orientation: to calibrate relation between two cameras for stereo \\
relative orientation: relates coordinate systems of two cameras to each other,\\
\hspace*{1cm} not knowing 3D points themselves, only their projections 
              in image \\
relative orientation of one camera to another constitutes stereo model:\\
\hspace*{1cm} specified by 5 parameters: 3 rotation angles, 2 translations \\
relative orientation: assumes interior orientation of each camera known 

\vspace{1cm}
\noindent
absolute orientation: determines transformation between 2 coordinate systems \\
\hspace*{1cm} or position and orientation of range sensor in absolute \\
\hspace*{1cm} coordinate system from coordinates of calibration points \\
absolute orientation: to convert depth measurements in viewer-centered \\
\hspace*{1cm} coordinates to absolute coordinate system for the scene \\
absolute orientation: orientation of stereo model in world reference frame \\
absolute orientation: determines scale, 3 translations, 3 rotations \\
absolute orientation: recovery of relation between two coordinate systems 

\vspace{1cm}
\noindent
$\left ( \begin{array}{c} x \\ y \\ z \end{array} \right )$: 
a point in the world reference frame \\
$\left ( \begin{array}{c} x_0 \\ y_0 \\ z_0 \end{array} \right )$: 
camera lens position 

\vspace{1cm}
\noindent
left-hand coordinate system: clockwise angle of rotation \\
$\omega$: angle of rotation around $x$-axis of camera reference frame \\
$\omega$: tilt angle \\
$\phi$: angle of rotation around $y$-axis \\
$\phi$: pan angle \\
$\kappa$: angle of rotation around $z$-axis \\
$\kappa$: swing angle  \\
3 $\times$ 3 rotation matrix:
\[
R(\omega)= \left (
\begin{array}{ccc}
1 & 0 & 0 \\
0 & \cos \omega & \sin \omega \\
0 & -\sin \omega & \cos \omega
\end{array}
\right )
\]
\[
R(\phi)= \left (
\begin{array}{ccc}
\cos \phi & 0 & -\sin \phi \\
0 & 1 & 0 \\
\sin \phi & 0 & \cos \phi
\end{array}
\right )
\]
\[
R(\kappa)= \left (
\begin{array}{ccc}
\cos \kappa & \sin \kappa & 0 \\
-\sin \kappa & \cos \kappa & 0 \\
0 & 0 & 1 
\end{array}
\right )
\]
$R(\omega, \phi, \kappa)=R(\kappa)R(\phi)R(\omega)
= \left (
\begin{array}{ccc}
r_{11} & r_{12} & r_{13} \\
r_{21} & r_{22} & r_{23} \\
r_{31} & r_{32} & r_{33} 
\end{array}
\right )=$
\Large
\[
\left (
\begin{array}{ccc}
\cos \phi \cos \kappa & \sin \omega \sin \phi \cos \kappa 
+\cos\omega\sin\kappa & -\cos\omega\sin\phi\cos\kappa +\sin\omega\sin\kappa \\
-\cos\phi\sin\kappa & -\sin\omega\sin\phi\sin\kappa+\cos\omega\cos \kappa 
& \cos\omega\sin\phi\sin\kappa+\sin\omega\cos\kappa \\
\sin\phi & -\sin\omega\cos\phi & \cos\omega\cos\phi 
\end{array}
\right )
\]
\LARGE
angles can be obtained directly from values $r_{ij}$:
\[
\sin\phi=r_{31}
\]
\[
\tan \omega = (-r_{32})/r_{33}
\]
\[
\tan \kappa = (-r_{21})/r_{11}
\]

\vspace{1cm}
\noindent
point $(x,y,z)'$ in world frame represented by $(p,q,s)'$ in camera frame: \\
$'$: denotes matrix or vector transpose \\
$R^{-1} = R'$
\[
\left (
\begin{array}{c}
p \\
q \\
s
\end{array} \right ) = R(\omega, \phi, \kappa) \left (
\begin{array}{c}
x - x_0 \\
y-y_0 \\
z-z_0
\end{array} \right )
\]
pinhole camera with image at distance $f$ from camera lens, projection:
\[
\left (
\begin{array}{c}
u \\
v 
\end{array} \right ) = \frac{f}{s} \left (
\begin{array}{c}
p \\
q
\end{array} \right )
\]
$f$: camera constant, related to focal length of lens

