Photogrammetry and Remote Sensin

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Photogrammetry and Remote Sensing
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A measurement process of collecting spatial
information remotely

• Our initial focus is on use of aerial photography

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An aerial photo is not a map.A map has one
scale.An photos scale changes as the distance
from the exposure to the ground changes
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Example

• Two football fields at two elevations will not
  have the same length on an aerial photo.
• Similarly, a ground based exposure of two six
  foot tall people different distances from the
  exposure will be of different length

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An aerial photo is not a map

• A map is a projection of points to a defined
  elevation (often sea level).
• A photo is a view from a single vantage point
  (the exposure).
• A single vantage point does not allow projection
  of points to a defined elevation.

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An aerial photo is not a map

• A map is an orthographic view. You are always
  looking straight down at a feature.
• A photo is a perspective view. It is what a part
  of the earth looks like from a unique position
  (the exposure).

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A photograph is not a map.

• A map cannot be tilted as it is in a defined
  projection.
• An aerial photo can contain tip, tilt, or crab
  which can cause distortion in an image.

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A photograph is not a map.

• A map contains a finite amount of detail
  points, lines, and text.
• A photo contains an almost infinite amount of
  detail (to the pixel level)

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A photograph is not a map.

• Symbology is used to define what points and lines
  are on a map.
• A human being defines what objects are on a photo.

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A photo is not a map.

• A map has a defined projection usually state
  plane or Universal Transverse Mercator (UTM)
• A user must define a coordinate system for a
  photo.

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A photo is not a map

• A map does not contain relief displacement as all
  points are projected to a defined elevation.
• A photo contains relief displacement.

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What is evidence of relief displacement?

• On a map a vertical object (power pole, building
  corner, etc.) has only one position.
• On an aerial photo it is often possible to see
  both the bottom and top of vertical objects. As
  if different points on a vertical object have
  different horizontal coordinates.

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But regarding relief displacement

• It exists at all points relative to a defined
  elevation.
• But it is very apparent on vertical objects
  unless the object is directly below the center of
  the photo.

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Typical metric film based aerial camera

• 6 in. focal length (longer than hand held
  camera) sometimes 3.5 or 12 in.
• 9 x 9 in. format (hand held has 35 mm format
  note a 11 contact print is 9 in. x 9 in. (no
  enlargement)
• Large film magazine (storage)
• Vacuum film flattening minimal distortion due
  to film unflatness
• All photogrammetric equations assume the negative
  is flat!

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Typical metric film based aerial camera

• Fiducial marks artificial marks in sides or
  corners of negative frame that appear in every
  exposure
• Fiducial marks are used to measure film shrinkage
  or expansion
• Fiducial marks are used to define x,y
  photocoordinate axes

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Typical metric film based aerial camera

• The intersection of lines connecting opposite
  fiducial marks estimates the location of the
  principal point
• The principal point is where a a line from the
  rear nodal point of the lens intersects the
  negative plane at a perpendicular angle.
• Nodal point is where all light rays intersect in
  the lens

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Typical metric film based aerial camera

• Forward motion image compensation (FMC)
• Since the airplane is moving quickly when
  exposures are being made, the negative can move
  slightly forward during exposure to account for
  the movement.

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Focusing an aerial camera

• Not important as at 100 ft. distance to an object
  a hand held camera is focused for infinity
• Lens law is 1/(image distance) 1/(object
  distance) 1/(focal length)
• Note image distance is the actual focusing
  distance negative to lens nodal point (where all
  light rays come together)
• At 100 ft. object distance and 6 in. (0.5 ft.)
  aerial camera focal length lens law gives image
  distance of 6.03 in. (insignificant change)

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Image products

• Contact Print a 11 positive on paper
• Diapositive a 11 positive on glass (old) or
  plastic for precise photogrammetric measurement
• Scanned image A typical digital image where the
  negative or diapositive was processed with a
  precise scanner capable of preserving
  photogrammetric accuracy

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Aerial camera calibration

• Usually performed at a federal government
  facility
• Calibrated focal length (6 in. is not a perfect
  value)
• Lens distortion radial and tangential
• Film unflatness
• Image resolution
• Shutter efficiency

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Aerial camera calibration

• Fiducial point coordinates
• Principal point coordinates
• 2 types of principal points
• (1) autocollimation perpendicular from rear
  nodal point of lens
• (2) symmetry the point radial lens distortion
  is symmetric about

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Simple calculations

• If an aerial photo can be assumed to be vertical
  (no tip or tilt) similar triangles can be used to
  solve for several important types of information
• Note these equations are thus only
  approximations, but have important uses.

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The scale equationab/AB f/H
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The scale equation

• ab f
• S ----- -------
• AB H'
• where
• S scale
• ab photo distance
• AB ground distance (horizontal)
• f focal length (6 inches for most film based
  aerial cameras)
• H' flying height above line AB

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Note

• H H h
• Where
• H flying height above datum (usually sea level)
• h elevation above datum

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The scale equation can be rewritten many ways.

• ab f
• ----- -------
• AB H
• Can be AB H ab /f
• Or ab AB f / H
• Or H AB f / ab
• Assume f 6 in. unless explicitly stated
  otherwise. (6 in. 152.4 mm.)

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• Let's assume a relatively flat area is on a
  photo. Measuring a photographic distance of a
  known ground horizontal distance allows
• us to solve for flying height.
• AB
• H' f ----
• ab

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• Once a flying height is determined other photo
  distances can be measured and one can solve for a
  ground distance.
• H' ab
• AB -------
• f

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Example

• A football field (goal line to goal line)
  measures 0.6 in. on an aerial photo. What is the
  flying height?
• H AB f / ab 300 ft. 6 in. / 0.6 in.
• H 3000 ft.

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Example

• At a flying height of 1200 ft., a building edge
  measures 0.15 in. on an aerial photo. What is
  its ground length?
• AB H ab /f 1200 ft. 0.15 in. / 6 in.
• AB 30 ft.

