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<title>第6章 腐蚀，膨胀，细化算法</title>
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<h1><a name="_Toc486331891"></a><a name="_Toc486332891"></a><a
name="_Toc486339000"></a><a name="_Toc454810865"></a><a name="_Toc454856639"><span><span>第<span
lang=EN-US>6</span></span></span></a><span><span><span style='font-family:黑体;"Times New Roman"'>章</span>
</span></span><span><span><span
style='font-family:黑体;"Times New Roman"'>腐蚀，膨胀，细化算法</span></span></span></h1>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>这一章的内容我认为是最有趣的。还记得前言中那个抽取骨架的例子吗？现在我们就来看看它是如何实现的。</span></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>今天所讲的内容属于一门新兴的学科：数学形态学</span><span lang=EN-US>(Mathematical
Morphology)</span><span style='font-family:宋体;
"Times New Roman"'>。说起来很有意思，它是法国和德国的科学家在研究岩石结构时建立的一门学科。形态学的用途主要是获取物体拓扑和结构信息，它通过物体和结构元素相互作用的某些运算，得到物体更本质的形态。在图象处理中的应用主要是：</span><span
lang=EN-US>(1)</span><span style='font-family:宋体;
"Times New Roman"'>利用形态学的基本运算，对图象进行观察和处理，从而达到改善图象质量的目的；</span><span
lang=EN-US>(2)</span><span style='font-family:宋体;
"Times New Roman"'>描述和定义图象的各种几何参数和特征，如面积、周长、连通度、颗粒度、骨架和方向性等。</span></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>限于篇幅，我们只介绍二值图象的形态学运算，对于灰度图象的形态学运算，有兴趣的读者可以阅读有关的参考书。在程序中，为了处理的方便，还是采用</span><span
lang=EN-US>256</span><span style='font-family:宋体;
"Times New Roman"'>级灰度图，不过只用到了调色板中的</span><span
lang=EN-US>0</span><span style='font-family:宋体;
"Times New Roman"'>和</span><span lang=EN-US>255</span><span
style='font-family:宋体;"Times New Roman"'>两项。</span></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>先来定义一些基本符号和关系。</span></p>
<p style='line-height:
18.0pt;'> <b><span
lang=EN-US>1.<span style='font:7.0pt "Times New Roman"'>
</span></span></b> <b><span style='font-family:宋体;"Times New Roman";"Times New Roman"'>元素</span><span
lang=EN-US></span></b></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>设有一幅图象</span><span lang=EN-US>X</span><span
style='font-family:宋体;"Times New Roman"'>，若点</span><span lang=EN-US>a</span><span style='font-family:
宋体;"Times New Roman"'>在</span><span
lang=EN-US>X</span><span style='font-family:宋体;
"Times New Roman"'>的区域以内，则称</span><span lang=EN-US>a</span><span
style='font-family:宋体;"Times New Roman"'>为</span><span lang=EN-US>X</span><span style='font-family:
宋体;"Times New Roman"'>的元素，记作</span><span
lang=EN-US>a</span><span style='font-family:宋体;
"Times New Roman"'>∈</span><span lang=EN-US>X</span><span
style='font-family:宋体;"Times New Roman"'>，如图</span><span lang=EN-US>6.1</span><span style='font-family:
宋体;"Times New Roman"'>所示。</span></p>
<p style='line-height:
18.0pt;'> <b><span
lang=EN-US>2.<span style='font:7.0pt "Times New Roman"'>
</span></span></b> <b><span lang=EN-US>B</span></b><b><span
style='font-family:宋体;"Times New Roman"'>包含于</span><span lang=EN-US>X</span></b></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>设有两幅图象</span><span lang=EN-US>B</span><span
style='font-family:宋体;"Times New Roman"'>，</span><span lang=EN-US>X</span><span style='font-family:
宋体;"Times New Roman"'>。对于</span><span
lang=EN-US>B</span><span style='font-family:宋体;
"Times New Roman"'>中所有的元素</span><span lang=EN-US>ai</span><span
style='font-family:宋体;"Times New Roman"'>，都有</span><span lang=EN-US>ai</span><span style='font-family:
宋体;"Times New Roman"'>∈</span><span
lang=EN-US>X</span><span style='font-family:宋体;
"Times New Roman"'>，则称</span><span lang=EN-US>B</span><span
style='font-family:宋体;"Times New Roman"'>包含于</span><span lang=EN-US>(included
in)X</span><span
style='font-family:宋体;"Times New Roman"'>，记作</span><span lang=EN-US>B<sub> <img width=16 height=13
