<p><strong>上学期修了数字图像处理这门课程，想着正好趁这个机会写（shui）几篇文章，告诉自己没有白学。傅里叶变换，是图像处理中的一个重要内容，频率域处理的操作都要建立在傅里叶变换的基础上，所以作为这个专栏的开篇，不如就简单介绍和实现一下离散傅里叶变换（DFT）。</strong></p>
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<h3>一、傅里叶级数和变换</h3>
<p style="text-indent:33px;">法国数学家傅里叶提出，任何周期函数都可表示为不同频率的正弦函数和/或余弦函数之和，其中每个正弦函数和/或余弦函数都要乘以不同的系数，这个和就称为傅里叶级数。按照这个思想，周期为<img alt="T" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-f47c973e8acf7bf94140e3461e627873.latex">的连续变量<img alt="t" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-b5eeaedf6ad956f27dfd8421ea1c8e45.latex">的周期函数<img alt="f(t)" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-bc9f6029f9e422773b3c8c70d1d6da1a.latex">，可表示为乘以适当系数的正弦函数和余弦函数之和，即</p>
<p style="text-indent:33px;"><img alt="f(t)=\sum_{n=-\infty }^{\infty }c_{n}e^{j\tfrac{2\pi n}{T}t}" class="mathcode" src="https://private.codecogs.com/gif.latex?f%28t%29%3D%5Csum_%7Bn%3D-%5Cinfty%20%7D%5E%7B%5Cinfty%20%7Dc_%7Bn%7De%5E%7Bj%5Ctfrac%7B2%5Cpi%20n%7D%7BT%7Dt%7D">，系数为</p>
<p style="text-indent:33px;"><img alt="c_{n}=\frac{1}{T}\int_{-T/2}^{T/2}f(t)e^{-j\tfrac{2\pi n}{T}t}dt, n=0,\pm 1,\pm 2,..." class="mathcode" src="https://private.codecogs.com/gif.latex?c_%7Bn%7D%3D%5Cfrac%7B1%7D%7BT%7D%5Cint_%7B-T/2%7D%5E%7BT/2%7Df%28t%29e%5E%7B-j%5Ctfrac%7B2%5Cpi%20n%7D%7BT%7Dt%7Ddt%2C%20n%3D0%2C%5Cpm%201%2C%5Cpm%202%2C..."></p>
<p style="text-indent:33px;">另外根据欧拉公式有<img alt="e^{j\theta }=cos\theta+jsin\theta" class="mathcode" src="https://private.codecogs.com/gif.latex?e%5E%7Bj%5Ctheta%20%7D%3Dcos%5Ctheta&plus;jsin%5Ctheta">，所以上述式子便可展开为正弦函数和余弦函数之和（其中涉及到复数的知识请读者自行了解，这里不作过多说明）。</p>
<p style="text-indent:33px;">而一些（曲线下方面积有限的）非周期函数也能用正弦函数和/或余弦函数乘以加权函数的积分来表示。这种情况下的公式就是傅里叶变换，其在许多理论和应用科学中起到非常大的作用。连续单变量函数的傅里叶变换和反变换表示为</p>
<p style="text-indent:33px;"><img alt="\left\{\begin{matrix} F(\mu )=\int_{-\infty }^{\infty }f(t)e^{-j2\pi \mu t}dt\\ \\ f(t) =\int_{-\infty }^{\infty }F(\mu)e^{j2\pi \mu t}d\mu \end{matrix}\right." class="mathcode" src="https://private.codecogs.com/gif.latex?%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%20F%28%5Cmu%20%29%3D%5Cint_%7B-%5Cinfty%20%7D%5E%7B%5Cinfty%20%7Df%28t%29e%5E%7B-j2%5Cpi%20%5Cmu%20t%7Ddt%5C%5C%20%5C%5C%20f%28t%29%20%3D%5Cint_%7B-%5Cinfty%20%7D%5E%7B%5Cinfty%20%7DF%28%5Cmu%29e%5E%7Bj2%5Cpi%20%5Cmu%20t%7Dd%5Cmu%20%5Cend%7Bmatrix%7D%5Cright."></p>
<p style="text-indent:33px;">上述两个式子共同构成傅里叶变换对，可以表示为<img alt="f(t)\Leftrightarrow F(\mu)" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-d2e65af33caca0d402e2af75189b0abc.latex">。</p>
<p style="text-indent:33px;">同样，令<img alt="f(t,z)" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-b05e3643f2980b6b58c6556cb6da8a55.latex">是两个连续变量<img alt="t" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-b5eeaedf6ad956f27dfd8421ea1c8e45.latex">和<img alt="z" class="mathcode" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-9020b55f2dda288b75c3151551d5bbc8.latex">的连续函数，则其二维连续傅里叶变换对为</p>
<p style="text-indent:33px;"><img alt="\left\{\begin{matrix} F(\mu,\nu )=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(t,z)e^{-j2\pi (\mu t+\nu z)}dtdz \\ \\f(t,z)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}F(\mu,\nu)e^{j2\pi (\mu t+\nu z)}d\mu d\nu \end{matrix}\right." class="mathcode" src="https://private.codecogs.com/gif.latex?%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%20F%28%5Cmu%2C%5Cnu%20%29%3D%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7D%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7Df%28t%2Cz%29e%5E%7B-j2%5Cpi%20%28%5Cmu%20t&plus;%5Cnu%20z%29%7Ddtdz%20%5C%5C%20%5C%5Cf%28t%2Cz%29%3D%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7D%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7DF%28%5Cmu%2C%5Cnu%29e%5E%7Bj2%5Cpi%20%28%5Cmu%20t&plus;%5Cnu%20z%29%7Dd%5Cmu%20d%5Cnu%20%5Cend%7Bmatrix%7D%5Cright."></p>
<h3>二、离散傅里叶变换（DFT）</h3>
<p style="text-indent:33px;">有了上一节中所讲的数学基础，我们这里直接给出离散傅里叶的变换对</p>
<p style="text-indent:33px;"><img alt="\left\{\begin{matrix} F(u)=\sum_{x=0}^{M-1}f(x)e^{-j2\pi ux/M}, u=0,1,2,...,M-1\\ \\ f(x)=\frac{1}{M} \sum_{x=0}^{M-1}F(u)e^{j2\pi ux/M},x=0,1,2,...,M-1 \end{matrix}\right." class="mathcode" src="https://private.codecogs.com/gif.latex?%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D%2 |
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