Triangulation Reports

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This document describes the data presented in a triangulation report. The triangulation report displays in the following situations:

There are two types of triangulation report depending upon the source of data: the Camera-based Triangulation Report and the Sensor-based Triangulation Report.

Depending upon the options selected, some of the elements described here may or may not appear in the report.

Camera-based Triangulation Report

  1. Triangulation Report Unit Definition

    The units for the image coordinates, exterior orientation rotation angles, and ground control points display in the first section. The positional elements of exterior orientation use the same units as the ground control points. Image coordinate residuals display in the units similar to input image coordinates.

    The Triangulation Report With IMAGINE Photogrammetry

    The output image x, y units:

    The output angle unit:

    The output ground X, Y, Z units: meters

  2. Image Coordinates and Result of Interior Orientation

    The image coordinates representing the image positions of GCPs, check points, and tie points display for each image in the block. The Image ID as specified in the IMAGINE Photogrammetry Project Manager CellArray is used to represent the individual images.

    Six affine transformation parameters display for each image. These coefficients are derived using a 2D affine transformation during interior orientation. The six coefficients represent the relationship between the file or pixel coordinate system of the image and the film or image space coordinate system. The file coordinate system has its origin in the upper-left corner of the image (that is, 0, 0). The origin of the image space coordinate system for each image is the principal point. Using the 2D affine transformation, the principal point position is determined mathematically by intersecting opposite fiducial mark positions.

    The six transformation coefficients are calculated once interior orientation has been performed. The six transformation parameters define the scale and rotation differences between two coordinate systems.

    Once the image position of a ground point has been measured automatically or manually, the six coefficients are used to transform the pixel coordinates of the ground point to image (or film) coordinates.

    Thus, the image coordinates and the six affine transformation coefficients display for each image in the block.

    The Input Image Coordinates

    image ID = 90

    Point ID

    x

    y

    1002

    952.625

    819.625

    1003

    1857.875

    639.125

    1005

    1769.450

    1508.430

    1006

    1787.875

    2079.625

    2001

    915.020

    2095.710

    2003

    846.530

    208.330

    Affine coefficients from file (pixels) to film (millimeters)

    A0

    A1

    A2

    B0

    B1

    B2

    -114.3590

    0.100015

    -0.001114

    116.5502

    -0.001117

    -0.100023

    image ID = 91

    Point ID

    x

    y

    1002

    165.875

    846.625

    1003

    1064.875

    646.375

    1005

    1007.250

    1518.170

    1006

    1023.625

    2091.390

    2001

    160.900

    2127.840

    2002

    2032.030

    2186.530

    1004

    1839.520

    1457.430

    2003

    49.303

    237.343

    ...

    ..

    .

    Affine coefficients from file (pixels) to film (millimeters)

    A0

    A1

    A2

    B0

    B1

    B2

    -114.1478

     0.100028

    -0.001025

    116.2826

    -0.001008

    -0.100025

    image ID = 92

    Point ID

    x

    y

    2002

    1227.375

    2199.125

    1006

    215.125

    2083.790

    1005

    224.670

    1510.670

    1004

    1050.600

    1465.230

    1003

    286.875

    639.125

    2007

    109.434

    352.269

    ...

    ..

    .

    Affine coefficients from file (pixels) to film (millimeters)

    A0

    A1

    A2

    B0

    B1

    B2

    -116.6171

    0.100021

    0.001458

    113.7281

    0.001475

    -0.100014

  3. Triangulation Results using the Bundle Block Adjustment

Aerial triangulation uses a bundle block adjustment. This approach uses an iterative least squares solution. The unknown parameters are either estimated or adjusted. The estimated or adjusted parameters include:

The corresponding accuracy estimates for each set of estimated or adjusted parameters is also provided if you select the Compute Accuracy For Unknowns radio button within the Aerial Triangulation dialog.

Input parameters are adjusted (rather than estimated) if initial approximations have been provided and statistically weighted. For example, if initial approximations for exterior orientation were input and statistically weighted, a new set of adjusted values would be computed after aerial triangulation has been performed.

