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<title>第2章 图象的几何变换</title>
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<h1><a name="_Toc486331867"></a><a name="_Toc486332867"></a><a
name="_Toc486338976"></a><a name="_Toc454810841"></a><a name="_Toc454856615"><span><span>第<span
lang=EN-US>2</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"'>图象的几何变换</span></span></span></h1>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>这一章我们将介绍图象的几何变换，包括图象的平移、旋转、镜象变换、转置、放缩等。如果你熟悉矩阵运算，你将发现，实现这些变换是非常容易的。</span></p>
<h2> <span
lang=EN-US>2.1</span> <span lang=EN-US> </span><a name="_Toc486331868"></a><a
name="_Toc486332868"></a><a name="_Toc486338977"></a><a name="_Toc454810842"></a><a
name="_Toc454856616"><span><span>平移</span></span></a></h2>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>平移</span><span lang=EN-US>(translation)</span><span
style='font-family:宋体;"Times New Roman"'>变换大概是几何变换中最简单的一种了。</span></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>如图</span><span lang=EN-US>2.1</span><span style='font-family:
宋体;"Times New Roman"'>所示，初始坐标为</span><span
lang=EN-US>(x<sub>0</sub>,y<sub>0</sub>)</span><span style='font-family:宋体;
"Times New Roman"'>的点经过平移</span><span
lang=EN-US>(t<sub>x</sub>,t<sub>y</sub>)(</span><span style='font-family:宋体;
"Times New Roman"'>以向右，向下为正方向</span><span
lang=EN-US>)</span><span style='font-family:宋体;
"Times New Roman"'>后，坐标变为</span><span lang=EN-US>(x<sub>1</sub>,y<sub>1</sub>)</span><span
style='font-family:宋体;"Times New Roman"'>。这两点之间的关系是</span><span lang=EN-US>x<sub>1</sub>=x<sub>0</sub>+t<sub>x
</sub></span><span style='font-family:宋体;
"Times New Roman"'>，</span><span lang=EN-US>y<sub>1</sub>=y<sub>0</sub>+t<sub>y</sub></span><span
style='font-family:宋体;"Times New Roman"'>。</span></p>
<p class=a><span lang=EN-US> <img width=195 height=203
src="chap02.files/image002.jpg" v:shapes="_x0000_i1025"> </span></p>
<p align=center style='text-align:center'><b><span
style='font-family:宋体;"Times New Roman"'>图</span>2.1 </b><b><span
style='font-family:宋体;"Times New Roman"'>平移的示意图</span><span lang=EN-US></span></b></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>以矩阵的形式表示为</span></p>
<p align=center style='text-align:center;line-height:18.0pt'><span
lang=EN-US><sub> <img width=244 height=77
src="chap02.files/image004.gif" v:shapes="_x0000_i1067"> </sub> </span></p>
<p align=right style='text-align:right;line-height:18.0pt'><span>
</span>(2.1)</p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>我们更关心的是它的逆变换：</span></p>
<p align=center style='text-align:center;line-height:18.0pt'><span
lang=EN-US><sub> <img width=268 height=77
src="chap02.files/image006.gif" v:shapes="_x0000_i1038"> </sub> </span></p>
<p align=right style='text-align:right;line-height:18.0pt'><span
lang=EN-US>(2.2)</span></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>这是因为：我们想知道的是平移后的图象中每个象素的颜色。例如我们想知道，新图中左上角点的</span><span
lang=EN-US>RGB</span><span style='font-family:宋体;
"Times New Roman"'>值是多少？很显然，该点是原图的某点经过平移后得到的，这两点的颜色肯定是一样的，所以只要知道了原图那点的</span><span
lang=EN-US>RGB</span><span style='font-family:宋体;
"Times New Roman"'>值即可。那么到底新图中的左上角点对应原图中的哪一点呢？将左上角点的坐标</span><span
lang=EN-US>(0,0)</span><span style='font-family:宋体;
"Times New Roman"'>入公式</span><span lang=EN-US>(2.2)</span><span
