图像的处理

Image processing basics

• Contrast：S curve，$\mathrm{output}(x,y)=f(\mathrm{input}(x,y))$
• Invert：$\mathrm{output}(x,y)=1-\mathrm{input}(x,y)$
• Blur 模糊
• Sharpen 锐化
• Edge detection 边缘检测

Gaussian Blur 高斯模糊

Obtain filter kernel by 2D Gaussian function:

$$f(i,j)=\frac{1}{2\pi {\sigma }^2}{e}^{-\frac{i^2+j^2}{2{\sigma }^2}}$$

3x3 Gaussian kernel:

$$\left[\begin{array}{ccc}.075& .124& .075\\ .124& .204& .124\\ .075& .124& .075\end{array}\right]$$

Sharpen Filter 锐化滤波

$$\left[\begin{array}{ccc}0& -1& 0\\ -1& 5& -1\\ 0& -1& 0\end{array}\right]$$

Sharpening is adding high frequencies.

Bilateral Filter 双边滤波

Kernel depends on both spatial distance and intensity difference.

Image Sampling 图像采样

Reducing image size – down-sampling

Aliasing 混叠

产生原因：Signals are changing too fast but sampled too slow. 信号变化太快，采样太慢。

Higher frequency signals need faster sampling.

Under-sampling：高频信号采样不足，采样结果错误地显示低频信号。

Fourier Transform 傅里叶变换

Represent a function as a weighted sum of sines and cosines. 将函数表示为正弦和余弦的加权和。

$$F(u)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)e^{-2\pi iux}\mathrm{d}x$$