1. Close Range Photogrammetry
Saju John Mathew
EE 5358
Monday, 24th March 2008
University of Texas at Arlington
Monday, March 24, 2008
EE 5358 Computer Vision

2. Overview Definitions Equipment Mathematical Explanations Working
3/24/2008
Overview
Definitions
Equipment
Mathematical Explanations
Working
Applications
On the application side try focussing on the biostereometric side of it.
Monday, March 24, 2008
EE 5358 Computer Vision

3. Close Range Photogrammetry(CRP)
3/24/2008
Close Range Photogrammetry(CRP)
Photogrammetry is a measurement technique where the coordinates of the points in 3D of an object are calculated by the measurements made in two photographic images(or more) taken starting from different positions.
CRP is generally used in conjunction with object to camera distances of not more than 300 meters (984 feet).
The image of each object point be precisely identified between photos so that the measurements will be consistent.
Monday, March 24, 2008
EE 5358 Computer Vision

4. Vertical Aerial Photographs
University of Texas at Arlington at approx. 30 meters
University of Texas at Arlington at approx meters
Monday, March 24, 2008
EE 5358 Computer Vision

5. CRP It can be broadly divided into two main parts:
Acquiring data from the object to be measured by taking the necessary photographs.
Reducing the photographs (perspective projection) into maps or spatial coordinates (orthogonal projection).
Monday, March 24, 2008
EE 5358 Computer Vision

6. Acquisition of Data: Camera
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Acquisition of Data: Camera
Cameras can be broadly classified into two:
Metric
Single Cameras
Stereometric Cameras
Non-metric
There is the inclusion of semi-metric cameras also
Monday, March 24, 2008
EE 5358 Computer Vision

7. Metric Cameras of the object scene from its stereo photographs
3/24/2008
Metric Cameras
Photogrammetric Camera that enables geometrically accurate reconstruction of the optical model
of the object scene from
its stereo photographs
Single Cameras
Total depth of field
Photographic material
Nominal focal length
Format of photographic material
Tilt range of camera axis and number of intermediate stops
Orientable camera support which can be mounted on a tripod and a tiltable metric chamber – which can be separated for transport
Monday, March 24, 2008
EE 5358 Computer Vision

8. Metric Cameras (contd.)
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Metric Cameras (contd.)
Stereometric Cameras
Base Length
Nominal Focal Length
Operational Range
Photographic Material
Format of photographic material
Tilt range of optical axes and
number of intermediate tilt stops
In effect, two metric cameras …find pictures or scan the pictures and upload it
Monday, March 24, 2008
EE 5358 Computer Vision

9. Non-metric Cameras
Cameras that have not been designed especially for photogrammetric purposes:
A camera whose interior orientation
is completely or partially unknown
and frequently unstable.
Monday, March 24, 2008
EE 5358 Computer Vision

10. Non-metric Cameras Advantages Disadvantages General availability
Flexibility in focusing range
Price is considerably less than for metric cameras
Can be hand-held and thereby oriented in any direction
Disadvantages
Lenses are designed for high resolution at the expense of high distortion
Instability of interior orientation (changes after every exposure)
Lack of fiducial marks
Absence of level bubbles and orientation provisions precludes the determination of exterior orientation before exposure
Monday, March 24, 2008
EE 5358 Computer Vision

11. Data Reduction Analog 1900 to 1960 Analytical 1960 onwards
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Data Reduction
Analog 1900 to 1960
Analytical 1960 onwards
Semi-analytical
Digital 1980 onwards
Analog data reduction is not used anymore because of the large and irregular lens and film distortions involved with non-metric photogrammetry
Monday, March 24, 2008
EE 5358 Computer Vision

12. Analytical Photogrammetry
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Analytical Photogrammetry
Based on camera parameters, measured photo coordinates and ground control
Accounts for any tilts that exist in photos
Solves complex systems of redundant equations by implementing least squares method
It was not used for a long time because of heavy computational efforts
Monday, March 24, 2008
EE 5358 Computer Vision

13. Review Collinearity Condition The exposure station
of a photograph, an object
point and its photo image
all lie along a straight
line. L y Z f xa a
Tilted photo plane ya x O Y ZL A Za Xa XL Ya YL X
Monday, March 24, 2008
EE 5358 Computer Vision