Input parameters are estimated if they are unknown prior to executing the aerial triangulation. For example, tie point coordinates are commonly unknown prior to aerial triangulation. Their X, Y, and Z coordinates are estimated using the bundle block adjustment.

Iterative Results

A global indicator of quality is computed for each iteration of processing. This is referred to as the standard error (also known as the standard deviation of image unit weight). This value is computed based on residuals computed for the estimated and adjusted observations for that particular iteration of processing. This includes residuals for image coordinates, GCPs, exterior orientation, interior orientation, and additional parameters. The units for the standard error are defined within the General tab of the Aerial Triangulation dialog.

After each iteration of processing, the software estimates the exterior orientation parameters of each camera/sensor station and X, Y, and Z tie point coordinates. The newly estimated exterior orientation parameters are then used along with the GCP and tie point coordinates to compute new x and y image coordinate values. The newly computed image coordinate values are then subtracted from the original image coordinate values. The differences are referred to as the x and y image coordinate residuals.

If the exterior orientation parameters are incorrect, then the newly computed image coordinate values are also incorrect. Incorrect estimates for exterior orientation may be attributed to erroneous GCPs, data entry blunders, or mismeasured image positions of GCPs or tie points. Any error in the photogrammetric network of observations is reflected in the image coordinate residuals.

The computed standard error for each iteration accumulates the effect of each image coordinate residual to provide a global indicator of quality. The lower the standard error, the better the solution.

SHARED Tip The number of iterations using the least squares adjustment continues until the corrections to the ground points (that is, GCPs, tie points, and check points) are less than the user-specified convergence value (that is, 0.001 default). After each iteration of processing, a new set of X, Y, and Z coordinates is computed for the GCPs, tie points, and check points. The new set of coordinates is subtracted from the previous set of coordinates (the previous iteration). The differences between the coordinates are also referred to as corrections to the unknowns. If the differences are greater than the convergence value, the iterations continue.

Automated Error Checking Results

IMAGINE Photogrammetry provides two blunder checking models that automatically identify and remove erroneous image measurements from the block. The blunder checking models can be specified by selecting the appropriate model within the blunder checking model dropdown list contained in the Advanced Options tab of the Aerial Triangulation dialog.

The results of the blunder checking model are display as follows:

The residuals of blunder points

Point

Image

Vx

Vy

1005

90

0.4224

-0.5949

x wrong

y wrong

1005

91

0.7458

-0.6913

x wrong

y wrong

1005

92

0.9588

-0.6326

x wrong

y wrong

Point 1005 will excluded in the further adjustment !

The x and y image coordinate residuals of the blunder points display. You are notified if a point is excluded from the aerial triangulation solution.

THE OUTPUT OF SELF-CALIBRATING BUNDLE BLOCK ADJUSTMENT

the no. of iteration =1

the standard error = 0.1438

the maximal correction of the object points = 40.30844

the no. of iteration =2

the standard error = 0.1447

the maximal correction of the object points = 0.74843

the no. of iteration =3

the standard error = 0.1447

the maximal correction of the object points = 0.00089

Exterior Orientation Parameters

The exterior orientation parameters

image ID

Xs

Ys

Zs

OMEGA

PHI

KAPPA

90

666724.3686

115906.5230

8790.3882

0.1140

0.0272

90.3910

91

666726.8962

119351.8150

8790.0182

0.2470

0.1874

89.0475

92

666786.3168

122846.5488

8787.5680

0.0923

0.1232

88.6543

A corresponding accuracy estimate is provided for each exterior orientation parameter. The accuracy estimates are computed from the covariance matrix of the final solution. The accuracy estimates reflect the internal accuracy of each parameter.

The accuracy of the exterior orientation parameters

image ID

mXs

mYs

mZs

mOMEGA

mPHI

mKAPPA

90

2.7512

3.4383

1.8624

0.0266

0.0207

0.0112

91

2.7014

3.1999

1.5232

0.0252

0.0209

0.0109

92

2.7940

3.3340

2.4975

0.0263

0.0204

0.0115

Interior Orientation Parameters

The interior orientation parameters associated with each camera station in the block display in this section. Since a self-calibrating bundle adjustment has not been performed, the interior orientation parameters for each camera remain the same.

