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多尺度分析是信号处理中一种方兴未艾的方法，它要求信号在不同尺度的算子作用下具有某种单调特性．针对只含有限个不可微点的连续曲线，本文证明了膨胀（腐蚀）算子具有一种单调性（极值点数目随尺度增加而单调减少），给出了极值点产生的条件，并且指出随尺度增加，极值点数变化率有变慢的趋势，从而为确定形态滤波器最大尺度提供了理论依据．
Multiscale analysis is an emerging approach to signal processing that requires signals to have some monotonic behavior under the influence of operators at different scales. For a continuous curve containing only a limited number of non-differentiable points, this paper proves that the expansion (erosion) operator has a monotonicity (the number of extreme points decreases monotonically with increasing scale) and gives the conditions for the occurrence of extreme points, and It is pointed out that with the increase of scale, the rate of change of extremum points tends to slow down, which provides a theoretical basis for determining the maximum scale of morphological filter.