Introduction to Photogrammetry

1
Introduction to Photogrammetry
Construction implementing design plan
Artists Impression
Concept
Finished
Plan
Exact Dimensions and Positions determined
Dimensions and Positions
finalised
2
Introduction to Photogrammetry
The Photograph on the left appears geometrically
correct in relative positioning and dimensions of
object features shown. A Photograph contains
(relative) POSITIONAL, DIMENSIONAL and ANGULAR
information. Measurements can be taken on the
photograph (eg. using a ruler with mm-scale) for
example to obtain window sizes or floor
heights in the picture. However, these
Photo-Measurements are effected by the
perspective nature of the Photo, eg. same-sized
windows appear smaller in the photo-background as
compared to the photo-foreground.
In order to resolve this problem we need to
understand the photographic image creation
process. Photos are taken, using optical
photographic cameras. Through a lens-system
light-rays from the object are guided onto the
photographic film. Geometrically this can be
modelled as a central perspective system of
straight lines passing through a single point
projection centre starting a any object-point and
passing to the flat image- (film) plane. This
will cause an image to be produced that is
up-side-down and side-reversed (as it truly
occurs in cameras). Photographs as we know them
(see example above) are produced through a
secondary process. This involves re-projection
(central perspective) of the original (negative)
photograph onto flat paper-print material. This
allows for more convenient use of the resulting
positive photo paper-print when taking
photo-measurements. Thus, when modelling the
imaging-process we need to consider both stages.
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Introduction to Photogrammetry
The Photographic Imaging Process a simplified
central perspective model
Camera Coordinate System (x, y, c) The
origin of this coordinate system is the
projection centre O. Image points only occur
in the image plane, which is perpendicular to
the optical axis of the lens-system, the z-axis.
The point at which the optical axis intersects
the image plane is called O, the Principal
Point. The shortest distance between the image
plane and O is called the Camera Constant c.
This constant equals the effective focal length
f of the lens-system. The coordinates of
O in the camera system are O (x 0, y
0, z c) Note For negative images c -f
for positive images c f The image
position of any point shown in the photograph is
given by P (xp, yp c) Co-linearity
Equation Due to the straight line geometry of
central perspective projections, the two
vectors PO (in object space) and OP (in camera
space) must be pointing in the same direction,
they are co-linear, but of different length.
OP is a scaled down version of PO OP l
PO (coordinate-systems assumed parallel) OP
(xp, yp, c) transposed PO
((XP-X0),(YP-Y),(ZP-Z0)) transp.
World Coordinate System (X, Y, Z) This is the
earth-bound reference coordinate system in
which we define object positions (XP, YP, ZP)
and also the position of the Camera Projection
Centre O as (X0, Y0, Z0).
P
Z
y
ZP
z
O
x
c
YP
Z0
xp
O
yp
P
Y
Y0
XP
X0
X
4
Introduction to Photogrammetry
The Photographic Imaging Process a simplified
central perspective model
Co-linearity Equation Due to the straight
line geometry of central perspective
projections, the two vectors PO (in object
space) and OP (in camera space) must be
pointing in the same direction, they are
co-linear, but of different length. OP is a
scaled down version of PO OP l
PO (coordinate-systems assumed parallel) OP
(xp, yp, c) transposed PO
((XP-X0),(YP-Y),(ZP-Z0)) transp. However, the
camera is rarely pointing parallel to the World
Coordinate Y-Axis, neither to any of the other
axis-combinations. Rotations between the two
systems must therefore be introduced and applied
before we can finalise the co-linearity
equation. Camera Rotations There are three
possible rotations which can characterise the
directional orientation of the camera (1) a
rotation about the y axis called phi
f (2) a rotation about the x axis called
omega w (3) a rotation about the z axis
called kappa k These three rotations lead
to one 3 by 3 element rotation matrix
containing combinations of sin and cos of these
rotation angles and depending on the direction
of rotation, eg. object to image space or image
to object space. Object to Image Space
Rotation Rotation Matrix M Image to Object
Space Rotation Rotation Matrix M-transp.
World Coordinate System (X, Y, Z) This is the
earth-bound reference coordinate system in
which we define object positions (XP, YP, ZP)
and also the position of the Camera Projection
Centre O as (X0, Y0, Z0).
P
Z
y
ZP
z
O
x
c
YP
Z0
xp
O
yp
P
Y
Y0
XP
X0
X
5
Introduction to Photogrammetry
The Photographic Imaging Process a simplified
central perspective model
World Coordinate System (X, Y, Z) This is the
earth-bound reference coordinate system in
which we define object positions (XP, YP, ZP)
and also the position of the Camera Projection
Centre O as (X0, Y0, Z0).
