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style='font-family:"Times New Roman"'>3.1 </span><span lang=ZH-CN
style='mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>平滑</span><span
style='font-family:"Times New Roman"'><o:p></o:p></span></h2>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>先举个例子说明一下什么是平滑</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>(smoothing)</span><span
lang=ZH-CN style='font-size:10.5pt'>，如下面两幅图所示：可以看到，图</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>3.2</span><span
lang=ZH-CN style='font-size:10.5pt'>比图</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>3.1</span><span lang=ZH-CN style='font-size:
10.5pt'>柔和一些</span><span style='font-size:10.5pt;font-family:"Times New Roman"'>(</span><span
lang=ZH-CN style='font-size:10.5pt'>也模糊一些</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>)</span><span lang=ZH-CN style='font-size:10.5pt'>。是不是觉得很神奇？其实实现起来很简单。我们将原图中的每一点的灰度和它周围八个点的灰度相加，然后除以</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>9</span><span
lang=ZH-CN style='font-size:10.5pt'>，作为新图中对应点的灰度，就能实现上面的效果。</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
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<p class=a style='line-height:18.0pt'><b><span lang=ZH-CN style='font-family:
宋体'>图</span>3.1 </b><b><span lang=ZH-CN style='font-family:
宋体'>原图</span></b></p>
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<p align=center style='text-align:center;line-height:18.0pt'><b><span
lang=ZH-CN style='font-size:10.5pt'>图</span></b><b><span style='font-size:
10.5pt;font-family:"Times New Roman"'>3.2 </span></b><b><span
lang=ZH-CN style='font-size:10.5pt'>经过平滑处理后的图</span></b><span
style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
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<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>这么做并非瞎蒙，而是有其道理的。大概想一想，也很容易明白。举个例子，就象和面一样，先在中间加点水，然后不断把周围的面和进来，搅拌几次，面就均匀了。</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>用信号处理的理论来解释，这种做法实现的是一种简单的低通滤波器</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>(low pass filter)</span><span
lang=ZH-CN style='font-size:10.5pt'>。哇，好深奥呀！不要紧，这些理论的内容并不多，而且知道一些理论也是很有好处的。在灰度连续变化的图象中，如果出现了与相邻象素的灰度相差很大的点，比如说一片暗区中突然出现了一个亮点，人眼能很容易觉察到。就象看老电影时，由于胶片太旧，屏幕上经常会出现一些亮斑。这种情况被认为是一种噪声。灰度突变在频域中代表了一种高频分量，低通滤波器的作用就是滤掉高频分量，从而达到减少图象噪声的目的。</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>为了方便地叙述上面所说的“将原图中的每一点的灰度和它周围八个点的灰度相加，然后除以</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>9</span><span
lang=ZH-CN style='font-size:10.5pt'>，作为新图中对应点的灰度”这一操作，我们采用如下的表示方法：</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
<p align=center style='margin:0cm;margin-bottom:.0001pt;text-align:center;
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style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
<p align=right style='margin:0cm;margin-bottom:.0001pt;text-align:right;
line-height:18.0pt'><span style='font-size:10.5pt;font-family:"Times New Roman"'>(3.1)<o:p></o:p></span></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>这种表示方法有点象矩阵，我们称其为模板</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>(template)</span><span
lang=ZH-CN style='font-size:10.5pt'>。中间的黑点表示中心元素，即，用哪个元素做为处理后的元素。例如</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>[2. 1]</span><span
lang=ZH-CN style='font-size:10.5pt'>表示将自身的</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>2</span><span lang=ZH-CN style='font-size:10.5pt'>倍加上右边的元素作为新值，而</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>[2 1.]</span><span
lang=ZH-CN style='font-size:10.5pt'>表示将自身加上左边元素的</span><span style='font-size:
10.5pt;font-family:"Times New Roman"'>2</span><span lang=ZH-CN
style='font-size:10.5pt'>倍作为新值。</span><span style='font-size:10.5pt;font-family:
"Times New Roman"'><o:p></o:p></span></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>通常，模板不允许移出边界，所以结果图象会比原图小，例如模板是</span><sub><span
lang=ZH-CN style='font-size:10.5pt;font-family:"Times New Roman"'> </span></sub><sub><span