The interior orientation parameters of photos

image ID

f(mm)

xo(mm)

yo(mm)

90

153.1240

-0.0020

0.0020

91

153.1240

-0.0020

0.0020

92

153.1240

-0.0020

0.0020

Interior Orientation and Self-Calibration

If a self-calibrating bundle adjustment had been performed, the focal length and principal point values would be estimated. The output would appear as follows:

The interior orientation parameters of photos

image ID

f(mm)

xo(mm)

yo(mm)

90

153.1220

-0.0097

-0.0039

91

153.1220

-0.0097

-0.0039

92

153.1220

-0.0097

-0.0039

The accuracy of the interior orientation parameters

image ID

mf(mm)

mxo(mm)

myo(mm)

all

0.0586

0.0664

0.0650

In this case, it was assumed that the interior orientation parameters were common for each image in the block. For this reason, the interior orientation values are common for each image. Additionally, one set of accuracy estimates was computed. The mf, mxo, and myo values represent the accuracy for the calibration parameters.

If each camera station had a unique set of interior orientation parameters estimated, the output would appear as follows:

The interior orientation parameters of photos

image ID

f(mm)

xo(mm)

yo(mm)

90

153.1164

-0.0190

-0.0156

91

153.1230

-0.0135

0.0003

92

153.1288

-0.0018

0.0038

The accuracy of the interior orientation parameters

image ID

mf(mm)

mxo(mm)

myo(mm)

90

0.0946

0.1020

0.0953

91

0.0936

0.0995

0.0954

92

0.0985

0.1031

0.0985

In this scenario, accuracy estimates were computed for each camera station.

Ground and Image Point Residuals and Statistical Parameters

Once the bundle block adjustment is complete, the adjusted values of the ground point coordinates, including ground control, check and tie points, are available. The new ground point coordinates are computed by applying least square principles in the triangulation system. They are the optimized estimation of the ground location of these points based on all inputs and settings you have chosen. The residuals are defined as the adjusted coordinates minus the original measured coordinates for both ground and image points. The root mean square error (RMSE) is computed from these residuals. The mean error is the numeric sum of the residuals divided by the total number of items.

In addition to RMSE and the mean errors, CE90 and LE90 defined by the US government agency are also computed. The method used here for computing CE90 is from Tom Ager with improvement from Kenton Ross (http://calval.cr.usgs.gov/JACIE_files/JACIE04/files/1Ross16.pdf). The LE90 method used here is a common method recommended by both Tom Ager and Tim Nagy (An Analysis of Metric Accuracy: Definitions and Methods of. Computation, 29 March 2004, by Thomas P. Ager, NGA/IXS).

For the measured ground control and check points, in addition to the optimally adjusted values from the triangulation system, there is also an intersected location available by using multiple image forward intersection principle with the measured image points and adjusted image parameters, just like in the stereo plotter situation, if the control or check point is measured on more than one image. In order to give report readers some reference if they want to compare the stereo model coordinates and residuals, the difference between these intersected point coordinates and the original measured point coordinates is also reported.

Check Point Residuals

Check points are used to independently verify the quality of the bundle block adjustment. Check points residuals and statistics are more important and useful to judge the quality of your triangulation than control points residuals. Once the triangulation iteration is converged, the adjusted check point coordinates are available from the triangulation computation. These computed coordinates are subtracted from the original input coordinates to compute the check point residuals. Check points serve as the best source for determining the accuracy of the bundle block adjustment.

The aX, aY, and aZ values reflect the average residual values for the X, Y, and Z check point coordinates, respectively. The mX, mY, and mZ values reflect the root mean square errors (standard deviation) of all X, Y, and Z coordinates, respectively. The CE90 indicates the 90% horizontal error estimation and LE90 indicates the 90% vertical error estimation for these check points according to the computation methods discussed in the beginning of this section.