src="chap06.files/image002.gif" v:shapes="_x0000_i1025"> </sub> X</span><span style='font-family:宋体;"Times New Roman";"Times New Roman"'>，如图</span><span
lang=EN-US>6.2</span><span style='font-family:宋体;
"Times New Roman"'>所示。</span></p>
<p style='line-height:
18.0pt;'> <b><span
lang=EN-US>3.<span style='font:7.0pt "Times New Roman"'>
</span></span></b> <b><span lang=EN-US>B</span></b><b><span
style='font-family:宋体;"Times New Roman"'>击中</span><span lang=EN-US>X</span></b></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>设有两幅图象</span><span lang=EN-US>B</span><span
style='font-family:宋体;"Times New Roman"'>，</span><span lang=EN-US>X</span><span style='font-family:
宋体;"Times New Roman"'>。若存在这样一个点，它即是</span><span
lang=EN-US>B</span><span style='font-family:宋体;
"Times New Roman"'>的元素，又是</span><span lang=EN-US>X</span><span
style='font-family:宋体;"Times New Roman"'>的元素，则称</span><span lang=EN-US>B</span><span
style='font-family:宋体;"Times New Roman"'>击中</span><span lang=EN-US>(hit)X</span><span
style='font-family:宋体;"Times New Roman"'>，记作</span><span lang=EN-US>B</span><span style='font-family:
宋体;"Times New Roman"'>↑</span><span
lang=EN-US>X</span><span style='font-family:宋体;
"Times New Roman"'>，如图</span><span lang=EN-US>6.3</span><span
style='font-family:宋体;"Times New Roman"'>所示。</span></p>
<p style='line-height:
18.0pt;'> <b><span
lang=EN-US>4.<span style='font:7.0pt "Times New Roman"'>
</span></span></b> <b><span lang=EN-US>B</span></b><b><span
style='font-family:宋体;"Times New Roman"'>不击中</span><span lang=EN-US>X</span></b></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>设有两幅图象</span><span lang=EN-US>B</span><span
style='font-family:宋体;"Times New Roman"'>，</span><span lang=EN-US>X</span><span style='font-family:
宋体;"Times New Roman"'>。若不存在任何一个点，它即是</span><span
lang=EN-US>B</span><span style='font-family:宋体;
"Times New Roman"'>的元素，又是</span><span lang=EN-US>X</span><span
style='font-family:宋体;"Times New Roman"'>的元素，即</span><span lang=EN-US>B</span><span style='font-family:
宋体;"Times New Roman"'>和</span><span
lang=EN-US>X</span><span style='font-family:宋体;
"Times New Roman"'>的交集是空，则称</span><span lang=EN-US>B</span><span
style='font-family:宋体;"Times New Roman"'>不击中</span><span lang=EN-US>(miss)X</span><span
style='font-family:宋体;"Times New Roman"'>，记作</span><span lang=EN-US>B</span><span style='font-family:
宋体;"Times New Roman"'>∩</span><span
lang=EN-US>X=</span><span style='font-family:宋体;
"Times New Roman"'>Ф；其中∩是集合运算相交的符号，Ф表示空集。如图</span><span
lang=EN-US>6.4</span><span style='font-family:宋体;
"Times New Roman"'>所示。</span></p>
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<p align=center style='text-align:center;line-height:18.0pt'><b><span
lang=EN-US> <img width=104 height=64
src="chap06.files/image004.gif" v:shapes="_x0000_i1030"> </span></b></p>
<p align=center style='text-align:center;line-height:18.0pt'><b><span
style='font-family:宋体;"Times New Roman"'>图</span>6.1
</b><b><span style='font-family:
宋体;"Times New Roman"'>元素</span></b></p>
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<p align=center style='text-align:center;line-height:18.0pt'><span
lang=EN-US> <img width=106 height=65
src="chap06.files/image006.gif" v:shapes="_x0000_i1031"> </span></p>
<p align=center style='text-align:center;line-height:18.0pt'><b><span
style='font-family:宋体;"Times New Roman"'>图</span>6.2
</b><b><span style='font-family:
宋体;"Times New Roman"'>包含</span></b></p>
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<p align=center style='text-align:center;line-height:18.0pt'><span
lang=EN-US> <img width=136 height=65
src="chap06.files/image008.gif" v:shapes="_x0000_i1032"> </span></p>
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