brightness value will again be affected. 
Three methods of resampling are commonly used. The 
first, the nearest neighbour method, takes the 
value of the pixel in the input image that is 
closest to the newly-computed centres. The second, 
bilinear interpolation, takes the average of the 
four pixels nearest to the new coordinates. The 
third method, cubic convolution, computes the new 
pixel value taking into account the 16 nearest 
pixels in the input image. 
A more complete description of the traditional 
techniques for geometric rectification as well for 
resampling can be found in Jensen (1986) and Mather 
(1987). 
2.3 Application of Traditional Techniques 
A geometric rectification was carried out on the 
ATM image of Antequera using the least squares 
polynomial fit method described in section 2.2. It 
was based on the selection of 71 ground control 
points reasonably well distributed in the image. 
The UTM geographic coordinates of the points were 
taken from the 1:10 000 topographic maps and the 
image coordinates from a display of the image on 
the Nottingham Image Processing System, NIPS. 
The results from the first order polynomial 
transformation showed some large residuals, of the 
order of 13 pixels, for both column and row. The 
Root Mean Square Error (RMSE) for columns was 5.8 
and for rows was 5.7 pixels. The results from the 
second order polynomial showed only a small 
improvement with RMSE values of 5.6 and 4.5 pixels 
for column and row respectively. A substantial 
improvement was achieved with the third order 
polynomial with the majority of residuals being 
within £4 pixels with RMSE values of 2.5 and 1.8 
for column and row respectively. 
A visual assessment of the effects of the 
transformations on the image was obtained by 
producing a corrected image for each order of 
polynomial and then overlaying the digitized map. 
The overlay was accomplished by clipping each 
corrected image by a number of rows and columns in 
order to get the best match with the map centre. 
As might be expected, the visual inspection showed 
increasing displacements of the corresponding 
features in the two images from the centre to the 
edges. The displacements were, as expected, bigger 
in the first and second order corrected images and 
smaller for the third order corrected image. The 
third order corrected image showed larger residuals 
than were acceptable. 
2.4 Geometric Characteristics of the Image 
The basic geometric operation of the ATM is given 
in section 3.2. 
In addition to the instrumental characteristics 
that affect the geometry of the image there are a 
number of external influences. These fall into two 
groups: those that are inherent in the geometry of 
the system and those that are due to operational 
difficulties and problems. 
The velocity to height ratio (V/H), the scan rate 
of the scanning mirror and the ground resolution, 
as well as the swath width are defined by the 
flying height above the ground level. Incorrect 
adjustment of the scanning mirror speed will cause 
under or over scanning and thus influence the 
geometry. There will also be a change in the 
ground resolution element size with different scan 
angles (see section 3.2). A variation in the 
ground relief will result in image displacement, 
which is a similar problem to that experienced in 
conventional aerial photography. 
has 
The sensor platform, being aircraft-based, 
16 
significant inherent instabilities, due to the 
characteristics of the aircraft motion and the 
atmospheric conditions in which it operates. To 
assist with minimising the effects of aircraft 
roll, up to a maximum of #15 degrees, a gyro 
assembly monitors the aircraft and only allows the 
image to be collected within an angle of 
approximately 37 degrees either side of the 
vertical. 
As mentioned above, for the polynomial technique to 
be fully effective dt not only "performs à 
transformation but corrects for the systematic 
errors present. It is extremely difficult to model 
the effect on the image geometry of all the above 
mentioned influences. Therefore, a more direct 
modelling technique could be employed. 
3. TIME-DEPENDENT GEOMETRY METHOD FOR 
RECTIFICATION OF IMAGES 
3.1 Introduction 
The basic principle of the technique discussed so 
far is that of modelling systematic errors in the 
image. The model is defined by a polynomial which 
should describe the systematic errors present. The 
polynomial should be based on an analysis of the 
influences affecting the geometry of the imaging 
system. Therefore the choice of polynomial is 
important to obtain the optimum results. An 
alternative approach to solving the problem is to 
attempt to model the cause rather than, as in the 
polynomial case, the effects on the image. The use 
of the collinearity equations in digital image 
processing is not new (Konecny, 1979), also the use 
of time-dependent geometry is not a new concept (El 
Hassin, 1981; Smith, 1989) although it is becoming 
more popular with developments in analytical and 
digital techniques. For this reason the solution 
presented here attempts to remain as general, for 
any similar sensor, as possible. The geometry of 
the imaging system must be analysed and modelled, 
which, in the case being considered here, involves 
the sensor geometry and the aircraft motion during 
the period of imaging. The principles of the 
sensor geometry are reasonably well documented 
where as the aircraft motion is not accurately 
known and requires certain assumptions to be made. 
3.2 Analysis of the Imaging System Geometry 
The sensing system consists of a rotating mirror 
that scans a swath of ground to either side of 
nadir. A scan line of data is collected as à 
series of 2.5 mrad instantaneous fields of view 
(IF0V), which therefore defines the resolution. 
Since the image data is captured in relation to a 
constant angular measure and constant pixel unit 
size, the ground area covered by the IFOV will vary 
depending on the angle from nadir, resulting in 
compression of the image towards the edges. This 
is removed from the image by the S-bend correction 
in the digitization process. So the fundamental 
Scanning geometry being described is therefore 
perspective geometry if the sensor was stationary. 
Successive scan lines of adjoining ground swath are 
produced by the forward motion of the aircraft. 
The butting together of the scan lines is dependent 
largely on the velocity-to-height ratio of the 
aircraft, although this is controlled by the 
allowable scan rates and the required resolution 
(as mentioned above). Normally these are selected 
to ensure some overlap, typically 10%. So the 
aircraft motion has an effect on the scan geometry 
(perspective geometry) and is used to create the 
series of scan lines to produce the image. It is 
therefore necessary to attempt to describe the 
aircraft motion during the image capture. 
Considering the rate at which the image is 
captured, scan rates available are 12.5, 25 or 50 
scans/sec, for an image of 1000 scan lines (the 
approximate image size being considered), the time 
to capture the image would be 80, 40 or 20 seconds