s/(.)(\.+)/$1 x (1+length $2)/seg;
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For example,

$ perl -wle'
    $_ = "III...MMMMMOOOO....MMMIIIII";
    print;
    s/(.)(\.+)/$1 x (1+length $2)/seg;
    print;
'
III...MMMMMOOOO....MMMIIIII
IIIIIIMMMMMOOOOOOOOMMMIIIII
[download]

Update: Oops, this window must have been open a while. keszler already gave basically the same answer a while ago. As punishment, I'll give an alternative that might be faster for short strings:

$ perl -wle'
    $_ = "III...MMMMMOOOO....MMMIIIII";
    print;
    1 while s/(.)(\.)/$1$1/sg;
    print;
'
III...MMMMMOOOO....MMMIIIII
IIIIIIMMMMMOOOOOOOOMMMIIIII
[download]