The residuals of the check points

Point ID

rX

rY

rZ

2001

-4.0786

0.0865

-1.9679

2002

1.6091

-3.0149

3.5757

aX

aY

aZ

-1.2348

-1.4642

0.8039

mX

mY

mZ

3.1003

2.1328

2.8860

CE90

LE90

8.8875

8.4344

The next section shows the image residuals of measured check points. It is the difference between the backward projected image location from the measured ground location and the measured image location of the check points. After the table listing for each image check point, a summary for mean errors and root mean square errors are listed for these image check point residuals in x and y respectively.

The image residuals of measured check points

Point

Image

Vx

Vy

2001

90

-0.020

0.351

2002

92

0.620

0.299

Point

Image

Vx

Vy

2002

91

1.009

-0.041

2002

92

0.779

-0.032

Mean error of 4 image points: ax=0.959, ay=-0.007

RMSE of 4 image points: mx=0.706, my0.242

Control Point Residuals

There are two groups of ground point residuals displayed in the triangulation report. The first one is the normal residuals between optimally adjusted ground point coordinates and original measured ground control point coordinates. It is called "The residuals of the control points". In the case the ground control points are chosen as "fixed values", all the residuals here are zero, since by definition no change to ground control points are allowed here, so the measured point location is the adjusted point location.

The second section of control point residuals is called "The difference of intersected and measured control points". It shows the ground X, Y, Z difference between the forward intersected ground location by using multiple image forward intersection principle with the measured image points and adjusted image parameters, and the original measured ground location. If you use intersection tool in TE or SPM, the residuals you would get should be same as here if you get the same image location and same number of images.

Both ground control point residual sections are followed by the mean errors, root mean square errors and CE90 and LE90 computed from these residuals.

The residuals of the control points

Point ID

rX

rY

rZ

1002

1.4390

5.6596

6.9511

1003

-3.0263

-0.5553

-2.2517

1004

3.0679

2.0052

-11.1615

1005

2.9266

-4.0629

-2.3434

1006

-2.0842

2.5022

2.6007

aX

aY

aZ

0.4646

1.1098

-1.2410

mX

mY

mZ

2.5904

3.4388

6.1680

CE90

LE90

2.9578

0.5146

The Difference of intersected and measured control points

Point ID

rX

rY

rZ

1002

1.5212

5.1571

4.8152

1003

-3.2724

-1.2353

-0.4763

1004

3.0171

2.0452

-8.3759

1005

2.7993

-4.2927

-1.8881

1006

-2.1118

2.4010

2.3213

aX

aY

aZ

0.3907

0.8150

-0.7207

mX

mY

mZ

2.6238

3.3614

4.5282

CE90

LE90

7.2011

8.1230

The next section shows the image residuals of intersected ground control points. It is the difference between the back projected image location from the intersected ground control point location and the measured image point location of the ground control point. These image residuals are listed per ground point. At the end, it lists the mean errors and RMSE of all above image residuals.

The image residuals of intersected GCP

 Point

Image

Vx

Vy

 1002

90

-0.000

-0.061

 1002

91

0.002

0.060

 Point

Image

Vx

Vy

 1003

90

0.043

-0.029

1003

91

-0.083

0.102

1003

92

0.043

-0.071

Point

Image

Vx

Vy

1004

91

-0.000

-0.039

1004

92

-0.000

0.039

Point

Image

Vx

Vy

1005

90

0.003

0.009

1005

91

-0.005

-0.020

1005

92

0.002

0.010

Point

Image

vX

vY

1006

90

0.007

0.028

1006

91

-0.017

-0.161

1006

92

0.006

0.134

The control point and image point measurement accuracy, the weighting and triangulation network configuration can affect triangulation accuracy and the ground control and check point residuals. Under same conditions, such as same weight and redundancy, relatively large residual values are indicative of error in the photogrammetric network of observations. Large errors can be attributed to mismeasured control points, data entry error, and poor quality of control points. In the following example, the Z coordinate for control point 1004 is relatively larger than the remaining residuals, thus inferring a possible error.