style='font-family:宋体;"Times New Roman"'>，得到</span><span lang=EN-US>x<sub>0=</sub>-t<sub>x
</sub></span><span
style='font-family:宋体;"Times New Roman"'>，</span><span lang=EN-US>y<sub>0</sub>=-t<sub>y</sub></span><span
style='font-family:宋体;"Times New Roman"'>；所以新图中的</span><span lang=EN-US>(0,0)</span><span
style='font-family:宋体;"Times New Roman"'>点的颜色和原图中</span><span lang=EN-US>(-t<sub>x</sub>
, -t<sub>y</sub>)</span><span
style='font-family:宋体;"Times New Roman"'>的一样。</span></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>这样就存在一个问题：如果新图中有一点</span><span lang=EN-US>(x<sub>1</sub>,y<sub>1</sub>)</span><span
style='font-family:宋体;"Times New Roman"'>，按照公式</span><span lang=EN-US>(2.2)</span><span
style='font-family:宋体;"Times New Roman"'>得到的</span><span lang=EN-US>(x<sub>0</sub>,y<sub>0</sub>)</span><span
style='font-family:宋体;"Times New Roman"'>不在原图中该怎么办？通常的做法是，把该点的</span><span lang=EN-US>RGB</span><span
style='font-family:宋体;"Times New Roman"'>值统一设成</span><span lang=EN-US>(0,0,0)</span><span
style='font-family:宋体;"Times New Roman"'>或者</span><span lang=EN-US>(255,255,255)</span><span
style='font-family:宋体;"Times New Roman"'>。</span></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>另一个问题是：平移后的图象是否要放大？一种做法是不放大，移出的部分被截断。例如，图</span><span
lang=EN-US>2.2</span><span style='font-family:宋体;
"Times New Roman"'>为原图，图</span><span lang=EN-US>2.3</span><span
style='font-family:宋体;"Times New Roman"'>为移动后的图。这种处理，文件大小不会改变。</span></p>
<p class=a style='line-height:18.0pt'><span lang=EN-US> <img width=306 height=193
src="chap02.files/image008.gif" v:shapes="_x0000_i1068"> </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>2.2 </b><b><span
style='font-family:宋体;"Times New Roman"'>移动前的图</span><span lang=EN-US></span></b></p>
<p align=center style='text-align:center;line-height:18.0pt'><b><span
lang=EN-US> <img width=308 height=195
src="chap02.files/image010.gif" v:shapes="_x0000_i1069"> </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>2.3 </b><b><span
style='font-family:宋体;"Times New Roman"'>移动后的图</span><span lang=EN-US></span></b></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>还有一种做法是：将图象放大，使得能够显示下所有部分，如图</span><span lang=EN-US>2.4</span><span
style='font-family:宋体;"Times New Roman"'>所示。</span></p>
<p class=a style='line-height:18.0pt'><span lang=EN-US> <img width=362 height=244
src="chap02.files/image012.gif" v:shapes="_x0000_i1070"> </span></p>
<p class=a style='line-height:18.0pt'><b><span style='font-family:宋体;
"Times New Roman"'>图</span>2.4 </b><b><span
style='font-family:宋体;"Times New Roman"'>移动后图象被放大</span><span lang=EN-US></span></b></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>这种处理，文件大小要改变。设原图的宽和高分别是</span><span lang=EN-US>w<sub>1</sub>,h<sub>1</sub></span><span
style='font-family:宋体;"Times New Roman"'>则新图的宽和高变为</span><span lang=EN-US>w<sub>1</sub>+|t<sub>x</sub>|</span><span
style='font-family:宋体;"Times New Roman"'>和</span><span lang=EN-US>h<sub>1</sub>+|t<sub>y</sub>|</span><span
style='font-family:宋体;"Times New Roman"'>，加绝对值符号是因为</span><span lang=EN-US>t<sub>x</sub>,
t<sub>y</sub></span><span
style='font-family:宋体;"Times New Roman"'>有可能为负</span><span lang=EN-US>(</span><span style='font-family:
宋体;"Times New Roman"'>即向左，向上移动</span><span
