Ecole Nationale Supérieure Agronomique de Montpellier ... - CIAM

ule which transforms the tangent points of the boolean model to
an homogeneous Poisson process with the same intensity as the underlying
boolean model (Stoyan et al. 1995, Molchanov 1997). The
test is then based on a Poisson test of the transformed process. But
Laslett’s transformation explicitely uses the positions of the extrema
of the grains and in practice these are often unknown. Moreover, when
these positions are available it is only with an error. How such errors
will affect the transformation is unknown but clearly errors add up and
may seriously affect the conclusion of the test. Indeed, in practice, the
convexity assumption of the grain appears as a serious limitation. To
get around this problem, one idea is to dilate the grain, but dilation
allows to tend towards the convexity without any garantee to reach
it. As an example, a regular star will need a large dilation before nonconvexity
becomes negligeable. As shown before, our test approach
only needs a bounded grain assumption and as a consequence allows
to work in more general contexts.
3 simulation examples
Two simulated examples were performed in order to check the procedure.
In the first one, we used a boolean model with non-convex
grain to see if the boolean assumption was well detected when taking
into account the non-convexity as proposed above, and rejected if not
taken into account. In the second one, we considered a Strauss point
process, on which we attached the same non-convex grains, in order to
test if the boolean assumption was rejected when taking into account
the non-convexity.
3.1 boolean model
Figure 1(a) presents the boolean model realization. The Poisson process
intensity is λ = 24.3 whereas each grain is formed of 3 discs of
radius r = 0.044 disposed on an horizontal line and separated by a
distance d = 0.066. Observation is made on a 5×5 window.
Bound was chosen equal to b = 0.244, distance between consecutive
horizontal transects was equal to b. Figure 1(b) illustrates the result
of the dilation on the original process observed on the intersection
with the transects and Wr.
Figure 1(c) presents the result of the test when no dilation of the