\vspace{1cm}
\noindent
principal point: origin of measurement image plane coordinate \\
principal point $(u_0, v_0)$: in image measurement plane coordinate system 
\[
\left (
\begin{array}{c}
u \\
v 
\end{array} \right ) = \left (
\begin{array}{c}
u_0 \\
v_0 
\end{array} \right ) 
+ \frac{f}{s} \left (
\begin{array}{c}
p \\
q
\end{array} \right )
\]

\vspace{1cm}
\noindent
general fundamental perspective projection equations: collinearity equation:
\[
\frac{x-x_0}{z-z_0}=\frac{r_{11}(u-u_0)+r_{21}(v-v_0)+r_{31}f}{
r_{13}(u-u_0)+r_{23}(v-v_0)+r_{33}f}
\]
\[
\frac{y-y_0}{z-z_0}=\frac{r_{12}(u-u_0)+r_{22}(v-v_0)+r_{32}f}{
r_{13}(u-u_0)+r_{23}(v-v_0)+r_{33}f}
\]
They show that the relationship between the measured 2D-perspective projection
coordinates and the 3D coordinates is a nonlinear function of  \\
$u_0, v_0, x_0, y_0, z_0, \omega, \phi,$ and $\kappa$. \\
=====Garfield 17:27=====

\vspace{1cm}
14.2 Nonlinear Least-Squares Solutions \\
noise model:
\[
\alpha_k = g_k(\beta_1, ..., \beta_M)+\xi_k, \ \ \ k = 1,...,K
\]
$\beta_1, ..., \beta_M$: unknown parameters governing nonlinear transformations \\
$g_1,..., g_K$: nonlinear transformations \\
$\alpha_1,...,\alpha_K$: observed values of $g_1,..., g_k$ \\
$\xi_1,...,\xi_K$: additive mean zero Gaussian random variables with covariance
$\not\Sigma$

\vspace{1cm}
\noindent
maximum likelihood solution: $\beta_1, ..., \beta_M$ maximize $Prob(\alpha_1,...,\alpha_K|\beta_1, ..., \beta_M)$ \\
maximum likelihood solution minimizes least-squares criterion
\[
\epsilon^2 = (\alpha - g)'\not\Sigma^{-1}(\alpha -g)
\]
where
\[
\alpha = \left (
\begin{array}{c}
\alpha_1 \\
\alpha_2 \\
\vdots \\
\alpha_K
\end{array} \right ) {\rm \ and \ } g = \left (
\begin{array}{c}
g_1(\beta_1, ... , \beta_M) \\
g_2(\beta_1, ... , \beta_M) \\
\vdots \\
g_K(\beta_1, ... , \beta_M) 
\end{array} \right )
\]
assumption: overconstrained and unique solution, $K>M$

\vspace{1cm}
\noindent
first-order Taylor series expansion of $g_k$ taken around $\beta^t$
\[
g_k(\beta^t+\Delta\beta)=g_k(\beta^t)+\Delta g_k(\Delta\beta;\beta^t)
\]
$\beta^t=(\beta_1^t,...,\beta_M^t)'$: $t$th iteration approximate solution \\
$\Delta\beta=(\Delta\beta_1,...,\Delta\beta_M)'$: adjustments added for a better
approximate solution \\
$\Delta g_k$: total derivative of $g_k$: linear function of adjustment vector
$\Delta\beta$
\[
\Delta g_k(\Delta\beta; \beta^t)=\left (\frac{\partial g_k}{\partial \beta_1}
(\beta^t),..., \frac{\partial g_k}{\partial \beta_M}(\beta^t)\right)\Delta\beta
\]
$G^t$: Jacobian given by
\[
G^t = \left ( 
\begin{array}{cccc}
\frac{\partial g_1}{\partial \beta_1} & \frac{\partial g_1}{\partial \beta_2} &
\cdots & \frac{\partial g_1}{\partial \beta_M} \\
\frac{\partial g_2}{\partial \beta_1} & \frac{\partial g_2}{\partial \beta_2} &
\cdots & \frac{\partial g_2}{\partial \beta_M} \\
\vdots & \vdots & \vdots & \vdots \\
\frac{\partial g_K}{\partial \beta_1} & \frac{\partial g_K}{\partial \beta_2} &
\cdots & \frac{\partial g_K}{\partial \beta_M} 
\end{array} \right )
\]
adjustments added to the current $\beta$ to form the next $\beta$:
\[
\beta^{t+1}=\beta^t+\Delta\beta=\beta^t+(G^{t'}\not\Sigma^{-1}G^t)^{-1}G^{t'}\not\Sigma^{-1}(\alpha - g^t)
\]

