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expressed by (9c) and as corrected by x; and Yi by any of thesuggested interpolation prodedures, 
e. g. (9a), (11d). 
5.2. Indirect Method 
In the indirect (inverse) mode collinearity equations of the form (2a) and(2b) can be applied in 
the following manner: 
A rectangular output pixel center X: Yi is selected. This position will also easily related to 
a D.T.M. for which Z; coordinates are usually known as functions of X. Yı + If the D.T.M. mesh 
is coarser than the pixel size, the z, may be interpolated within that mesh in a simple manner. 
For images with a constant orientation for all image points, as in aerial photography, this is 
the simplest differential rectification procedure. For images with a time varying orientation 
(Scanners), the solution is iterative, since the image point can only be properly computed with 
orientation elements expressed as functions of the unknown tim or x: . Therefore an iteration 
must be started using (8d). This value may be corrected by corrected using the correction 4x; by 
any of the interpolation methods for which the coefficients have been calculated in a previous 
calculation according to any of the methods (8a) and (8b) or (10a) to (10d) . This already con- 
stitutes the simplified approach for the indirect method. 
If collinearity equations ar subsequently applied the corrected xi +dx; is used to determine 
ts and the appropriate orientation parameters for the equation. 
A formula for a direct iteration procedure is stated by Schuhr /78/. 
5.3. Grey Level Assignment 
The direct method yields an output image with irregularly spaced grey shades. For output in a 
rectangular fashion each rectangular output pixel must be assigned a grey value by aid of the 
irregularly spaced grey shades. The most common form of assignment is by nearest neighbourhood; 
for each output pixel a search must be made for the nearest irreglarly spaced grey shade value. 
Then its grey schade must be adopted. 
The indirect method does not require a search prodedure. A computed image coordinat belongs to 
integer values of image pixels. The identification of the input pixel and the transfer of its grey 
shade to the output pixel constitutes the nearest neighbourhood assignment. 
The computational problems of both methods are addressed in /50/ : the direct method requires a 
large output storage area in core. The indirect method requires a large input storage area. For 
the digital rectification of an entire image the stored output or input areas only permit the 
assignment of grey shades within an area of 1/3 x 1/3 the core storage allocation. For the grey 
level assignment within the next area the core storage content must be shifted by 1/3 of the 
linear storage dimension. The transfer of the input or output information into core and back is 
best done from or onto disk or tape. 
In order to make the rectification procedure computationally more efficient IBM has devised a 
shortcut to the solution of the geometric transformation equation (only the simplified relations 
are used, not the colliearity equations). A wide interpolation grid is established (for the in- 
direct method) on the output image. For this grid correctionsd x! are computed for the input. All 
pixels Xi Signated between the interpolation grid intersection receive interpolated corrections 
according to 
' = 
4 x: at a,x + ay + a xy 
4 Y; b * b,x + boy + b,xy