lang=EN-US>)</span><span style='font-family:宋体;
"Times New Roman"'>。</span></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>下面的函数</span><span lang=EN-US>Translation</span><span
style='font-family:宋体;"Times New Roman"'>采用的是第一种做法，即移出的部分被截断。在给出源代码之前，先说明一个问题。</span></p>
<p style='line-height:18.0pt'><span
style='font-family:宋体;"Times New Roman"'>如果你用过</span><span lang=EN-US>Photoshop,Corel
PhotoPaint</span><span
style='font-family:宋体;"Times New Roman"'>等图象处理软件，可能听说过“灰度图”</span><span lang=EN-US>(grayscale)</span><span
style='font-family:宋体;"Times New Roman"'>这个词。灰度图是指只含亮度信息，不含色彩信息的图象，就象我们平时看到的黑白照片：亮度由暗到明，变化是连续的。因此，要表示灰度图，就需要把亮度值进行量化。通常划分成</span><span
lang=EN-US>0</span><span style='font-family:宋体;
"Times New Roman"'>到</span><span lang=EN-US>255</span><span
style='font-family:宋体;"Times New Roman"'>共</span><span lang=EN-US>256</span><span style='font-family:
宋体;"Times New Roman"'>个级别，其中</span><span
lang=EN-US>0</span><span style='font-family:宋体;
"Times New Roman"'>最暗</span><span lang=EN-US>(</span><span
style='font-family:宋体;"Times New Roman"'>全黑</span><span lang=EN-US>)</span><span style='font-family:
宋体;"Times New Roman"'>，</span><span
lang=EN-US>255</span><span style='font-family:宋体;
"Times New Roman"'>最亮</span><span lang=EN-US>(</span><span
style='font-family:宋体;"Times New Roman"'>全白</span><span lang=EN-US>)</span><span style='font-family:
宋体;"Times New Roman"'>。</span><span
lang=EN-US>.bmp</span><span style='font-family:宋体;
"Times New Roman"'>格式的文件中，并没有灰度图这个概念，但是，我们可以很容易在</span><span
lang=EN-US>.bmp</span><span style='font-family:宋体;
"Times New Roman"'>文件中表示灰度图。方法是用</span><span lang=EN-US>256</span><span
style='font-family:宋体;"Times New Roman"'>色的调色板，只不过这个调色板有点特殊，每一项的</span><span lang=EN-US>RGB</span><span
style='font-family:宋体;"Times New Roman"'>值都是相同的。也就是说</span><span lang=EN-US>RGB</span><span
style='font-family:宋体;"Times New Roman"'>值从</span><span lang=EN-US>(0</span><span style='font-family:
宋体;"Times New Roman"'>，</span><span
lang=EN-US>0</span><span style='font-family:宋体;
"Times New Roman"'>，</span><span lang=EN-US>0)</span><span
style='font-family:宋体;"Times New Roman"'>，</span><span lang=EN-US>(1</span><span style='font-family:
宋体;"Times New Roman"'>，</span><span
lang=EN-US>1</span><span style='font-family:宋体;
"Times New Roman"'>，</span><span lang=EN-US>1)</span><span
style='font-family:宋体;"Times New Roman"'>一直到</span><span lang=EN-US>(255</span><span
style='font-family:宋体;"Times New Roman"'>，</span><span lang=EN-US>255</span><span style='font-family:
宋体;"Times New Roman"'>，</span><span
lang=EN-US>255)</span><span style='font-family:宋体;
"Times New Roman"'>。</span><span lang=EN-US>(0</span><span
style='font-family:宋体;"Times New Roman"'>，</span><span lang=EN-US>0</span><span style='font-family:
宋体;"Times New Roman"'>，</span><span
lang=EN-US>0)</span><span style='font-family:宋体;
"Times New Roman"'>是全黑色，</span><span lang=EN-US>(255</span><span
style='font-family:宋体;"Times New Roman"'>，</span><span lang=EN-US>255</span><span style='font-family:
宋体;"Times New Roman"'>，</span><span
lang=EN-US>255)</span><span style='font-family:宋体;
"Times New Roman"'>是全白色，中间的是灰色。这样，灰度图就可以用</span><span
lang=EN-US>256</span><span style='font-family:宋体;
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