\vspace{1cm}
14.3 The Exterior Orientation Problem \\
one-camera exterior orientation problem: \\
known:
\begin{itemize}
\item $(x_n,y_n,z_n)'$: set of $N$ 3D points with known positions, $n=1,...,N$ 
\item $(u_n,v_n)'$: corresponding set of 2D-perspective projections, $n=1,...,N$
\end{itemize}
unknown:
\begin{itemize}
\item rotation $(\omega, \phi, \kappa)$ and translation $(x_0, y_0, z_0)'$ 
   that put the camera reference frame in the world reference frame
\end{itemize}
camera calibration problem: unknown position of camera in object frame \\
object pose estimation problem: unknown object position in camera frame \\
spatial resection problem in photogrammetries: 3D positions from 2D orientation

\vspace{1cm}
$(p_n,q_n,s_n)$: point $(x_n,y_n,z_n)$'s coordinate in camera reference frame
\noindent
\[
\left (
\begin{array}{c}
p_n \\
q_n \\
s_n
\end{array} \right ) = R(\omega, \phi, \kappa) \left (
\begin{array}{c}
x_n - x_0 \\
y_n -y_0 \\
z_n -z_0
\end{array} \right ), \ \ \ n=1,...,N
\]
observed perspective projection
\[
\left (
\begin{array}{c}
u_n \\
v_n 
\end{array} \right ) = \frac{f}{s_n} \left (
\begin{array}{c}
p_n \\
q_n
\end{array} \right ), \ \ \ n=1,...,N
\]

\vspace{1cm}
\noindent
in the notation of the previous section, $\beta=(x_0,y_0,z_0,\omega,\phi,\kappa)$
\[
g_{2n-1}(\beta)=u_n=f p_n/s_n
\]
\[
g_{2n}(\beta)=v_n=f q_n/s_n
\]

\vspace{1cm}
14.3.1 Standard Solution \\
by chain rule for partial differentiation
\[
\Delta u_n = f\frac{s_n \Delta p_n-p_n \Delta s_n}{s_n^2}
\]
\[
\Delta v_n = f\frac{s_n \Delta q_n-q_n \Delta s_n}{s_n^2}
\]
in matrix form
\[
\left (
\begin{array}{c}
\Delta u_n \\
\Delta v_n 
\end{array} \right ) = \frac{f}{s_n} \left (
\begin{array}{ccc}
1 & 0 & -p_n/s_n \\
0 & 1 & -q_n/s_n 
\end{array} \right ) \left (
\begin{array}{c}
\Delta p_n \\
\Delta q_n \\
\Delta s_n 
\end{array} \right )
\]
=====p. 134===== \\
..... (tedious mathematics, study by yourself)

\vspace{1cm}
14.3.2 Auxiliary Solution \\
Instead of iteratively adjusting the angles directly, we can reorganize the
calculation such that we iteratively adjust the three auxiliary parameters of
a skew symmetric matrix associated with the rotation matrix. \\
Then we determine the adjustments of the angles in terms of the adjustments of
the parameters of the skew symmetric matrix associated with the rotation
matrix.

\vspace{1cm}
14.3.3 Quaternion Representation \\
quaternion: one other representation for rotation matrices \\
from any skew symmetric matrix
\[
S = \left (
\begin{array}{ccc}
0 & -c & b \\
c & 0 & -a  \\
-b & a & 0
\end{array} \right )
\]
a rotation matrix R can be constructed by choosing scalar $d$
\[
R = (dI+S)(dI-S)^{-1}
\]
this definition guarantees that $R'R=I$:
\[
R'R=[(dI+S)(dI-S)^{-1}]'[(dI+S)(dI-S)^{-1}]= \cdots = I
\]
expanding the equation for $R$, we have $R=$
\[
\left (
\begin{array}{ccc}
d^2+a^2-b^2-c^2 & 2(ab-cd) & 2(ac+bd) \\
2(ab+cd) & d^2-a^2+b^2-c^2 & 2(bc-ad)  \\
2(ac-bd) & 2(bc+ad) & d^2-a^2-b^2+c^2
\end{array} \right ) \frac{1}{a^2+b^2+c^2+d^2}
\]
parameters $a,b,c,$ and $d$ can be constrained to satisfy
\[
a^2+b^2+c^2+d^2=1
\]
in terms of quaternion parameters
\[
\Delta R = \frac{\partial R}{\partial a}\Delta a + \frac{\partial R}{\partial b}
\Delta b+\frac{\partial R}{\partial c}\Delta c+\frac{\partial R}{\partial d}
\Delta d
\]
so that 
\[
\Delta R R'= \frac{\partial R}{\partial a}R'\Delta a + \frac{\partial R}{\partial b}R'
\Delta b+\frac{\partial R}{\partial c}R'\Delta c+\frac{\partial R}{\partial d}R'
\Delta d
\]
calculating the matrices directly, we obtain
\[
\frac{\partial R}{\partial a}R'=2 \left (
\begin{array}{ccc}
a & b & c \\
-b & a & -d  \\
-c & d & a
\end{array} \right )
\]
..... (tedious mathematics, study by yourself) \\
=====Oldie 33:76=====

