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2. THEORETICAL BACKGROUND
In practice a few assumption have to be made in order to be
able to make a correction model feasible. The most
important prerequisite is defining the object as an
approximate Lambertian surface. This comes close to the
actual reflection behaviour in many cases, in particular for
imagery taken with sensors whose “poor” resolving power is
not able to distinguish between small object details.
Minnaert [1941] developed a simple BRDF (1) for correcting
the shading and illumination effects of celestial bodies by
slightly modifying the rules that would apply for ideal
Lambertian surfaces.
g orig
g =
7" . cos*()-cos*(e)
(1)
where g are the greyvalues of the original image and the
corrected image, respectively. k is the so-called Minnaert
constant, the unknown that has to be determined. i is the
incidence angle of the illuminating light beam and e is the
exitance angle, which is approximately the terrain slope
assuming a very narrow field of view, so that the observing
direction is more or less constant over the entire sensed
area and perpendicular to a horizontal plane. In the case of
the common Earth observation satellites, such as Landsat
TM, SPOT or IRS this assumption is justified. If the sun
position is known for the given acquisition time and date, /
as well as e can be calculated with the help of a terrain
model. One can see that for k=1 Minnaert's formula
corresponds to the Lambertian reflection behaviour.
Though this simple approach cannot fulfil the reality, it is
commonly used and/or slightly modified on the one hand,
on the other hand it is rejected by those researches who
clearly see the limitations and who aim at a more universal
and physically based solution. Our oppinion is that a
satisfying and thorough model is still far from realisation.
Too many influences with hardly determinable parameters,
such as mutual illumination of objects, atmospheric effects
like skylight or airlight, deteriorate the radiance of the object
surface leading to the impossibility of obtaining the
characteristic spectral properties from the acquired grey-
values with feasible effort.
3. THE EXTENDED MINNAERT MODEL
In order to find a rather primitive and generally applicable,
though still approximate, algorithm we propose an extended
Minnaert approach, whose parameters can be determined
from the original greyvalues of the image, the digital terrain
model, and a few assumptions that might look obscure at
the first sight although, as the experience shows, are
fulfilled in many cases.
Firstly, a few considerations should give some hints how the
Minnaert model could be extended:
a. Satellite images show low contrast in the short wave
channels due to skylight. Even not directly illuminated
regions (i.e. areas with sun incidence angles greater
than 90°) appear to be illuminated. This illumunination is
not that apparent in infrared, for instance.
b. The above Minnaert model applies a multiplication factor
(in other words a contrast enhancement factor) to the
International Archives of Photogrammetry and Remote Sensing. Vol. XXXII, Part 7, Budapest, 1998
original greyvalues that is inversely related to the cosine
of the incidence angle. In any case this factor for
incidence angles towards 90? becomes infinite very
rapidly independent of the actual contrast between
directly illuminated parts and unilluminated regions.
c. In (1) the correcting influence of the exitance or ob-
.servation angle is strictly connected to the incidence
angle and is solely driven by the magnitude of the
Minnaert constant k. Only in case k=1 an influence of
the observation direction does not exist.
ad a: The skylight influence must not be neglected even by
a simple model, otherwise the result will never be
satisfactory in particular in the short wave ranges. (We
shall see later that the entire visible spectrum is more or
less heavily subjected to skylight). Therefore, the
improved model must contain a parameter that is able to
adjust to the actual influence of skylight or ambient light.
ad b: The k-parameter controls the steepness of the cosine
function. The less k the wider the range of incidence
angles with low contribution to the radiometric
correction. In other words, for k-values near 0 there is a
marginal contrast enhancement from i = 0° up to rather
great indicent angles. Then, close to i = 90° the
steepness increases rapidly and the contrast
enhancement during correction will be overrated
significantly. For that reason many known approaches
exclude areas with / near 90° from correction.
ad c: Although the observation angle of the object has
certainly some influence to the detected radiance, we
are convinced that the sole modelling by the parameter
k is not - even not approximately - satisfactory for the
great majority of object classes. (We should always
bear in mind, that Minnaert developed his formula for a
very different purpose). If taken into consideration, the
effect of the observing direction must be modelled inde-
pendently. Our suggestion is therefore to exclude e (or
possibly another more appropriate derivative) from the
global correction we are aiming to, at least in the first
step of our development.
The new augmented approach has still the same basic form
of Minneart's formula, ie:
9cor = orig‘ Ki,8,0,5)
Wil Kom ln ne (2)
f(cosi): f(cose,s)
f(cosit) - t«(1-0):cos*i
f(cose,s) = $«(1-S):cos*e - (1-s)+s-cos*e 3)
Let us call ¢ skylight term and s slope term. Both are within
the interval [0,...,1]. They describe the percentage of the
influence of the respective error source for incidence angles
around 90°. Figure 1 shows the graph of the above
functions. The idea behind is, that the contribution of
ambient light to directly lit areas remains negligibly small,
while it increases with growing / but never so much that the "
function exceeds its maximum of 1.The constant extension
to angles greater than 90° is justified because there is
primarily the influence of the diffuse skylight illumination and
therefore no deciding necessity to take into account any
incidence angle.