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• Let's use these simple equations to estimate how
  well you could derive ground distances on a
  vertical photograph taken at a H' 1000 ft. (low
  altitude) with a scale or digitizing tablet which
  will be assumed to have a measuring resolution of
  0.01 inch.
• AB H ab /f 1000 ft. 0.01 in. / 6 in.
• AB 1.67 ft.
• Note at a flying height of 10000 ft. you could
  only measure to 16.67 ft.!
• The most logical route to improving ground
  accuracy is improving photogrammetric measuring
  accuracy.

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• Modern photogrammetric measurement has a
  resolution/least count of 0.001 mm (1 micron) and
  a measuring ability of 10 microns on distinct
  features. Change the previous problem to this
  measuring ability.
• AB H ab /f
• 1000 ft. 0.01 mm. / 152.4 mm
• 0.07 ft.
• Which approaches our accuracy achievable with
  ground based techniques!

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Flying height limits

• Fixed wing aircraft are limited to 1200 ft. in
  urban areas and 1000 ft. in rural areas.
• A helicopter can be used to obtain lower flying
  heights for higher accuracy.
• A helicopter is more expensive to use than a
  fixed wing aircraft.
• Lower flying height means less coverage on one 9
  x 9 in. format exposure.

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Example

• How many acres (1 acre 43560 sq. ft.) are on a
  photo at a flying height of 10000 ft.?
• AB H ab /f 10000 ft. 9 in. / 6 in.
• AB 15000 ft.
• acres 15000 ft. 15000 ft. / 43560
• acres 5200 acres

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Measuring heights of object by relief displacement

• Assuming vertical photography
• Since both the top and bottom of a vertical
  object appear on an aerial photo, the height of
  the object can be estimated.
• Note relief displacement is the photographic
  distance of a vertical object.
• It will grow as the object is displaced further
  from the center of the photo.

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Relief displacement side viewh height of
vertical object
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Relief displacement on photor radial dist.
Prin. Pt. to top of objectd relief
displacement photo dist. Bottom to top of
objectNote relief displacement is along a radial
line.
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Relief displacement

• By similar triangles
• d/h r/H
• Or
• h dH/r
• h actual height of object
• H flying height above bottom of object
• Note d 0 at principal point so relief disp.
  cannot be measured unless offset from it.

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Relief displacement

• h dH/r
• Given flying height and the ability to measure d
  and r on a vertical photo
• An objects height can be determined
• Note the equation can also be written
• d hr/H or rdH/h or Hhr/d

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• A building edges top is 3.5 in. from the center
  of a photo and its vertical edge measures 0.05
  in. on the photo. If the flying height of the
  photo is 3000 ft. what is the height of the
  building?
• h dH/r 0.05 in. 3000 ft. / 3.5 in.
• h 43 ft.
• Note if the same dimensions existed except the
  top of the building was half the distance from
  the photo center, h would double in magnitude!

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• The flying height is 1200 ft. If ones measuring
  ability on a photo is 0.01 in., and once desires
  to measure vertical objects to a resolution of 5
  ft., how far does the top of the object need to
  be displaced from the center of the photo?
• rdH/h 0.01 in. 1200 ft. / 5 ft.
• r 2.4 in.

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Error in horizontal location due to relief
displacement

• Remember points on a photo need to be projected
  to a map projections datum that is usually sea
  level. This creates the orthographic projection
• To perform the projection one needs to know the
  elevation of the point.
• Error in elevation results in error in proper
  projection horizontal location

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• Example
• . A 20 ft. error in elevation is located 4
  inches from the center of a photo whose flying
  height is 1200 ft.
• The 20 ft. error can be considered relief
  displacement.
• 4 in 20 ft.
• d --------------- 0.07 in.
• 1200 ft.
• The 0.07 inches represents a horizontal distance
  error. Using the scale equation its horizontal
  ground distance would be
• 1200 ft 0.07 in.
• AB ------------------ 14 ft.
• 6 in.
• In other words a 14 ft. difference in ground
  position is attributable to a 20 ft. elevation
  difference! This shows why elevation difference
  must be accounted for.

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• Thoughts on the horizontal error due to elevation
  error example
• Points nearer the principal point have less error
  as r (radial distance to top of object) is
  small
• Higher flying heights can relate to greater map
  positional error due to measuring error relating
  to larger ground distance
• Error in the elevation model being used is very
  difficult to determine, and will change value for
  different points

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• Measurement of elevation difference by parallax
• Photogrammetry is capable of measuring elevation
  differences through the use of parallax.
• Parallax is defined as the apparent displacement
  of a point due to a change in view of the point.

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• Parallax human eye example
• Hold your finger out in front of you and look at
  where it is relative to a wall in the background
  with your right eye.
• Then look at it with your left eye and its
  appearance relative to the wall has changed.
• The relative change in appearance is due to
  parallax.

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• Parallax in photogrammetry is the change in
  position of the same point on two overlapping
  photos due to the change in position of the
  exposures.
• The change is along the flight line between the
  exposures of the overlapping photos which is
  roughly defined as the X axis for photocoordinate
  measurement.
• Parallax x left photo coor. x right photo
  coor. (assuming x axis parallel to flight line)

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• Epipolar line any line on a photo parallel with
  the flight line axis parallax occurs along
  epipolar lines
• Defining the flight line requires locating a
  conjugate principal point which is a principal
  point image transferred to its location on an
  overlapping photo.
• The line between the principal point and a
  conjugate principal point is the flight line axis

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• Now think of a point close to the overlapping
  exposures and a point far away.
• The closer point will "shift" more than the point
  further away.
• In aerial photography points of higher of
  elevation will have larger parallaxes than points
  of lower elevation (further from the exposures).

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• Assuming vertical photography and exposures from
  the same flying height, elevation difference is
  determined by
• dp H'
• dh ha - hc -----------
• pc
• where
• dh change in elevation between two points a
  and c
• dp parallax point a - parallax point c
• H' flying height
• pc parallax of point c
• where parallax of a point is the change in x
  coordinates where the x axis is parallel to the
  flight line.