Co-linearity Equation incorporating
Camera-Rotations OP l M PO OP
(xp, yp, c) transposed PO
((XP-X0),(YP-Y),(ZP-Z0)) transp. M Rotation
Matrix for object to image space rotation l
object to image vector scale factor With m11
m12 m13 M1 M
m21 m22 m23 M2
m31 m31 m33 M3 and (XP -
X0) PO (YP - Y0) (ZP -
Z0) the co-linearity equation can be written in
long-hand as xp l M1 PO yp l
M2 PO c f l M3
PO (c f for positive image)
P
Z
y
ZP
z
O
x
c
YP
Z0
xp
O
yp
P
Y
Y0
XP
X0
X
6
Introduction to Photogrammetry
The Photographic Imaging Process a simplified
central perspective model
Co-linearity Equation incorporating
Camera-Rotations OP l M PO OP
(xp, yp, c) transposed PO
((XP-X0),(YP-Y),(ZP-Z0)) transp. M Rotation
Matrix for object to image space rotation l
object to image vector scale factor xp
l M1 PO yp l M2 PO c
f l M3 PO (c f for positive
image) Note The camera constant c is known
(focal length of the camera used),
while xp and yp on the left-hand side are to be
found. Note also the scale factor l is unknown
and varies from object space point to point.
Dividing the two first equations by the third
and multiplying by c takes care or this Imaging
Equations xp c (M1 PO) / (M3 PO) yp
c (M2 PO) / (M3 PO) Thus, image point
positions can be computed for object points, when
camera position and orientation is known.
World Coordinate System (X, Y, Z) This is the
earth-bound reference coordinate system in
which we define object positions (XP, YP, ZP)
and also the position of the Camera Projection
Centre O as (X0, Y0, Z0).
P
Z
y
ZP
z
O
x
c
YP
Z0
xp
O
yp
P
Y
Y0
XP
X0
X
7
Introduction to Photogrammetry
The Photographic Imaging Process a simplified
central perspective model
Imaging Equations xp c (M1 PO) / (M3
PO) yp c (M2 PO) / (M3 PO) Exterior
Orientation The six parameters defining the
position and angular orientation of the camera
are call exterior Orientation parameters Rotati
ons f, w and k Camera Position O (X0, Y0 and
Z0) Interior Orientation The camera internal
parameters defining its geometry are called
interior Orientation parameters. They are (1)
the effective focal length of the camera, which
could be slightly different to the nominal
focal length of the lens, (eg. when
changing focus) and (2) The position (x0, y0)
of the Principal Point O within the
coordinate system used for measuring image
positions. Since measuring image positions
is a process independent of the camera, it
must be assumed that the origin of
measured image coordinates is not identical
with O, the geometric image coordinate
system. Final Imaging Equations (xp - x0)
c (M1 PO) / (M3 PO) (yp - y0) c (M2
PO) / (M3 PO)
World Coordinate System (X, Y, Z) This is the
earth-bound reference coordinate system in
which we define object positions (XP, YP, ZP)
and also the position of the Camera Projection
Centre O as (X0, Y0, Z0).
P
Z
y
ZP
z
O
x
c
YP
Z0
xp
O
yp
P
Y
Y0
XP
X0
X
8
Introduction to Photogrammetry
The Photographic Imaging Process a simplified
central perspective model
Final Imaging Equations xp x0 c (M1
PO) / (M3 PO) yp y0 c (M2 PO) / (M3
PO) Utilising Geometric Information contained in
Photographs Once Photographs are obtained for a
project, we can attempt to extract dimensional
and positional information from these photos. It
is most convenient to work here with positive
images, rather than with negatives. However,
this usually will change the interior
orientation parameters or they will be lost in
the process. Thus, we must be Recovering the
Interior Orientation Assume at this stage that
the exterior orientation parameters
(camera position and rotations) and positions for
some object points (XP,YP,ZP) shown on the photo
are known and their corresponding image
positions (xp and yp) have been measured in
some image coordinate measurement system, whose
axis are parallel to the ideal image coordinate
axis. Then, analogous to solving algebraic
equation systems, we can place this information
into the imaging equations and solve for the
unknown interior orientation parameters The
equations have the form (measured image
coordinate) f ( unknown parameters, some known
factors), eg. 15.3 0.6 3.1 x -
7.9 y 13.4 -0.3 2.7 x - 5.6 y etc. one
pair of xp, yp equations per measured image
point coordinate pair. Note This has to be done
only once, if the camera and the positive
process stay unchanged for numerous images.
World Coordinate System (X, Y, Z) This is the
earth-bound reference coordinate system in
which we define object positions (XP, YP, ZP)
and also the position of the Camera Projection
Centre O as (X0, Y0, Z0).
