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| father-pixel.
4 son-pixels /
‘Figure 2: REDUCE and EXPAND operations. The lowest gray level son-pixel used to build father-pixel is represented
by shaded area.
1. ¥(X) C X, anti-extensivity,
n3
.XcCcY 2 y(X)c y(Y) increasing,
9»
. V o y» — vy, idempotence
The last two rules mean, respectively, that 3» maintains inclusion relationships on the output image, and the result of the
filter operation is unchanged when % is reapplied.
As one can imagine, the alteration of the input image by such a transformation is highly dependent upon the shape and the
size of the structuring element. In practice, openings are used to obtain a structural decomposition of an image by using
structuring elements of same shape but with increasing size. By this way, larger image details are filtered out. It gives a
multidimensionnal analysis of the input texture usually called opening distribution.
3 MORPHOLOGICAL PYRAMID
The main drawback of morphological filters is the difficult choice of the appropriate structuring element needed to probe
texture objects. To avoid this problem, we define an original morphological pyramid, whose conctruction is based on
usual quadtree decomposition. The REDUCE process, which generates each image I; from its predecessor I;_1, is based
on the set of non overlapping 2 x 2 pixels blocks extracted from I;.
An important aspect for road surface characterisation is directly linked to low gray tone regions located in the wearing
course. Our aim is to detect and identify background regions by rising up low gray levels through a morphological
pyramid. The REDUCE operation is then definied as :
GL(I)(1,j) = min{GL(I;—1)(u.v)}, (4)
where 2; € « € 2i -- 1 and 2j € v € 2j 4- 1.
Equation (4) means that for each non overlapping pixel blocks, only pixels having the lowest gray level value is taken into
account. The min operator make REDUCE operation similar to an erosion step. Moreover, we can note that a pixel in I;
is linked to 4 son-pixels in I;_; and to one and only one father-pixel in 1,41. By this way, a hierarchical structure is then
developped.
À reverse operation can be defined. EXPAND operation consists in expanding an image of size NxN to an image of size
2Nx2N. This step is realized by affecting to each son-pixel its father-pixel gray level. A new image is then constructed :
NI; — EXPAND(I;+1) (5)
Operation 7 (see figure (2)), which consists in transforming image 7; into image N 7; is clearly anti-extensive, increasing
and idempotent. So 1 is an opening and removes all small objects of high intensity and then rises up low gray levels
related to background pixels.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 687