14. Image Coordinate System
Ground Coordinate System - X, Y, Z
wrt Ground Coordinate System
Exposure Station Coordinates – XL, YL, ZL
Object Point (A) Coordinates – Xa, Ya, Za
Rotated coordinate system parallel to ground
coordinate system (XYZ) – x’, y’, z’
wrt Rotated Coordinate System
Rotated image coordinates – xa’, ya’, za’
xa’ , ya’ and za’ are related to the measured
photo coordinates xa, ya, focal length (f) and the
three rotation angles omega, phi and kappa. z’ y’ L Z x’
za’
ya’
Xa’ ZL Y A Za Xa Ya XL YL X
Monday, March 24, 2008
EE 5358 Computer Vision

15. 3/24/2008
Rotation Formulas
Developed in a sequence of three independent two-dimensional rotations.
ω rotation about x’ axis
x1 = x’
y1 = y’Cosω + z’Sinω
z1 = -y’Sinω + z’Cosω
φ rotation about y’ axis
x2 = -z1Sinφ + x1Cosφ
y2 = y1
z2 = z1Cosφ + x1Sinφ
κ rotation about z’ axis
x = x2Cosκ + y2Sin κ
y = -x2Sinκ + y2Cosκ
z = z2
rotation about x’ axis means that the x’ and x1 axes are coincident (d1 = cos (theta)d2; where d is the distance that the plane is offset from the origin, times the length of vector(x,y,z))
Monday, March 24, 2008
EE 5358 Computer Vision

16. 3/24/2008
Rotation Matrix
X = MX’ Rotation Matrix The sum of the squares of the three “direction cosines” in any row or in any column is unity. M -1 = MT X’ = MTX
x = x’(CosφCosκ) + y’(SinωSinφCosκ + CosωSinκ) + z’(-CosωSinφCosκ + SinωSinκ)
y = x’(-CosφSinκ) + y’(-SinωSinφSinκ + CosωCosκ) + z’(CosωSinφSinκ + SinωCosκ )
z = x’(Sinφ) + y’(-SinωCosφ) + z’(CosωCosφ)
x = m11x’ + m12y’ + m13z’
y = m21x’ + m22y’ + m23z’
z = m31x’ + m32y’ + m33z’
Direction Cosines are the cosines of the angles in space between the respective axes, the angles being taken between 0 and 180 degrees.
Rotation Matrix is an orthogonal matrix which has the property that its inverse is equal to its transpose
Monday, March 24, 2008
EE 5358 Computer Vision

17. Collinearity Condition Equations
Collinearity condition equations developed from similar triangles * Dividing xa and ya by za * Substitute –f for za * Correcting the offset of Principal point (xo, yo)
x = m11x’ + m12y’ + m13z’
y = m21x’ + m22y’ + m23z’
z = m31x’ + m32y’ + m33z’
Monday, March 24, 2008
EE 5358 Computer Vision

18. Collinearity Equations
3/24/2008
Collinearity Equations
Nonlinear
Nine unknowns
ω, φ, κ
XA, YA and ZA
XL, YL and ZL
Taylor’s Theorem is used to linearize the nonlinear equations
substituting
xa, ya, xo, yo and f are the constants
Because higher order terms are ignored in linearization by Taylor’s theorem, the linearized forms of the equations are approximations
Hence they must be solved iteratively until the magnitudes of corrections to initial approximations become negligible.
Monday, March 24, 2008
EE 5358 Computer Vision

19. Linearizing Collinearity Equations
Rewriting the Collinearity Equations
F0 and G0 are functions F and G evaluated at the initial approximations for the nine unknowns
dω, dφ, dκ are the unknown corrections to be applied to the initial approximations
The rest of the terms are the partial derivatives of F and G wrt to their respective unknowns at the initial approximations
Taylor’s Theorem
Monday, March 24, 2008
EE 5358 Computer Vision

20. Applying LLSM to Collinearity Equations
3/24/2008
Applying LLSM to Collinearity Equations
Residual terms must be included in order to make the equations consistent
J = xa – Fo ; K = ya – Go
b terms are coefficients equal to the partial derivatives
Numerical values for these coefficient terms are obtained by using initial approximations for the unknowns.
The terms must be solved iteratively (computed corrections are added to the initial approximations to obtain revised approximations) until the magnitudes of corrections to initial approximations become negligible.
Sum of the squares of the residuals is minimized
Conditions: 1) Number of observations being adjusted is large
2) Frequency distribution of the errors is normal(gaussian)
Monday, March 24, 2008
EE 5358 Computer Vision