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• Note any point being measured has to appear on
  both photos that overlap!!!
• If point c has a known elevation (benchmark) and
  its parallax can be measured
• Any point a whose parallax can be measured can
  have an elevation difference from c to a computed
  for it thus
• Any point as elevation can be computed relative
  to point cs known elevation
• Conclusion parallax measurement enables
  computation of an elevation model

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• Example
• Benchmark c has an elevation of 1545.32 ft., x
  coor. on left photo of 74.12 mm and on right
  photo of -18.41 mm. Unknown point a has x coor.
  On left photo of 65.78 mm and on right photo of
  -24.38 mm. If flying height above average ground
  is 3000 ft. what is the elevation of point a?
• Parallax is the change in x coordinates defined
  as parallel to the flight line.
• Parallax c 74.12 (-18.41) 92.53 mm
• Parallax a 65.78 (-24.38) 90.16 mm

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• dp H'
• dh ha - hc -----------
• pc
• dp pa pc 90.16 92.53 -2.37 mm
• dh (-2.37 mm) (3000 ft.) / 92.53 mm
• dh -76.84 ft.
• Elev a elev c dh 1545.32 (-76.84)
• Elev a 1468.48 ft.
• Note point a has less parallax so it is at a
  lower elevation than point c.

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• The standard format of an aerial photo is 9 in.
  by 9 in.
• standard overlap between successive photos in a
  flight line is 60, which means an advance of 40
  of the format.
• 40 of 9 in. is 3.6 in. and can be assumed to be
  the "average" parallax for points on a photo.
• The ground distance related to the advance
  between photos is known as the air base (3.6 in.
  H / focal length)

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• Let's use this parallax of 3.6 in. as point c and
  assume a 1200 ft. flying height.
• Assuming the accomplished stereoplotter operator
  can measure parallax differences pessimistically
  to 0.010 mm
• 0.010 mm 1200 ft
• dh --------------------------- 0.13 ft.
• 3.6 in 25.4 mm/in

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• This same type of computation at the same flying
  height was performed for horizontal ground
  measuring resolution and resulted in 0.08 ft.
• Thus photogrammetry is more capable of producing
  higher horizontal than vertical accuracies. This
  is offset in production by utilizing more
  vertical control points when compared to required
  horizontal control points.

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Ground X,Y coordinates from parallax

• (0,0) is the left exposure and X is in the
  flight line direction
• X (B/p) x and Y (B/p) y
• Where X,Y ground coor.
• x,y photo coor. on left photo based on flight
  line axis being the x axis
• p is the measured parallax of the point
• B is the air base

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Air Base calculation

• Find a conjugate principal point as previously
  discussed.
• Measure photo distance from principal point to
  conjugate principal point (o-o)
• Multiple o-o by H/f as in the scale equation
• Note now you have ground X,Y,Z coordinates from
  parallax!!

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• Note it is very easy to enter different flying
  heights and measuring resolutions in the scale,
  relief displacement, and parallax equations.
• Also realize since these equations make several
  assumptions they are only useful for rough
  computations, and not final map production.

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• The elevation difference accuracy of 0.13 ft.
  from 1200 ft. flying height illustrates the limit
  of fixed wing aircraft photogrammetry
• If elevation accuracies of 0.10 ft. or less are
  required for a project, one has to consider
  helicopter photogrammetry as it allows lower
  flying heights
• But lower flying height means more photos per
  project and thus higher costs

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• Elevation difference by parallax vs. relief
  displacement
• Relief displacement can only measure vertical
  objects
• Parallax measures elevation difference between
  any two points in same overlap region between two
  photos
• Relief displacement uses only one photo but needs
  the vertical object displaced from the center of
  the photo.
• Both simplified equations assume vertical
  photographs

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• Scale, relief displacement and parallax equations
• All simple equations based on vertical
  photography and similar triangles
• Have excellent use for rough measurements with
  a scale
• Have excellent use in estimating accuracy of
  product by placing measuring error in photo
  distance, relief displacement photo distance, or
  parallax difference unknowns.

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Stereo Viewing

• 3-D movies the latest!
• 2 cameras offset in position film the set
• Both images are displayed on the movie screen or
  TV at different frequencies
• Your glasses force the left eye to only see the
  left image, and the right eye only see the right
  image
• This creates a stereo image (3-D effect)

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Stereo Viewing Aerial Photography

• Camera axes are near vertical
• Exposure stations are taken so two successive
  photos overlap approximately by 60
• Forcing your left eye to view the left image and
  right eye view the same portion of the photo on
  the right image creates stereo viewing

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Possible ways to view in stereo

• Forced viewing in oculars 1 eye1
  image-stereoscope
• Color red vs. blue/green image glasses
• Shutter Image Alternators SIA shutters move
  very quickly in sync so left open when right
  closed, and vice versa
• Polarized viewing same polarized in left image
  left viewing and vice versa for right

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Is stereo vision required to measure parallax?

• If a point is monoscopically identifiable on both
  photos (manhole, end of paint stripe, sidewalk
  intersection, driveway corner, etc.) one can
  measure without stereo viewing
• A monoscopically identifiable image can have
  coordinates measured for it on any photo,
  enabling a parallax computation

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Is stereo vision required to measure parallax?

• If a point is not monoscopically identifiable
  (point in grass or dirt, random point on
  pavement, etc.) its parallax measurement requires
  stereoviewing
• The big question how can the same exact point
  on a grass lawn be identified on multiple photos?

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General parallax measurement requires
superimposition!

• Superimposition if an artificial mark is placed
  on one photo (an X in the grass) and that photo
  is viewed in stereo with an overlapping photo not
  containing the X
• In stereo the X will appear to be in both
  photos. Therefore it can be located on the photo
  where it does not appear. An example would be
  placing a mark with a pencil in that location on
  the photo where it does not appear.

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General parallax measurement requires
superimposition!