The aX, aY, and aZ values reflect the average residual values for the X, Y, and Z control point

coordinates, respectively. The mX, mY, and mZ values reflect the root mean square errors (standard deviation) of all X, Y, and Z coordinates, respectively.

Control Point, Check Point, and Tie Point Coordinates and Accuracy Estimates

Once the bundle block adjustment is complete, new ground control point, check point, and tie point coordinates are computed based on the estimated or adjusted exterior orientation parameters. The computation of X, Y, and Z coordinates for tie points is referred to as ground point determination. If the tie points are acceptable, they can be converted to control points within the Classic or Stereo Point Measurement Tool. This process is referred to as control point extension.

SHARED Tip The original GCP and check point values as displayed within the point measurement tool do not change. If the Update or Accept buttons have been selected, the tie point coordinates are populated in the point measurement tool.

The coordinates of object points

Point ID

X

Y

Z

Overlap

1002

665230.0078

115015.7356

1947.0091

2

1003

664452.5217

119050.0976

1990.0849

3

1004

668152.5139

122406.1013

1971.3625

2

1005

668340.3906

118681.5541

1885.8520

3

1006

670840.0509

118698.6034

2014.9514

3

2001

670966.3714

114815.3165

1889.9201

2

2002

671410.3391

123163.5051

1987.3377

2

2003

662482.9556

114550.1580

1927.2971

2

2004

662662.3529

116234.8134

2064.6330

2

...

..

.

2041

671068.5733

123961.8788

1971.4592

2

The total object points = 46

The Overlap column specifies the number of images on which the point has been measured. This is also referred to as redundancy.

The accuracy of each control point, check point, and tie point is computed. The accuracy estimates are computed based on the respective diagonal elements of the covariance matrix for the final solution.

The accuracy estimates are computed by multiplying the standard deviation of unit weight by the variance of the individual parameters. The variance of the computed parameters is determined and contained within an output covariance matrix. Accuracy estimates are computed for:

  • The exterior orientation parameters (X, Y, Z, omega, phi, kappa) for each camera exposure station
  • The X, Y, and Z coordinates of the tie points
  • Interior Orientation parameters (focal length, principal point xo, and principal point yo)
  • Additional Parameters (AP)
  • exterior orientation parameters for each image
  • X, Y, and Z coordinates of the tie points
  • interior orientation parameters if SCBA is performed
  • AP if a functional model is selected

The accuracy of object points

Point ID

mX

mY

mZ

mP

Overlap

1002

1.0294

1.0488

1.2382

1.9217

2

1003

0.8887

0.8731

1.1739

1.7117

3

1004

0.9693

0.9593

1.2219

1.8311

2

1005

0.7181

0.7063

0.9397

1.3775

3

1006

0.9534

0.9119

1.1841

1.7727

3

2001

1.8123

1.5641

3.1611

3.9653

2

2002

1.9436

1.5091

2.8482

3.7639

2

2003

1.9684

1.7646

3.0069

4.0037

2

2004

1.7427

1.3758

2.6308

3.4426

2

...

..

.

2041

1.9419

1.6884

3.0368

3.9804

2

amX

amY

amZ

1.2681

1.1773

2.1932

The mX, mY, and mZ values represent the geometric mean residual for the X, Y, and Z coordinates of GCPs, check points, and tie points. The mP values represent the mean square error for a particular point by considering the mean error for each coordinate of a given point.

The amX, amY, and amZ values represent the average mean square error for the X, Y, and Z coordinates of the control, check, and tie points.

The Overlap column specifies the number of images on which the point has been measured. This is also referred to as redundancy.

Image Coordinate Residuals

Once the bundle block adjustment is complete, image coordinate residuals are computed. The residuals are computed based on the estimated or adjusted exterior orientation parameters, GCP, check point, and tie point coordinates, and their respective image measurements. During the iterative least square adjustment, the results from the previous iteration are compared to results of the most recent iteration. During this comparison, image coordinate residuals reflecting the extent of change between the original image coordinates and the new values are computed. The values of the new image coordinates are dependent on the estimated or adjusted parameters of the bundle block adjustment. Therefore, errors in the estimated or adjusted parameters are reflected in the image coordinate residuals.