21. Analytical Stereomodel
3/24/2008
Analytical Stereomodel
Mathematical calculation of three-dimensional ground coordinates of points in the stereomodel by analytical photogrammetric techniques
Three steps involved in forming an Analytical Stereomodel:
Interior Orientation
Relative Orientation
Absolute Orientation
Monday, March 24, 2008
EE 5358 Computer Vision

22. Analytical Interior Orientation
3/24/2008
Analytical Interior Orientation
Requires camera calibration information and quantification of the effects of atmospheric refraction.
2D coordinate transformation is used to relate the comparator coordinates to the fiducial coordinate system to correct film distortion.
Lens distortion and principal-point information from camera calibration are used to refine the coordinates so that they are correctly related to the principal point and free from lens distortion.
Atmospheric refraction corrections are applied.
Interior Orientation: Comprises of affine coordinate transformation to relate comparator coordinates to the fiducial coordinate system and to correct film distortion; Lens distortion and principal-point distortion are corrected; atmospheric refraction corrections
Monday, March 24, 2008
EE 5358 Computer Vision

23. Analytical Relative Orientation
3/24/2008
Analytical Relative Orientation
Process of determining the elements of exterior orientation
Fix the exterior orientation elements of the left photo of the stereopair to zero values
Common method in use to find these elements is through Space Resection by Collinearity(see slide below)
Each object point in the stereomodel contributes 4 equations
5 unknown orientation elements + 3 unknowns(X, Y & Z)
Relative Orientation: omega1=phi1=kappa1=XL1=YL1=0; ZL2=f, XL2=b and ZL1=ZL2 i.e. there are 5 unknowns, so we need atleast 5 control points or 6 control points(improved soln using LSM) are required.
Each point results in a net gain of one equation for the overall solution.
Monday, March 24, 2008
EE 5358 Computer Vision

24. Space Resection by Collinearity
3/24/2008
Space Resection by Collinearity
Formulate the collinearity equations for a number of control points whose X, Y and Z ground coordinates are known and whose images appear in the tilted photo. The equations are then solved for the six unknown elements of exterior orientation which appear in them.
Space Resection collinearity equations for a point A
A two dimensional conformal coordinate transformation is used
X = ax’ – by’ + Tx X, Y – ground control coordinates for the point
Y = ay’ + bx’ + Ty x’, y’ – ground coordinates from a vertical photograph
a, b, Tx, Ty – transformation parameters
Uses redundant amounts of ground control, hence least squares computational techniques can be used to determine most probable values for the six elements.
Mention Space Intersection and the fact that it is not as popular as Space Intersection
Monday, March 24, 2008
EE 5358 Computer Vision

25. Analytical Absolute Orientation
3/24/2008
Analytical Absolute Orientation
Utilizes a 3D conformal coordinate transformation
Requires a min. of 2 horizontal and 3 vertical control points
Stereomodel coordinates of control points are related to their 3D coordinates in a cartesian coordinate system
Coordinates of all stereomodel points in the ground system can be computed by applying the transformation parameters
Additional control points provide redundancy, which enables least squares solution
Monday, March 24, 2008
EE 5358 Computer Vision

26. Bundle Adjustment Adjust all photogrammetric measurements to ground
3/24/2008
Bundle Adjustment
Adjust all photogrammetric measurements to ground
control values in a single solution
Unknown quantities
X, Y and Z object space coordinates of all object points
Exterior orientation parameters of all photographs
Measurements
x and y photo coordinates of images of object points
X, Y and/or Z coordinates of ground control points
Direct observations of the exterior orientation parameters of the photographs
Extension of analytical photogrammetry applied to an unlimited number of overlapping photographs
put diagram of block of photos
Monday, March 24, 2008
EE 5358 Computer Vision

27. Bundle Adjustment-Observations
Photo Coordinates - Fundamental Photogrammetric Measurements made with a comparator or analytical plotter. According to accuracy and precision the coordinates are weighed
Control Points – determined through field survey
Exterior Orientation Parameters – especially helpful in understanding the angular attitude of a photograph
Regardless of whether exterior orientation parameters were observed, a least squares solution is possible since the number of observations is always greater than the number of unknowns.
xij, yij – measured photo coordinates of the image
of point j on photo i related to fiducial axis system
xo, yo – coordinates of principal points in fiducial axis system
f - focal length/principal distance Xj, Yj, Zj – coordinates of point j in object space
m11i, m12i, …….,m33i – rotation parameters for photo i XLi, YLi, ZLi – coordinates of incident nodal point
of camera lens in object space
Monday, March 24, 2008
EE 5358 Computer Vision

