@Article{BarreraTeraHiraHira:2000:AuPrMo,
               author = "Barrera, Junior and Terada, Routo and Hirata Junior, Roberto and 
                         Hirata, Nina Sumiko Tomita",
                title = "Automatic programming of morphological machines by PAC learning",
              journal = "Fundamenta Informaticae",
                 year = "2000",
               volume = "41",
               number = "1",
                pages = "229--258",
                month = "January",
                 note = "{}",
             keywords = "mathematical morphology, operator decomposition, PAC learning.",
             abstract = "An important aspect of mathematical morphology is the description 
                         of complete lattice operators by a formal language, the 
                         Mophological Languange (ML), whose vocabulary is composed of 
                         infimum, supremum, dilations, erosions, anti-dilations and 
                         anti-erosions. This language is complete (i.e., it can represent 
                         any complete lattice operator) and expressive (i.e., many useful 
                         operators can be represented as phrases with relatively few 
                         words). Since the sixties special machines, the Morphological 
                         Machines (MMachs), have been built to implement the ML restricted 
                         to the lattices of binary and gray-scale images. However, 
                         designing useful MMach programs is not an elementary task. 
                         Recently, much research effort has been addressed to automate the 
                         programming of MMachs. The goal of the different approaches for 
                         this problem is to find suitable knowledge representation 
                         formalisms to describe transformations over geometric structures 
                         and to translate them automatically into MMach programs by 
                         computational systems. We present here the central ideas of an 
                         approach based on the representation of transformations by 
                         collections of observed-ideal pairs of images and the estimation 
                         of suitable operators from these data. In this approach, the 
                         estimation of operators is based on statistical optimization or, 
                         equivalently, on a branch of Machine Learning Theory known as PAC 
                         Learning. These operators are generated as standard form 
                         morphological operators that may simplified (i.e., transformed 
                         into equivalent morphological operators that use fewer vocabulary 
                         words) by syntatical transformations.",
           copyholder = "faria - tese",
        urlaccessdate = "03 maio 2024"
}