\vspace{1cm}
14.4 Relative Orientation \\
The transformation from one camera station to another can be represented by
a rotation and a translation. \\
The relation between the coordinates, $r_l$ and $r_r$ of a point $P$ can be
given by means of a rotation matrix and an offset vector. \\
=====Horn, {\em Robot Vision}, Fig. 13.3===== \\
Relative orientation is typically concerned with the determination of the
position and orientation of one photograph with respect to another, given a set
of corresponding image points. \\
$(x_L,y_L,z_L), (x_R, y_R,z_R)$: 3D positions of left and right camera lenses\\
$(\omega_L,\phi_L,\kappa_L), (\omega_R,\phi_R,\kappa_R)$: rotation angles 
specifying exterior orientation \\
$\{(u_{Ln},v_{Ln})\}_{n=1}^N$: set of $N$ image points from left image \\
$\{(u_{Rn},v_{Rn})\}_{n=1}^N$: corresponding set of $N$ image points from right image \\
separation of two camera lenses $(x_R-x_L)$: as constant controlling scale \\
relative orientation: specified by five parameters \\
\hspace*{1cm} $(y_R-y_L),(z_R-z_L),(\omega_R-\omega_L),
(\phi_R-\phi_L),(\kappa_R-\kappa_L)$ \\
assumption: 
\begin{itemize}
\item camera interior orientation known 
\item image positions expressed to identical scale and w.r.t. principal point
\end{itemize}

\vspace{1cm}
14.4.1 Standard Solution \\
rotation matrix associated with exterior orientation of left image 
\[
Q_L'=R(\omega_L, \phi_L, \kappa_L)
\]
rotation matrix associated with exterior orientation of right image 
\[
Q_R'=R(\omega_R, \phi_R, \kappa_R)
\]
$f_R$: distance between right image plane and right lens \\
$f_L$: distance between left image plane and left lens \\
from perspective collinearity equation
\[
\left (
\begin{array}{c}
u_{Ln} \\
v_{Ln} \\
f_L
\end{array} \right ) = \frac{1}{\lambda_{Ln}} Q'_L\left (
\begin{array}{c}
x_n - x_L \\
y_n - y_L \\
z_n - z_L
\end{array} \right ) {\rm \ and \ }
\left (
\begin{array}{c}
u_{Rn} \\
v_{Rn} \\
f_R
\end{array} \right ) = \frac{1}{\lambda_{Rn}} Q'_R\left (
\begin{array}{c}
x_n - x_R \\
y_n - y_R \\
z_n - z_R
\end{array} \right ) {\rm \ and \ }
\]
Hence
\[
\left (
\begin{array}{c}
x_n \\
y_n \\
z_n
\end{array} \right ) = 
\left (
\begin{array}{c}
x_L \\
y_L \\
z_L
\end{array} \right ) + \lambda_{Ln} Q_L
\left (
\begin{array}{c}
u_{Ln} \\
v_{Ln} \\
f_L
\end{array} \right ) =
\left (
\begin{array}{c}
x_R \\
y_R \\
z_R
\end{array} \right ) + \lambda_{Rn} Q_R
\left (
\begin{array}{c}
u_{Rn} \\
v_{Rn} \\
f_R
\end{array} \right )
\]
where $\lambda_{Ln}=\frac{s_{Ln}}{f_L}$ and $\lambda_{Rn}=\frac{s_{Rn}}{f_R}$ \\
..... (tedious mathematics, study by yourself)

\vspace{1cm}
14.4.2 Quaternion Solution \\
Hinsken sets up the problem in a slightly different way. \\
Instead of determining the relative orientation of the right image with respect
to the left image, he aligns a reference frame having its $x$-axis along the
line from the left image lens to the right image lens. \\
The relative orientation is then determined by the angles 
$(\omega_R, \phi_R, \kappa_R)$, which rotate the right image into
this reference
frame, and the angles $(\omega_L, \phi_L, \kappa_L)$, which rotate the 
left image into this reference frame. 