• Superimposition tricks our stereo perception
  ability into seeing the single mark on both
  photos.
• This allows the same undefined image location to
  be transferred to overlapping photos
• If the mark on the overlapping photos is
  erroneously located, in stereo one will see both
  marks instead of the superimposition of both marks

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Floating mark/half mark

• Artificial marks are superimposed on a left and
  right image on an epipolar line
• Holding the left mark fixed in position, the
  right mark is allowed to move along the epipolar
  line (or vice versa)
• When the two marks are close to the same image
  location they will appear to merge and rise and
  fall relative to the stereo image

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Floating mark/half mark

• When the merged mark appears to be on the
  ground in the stereo image you are measuring the
  same image point in both photos.
• This enables a parallax measurement.
• Measuring using the floating mark is very
  difficult at first as we are not used to stereo
  measurement. An accomplished photogrammetrist
  performs this task with routine ease due to hours
  of practice.

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Modern Photogrammetric Parallax Measuring Ability

• Modern photogrammetric techniques allow
  photocoordinates, and therefore parallax, to be
  measured to 0.006 0.030 mm (or 6 30 microns
  as 1 micron 0.001 mm)
• Measuring ability is affected by operator
  ability, image quality, and whether the image is
  distinctly monoscopically identifiable or
  measured through stereo viewing

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Coordinate Transformations

• Various parts of photogrammetry involve both 2-D
  and 3-D coordinate transformations.
• Many measurements, and some interim calculations,
  are performed in assumed (arbitrary) coordinates
  systems.
• The assumed coordinates need to be scaled,
  rotated, and translated into coordinate systems
  with defined references.
• If the same point has coordinates in both systems
  it can be used to determine the coefficients for
  the transformation

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Photocoordinate measuring

• Precise x,y coordinate measurement was performed
  on mechanical devices called comparators with
  detailed visual magnification for a user.
• Monocomparators measured one photo,
  stereocomparators allowed measurement of
  photocoordinates on two overlapping photos.
• Comparators are digitizing tablets on steroids,
  as their least count resolution was usually 1
  micron with actually coordinate measuring ability
  in the 3-30 micron range.

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Comparator evolution

• Comparators initially had dial type coordinate
  readouts
• Encoding the dials allowed the coordinate
  measurements to be stored on a computer
• Stereoplotters combined coordinate measuring,
  orientation of stereo photos, stereo vision, and
  map compilation
• Today an image (or images in stereo) on the
  computer screen is the comparator
• Pixels on a computer screen are an arbitrary
  coordinate system

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What does comparator mean?

• Comparators, including digitizing tablets,
  actually measure coordinate differences.
• The origin of the comparator system is very
  arbitrary as its location is not important to the
  ensuing measurement process.
• It is similar to assumed coordinates in plane
  surveying. The location of the origin is not
  important.

77
2-D photocoordinate measurement

• Fiducial marks have camera calibration derived
  photo coordinates relative to the principal
  point.
• Inner orientation is the process of measuring
  fiducial marks in the comparator arbitrary
  coordinate system
• These arbitrary coordinates are associated with
  the calibration fiducial coordinates.

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2-D photocoordinate measurement

• Film can shrink or expand example
• Monocomparator measurement of two fiducials (in
  mm.)
• x -0.246 y 114.921
• x -114.303 y 3.034
• And the equivalent calibration fiducial
  coordinates were
• x 0.028 y 113.029
• x -112.976 y -0.013

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2-D photocoordinate measurement

• A scale coefficient can be computed by
• Scale fiducial dist. / comparator dist.
• By Pythagoreum theorem
• Fiducial dist. 159.839
• Comparator dist. 159.774
• Scale 159.839 / 159.774 1.00041
• Greater than one means the photo has shrunk so it
  needs mathematical enlargement

80
2-D conformal transformation

• 4 unknowns
• (1) scale accounts for film shrinkage/expansion
• (2) rotation
• (3) x translation
• (4) y translation
• Conformal means a horizontal angle stays the same
  value before and after transformation as only one
  scale exists

81
2-D conformal transformation

• xp sxcos(t)-sysin(t)Tx
• yp sxsin(t)sycos(t)Ty
• Where xp, yp photocoordinates
• x,y comparator (assumed) coor.
• s scale, t rotation angle, Tx x
  translation, and Ty y translation

82
2-D conformal transformation

• Solving for the rotation angle mathematically
  turns the equation into non-linear form, which is
  harder to solve.
• Fortunately we can substitute ascos(t) and
  bssin(t) to turn the equation into a linear
  form of
• xp ax-byTx
• yp bxayTy
• which is linear and much easier to solve.

83
2-D conformal transformation

• Each coordinate is a measurement.
• To solve the transformation we need the number of
  measurements to be greater than or equal to the
  number of unknowns 4 in this case.
• Thus two measured fiducial marks are 4 measured
  coordinates so the transformation can be solved
  for

84
2-D conformal transformation

• One measured fiducial mark can only solve for an
  estimate of x and y translation. No scale or
  rotation can be resolved as thos quantities
  require distance and direction both require two
  points
• Two measured fiducial marks generate 4 equations
  that can be uniquely solved for the 4 unknowns.
• Once scale, rotation, and the two translations
  are solved for, any measured comparator
  coordinates can be converted to photo coordinates.

85
2-D conformal transformation

• BUT!!!!
• Two measured fiducial marks afford no check a
  blunder would be undetected mathematically
• Three or more measured fiducial marks (note 4 or
  8 fiducial marks exist on metric film cameras)
  afford a redundant solution
• Redundant solutions are generally solved by least
  squares, which minimizes the sum of the squares
  of the weighted residuals.

86
2-D conformal transformation

• In our case all measurements are treated as
  equally weighted as measured with the same
  equipment on a unique photo.
• Each fiducial coordinate will have a residual
  computed for it which is how it misfits the least
  squares best fit results.
• Residuals are estimates of data quality.
• With only 2 measured points residuals are zero as
  no redundancy exists.

87
Magnitude of residuals

• Reasonable magnitudes become logical through many
  measurements.
• Based on current measurement technologies, photo
  coordinate residuals should be in the 3 15
  micron range.
• Residuals larger than normal are a product of
  measurement error, incorrect entry of fiducial
  calibration coordinates, or a problem with film
  processing or scanning (example film flattening
  mechanism was not functioning)

88
Unique properties of film

• Film grain actually has a proven tendency to
  shrink/expand different amounts in the x and y
  directions.
• This means the assumption of one scale in the 2-D
  conformal transformation can be considered
  invalid for film.
• Therefore the 2-D affine transformation can be
  considered more valid for inner orientation of a
  film derived image (scanned data is derived from
  film)

89
2-D affine transformation

• 6 unknown transformation parameters
• 2 scales
• 2 rotations
• 2 translations
• Thus measurement of two fiducial marks does not
  afford a solution as 4 measurements cannot solve
  for 6 unknowns.

