.NTIyMQ.MjY1NTIw

16 i Fig. 1-4 (à and b) - Example of discontinuous transformation Conversely, one can wonder if these basic conditions are characteristic • •• • • •• - • • * • * of the i.e., if only this transformation satisfies them. The answer • • m m • is no. However, the combinations of erosions cover a very broad scope of . a • . possibilities, which show, in particular, the strong following theorem : "Any increasing set transformation invariant under translations is neces¬ sarily a union of erosions, and also an intersection of dilations" (G. MATHERON 29 ). % * According to this theorem, the erosion (or the dilation) appears as the • • simplest increasing set transformation translation invariant, the other map¬ pings, more sophisticated, being logical combinations of erosions or dila¬ tions. (One will notice that the general H.M.T. is not increasing : for 9 example in the transformation X -*• boundary of X (formula (1-10), X CY does not imply (boundary of x)d (boundary of Y).