28. Bundle Adjustment - Weights
Photo coordinates
σ02 – reference variance
σxij2 , σyij2 – variances in xij and yij resp.
σxijyij = σyijxij – covariance of xij and yij
Ground Control coordinates
σXj2, σYj2, σZj2 – variances in Xj00, Yj00, Zj00 resp.
σXjYj = σYjXj – covariance of Xj00 with Yj00
σXjZj = σZjXj – covariance of Xj00 with Zj00
σYjZj = σZjYj – covariance of Yj00 with Zj00
Exterior Orientation Parameters
Monday, March 24, 2008
EE 5358 Computer Vision

29. Direct Linear Transformation (DLT)
This method does not require fiducial marks and can be solved without supplying initial approximations for the parameters
Collinearity equations along with the correction for lens distortion
δx, δy – lens distortion
fx – pd in the x direction fy – pd in the y direction
Rearranging the above two equations
Monday, March 24, 2008
EE 5358 Computer Vision

30. DLT(contd.) The resulting equations are solved iteratively using LSM
Advantages
- No initial approximations are required
for the unknowns.
Limitations
- Requirement of atleast six 3D object space control points
- Lower accuracy of the solution as compared with a rigorous bundle
adjustment
Monday, March 24, 2008
EE 5358 Computer Vision

31. Analytical Self Calibration
The equations take into account adjustment of the calibrated focal length, principal-point offsets and symmetric radial and decentering lens distortion.
xa, ya – measured photo coordinates related to fiducials
xo, yo – coordinates of the principal point
= xa – xo where
= ya - yo
Monday, March 24, 2008
EE 5358 Computer Vision

32. Analytical Self Calibration(contd.)
3/24/2008
Analytical Self Calibration(contd.)
k1, k2, k3 = symmetric radial lens distortion coefficients p1, p2, p3 = decentering distortion coefficients f = calibrated focal length r, s, q = collinearity equation terms Provides a calibration of the camera under original conditions which existed when the photographs were taken. Geometric Requirements - Numerous redundant photographs from multiple locations are required, with sufficient roll diversity - Many well-distributed image points be measured over the entire format to determine lens distortion parameters The numerical stability of analytical self calibration is of serious concern.
Roll diversity is the condition in which the photographs have angular attitudes that differ greatly from each other.
Monday, March 24, 2008
EE 5358 Computer Vision

33. Applications Automobile Construction
Machine Construction, Metalworking, Quality Control
Mining Engineering
Objects in Motion
Shipbuilding
Structures and Buildings
Traffic Engineering
Biostereometrics
Monday, March 24, 2008
EE 5358 Computer Vision

34. Biomedical Applications
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Biomedical Applications
Linear tape and caliper measurements of inherently irregular three-dimensional biological structures are inadequate for many purposes.
Subtle movements produced by breathing, pulsation of blood, and reflex correction for control of postural stability.
Short patient involvement times, avoids contact with the patient and thereby avoiding risk of deforming the area of interest and spreading infection.
All medical photogrammetric measurements require further interpretation and analysis to allow meaningful information to be given to the end-user.
Non-metric cameras are used mostly because of the very high cost of metric cameras
Monday, March 24, 2008
EE 5358 Computer Vision

35. Bibliography
Kamara, H.M. (1979). Handbook of Non-Topographic Photogrammetry, American Society of Photogrammetry.
Wolf , Paul R., Dewitt, Bon A. (2000). Elements of Photogrammetry, McGraw Hill.
Devarajan, Venkat and Chauhan, Kriti (Spring 2008). Lecture Notes: Mathematical Foundation of Photogrammetry, EE 5358 University of Texas at Arlington.
Karara, H.M. (1989). Non-Topographic Photogrammetry, American Society for Photogrammetry and Remote Sensing.
Mitchell, H.L. and Newton, I. (2002). Medical photogrammetric measurement: overview and prospects. ISPRS Journal of Photogrammetry & Remote Sensing, 56,
Monday, March 24, 2008
EE 5358 Computer Vision

36. Acknowledgments Dr. Venkat Devarajan Kriti Chauhan
Monday, March 24, 2008
EE 5358 Computer Vision