90
2-D affine transformation

• 3 measured fiducial points results in 6 measured
  coordinates thus a unique solution (no
  redundancy)
• 4 or more fiducial points results in a redundant
  solution, a least squares best fit solution, and
  analysis of residuals
• A horizontal angle may not be preserved
  before/after transformation as two scales are
  being utilized in the transformation

91
3-D coordinate transformation

• Uses
• (1) convert assumed ground coordinates (such as
  derived from parallax equation) into ground
  survey based coordinates (usually in a defined
  map projection system)
• (2) Merge 2 arbitrary coordinates systems (2
  overlapping stereomodels) into one system
• Arbitrary 3-D coordinates derived from a
  stereomodel as usually called model coordinates

92
3-D coordinate transformation

• A 3-D coordinate system is defined by
• (1) an origin (fixes 3 coordinates or 3
  coordinate translations)
• (2) direction of the 3 coordinate axes (fixes 3
  rotations about each axis)
• (3) Scale Photogrammetry does not use a EDM or
  tape which defines scale in ground surveying
• 3 translations 3 rotations 1 scale the 7
  unknowns of the 3-D coordinate transformation

93
3-D coordinate transformation

• 7 unknown transformation parameters require 7
  coordinate measurements to be made
• 2 common 3-D points only yields 6 measured
  coordinates one measurement short of a unique
  solution
• Two 3-D points does not define a vertical datum
  so the third point for a unique solution only
  requires an elevation

94
3-D coordinate transformation

• 3 or more 3-D points measured in both systems
  enables a redundant solution.
• Therefore a least squares solution yields
  residuals for each measured point with
  coordinates in both systems
• Blunders in image identification, assigning
  coordinates to the wrong control point
  identifier, and errors in the field survey can
  all lead to residuals which indicate blunders

95
3-D coordinate transformation

• Could a 3-D affine transformation exist?
• Mathematically yes.
• Logically no as 3-D coordinate transformations
  are not used for measurements of photo
  coordinates
• The unique scale property of film thus does not
  apply to 3-D coordinate transformations in
  photogrammetry

96
Collinearity the reality of analytical
photogrammetry

• Collinearity the ground point, the nodal point
  of the lens, and the image point all lie in a
  straight line
• The ground point is defined by 3 unknowns X, Y,
  Z
• The image point is defined by 2 measurements
  x,y photocoordinates
• The nodal point of the lens is called the
  exposure station. It has six unknowns 3
  coordinates X,Y,Z (in the same system as the
  ground point) and 3 rotations defining the
  direction of the camera axis relative to the
  ground coordinate system. The rotations are
  historically Greek letters omega phi kappa about
  X, Y, and Z respectively.
• We will let W, P, K represent omega, phi, kappa
  respectively. Note kappa (about Z) relates
  mostly to the direction of flight relative to the
  ground coordinate system.

97
Collinearity

• For a point a on a photo derived from exposure L
• xa,ya as photocoordinates
• XA,YA,ZA As ground coordinates
• WL, PL, KL, XL, YL, ZL exposure Ls unknown
  camera angles and coordinates

98
The Collinearity condition

• xa f (WL, PL, KL, XL, YL, ZL, XA,YA,ZA)
• ya g (WL, PL, KL, XL, YL, ZL, XA,YA,ZA)
• Where f and g represent mathematical functions
• The measured photo coordinates are a function of
  the exposure station unknowns and the ground
  station unknowns.
• This is an observation equation the measurement
  is described in terms of the unknowns.

99
Ground control and collinearity

• Ground control is either
• (1) targeted (a plastic, fabric, or painted cross
  or T) in an open location such as a road
  intersection or parking lot. This requires
  marking the control point before the flight.
• (2) Photo ids or Picture points
  monoscopically identifiable points are located
  after the flight has occurred manholes,
  sidewalk intersections, end of paint stripes, etc.

100
Exposures and collinearity

• Historically the exposure station unknowns had to
  be solved based on measured ground control
  coordinates.
• Today airborne GPS and IMU (Inertial Measuring
  Unit, i.e. gyroscopes) are used to measure the
  exposure station position and orientation.
  GPS-IMU has accuracy limitations and thus cannot
  be relied on for precise engineering product.

101
Collinearity solving it

• Collinearity is what is called a non-linear
  solution which is difficult to solve.
• An example of two linear equations are
• X 2Y 5
• X - Y -1
• An example of two non-linear equations are
• Sqrt (X) Sine (Y) .12
• Cosine (X) - .2 Y -.17

102
Collinearity solving it

• Linear equations have no powers or trig
  functions, and thus can be solved by adding
  multiples of equations to each other to eliminate
  terms
• Non-linear equations due to powers and/or trig
  functions cannot be solved by adding multiples of
  equations to each other the unknown terms
  cannot be eliminated

103
Collinearity solving it

• Non-linear equations are solved through a
  calculus process called linearization. To use
  this process
• (1) approximations must exist for all unknown
  ground coordinates and exposure unknowns
• (2) the solution solves for updates to the
  approximations, and iterates until the updates to
  the approximations become insignificant

104
Collinearity applications

• (1) Relative orientation solving for the
  relative relationship of a right photo to an
  arbitrarily held fixed left photo
• (2) Bundle adjustment simultaneous solve for
  all ground coordinates of all points with
  measured photocoordinates along with all exposure
  unknowns

105
Historical Aerotriangulation (AT)

• Control densification by photogrammetry to
  minimize ground control requirements and validate
  harmony of ground control and photocoordinate
  measurements
• Today measured exposure station unknowns by
  GPS-IMU are included
• Requiring multiple control points in every
  stereomodel would be cost and time ineffective if
  AT can provide bridged control between a
  sparser ground control network