The software computes the image coordinate residuals for each image measurement in the block.

The residuals of image points

Point

Image

Vx

Vy

1002

90

-0.176

-0.040

1002

91

-0.001

-0.016

Point

Image

Vx

Vy

1003

90

0.053

-0.001

1003

91

0.017

0.149

1003

92

0.045

0.050

Point

Image

Vx

Vy

1004

91

0.103

-0.094

1004

92

-0.182

-0.010

Point

Image

Vx

Vy

1005

90

0.164

-0.021

1005

91

0.102

-0.078

1005

92

-0.031

-0.023

Point

Image

Vx

Vy

1006

90

-0.133

0.044

1006

91

-0.035

-0.104

1006

92

0.074

0.138

Point

Image

Vx

Vy

2001

90

-0.000

-0.016

2001

91

0.000

0.016

Point

Image

Vx

Vy

2002

91

0.000

0.017

2002

92

0.000

-0.017

Point

Image

Vx

Vy

2003

90

-0.000

-0.094

2003

91

0.002

0.095

Point

Image

Vx

Vy

2004

90

0.000

0.074

2004

91

-0.002

-0.074

Image point residuals here are computed with the adjusted optimal ground point location and adjusted optimal image orientation. It is the difference between back projected image location from the adjusted optimal ground point and the measured image point location. It is listed in group for each ground points. After all image point residuals are listed, the mean errors and RMSEs of all image residuals in x and y are computed and listed.

After image residuals are listed in group per ground point, the image residuals of ground control points are listed again in group per image followed by RMSE for each image. After all images are listed, the total RMSEs for all image ground control points are computed and listed. These RMSEx and RMSEy should be same as reported in the triangulation short report.

If there are ground check points, the image residuals of ground check points are then listed again in group per image, same as for ground control points. After all images are listed, the total RMSEs for all image check points are listed. These RMSEx and RMSEy should be same as reported in the triangulation short report.

(RPC triangulation has been changed in the same way as for frame camera, so please update the control, check and image point residual parts in the report in the same way as described here, if there is one.)

Sensor-based Triangulation Report

  1. Triangulation Report Unit Definition

    The units for the image coordinates, exterior orientation rotation angles, and ground control points are defined here. The positional elements of exterior orientation use the same units as the ground control points. Image coordinate residuals display in the same units as the input image coordinates.

    The Triangulation Report With IMAGINE Photogrammetry

    The output image x, y units:

    The output angle unit:

    The output ground X, Y, Z units: meters

  2. Automated Error Checking Model

    The software provides an error checking model that automatically identifies and removes erroneous image measurements from the block. The error checking model is specified by selecting the Simple Gross Error Check options within the Advanced Options tab of the Triangulation dialog. Once the radio button has been selected, the times of unit multiplier can be specified. This value is multiplied by each image coordinate residual value. If an image coordinate residual value is larger than the multiplier value, the ground point is excluded from the triangulation.

    The results of the gross error checking model display as follows:

    The x and y image coordinate residuals of the erroneous points display here. You are notified if a point is excluded from the triangulation solution. The pid designation defines the point id of a ground point.

    For sensors with RPC models, triangulation determines which points have parallel rays and uses these parallel rays as an additional constraint. Points may have parallel rays if they are measured on the overlapping area of the subset of images clipped from the same image. The rays of these points do not intersect and cannot give a unique 3D coordinate on the terrain surface. These points are listed on the Triangulation Report with a notation of "single ray, X, Y is relative to Z." The Z coordinate is locked to an internally computed value and is for reference only. It cannot be used to check against any ground points from other sources such as a GPS survey or LiDAR. The threshold for determining which points have parallel rays is set in the Angle threshold for detecting ray convergence in RPC refinement preference in the IMAGINE Photogrammetry category.