P
Z
y
ZP
z
O
x
c
YP
Z0
xp
O
yp
P
Y
Y0
XP
X0
X
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Introduction to Photogrammetry
The Photographic Imaging Process a simplified
central perspective model
Final Imaging Equations xp x0 c (M1
PO) / (M3 PO) yp y0 c (M2 PO) / (M3
PO) Recovering the Interior Orientation (....
Continued) Since there are three interior
orientation parameters (x0, y0 and c) to
be recovered, we need at least three (3)
equations in our solution. Every point measured
in the photo will produce two equations, one for
xp and one for yp, two measured image points
will suffice. However, note that it was assumed
that the exterior orientation parameters where
exactly known. THIS IS RARELY THE CASE! Thus, we
will also have a need for Recovering the
External Orientation There are six (6) exterior
Orientation parameters the position of the
camera projection centre (the lens system) as
(X0, Y0 and Z0) and the three camera rotations
(f, w, k). Hence, we require at least six
separate equations to solve for these six unknown
parameters. Analogous to the recovery of the
interior orientation, we require that at least
three (3) image points are measured in (xp, yp)
and that their object space coordinates are also
known. Note, however, since the rotations (f,
w,k) are represented as combinations of sin and
cos in the imaging equations, the resulting
equation system is NON-LINEAR. To find a valid
solution, the original imaging equations will
have to be lienarised by Taylor development and
appropriate approximation need to be introduced.
This also is a requirement of solving non-linear
systems by Least Squares. Therefore, the recovery
of external orientation parameters is done
by Least Squares Adjustment of Imaging Equations,
which also allows for the use of more than the
required number of equations in the presence of
measurement errors..
World Coordinate System (X, Y, Z) This is the
earth-bound reference coordinate system in
which we define object positions (XP, YP, ZP)
and also the position of the Camera Projection
Centre O as (X0, Y0, Z0).
P
Z
y
ZP
z
O
x
c
YP
Z0
xp
O
yp
P
Y
Y0
XP
X0
X
10
Introduction to Photogrammetry
The Photographic Imaging Process a simplified
central perspective model
World Coordinate System (X, Y, Z) This is the
earth-bound reference coordinate system in
which we define object positions (XP, YP, ZP)
and also the position of the Camera Projection
Centre O as (X0, Y0, Z0).
Final Imaging Equations xp x0 c (M1
PO) / (M3 PO) yp y0 c (M2 PO) / (M3
PO) Combined Recovery of Interior and Exterior
Orientation The usual process in photogrammetry
is to recover both, interior and
exterior Orientation parameters simultaneously.
However, if the interior orientation parameters
are known and can be assumed unchanged, than only
the exterior orientation is determined for every
photograph to be used. In the case of combine
recovery, the following applies
determine three interior orientation parameters
per camera determine six
exterior orientation parameters per
photograph 3 equations to determine the
interior orientation sets and 6 equations to
determine the exterior orientation parameters In
total 9 equations formed from the imaging
equation system for at least 5 points with known
object space coordinates and measured image
coordinates need to be solved through Least
Squares Adjustment. Note Points for which
object coordinates (XP, YP, ZP) are known are
called CONTROL-POINTS. Control Points are
equivalent to fixed Survey Marks in conventional
Surveying.
P
Z
y
ZP
z
O
x
c
YP
Z0
xp
O
yp
P
Y
Y0
XP
X0
X
11
Introduction to Photogrammetry
The Photographic Imaging Process a simplified
central perspective model
World Coordinate System (X, Y, Z) This is the
earth-bound reference coordinate system in
which we define object positions (XP, YP, ZP)
and also the position of the Camera Projection
Centre O as (X0, Y0, Z0).
Final Imaging Equations xp x0 c (M1
PO) / (M3 PO) yp y0 c (M2 PO) / (M3
PO) Determining Object Coordinates for new
points In the above Imaging Equations vector PO
extends from any object point P to the camera
projection centre O. PO thus not only relates to
O(X0,Y0,Z0), but also to any object point P(XP,
YP, ZP) shown in the photograph. Thus, assuming
that exterior and interior Orientation are solved
for a given photograph, we can continue to
measure additional points in the image for
which object coordinates are not
known. Remember each point measured in the
image will add one xp and one yp
image coordinate and with that one pair of
imaging equations ( two equations per
point). However, each point measured relates to
a (XP, YP, ZP)-triplet Three Values to be
found. WE CANNOT DETERMINE THREE DIMENSIONAL
DATA FROM TWO MEASUREMENT VALUES (2D) !!!! SO
WHAT TO DO ???
P
Z
y
ZP
z
O
x
c
YP
Z0
xp
O
yp
P
Y
Y0
XP
X0
X