106
Historical Aerotriangulation (AT)

• Consisted of 3 primary mathematical steps after
  all photocoordinates were measured
• (1) relative orientation create unique assumed
  3-D model coordinates for each stereomodel
• (2) (a) strip/block adjustment combine all
  unique model coordinates systems into one
  combined unique coor. system using common points
  between stereomodels and flight lines
• Strip combines models along a flight line
• Block combines flight lines after joined by Strip
• (2) (b) strip/block adjustment convert the
  common assumed system into the ground coor.
  system using measured ground control points

107
Historical Aerotriangulation (AT)step 3

• Bundle adjustment the simultaneous least
  squares best fit of all photo coordinates,
  measured ground control coordinates , and
  measured exposure station coordinates solving for
  any remaining unknown parameters
• The bundle adjustment was too mathematically
  intense till main frame computers existed, and is
  of course routine on todays personal computers
• Relative orientation and the strip/block
  adjustment provide the initial approximations for
  all unknown parameters in the bundle adjustment

108
Historical Aerotriangulation (AT)

• Three unknown points are selected near the middle
  of a photo, one each near the top, middle, and
  bottom of the photo called pass points as they
  pass control
• These three points could be distinct images, or
  could be artificially marked by a small drill
  into the photo called a pug as the instrument
  is called a pug machine
• Due to 60 overlap, these three points will
  appear in the photo immediately left or right
  along the flight line (end photo only overlaps in
  one direction)
• If multiple flight lines the top and bottom
  points usually appear in the overlap between
  flight lines (20 overlap is common across flight
  lines) these pass points are also tie points as
  they tie across flight lines

109
Historical Aerotriangulation (AT)

• A point in the overlap region across flight lines
  could potentially be on 6 photos, 3 photos on
  each line, due to 60 endlap along the flight
  lines and 20 sidelap across flight lines
• Along a flight line points in the middle of a
  photo can be viewed in 2 stereomodels as it is a
  right photo in one stereomodel and a left photo
  in the second stereomodel
• It is imperative these points in the middle of
  the photo are not too far left or right to not be
  in two consecutive stereomodels

110
Historical Aerotriangulation (AT)

• A pug (artificial) mark takes advantage of
  superimposition
• When viewed in stereo, the pug mark appears to
  be in the photo in which it is not marked, and
  thus in stereo can be measured on the un-marked
  photo
• If the point is monoscopically identifiable
  measurement does not require stereo viewing
• In addition to pass points, ground control points
  are also measured when viewable

111
Historical Aerotriangulation (AT)

• Thus a stereomodel has six pass points in it,
  plus possibly some control points.
• The left three points overlap into the next
  stereomodel to the left.
• The right three points overlap into the next
  stereomodel to the right.
• A unique point is assigned the same point
  name/identifier no matter what photo it is
  measured on. Usually the point name is derived
  from what photos center the point is on.
  Control points usually use the point name
  assigned by the surveyor who put ground
  coordinates on it.

112
Relative Orientation

• Involves only one stereomodel (2 photos)
• The six exposure unknowns of the left photo are
  all held fixed at zero. In addition the X
  coordinate of the right photo is held fixed at an
  arbitrary distance to define scale.
• Note we have fixed 7 parameters six on the left
  exposure and one on the right exposure 7
  parameters uniquely describe a 3-D coordinate
  system.

113
Relative Orientation

• Each measured point has three unknowns in
  collinearity ground X,Y,Z though in relative
  orientation it is called model X,Y,Z as the
  ground control coordinates are not used
• Each measured point generates 4 collinearity
  measurements photo x, photo y on the left
  photo and photo x, photo y on the right photo

114
Relative Orientation

• n unknowns 5 unknowns for the right
  exposure 3 of measured points
• m measurements 4 points
• points 1 m 4 1 4 n 5 3 1 8
• 8 unknowns and 4 measurements cannot be solved as
  measurements be equal to or greater than
  unknowns

115
Relative Orientation

• 2 measured points m 8 n 11
• 3 measured points m 12 n 14
• 4 measured points m 16 n 17
• 5 measured points m 20 n 20 so uniquely
  solved (not redundant)
• 6 measured points m 24 n 23 redundant so
  residuals can be calculated!
• Note 6 points are measured in each stereomodel

116
Relative Orientation

• Thus after relative orientation is completed each
  stereomodel has X,Y,Z model coordinates for each
  measured point
• Unfortunately each stereomodel is its own unique
  arbitrary coordinate system
• Fortunately overlapping stereomodels have 3
  common points between them (the 3 points down the
  center of the common photo)

117
Strip Adjustment

• A series of 3-D conformal coordinate
  transformations to produce one common arbitrary
  coordinate system
• Three common points to overlapping stereomodels
  create 9 measurements (3X, 3Y, 3Z coordinates).
  The 3-D conformal transformation has 7 unknowns
  so a redundant solution enables the
  transformation to occur and residuals calculated
  to assess data quality

118
Strip Adjustment

• Sequentially all stereomodels are converted into
  one arbitrary coordinate system by one 3-D
  conformal transformation after another.
• The block adjustment uses tie points across
  flight lines and the 3-D conformal transformation
  to put all flight lines/all stereomodels into one
  common arbitrary coordinate system. Redundancy
  allows calculation of residuals in every
  transformation.

119
Strip/Block Adjustment to Ground

• Note control was also measured during the
  relative orientation/aerotriangulation collection
  process so these points have model X,Y,Z
  coordinates for them.
• These control points also have ground X,Y,Z
  coordinates. They thus allow a 3-D coordinate
  transformation of the arbitrary coordinate system
  into the ground coordinate system. Hopefully
  redundant control stations (3 or more) enable
  calculation of residuals that test the
  consistency of the ground control. At this point
  all measured AT points have ground X,Y,Z
  coordinates and exposure stations can all have
  values calculated for their unknown parameters.
• The initial approximations for the bundle
  adjustment have thus been calculated!