    Points excluded with gross errors:

    image

    pid

    image_x

    image_y

    residual_x

    residual_y

    1

    4

    869.5420

    2487.9960

    -1.8265

    0.2271

    1

    8

    2890.8800

    1258.8520

    1.1212

    0.4837

    1

    9

    1978.1380

    2919.0040

    2.1869

    0.1647

    THE OUTPUT OF BUNDLE BLOCK ADJUSTMENT

    the no. of iteration =1

    the standard error = 0.8856

    the maximal correction of the object points = 1.16032

    the no. of iteration =2

    the standard error = 0.8854

    the maximal correction of the object points = 0.06046

    Object Point Coordinates

    Point ID

    X

    Y

    Z

    Overlap

    1

    23.34109752

    49.91233930

    -50.0000

    2

    single ray, X, Y is relative to Z

    2

    23.41917906

    49.91598176

    -50.0000

    2

    single ray, X, Y is relative to Z

    3

    23.42044219

    49.91549649

    -50.0000

    2

    single ray, X, Y is relative to Z

  3. Iterative Triangulation Results

    The results for each iteration of processing are provided once the triangulation has been performed. A global indicator of quality is computed for each iteration of processing. This is referred to as the standard error (also known as the standard deviation of image unit weight). This value is computed based on the image coordinate residuals for that particular iteration of processing. This value is referred to as the Total RMSE within the triangulation report.

    The units for the standard error are defined within the General tab of the Triangulation dialog.

    After each iteration of processing, the software estimates the position and orientation parameters of the satellite sensor and X, Y, and Z tie point coordinates. The newly estimated parameters are used along with the GCP and tie point coordinates to compute new x and y image coordinate values. The newly computed image coordinate values are then subtracted from the original image coordinate values. The differences are referred to as the x and y image coordinate residuals.

    If the position and orientation information associated with a satellite is incorrect, then the newly computed image coordinate values are also incorrect. Incorrect estimates for the unknown parameters may be attributed to erroneous GCPs, data entry blunders, and/or mismeasured image positions of GCPs or tie points. Any error in the input observations is reflected in the image coordinate residuals.

    The computed standard error for each iteration accumulates the effect of each image coordinate residual to provide a global indicator of quality. The lower the standard error, the better the solution.

    IMAGINE Photogrammetry performs the first portion of the triangulation using the weighted iterative approach. In this scenario, the statistical weights associated with the GCPs are used. The solution continues until the corrections to the unknown parameters are less than the specified convergence value. The convergence value is specified within the General tab of the Triangulation dialog. The results of the normal weighted iterative solution appear as follows:

    In this scenario, the statistical weight assigned to the ground control points has been used in the triangulation. The statistical weights for the ground control points can be specified within the Point tab of the Triangulation dialog. The standard error associated with each iteration displays. The units of the standard error are the same as the image coordinate units. Additionally, the maximum image coordinate residual associated with a point displays for each iteration of processing. Information pertaining to the point ID and the image it is located on is also provided.

    We highly recommend that the number of Maximum Normal Iterations be set to 5. Fewer iterations may produce inadequate results.

    Triangulation using a free-weighted adjustment

    Once the weight-restrained iterative triangulation has been performed, the software also has the ability to process the triangulation using a free-weighted adjustment. In this case, each iteration of processing does not use the statistical weights associated with the GCPs. This is referred to as iterations with relaxation.

    If the Iterations with relaxation option is set to a value greater than zero, a free-weighted least squares adjustment is used. The iterative solution continues until the correction to the unknown parameters is less than the convergence value. For example, if correction to the sensor position and orientation is greater than the user-specified convergence value, the iteration continues. The unknown parameters include sensor position and orientation.

    This approach is advantageous in cases where GCPs are of poor quality and the triangulation solution cannot converge using the specified number of iterations. The results appear as follows:

    The results for the free-weighted triangulation display after the results of the normal weighted triangulation.

  4. Exterior Orientation Results

    Once the triangulation solution has converged, the resulting coefficients associated with the exterior orientation parameters display along with their corresponding precision estimates.