\vspace{1cm}
14.5 Interior Orientation \\
interior orientation: inner orientation: of a camera is specified by
\begin{itemize}
\item camera constant $f$: distance between image plane and camera lens
\item principal point $(u_p, v_p)$: intersection of optic axis with image plane
  in measurement reference frame located on image plane
\item geometric distortion characteristics of the lens; assuming isotropic 
around the principal point
\end{itemize}

\vspace{1cm}
14.6 Stereo \\
optical axes parallel to one another and perpendicular to baseline \\
simple camera geometry for stereo photography \\
=====Horn, {\em Robot Vision}, Fig. 13.1===== \\
stereo pair taken by Viking Lander I on the surface of Mars \\
=====Horn, {\em Robot Vision}, Fig. 13.2===== \\
parallax: displacement in perspective projection by position translation \\
$(x,y,z)$: 3D point position \\
$(u_L, v_L)$: perspective projection on left image of stereo pair \\
$(u_R, v_R)$: perspective projection on right image of stereo pair \\
$b_x$: baseline length in $x$-axis \\
position of the left camera:
\[
\left (
\begin{array}{c}
-b_x/2 \\
0 \\
0
\end{array} \right )
\]
position of the right camera:
\[
\left (
\begin{array}{c}
b_x/2 \\
0 \\
0
\end{array} \right )
\]
$v_L=v_R$, so that the $y$-parallax is zero
\[
\left (
\begin{array}{c}
u_L \\
v_L
\end{array} \right ) = \frac{f}{z}\left (
\begin{array}{c}
x+b_x/2 \\
y
\end{array} \right ) {\rm \ and \ }
\left (
\begin{array}{c}
u_R \\
v_R
\end{array} \right ) = \frac{f}{z}\left (
\begin{array}{c}
x-b_x/2 \\
y
\end{array} \right ) {\rm \ and \ }
\]
$x$-parallax: the difference $u_L-u_R$
\[
u_L-u_R=\frac{f}{z}[x+\frac{b_x}{2}-(x-\frac{b_x}{2})]=\frac{f}{z}(b_x)
\]
Hence 
\[
z = \frac{fb_x}{u_L-u_R}
\]
=====Example 14.3=====

\vspace{1cm}
relation $z=fb_x/(u_L-u_R)$ close to being useless in real-world, because
\begin{itemize}
\item observed perspective projections are subject to measurement errors \\
      so that $v_L \neq v_R$ for corresponding points
\item left and right camera frames may have slightly different orientations
\item when two cameras used, almost always $f_R \neq f_L$
\end{itemize}
derive least-squares solution as in the textbook

\vspace{1cm}
14.7 2D-2D Absolute Orientation \\
The relationship between two coordinate systems is easy to find if we can
measure the coordinates of a number of points in both systems. \\
It takes three measurements to tie two coordinate systems together uniquely. \\
(a) A single measurement leaves three degrees of freedom of motion. \\
(b) A second measurement removes all but one degree of freedom. \\
(c) third measurement rigidly attaches two coordinate systems to each other. \\
=====Horn, {\em Robot Vision}, Fig. 13.4===== \\
2D-2D absolute orientation problem: 2D-2D pose detection problem: \\
to determine from matched points more precise estimate of rotation matrix $R$
and translation $t$ such that $y_n=Rx_n+t, n=1,...,N$: \\
determine $R$ and $t$ that minimize weighted sum of residual errors:
\[
\epsilon^2=\sum_{n=1}^N w_n||y_n-(Rx_n+t)||^2
\]
weights $w_n, n=1,...,N$: satisfy $w_n \geq 0$ and $\sum_{n=1}^N w_n = 1$ \\
if no prior knowledge: can be defined to be equal, $w_n=1/N$ \\
..... (tedious mathematics, study by yourself)

\vspace{1cm}
14.8 3D-3D Absolute Orientation \\
$p_1,...,p_N$: $N$ 3D points relative to camera reference frame \\
$q_1,...,q_N$: $N$ 3D points relative to object model reference frame \\
we must determine rotation matrix $R$ and translation vector $t$ satisfying
\[
p_n = R q_n+t, \ \ \ n = 1,...,N
\]
constrained least-squares problem to minimize
\[
\sum_{n=1}^N w_n ||p_n-(R q_n +t)||^2
\]
The least-squares problem can be modeled by a mechanical system in which
corresponding points in the two coordinate systems are attached to each
other by means of springs. \\
The solution to the least-squares problem corresponds to the equilibrium
position of the system, which minimizes the energy stored in the springs. \\
=====Horn, {\em Robot Vision}, Fig. 13.5===== \\
Lagrange multipliers can be introduced and a solution found \\
..... (tedious mathematics, study by yourself) \\
=====joke=====

