As we have seen, differential operations on sampled images require the image to be first smoothed by filtering. The filtering operation introduces an arbitrary parameter- the scale of the filter, e.g., the standard deviation for the Gau&sian filter.
正如我们所看到的,对取样图像进行的微分运算首先需要用滤波将它变平滑。 滤波操作引入了一个随机参数-滤波的范围，比如高斯过滤器的标准偏差。
In computer vision, the advantages of using several scales of filtering was? realized quite early on, and this was supported by evidence suggesting the presence of filters of several sizes in the human visual system [33], [34], [43]. More recently, Witkin [55] introduced a scale-space description of zero crossings which gives the position of the zero crossing across a continuum of scales, i. e., sizes of the Gaussian filter (parametrized by the a of the Gaussian).
在计算机视觉领域，使用一系列尺度滤波的优势很早就被发现了,并且这是有证可依的：在人类视觉系统中同时有几个尺寸的过滤器的存在[33]、[34]、[43]。最近,Witkin[55]提出了零交点的位置在一个连续刻度一个刻度间距为零的想法,比如高斯滤波器尺寸(高斯参数化)。
The signal-or the result of applying to the signal a linear (differential) operator-is convolved with a Gaussian filter over a continuum of sizes of the filter. Zero or level crossings of the filtered signal are contours on the x-a plane and surfaces in the x, y, a space. Witkin proposed that this concise map can be effectively used to obtain a rich and qualitative description of the signal.
信号,或者用来作为线性(微分)的操作数的信号的结果是：这个信号会和高斯滤波对一个连续的过滤器的尺寸混在一起。零或水平的过滤信号的相交处在x,y,a 空间会 x-a轮廓面和表面会形成一条轮廓线条。Witkin提出,这种简洁的地图可以有效的获得丰富和量化的信号描述。
Yuille and Poggio [56], [57]-who called the maps of zero-crossings across scales fingerprints-have established interesting relationships between multiresolution analysis, the Gaussian filter and zero-crossings of filtered signals. Their main results are two theorems: 1) Zero and level crossings of an image filtered through a Gaussian filter have nice scaling properties, i.e., a simple behavior of zero crossings across scales. Zero crossings are not created as the scale increases. The Gaussian filter is the only filter that has this nice scaling behavior (see also [6]).
宣称穿过地图零交点刻度为指纹的Yuille和Poggio[56]、[57]，已经在多分辨率分析，高斯滤波器和零交点刻度信号之间建立起了有趣的联系。他们的主要成果是两个定理:1)在穿过高斯滤波器得到的零级相交图像具有较好的缩放性能,即跨越零刻度的简单的动作。零交点没有随着刻度的增大而生成。高斯滤波器是唯一有如此之好的滤波尺度功能的滤波器(请参见[6])。
2) The map of the zero crossings across scales determines the filtered signal uniquely for almost all signals in the absence of noise. The scale map obtained by Gaussian filters is thus a complete representation of the image. This result applies. to level crossings of any arbitrary linear differential operator of the Gaussian (modulus the null space of the differential operator and provided there are at least two zero-crossing contours), since it applies to functions that obey the diffusion equation.
2)对于几乎所有不受噪声影响的信号，零交点的地图决定了滤波信号的唯一性。高斯滤波器所获得的比例地图因而是一个完整的图象。这个结果应用到高斯的任意线形微分算子(减去无效的微分算子空间并且假设至少两个零焦点回路),因为它的范围也适用于服从扩散等式的函数。