    The results display for each image as follows:

    In the above example, the polynomial order specified for X, Y, and Z was 2. Three coefficients were computed including a0, a1, and a2. Just below each coefficient value is the associated quality. For the same example, the polynomial order specified for omega, phi, and kappa is 1. Thus, two coefficients were computed including a0 and a1. The polynomial coefficients can be used to compute the exterior orientation parameters associated with each scan line.

  5. Ground Point Results

    Once the triangulation solution has converged, ground point values for GCPs, tie points, and check points are computed along with their corresponding accuracy. The results appear as follows:

    The precision of each ground point displays in brackets. The ground point information displays as Point ID, Image Number, X (accuracy), Y (accuracy), and Z (accuracy).

  6. Image Coordinate Information

    Once the triangulation is complete, image coordinate residuals are computed. The residuals are computed based on the estimated exterior orientation parameters, GCP, check point, and tie point coordinates, and their respective image measurements. During the iterative least square adjustment, the results from the previous iteration are compared to results of the most recent iteration. During this comparison, image coordinate residuals reflecting the extent of change between the original image coordinates and the new values are computed. The values of the new image coordinates are dependent on the estimated or adjusted parameters of the triangulation. Therefore, errors in the estimated parameters are reflected in the image coordinate residuals.

    The software computes the image coordinate residuals for each image measurement in the block. The results appear as follows.

    The information for each image coordinate displays as image number, point ID, image X coordinate, image y coordinate, x image residual, and y image residual. Relatively large residuals indicate erroneous points that can be attributed to image mismeasurement, data entry, and/or incorrect input data.

    Normal weighted iterative adjustment:

    No.

    Total_RMSE

    Max_Residual

    at image

    pid

    1

    0.973941

    4.1854

    2

    21

    2

    0.951367

    4.0954

    2

    21

    3

    0.929783

    4.0077

    2

    21

    4

    0.909040

    3.9213

    2

    21

    5

    0.889105

    3.8372

    2

    21

    Iterative adjustment with weight relaxation:

    No.

    Total_RMSE

    Max_Residual

    at image

    pid

    4

    0.200501

    0.7890

    1

    11

    Image parameter value and precision:

    image id 1:

    x:

    7.85788145e+005

    1.01160973e+000

    -9.41131523e-006

    1.23371172e+003

    5.48444936e-001

    2.94370828e-006

    y:

    3.69353383e+006

    2.21320881e+000

    5.47762346e-006

    1.13621691e+003

    5.85348349e-001

    3.22204830e-006

    z:

    8.18549949e+005

    -2.73735010e-001

    -6.57266270e-005

    7.94892316e+002

    3.56703998e-001

    9.53403929e-006

    omega:

    7.04151443e-002

    9.23492069e-006

    1.39654734e-003

    7.20947218e-007

    phi:

    2.94805754e-001

    -1.31834309e-006

    1.64134729e-003

    7.29915374e-007

    kappa:

    1.60676667e-001

    -1.39612089e-007

    8.63131836e-005

    3.89509598e-008

    Ground point value and precision in parenthesis:

    point id 1:

    566190.7372

    ( 0.2989)

    3773588.3997

    ( 0.2982)

    996.6927

    ( 0.3002)

    point id 2:

    555691.3534

    ( 0.2977)

    3728387.0138

    ( 0.2972)

    228.0382

    ( 0.3000)

    point id 3:

    501918.8209

    ( 0.3004)

    3732593.1751

    ( 0.3000)

    483.9667

    ( 0.3007)

    ...

    ..

    .

    Image points and their residuals:

    image

    pid

    image_x

    image_y

    residual_x

    residual_y

    1

    1

    5239.4680

    337.3840

    -0.2213

    0.0647

    1

    2

    5191.5900

    4969.5460

    -0.7131

    0.2838

    1

    3

    230.9250

    5378.8230

    0.0798

    0.2689

    2

    1

    2857.2700

    753.8520

    -0.0267

    0.0850

    2

    2

    3003.7820

    5387.8920

    0.4579

    0.3762

    2

    6

    2736.1250

    3070.2270

    0.2412

    0.7679