Analytical Photogrammetry

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Analytical Photogrammetry
Hande Demirel, PhD, Assistant Prof. Istanbul
Technical University (ITU), Faculty of Civil
Engineering, Geodesy and Photogrammetry
Department Division of Photogrammetry
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Least Squares Solution

• The method of least squares, also known as
  regression analysis, is used to model numerical
  data obtained from observations by adjusting the
  parameters of a model so as to get an optimal fit
  of the data.
• The best fit is characterized by the sum of
  squared residuals have its least value, a
  residual being the difference between an observed
  value and the value given by the model.
• The method was first described by Carl Friedrich
  Gauss around 1794.
• Least squares corresponds to the maximum
  likelihood criterion if the experimental errors
  have a normal distribution.

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Least Squares Solution

• Gauss was able to state that the least-squares
  approach to regression analysis is optimal in the
  sense that in a linear model where the errors
  have a mean of zero, are uncorrelated, and have
  equal variances, the best linear unbiased
  estimators of the coefficients is the
  least-squares estimators. This result is known as
  the Gauss-Markov theorem.
• In a linear model in which the errors have
  expectation zero, are uncorrelated and have equal
  variances, the best linear unbiased estimator of
  any linear combination of the observations, is
  its least-squares estimator. "Best" means that
  the least squares estimators of the parameters
  have minimum variance. The assumption of equal
  variance is valid when the errors all belong to
  the same distribution.

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Least Squares Solution

• The objective consists of adjusting the
  parameters of a model function so as to best fit
  a data set.
• A simple data set consist of m points (data
  pairs) (xi, yi) , i1,m, where xi is an
  independent variable and yi is a dependent
  variable whose value is found by observation.
• The model function has the form f(x,ß) , where
  the n adjustable parameters are held in the
  vector ß . We wish to find those parameter values
  for which the model "best" fits the data.

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Least Squares Solution

• The least squares method defines "best" as when
  the sum, S, of squared residuals
• is a minimum. A residual is defined as the
  difference between the values of the dependent
  variable and the model.

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Solving the Least Squares Problem

• Least squares problems fall into two categories,
  linear and non-linear.
• The linear least squares problem has a closed
  form solution, but the non-linear problem has to
  be solved by iterative refinement at each
  iteration the system is approximated by a linear
  one, so the core calculation is similar in both
  cases.

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Least Squares Solution
For least squares solution we need linearized
observation equations
The equations (5) and (6) are nonlinear and must
be linearized.
(5)
(6)
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Least Squares Solution

• To linearize the equations we must expand
  them into the Taylor series and ignore the terms
  of degree 2 and higher. The differential
  quotients used in the Taylor series are as
  follows

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Non linear Equations
xf(X0,Y0,Z0, ?,?,?, X,Y,Z)
yf(X0,Y0,Z0 ,?,?,?,X,Y,Z)
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Non linear Equations
The Differential Quotients
.........................
.........................
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Observation Equations
The differential quotients are used to write the
linearized observation equations of a least
squares adjustment by indirect observations.
EACH MEASURED IMAGE POINT YIELDS TWO OBSERVATION
EQUATION.
iindex for the measured point, jindex for the
photograph
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Linearised Observation Equations
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Linearised Observation Equations
The unknowns are the six elements (X0,Y0,Z0
,?,?,?) of outer orientation of the photograph
with the index j and the three ground
co-ordinates (X,Y,Z) of a point Pi .
The differential quotients ( )0 are calculated
from approximate values of the unknowns. For
example the approximate values of the unknowns
are put in the equation below and the value of
the differential quotient ,
is obtained.
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Linearised Observation Equations
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Linearised Observation Equations
and
are computed values for the functions below using
approximate values of the unknowns.
xij and yij are the measured image coordinates
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The approximation of the Unknowns
There are various ways to derive approximations
Some of them are as follows

• For near vertical photographs
• ?0 ?0 0
• ?0 is known from the flight plan
• The co-ordinates of the projection center and the
  new points can be obtained by a block adjustment
  with independent models.

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Example
2
1
Observed Image coordinates
1
2
3
4
3
4
2x6x448
5
6
5
6
Photo 1
Photo 2
Unknowns
3x412 Projection centre co-ordinates 3x412
Rotations 3x412 New point co-ordinates Total
36 Redundancy 48-3612
3
4
3
4
5
6
5
6
7
8
7
8
Photo 3
Photo 4
Control Point
New point
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Observation Equations for the Example
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Normal Equations (in matrix notation)
vAx-l (observation Equa.)
ATAxATl
ATAN and ATln
Nxn (Normal equation)
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The Structure of Normal Equations
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In Matrix Notation

• N11 is a hyperdiagonal matrix with submatrices
  each of 6x6 elements
• N22 is similary a hyperdiagonal matrix with
  submatrices of 3x3

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In Matrix Notation
Vector of unknowns x is divided into two
subvectors x1 and x2
Subvector of the unknown outer orient. elements
Subvector of the unknown new point coordinates
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Solution of the Normal Equations
(1)
The system can be reduced by the new point
co-ordinates x2
From the system (1) we can write
(2)
(3)
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Solution of the Normal Equations
Multiply (3) with N22-1 from left we obtain
(4)
(5)
To eliminate x2 we replace it in Equ. (2) with
(5)
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Solution of the Normal Equations
(The reduced equation)
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Solution of the Normal Equations

• An adjustment yields corrections to the
  approximate values of the elements of outer
  orientation of each photograph and to the
  approximate co-ordinates of the new points.
• If the approximation are very poor, the corrected
  values must be treated as new approximations for
  a new adjustment.
• This process is repeated until there is no
  further significant change in the unknowns of
  the block adjustment.

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Accuracy of Bundle Adjustment
The accuracy can be estimated for a bundle block
adjustment with 60 overlap and 20 sidelap and
for signalised points as Planimetry sxy
3 µm in the photograph Height sz
0.03 of camera distance (NA-WA)
0.04 of camera distance
(SWA)
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Accuracy of Bundle Adjustment
History of ?0 (Grün )
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Advantages of Bundle Adjustment

• Most accurate method of aerial triangulation
• (direct relation between image and ground
  co-ordinates without of intermediate step of
  model formation)
• Simple possibility of extending the technique to
  compensate systematic error
• Simple possibility of incorporating external
  information in the adjustment
• (e.g known outer orient. elements, field
  surveyed observations such as ,lengths, angles
  and geometric information)

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Advantages of Bundle Adjustment

• Possibility of using unconventional camera
  dispositions amateur camera photographs such as
  are often necessary in close -range
  photogrammetry.
• Possibility of deriving the elements of outer
  orient. To be set in particular analogue or
  analytical plotters.
• Possibility of combined adjustment with
  co-ordinates of the projection centres determined
  by kinematic GPS positioning.
• Possibility of self calibration by additional
  parameters and determination of Systematic Image
  Error

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Disadvantages of Bundle Adjustment

• Nonlinear problem for which approximations can
  only be established after long procedures.
• Computer intensive method.
• Analogue instruments can not be used for the
  measurements.
• Always a spatial problem,so that separate
  adjustments in planimetry and height are not
  possible.

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Bundle Adjustment

• To obtain a band matrix the photograph must be
  numbered vertical to the flight direction. With
  the band matris structure we can avoid the
  multiplications with zero element and spare time
  by computing