3.6 Statistical Tools 
The weighted, leastsquares bundle adjustment performed by GIANT enforces the collinearity condition to mathematically relate the image space with the object space. Because of errors intrinsic in the measurements of image and object space coordinates, there are invariably departures from strict collinearity. The leastsquares principle supplies a theory to reconcile these disparities. To guide the leastsquares solution, weights are assigned to the measurements to indicate their relative uncertainties: less weight is assigned to measurements that are believed to be more uncertain. For a meaningful triangulation solution, it is very important that weights are assigned equitably. GIANT provides several tools to help in the statistical analysis of the triangulation solution. Generally speaking, photogrammetric triangulation enables the calculation of coordinates of points in the ground or object space from measurements of coordinates of the point in the image space. Since, as with any measurement, there is an inevitable uncertainty in the measurement of the image space coordinates, common sense suggests that any quantity computed from the uncertain coordinates must itself have some uncertainty. This is the principle of error propagation, as applied to photogrammetric triangulation. GIANT contains provisions for determining the uncertainty of any computed quantity, given the uncertainty of image coordinate measurement. To activate this facility, visit the Process Options screen and set the Error propagation Yes/No switch to Yes. When active, GIANT will determine, and output a variancecovariance matrix for all of the computed quantities. The variances lie along the diagonals of these matrices. The standard deviation, computed as the square root of the variance, has the same units as the computed quantity itself, and suggests the degree of uncertainty of the quantity. It is often more meaningful to assess the standard deviation with respect to the magnitude of the quantity, than in an absolute sense: a standard deviation of 10.0 would be large for a quantity whose expected value is 1.0, small for a quantity whose expected value is 100.0. Inspection of standard deviations for computed quantities can often indicate geometric weakness in control or block configurations. Also on the Process Options screen is a switch panel labeled Unit Variance with two options, Computed and Set to unity. After the leastsquares bundle adjustment converges to a solution, a single number, called the unit variance, is often computed from the weighted sum of squared measurement residuals. Using statistical arguments, it can be shown that this unit variance is expected to have a value of approximately 1.0. A unit variance significantly larger than 1.0 can result from large residuals and may suggest a problem with the bundle adjustment. Formal adjustment theory stipulates that the variancecovariance matrices of computed quantities should be scaled by the unit variance. Accordingly, a unit variance larger than 1.0 would lead to an increase in the sizes of standard deviations of measured quantities. However, since all variancecovariance matrices are scaled by the unit variance, the relative sizes of the standard deviations remain unchanged. On the basis of this discussion, the user can set the Unit Variance switch to Computed or Set to Unity. When the Computed setting is used, the standard deviations may be judged to be more accurate in their representation of the uncertainty in the computed quantities. This setting should be used for operational work. If the switch is set to Set to Unity, the standard deviations should only be assessed on their relative sizes. When studies are conducted with simulated data that contain no errors, the Set to Unity setting should be used. In this case, the computed variance, computed from perfect data, will be zero, as well as all propagated variances. Another useful statistical tool to help assess the quality of the bundle adjustment is the listing of image residuals. Image residuals represent, generally, the differences between the measured positions of images and their positions which best satisfied the collinearity conditions as determined by the leastsquares solution. Errorless image measurements would theoretically have residuals of zero. Large residuals typically indicate a problem with the adjustment, and often the presence of poor quality measurements or blunders. GIANT contains a flexible and useful feature for image residual analysis. Access the feature by using the mouse to click on the General Options tab to bring the General Options screen to the front of the display. A switch panel on the General Options display labeled Image Residual Listings offers three choices: Greater than ?, All, None. Use the mouse to set the switch to the desired position. Clearly, None means that no image residuals will be included in the GIANT output listing. In contrast, setting the switch to the ALL position will direct all image residuals to the output listing. For small projects, the ALL position may be suitable; for large projects the amount of data generated would prohibit convenient analysis. Most likely, the most useful position for analysis will be Greater than ? When this setting is selected, GIANT, will pop up a window labeled Microns. In this window the user can key in a threshold value. GIANT compares all image residuals against the specified threshold value and includes those images that exceed the threshold, and the frames on which the images were measured, in the output listing. GIANT has several provisions for the specification of standard deviations for each of the three classes of input information: image coordinates, frame position and attitude information and ground control points. For each class, the following table indicates the sequence GIANT will follow when assigning a standard deviation to an input quantity. 
Priority 
Image Coordinates 
Frame State Vector 
Ground Control 
Highest 
.IMG File Record 
.FRM File Record 
.CNT File Record 

.IMG Header Record 



Group Default 
Group Default 


Camera Default 
Camera Default 
General Option 
Lowest 
System Default 
System Default 
System Default 
For instance, when GIANT is to assign a standard deviation to an image coordinate measurement, it first consults the record of the .IMG file which contains the image coordinate. If a standard deviation is not specified, information from the header record of the .IMG is checked. If a standard deviation is not specified there, the default standard deviations for the group of which the frame is a member is checked next, followed by the default standard deviations for the appropriate camera system. As a last resort, GIANT has system default standard deviations, tabulated below. The same hierarchical approach to determining standard deviations applies to the frame position and attitude information and the ground control information except, as shown in the table, there are only four and three alternatives, respectively,  
GIANT System Default Standard Deviations 

Camera Station Position and Ground Control Coordinates 



Camera Station Orientation


Image Coordinates


Interior Orientation


Radial and Asymmetric Distortions


GPS Antenna Offsets


In the hierarchical chart notice that the middle option for ground control indicates General Options. Access this option by using the mouse to select the General Options tab to bring the General Options screen to the front of the display. The field to the lower left is a Yes/No switch labeled Control Default Std Dev. Set the switch to Yes to specify default standard deviations for control points. When set to Yes, meaning that default standard deviations for control points are to be set, three additional fields will appear, indicating units consistent with the chosen coordinate system (DMS for geographic, meters for rectangular). Key in default standard deviation values the control points= coordinates that are suitable for the project. Note from the table, however, that any standard deviation specified as part of a ground control point record in the .CNT file will take precedence over the control default standard deviation discussed here. The hierarchical assignment of standard deviations occurs on a itembyitem basis: if some data records contain standard deviations and others do not, GIANT will consult the hierarchical table to obtain default values for the unspecified standard deviations.  