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src="./chp3.files/image004.gif" v:shapes="_x0000_i1028"><![endif]></span></sub><span
lang=ZH-CN style='font-size:10.5pt'>，原图是</span><sub><span lang=ZH-CN
style='font-size:10.5pt;font-family:"Times New Roman"'> </span></sub><sub><span
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lang=ZH-CN style='font-size:10.5pt'>，经过模板操作后的图象为</span><sub><span lang=ZH-CN
style='font-size:10.5pt;font-family:"Times New Roman"'> </span></sub><sub><span
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lang=ZH-CN style='font-size:10.5pt'>；其中数字代表灰度，</span><span style='font-size:
10.5pt;font-family:"Times New Roman"'>x</span><span lang=ZH-CN
style='font-size:10.5pt'>表示边界上无法进行模板操作的点，通常的做法是复制原图的灰度，不进行任何处理。</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>模板操作实现了一种邻域运算</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>(Neighborhood Operation)</span><span
lang=ZH-CN style='font-size:10.5pt'>，即某个象素点的结果灰度不仅和该象素灰度有关，而且和其邻域点的值有关。在以后介绍的细化算法中，我们还将接触到邻域运算。模板运算的数学涵义是一种卷积</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>(</span><span
lang=ZH-CN style='font-size:10.5pt'>或互相关</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>)</span><span lang=ZH-CN style='font-size:10.5pt'>运算，你不需要知道卷积的确切含义，只要有这么一个概念就可以了。</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>模板运算在图象处理中经常要用到，可以看出，它是一项非常耗时的运算。以</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
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style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
<p align=right style='margin:0cm;margin-bottom:.0001pt;text-align:right;
line-height:18.0pt'><span style='font-size:10.5pt;font-family:"Times New Roman"'>(3.2)<o:p></o:p></span></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify;text-justify:
inter-ideograph;line-height:18.0pt'><span lang=ZH-CN style='font-size:10.5pt'>为例，每个象素完成一次模板操作要用</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>9</span><span
lang=ZH-CN style='font-size:10.5pt'>个乘法、</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>8</span><span lang=ZH-CN style='font-size:10.5pt'>个加法、</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>1</span><span
lang=ZH-CN style='font-size:10.5pt'>个除法。对于一幅</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>n</span><span lang=ZH-CN style='font-size:10.5pt'>×</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>n(</span><span
lang=ZH-CN style='font-size:10.5pt'>宽度×高度</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>)</span><span lang=ZH-CN style='font-size:10.5pt'>的图象，就是</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>9n<sup>2</sup></span><span
lang=ZH-CN style='font-size:10.5pt'>个乘法，</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>8n<sup>2</sup></span><span lang=ZH-CN
style='font-size:10.5pt'>个加法和</span><span style='font-size:10.5pt;font-family:
"Times New Roman"'>n<sup>2</sup></span><span lang=ZH-CN style='font-size:10.5pt'>个除法，算法复杂度为</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>O(n<sup>2</sup>)</span><span
lang=ZH-CN style='font-size:10.5pt'>，这对于大图象来说，是非常可怕的。所以，一般常用的模板并不大，如</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>3</span><span
lang=ZH-CN style='font-size:10.5pt'>×</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>3</span><span lang=ZH-CN style='font-size:10.5pt'>，</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>4</span><span
lang=ZH-CN style='font-size:10.5pt'>×</span><span style='font-size:10.5pt;
font-family:"Times New Roman"'>4</span><span lang=ZH-CN style='font-size:10.5pt'>。有很多专用的图象处理系统，用硬件来完成模板运算，大大提高了速度。另外，可以设法将二维模板运算转换成一维模板运算，对速度的提高也是非常可观的。例如，</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'>(3.2)</span><span
lang=ZH-CN style='font-size:10.5pt'>式可以分解成一个水平模板和一个垂直模板，即，</span><span
style='font-size:10.5pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
<p align=center style='margin:0cm;margin-bottom:.0001pt;text-align:center;
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