120
Strip/Block Adjustment to GroundHistorical

• Polynomial correction were often added during the
  adjustment of model coordinates to ground to
  attempt to correct the sequential error build-up
  of multiple 3-D conformal transformations
• The polynomial corrections were used because the
  bundle adjustment was too computationally intense
  before modern computers
• Today the bundle adjustment occurs next,
  eliminating any sequential error build-up of the
  multiple coordinate transformations. Thus
  polynomials are not required to simply generate
  initial approximations for the bundle adjustment.

121
Bundle adjustment

• The relative orientation and strip/block
  adjustment help find blunders and set up the
  bundle adjustment with initial approximations
• Possible input to bundle adjustment
• Photo coordinates collinearity
• Ground Control coordinates
• Exposure station coor. and angles (if airborne
  GPS-IMU is used)

122
Bundle adjustment

• Output of bundle adjustment
• Ground X,Y,Z of any points with measured photo
  coordinates
• W,P,K,X,Y,Z of all exposure stations
• It is a simultaneous least squares solution so
  does not contain the problem of systematic error
  buildup as in rel. or., strip, block adjustment
  process

123
Bundle adjustment

• Least squares is enhanced by proper user defined
  error estimation
• Photo coordinates error est. of 3 -20 microns
  depending on quality of imagery and abilities of
  photogrammetrist
• Ground control error estimates are desired to be
  fixed (0.001 ft. is mathematically the same as
  fixed) but in reality no survey product is
  perfect, and no photogrammetrist can measure
  exactly the same point the surveyor measured as
  images have to be visually interpreted
• Exposure station coordinates and angles are
  derived from the GPS-IMU processing that is
  measurement based and thus is not perfect

124
Bundle adjustment

• Ground control error estimation
• Larger error estimates could be place on picture
  points than targeted control as a target is
  better defined both in a ground survey and in
  measuring an image on a photograph
• Larger error estimates would be placed on
  traverse derived coordinates vs. fixed ambiguity
  GPS coordinates
• Larger error estimates would be placed on trig.
  Leveling derived elevations vs. those derived
  from differential leveling

125
Bundle adjustment

• Least squares minimizes the sum of the squares of
  (weight residual)
• Where weight 1 / error estimate
• Note residual / error estimate is unitless.
• This allows the proper mixing of different types
  of measurements (photo coordinates, ground
  control coor., exposure measurements) properly in
  one simultaneous analysis

126
Bundle adjustmentcontrol requirements -
Historical

• Pre airborne GPS-IMU
• Mathematically 2 X,Y,Z control points and 1 Z
  only control point define a 3-D coordinate system
  and will allow the bundle adjustment to function
• Unfortunately no check in the quality of the
  control coordinates would exist
• Errors would propagate significantly on a large
  job where the control is separated by significant
  amounts of photos

127
Bundle adjustmentcontrol requirements -
Historical

• Pre airborne GPS/IMU
• Mathematically bundle adjustment is weaker in
  vertical than horizontal because the triangles
  created by collinearity are longer and skinnier
  in the vertical dimension
• Thus rule of thumb realistic to prevent
  significant error propagation horizontal
  control every 4-6 photos, vertical control every
  3-4 photos. More vertical control attempts to
  tighten up the weaker vertical geometry of
  AT/collinearity
• Realistically using GPS all control is 3-D so
  concept of a 2-D or a 1-D only control point goes
  away.

128
Bundle adjustmentcontrol requirements using
airborne GPS-IMU

• Mathematically no ground control is required as
  the exposure unknowns have been determined
• Realistically control should be placed in the 4
  corners of the job to provide a realistic check
  on the GPS-IMU solutions
• Note GPS-IMU is not yet accurate enough to serve
  as control for precise engineering design
  photogrammetric projects

129
Bundle adjustmentcontrol requirements using
airborne GPS-IMU

• Note one of the reasons for relative orientation
  and strip adjustment was to generate
  approximations for exposure station unknowns
• If GPS-IMU is used, the exposure unknowns become
  measured. They can be used to solve for any
  ground coordinate approximations using
  collinearity, and thus the need for relative or.
  and strip adjustment prior to the bundle
  adjustment is eliminated.

130
Using error estimation in the bundle adjustment
to find blunder

• Robustness is this procedure
• (1) adjustment runs with user defined error
  estimation
• (2) new error estimate (old error estimate plus
  abs. value (residual)) /2
• Old error estimate and residual are averaged to
  create new error estimate
• Strong measurements are held tighter, weak
  measurements are held weaker, and the poorer
  measurements are filtered out as potential
  blunders. Larger error estimates allow larger
  residuals to exist on the weaker measurements.

131
Bundle adjustment

• Residual rule of thumb (all adjustments)
• If a residual is more than three times its
  corresponding error estimate you are 95
  confident something is suspect
• Residuals will be larger than its corresponding
  error estimate approximately 33 of the time in
  other words at one standard deviation residuals
  will be less than their error estimates 67 of
  the time
• So do not worry if a residual is larger than an
  error estimate start worry when it starts to
  get near three times the size of the error
  estimate

132
Bundle adjustment

• But be careful Example
• Unusually large photo coordinate residuals exist
  on a ground control point
• Re-measuring the photo coordinates on that point
  produced no different results
• If the control points ground coordinates were
  held fixed, it is very possible that is the
  source of the problem. By fixing the control it
  cannot adjust residuals are zero- so the misfit
  is converted into the photo coordinates as those
  were assigned reasonable error estimates.

133
Absolute orientation

• Before map compilation one additional
  mathematical step is required called absolute
  orientation
• It is a one stereo model coordinate conversion to
  ground thus a 3-D conformal transformation
• All bundle adjustment points will have model
  coordinates from relative or. and ground coor.
  from the bundle adjustment
• Thus at least six points from the bundle
  adjustment will exist in each stereomodel so
  absolute orientation is very redundant.

134
Absolute orientation

• Absolute orientation validates how the model
  coordinates fit the final bundle adjusted ground
  coordinates with a 3-D conformal transformation
  without polynomials
• Residuals are an indicator of how good mapped
  positions will be both horizontally and
  vertically
• More importantly it shows how mapped positions in
  successive stereomodel overlap regions, or in
  flight line overlap regions, fit each other.