\vspace{1cm}
14.9 Robust M-Estimation \\
least-squares techniques are ideal when random data perturbations \\
\hspace*{1cm} or measurement errors are Gaussian distributed \\
this section describes some robust techniques for nonlinear regression

\vspace{1cm}
14.9.1 Modified Residual Method \\
$r_i$: residual of $i$-th datum \\
$r_i$: difference between $i$-th observation and its fitted value \\
standard least-squares method: tries to minimize $\sum_i r_i^2$ \\
standard least-squares method: unstable if outliears present in data \\
outlying data: strong effect to distort estmated parameters \\
M-estimators: try to reduce outlier effect by replacing squared residuals \\
M-estimators: min $\sum_i \rho (r_i)$ \\
$\rho$: symmetric, positive-definite function with unique minimum at zero \\
$\rho$: chosen to be less increasing than square \\
=====p. 168===== \\
=====p. 169=====


\vspace{1cm}
14.9.2 Modified Weights Method 

\vspace{1cm}
14.9.3 Experimental Results

\vspace{1cm}
14.10 Error Propagation \\
$x_1,...,x_N$: input parameters \\
$\Delta x_1, ..., \Delta x_N$: random errors \\
quantity $y$ depends on input parameters through known function $f$:
\[
y = f(x_1,...,x_N)
\]
quantities $x_1+\Delta x_1, ..., x_N+\Delta x_N$ observed and quantity
$y+\Delta y$ calculated: 
\[
y+\Delta y=f(x_1+\Delta x_1, ..., x_N+\Delta x_N)
\]
error propagation analysis: determines expected value and variance of $y+\Delta y$ \\
known information about $\Delta x_1, ..., \Delta x_N$: mean, variance:
\[
E \left [ \left (
\begin{array}{c}
\Delta x_1 \\
\vdots \\
\Delta x_N
\end{array} \right ) \right ]= 0
\]
\[
E \left [ \left (
\begin{array}{c}
\Delta x_1 \\
\vdots \\
\Delta x_N
\end{array} \right ) (\Delta x_1, ..., \Delta x_N) \right ]= \left (
\begin{array}{cccc}
\sigma_{11} & \sigma_{12} & \cdots & \sigma_{1N} \\
\vdots & \vdots & \vdots & \vdots \\
\sigma_{N1} & \sigma_{N2} & \cdots & \sigma_{NN} 
\end{array} \right ) 
\]

\vspace{1cm}
14.10.1 Implicit Form \\
error propagation: can also be done in implicit form \\
known function $f$ has the form
\[
f(x_1,x_2,...,x_N,y)=0
\]
The quantities $(x_1+\Delta x_1,...,x_N+\Delta x_N)$ are observed, and the
quantity $y+\Delta y$ is determined to satisfy $f(x_1+\Delta x_1,...,x_N+\Delta x_N,y+\Delta y)=0$.

\vspace{1cm}
14.10.2 Implicit Form: General Case \\
general case: $y$ is not a scalar but a $L \times 1$ vector $\beta$ \\
$x_1,...,x_K$: are $K$ $N \times 1$ vectors representing true values \\
$x_1+\Delta x_1,...,x_K+\Delta x_K$: are $K$ $N \times 1$ vectors representing
noisy observed values \\
$\Delta x_1,...,\Delta x_K$: random perturbations \\
$\beta$: is a $L \times 1$ vector representing unknown true parameters \\
noiseless model:
\[
f(x_k,\beta)=0, \ \ \ k=1,...,K
\]
with noisy observations, the idealized model:
\[
f(x_k+\Delta x_k, \beta + \Delta \beta)=0, \ \ \ k=1,...,K
\]

\vspace{1cm}
14.11 Summary \\
we have shown how to 
\begin{itemize}
\item take a nonlinear least-squares problem
\item linearize it
\item solve by iteratively solving successive linearized least-squares problems
\end{itemize}
=====joke=====

 
\vspace{1cm}
Project due April 19: \\
camera calibration: interior orientation: \\
to determine principal point \\
do camera motion in z-axis, find focus of expansion/contraction \\
use lens of focal length: 16mm, 25mm, 50mm 





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