Abstract: Sieves decompose 1D bounded functions, e.g., f to a set of increasing scale granule functions $LFBC@d$-m$/, m $EQ 1 ...R$RTBC@, that represent the information in a manner that is analogous to the pyramid of wavelets obtained by linear decomposition. Sieves based on sequences of increasing scale open-closings with flat structuring elements (M and N filters) map f to $LFBC@d$RTBC and the inverse process maps $LFBC@d$RTBC to f. Experiments show that a more general inverse exists such that $LFBC@d$RTBC maps to f and back to $LFBC@d$RTBC@, where the granule functions $LFBC@d$RTBC@, are a subset of $LFBC@d$RTBC in which granules may have changed amplitudes, that may include zero but not a change of sign. An analytical proof of this inverse is presented. This key property could prove important for feature recognition and opens the way for an analysis of the noise resistance of these sieves. The resulting theorems neither apply to parallel open-closing filters nor to median based sieves, although root median sieves do `nearly' invert and offer better statistical properties. !12
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机译：摘要：筛将1D有界函数（例如f）分解为一组递增尺度的粒度函数$ LFBC @ d $ -m $ /，m $ EQ 1 ... R $ RTBC @，它们以以下方式表示信息：类似于通过线性分解获得的小波金字塔。基于具有平坦结构元素（M和N过滤器）的递增比例开闭序列的筛网将f映射到$ LFBC @ d $ RTBC，逆过程将$ LFBC @ d $ RTBC映射到f。实验表明，存在更一般的逆，例如$ LFBC @ d $ RTBC映射到f并返回到$ LFBC @ d $ RTBC @，其中颗粒函数$ LFBC @ d $ RTBC @是$ LFBC @ d的子集$ RTBC，其中颗粒的振幅可能已更改，其中可能包括零，但符号没有变化。给出了该逆的解析证明。该关键特性可能对特征识别非常重要，并为分析这些筛子的抗噪性开辟了道路。所得定理既不适用于并行开闭滤波器，也不适用于基于中值的筛子，尽管根中值筛子确实“几乎”反转并且提供了更好的统计特性。 ！12
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