135
Absolute orientation

• Fit across models example
• Vertical
• Point 121 in model 1-2 has a vertical residual of
  -0.20 ft.
• Point 121 in model 2-3 has a vertical residual of
  0.31 ft.
• Rule of thumb is control should fit to better
  than ½ the desired contour interval

136
Absolute orientation

• -0.20 and 0.31 residuals are less than ½ the
  contour interval BUT!!
• The difference in residuals on the same point
  (note one is negative and one is positive)
  indicate a misfit across stereomodels of 0.51 ft.
  in the region around that point!
• The ½ contour interval rule has been exceeded!

137
Absolute orientation

• Horizontal equivalent of overlap model misfit
• In model 1-2 point 121 has ground X and Y
  residuals of -0.23 and 0.27 ft.
• In model 2-3 point 121 has ground X and Y
  residuals of -0.02 and -0.31 ft.
• Horizontal misfit across stereomodels is the
  distance based on Pythagoreum Theorem
• Sqrt (-0.02 (-0.23))2(-0.31-(0.27))2
• 0.62 ft.
• Thus the difference in position between models is
  significantly more than the individual residuals.

138
Absolute Orientation

• If at least 2 X,Y,Z and 1 Z only control points
  exist in an stereomodel absolute orientation can
  be performed without prior aerotriangulation (the
  control in the model has to satisfy the 3-D
  conformal transformation)
• This is called a full fielded stereomodel.

139
Stereoplotter Orientation without
Aerotriangulation

• (1) Inner Orientation measure fiducials to
  convert comparator coordinates to photo x,y coor.
• (2) Relative Orientation measure at least 6
  common identifiable points across the entire
  stereomodel produces model x,y,z coordinates
• (3) Absolute Orientation measure control points
  3-D conformal of model x,y,z to ground X,Y,Z
• Control points can be measured as part of (2)
  combining (2) and (3) into what is called
  Exterior Orientation

140
What happens in map collection

• A point on the right photo can only be measured
  along an epipolar line given a left photo x,y
  (the original parallax concept)
• Comparator coordinates convert to photo
  coordinates. Left and right photo coordinates
  convert to model coordinates. Model coordinates
  convert to ground coordinates via the resolved
  orientations.

141
Vector (Map) compilation

• Vector (points, lines, and text) is one form of
  map product derived from photogrammetry
• A user digitizes vector information in stereo
  viewing at recognizing the feature code/attribute
  of an image
• Points (manholes, trees, light poles, power
  poles, hydrants, etc.) are point feature codes
  represented by a user defined symbology at a user
  defined scale.
• In computer aided drafting points normally are
  stored in a layer/level associated with the
  feature code name, and the symbol is a defined
  block/cell.

142
Vector (Map) compilation

• Lines (centerlines, pavement edges, curbs,
  sidewalks, power lines, etc.) are line feature
  codes that are associated with symbology of
  color, line width, line style, straight/curve,
  etc.)
• Lines are segregated in computer-aided drafting
  by a feature code being assigned with a
  layer/level.

143
Vector (Map) compilationTricks in software to
enhance mapping

• (1) close a line obvious use is buildings
  after digitizing last point automatically close
  the building (back to the first point)
• (2) make angles 90 degrees for certain features
  (buildings are a great example) if a corner is
  within a user defined angle of 90 degrees make it
  a 90 degree angle
• (3) parallel line offset great on roadways for
  centerline, pavement edge, curbs, sidewalks,
  ditches, etc, that are parallel should include
  option for a vertical offset (such as on curbs)
• This can include how certain features (sidewalks)
  intersect other features (driveways)

144
Vector (Map) compilationTricks in software to
enhance mapping

• (4) Extend undershoots and trim overshoots This
  is also usually associated with how certain
  feature codes interact with other feature codes
• Example a driveway should intersect the edge of
  a building but when digitizing the driveway will
  be short or past it. Automated software can fix
  these while collecting the data

145
Vector (Map) compilationTricks in software to
enhance mapping

• (5) Connect end points near each other - snap
  (near defined by user defined distance input)
• Example Two sidewalk edges were digitized but
  connect at a common point collected on two
  different lines If these two endpoints are
  within the user defined tolerance it should be
  snapped together
• Many line joins occur when features continue
  across distinct stereomodels or flight lines

146
The other superimposition

• The display of vector information superimposed on
  the raster stereomodel is the perfect way to see
  if all information has been digitized
• Superimposition is also used in map updating a
  new flight is viewed with an old digital map
  superimposed on it. Changes due to construction,
  etc. can be seen and updates made to the existing
  digital map information
• Prior to computerization the display of vector
  information overlaid on raster information was
  very impractical.

147
The traditional work flow

• (1) large format calibrated aerial cameras with
  a specific mount in an airplane
• Today this could be a digital camera.
• Today a film based or digital camera could have
  GPS-IMU also in the ariplane to solve for
  exposure station position and orientation

148
The traditional work flow

• (2) processing of photogrammetry in one-to-one
  production of images on film or glass
  diapositives specifically designed for
  minimization of distortion during production and
  due to temperature, pressure, and humidity
  changes
• Today film imagery uses a high precision scanner
  to convert to a digital format
• Image from a digital camera are already in a
  computer format (usually Mr. Sid, JPEG, TIFF,
  etc.)

149
The traditional work flow

• (3) a realistic amount of ground control which is
  either targeted or photo identifiable
• Control accuracy requirements are a function of
  flying height and desired product accuracy
• Ground control requirements can be minimized in
  higher altitude lower accuracy jobs by airborne
  GPS-IMU

150
The traditional work flow

• (4) a measurement and ensuing least squares
  analysis process called aerotriangulation which
  validates the ground control and densifies it to
  a suitable point for use in stereoplotter
  orientation and map compilation,
• Small jobs may be full fielded with control and
  aerotriangulation can be by-passed
• GPS-IMU may eliminate the need for
  aerotriangulation in lower accuracy jobs

151
The traditional work flow

• (5) stereoplotter orientation based on the
  densified ground control which resolves the
  relation of the photos to each other and the
  g