二维的傅里叶变换及其反变换
image.png
image.png
二维傅里叶变换的特性：
image.png
image.png
案例：
值得注意的是，进行FFT变换之后，往往要进行频谱平移，使得原点被移到图像中心。
image.png
image.png
image.png

值得注意的是，图中的方块位置发生改变，频谱图像并不会发生变换，但是相位谱会发生变换。所以在实际进行频域操作时，往往只在频谱上操作 => 保证位置不变。由于有平移操作，所以往往要保证图像的频域图像增强 - 图8都为偶数。

无法在空域和时域建立直接的关联。
The low frequencies correspond to the slowly varying components of an image, while the higher frequencies correspond to the fast gray level changes, such as edges and noise.
image.png

滤波器

image.png

空域滤波 VS 频域滤波

image.png
频域滤波比较直观，空域滤波可以选取小尺寸的卷积核。可以结合二者设计尺寸小的空域滤波器：在频域进行滤波器选取，反变换之后指导空域进行滤波器选择。

高斯低通滤波器

image.png

上面都只是一条曲线，实际上是一个曲面，在曲面上进行参数的选取。（旋转产生曲面）

平均就相当于低通滤波。

高斯高通滤波器

image.png
锐化就相当于高通。

  • The frequency domain may be viewed as a “laboratory” in which we take advantage of the correspondence between frequency content and image appearance, and carries with it a significant degree of intuitiveness regarding how to specify filters.
  • Some enhancement tasks that would be exceptionally difficult or impossible to formulate directly in the spatial domain become almost trivial in the frequency domain.
  • The frequency domain implementation runs fast enough.

  • Once selected a specific filter via experimentation in the frequency domain, the actual implementation of the method usually is done in the spatial domain.

    低通滤波器

    There are three types of lowpass (smoothing) filters:

  • Ideal

  • Butterworth
  • Gaussian

These three filters cover the range from very sharp (ideal) to very smooth (Gaussian) filter functions.
High order Butterworth filter approaches the form of the ideal filter, while the low order has a smooth form similar to the Gaussian filter.

理想低通

The 2-D ideal lowpass filter (ILPF) has the transfer function：
image.png
其中：image.png
One way to establish a set of standard cutoff frequency loci is to compute circles that enclose specified amounts of total image power PT :
image.png
image.png
环状效应原因：
image.png

Butterworth Lowpass Filters

其公式为：
image.png
image.png
不是垂直下降，有缓变过程。
For filters with smooth transfer functions, defining a cutoff frequency locus at points for H(u, v) is down to a certain fraction of its maximum value is customary. In this case, H(u, v) = 0.5 when D(u, v) = _D_0.
image.png

高斯低通滤波

高斯低通公式：
image.png
image.png

  • 高斯滤波器在频域变换时下降沿更加缓慢，所以低通滤波器平滑效果相对来说差一些；
  • 不会产生“环”效应；（振铃效应）

    低通滤波实例

    An example for character recognition from the field of machine perception。
    image.png
    An example of pre-processing for printing and publishing. The typical objective is to produce a smoother, soft-looking result from a sharp original.
    image.png
    An example of processing satellite/aerial images with different purposes.
    image.png

    低通滤波：去掉细节信息。

高通滤波器

Image sharpening can be achieved in the frequency domain by a highpass filtering process。
As before, we consider only zero-phase-shift, radially symmetric filters. Conceptually, the intended function of the filters is to perform precisely the reverse operation of the ideal lowpass filters:
image.png
仍旧讨论之前的三种滤波器形式的高通形式：
image.png
image.png

ButterWorth

image.png
image.png

Gaussian

image.png
image.png

The Laplace in the Frequency Domain

image.png
image.png

将 Lapalace 3D 绘图

image.png

Unsharp Masking, High-Boost Filtering, and High-Frequency Emphasis Filtering

image.png
之前在空域：原图减去低频成分。
image.png

高频增强案例

image.png

  • The final enhanced image is a little noisy, this is typical of X-ray images when their gray scale is expanded.

  • This is an example of how spatial domain processing can be complement frequency domain filtering.

    Homomorphic Filtering（同态滤波）

    Remember the illumination-reflectance（照明-反射）model mentioned before:
    image.png
    image.png
    image.png
    理论
    照射分量：反映的是图像中变换比较小的（通常是一些背景）
    反射分类：反映的是图像中的高频部分

  • The enhancement approach is based on a special case of a class of systems known as homomorphic systems.

  • The key to the approach is the separation of the illumination and reflectance components. The homomorphic filter function H(u, v) can then operate on these components separately.
  • The illumination component of an image generally is characterized by slow spatial variations and associated with the low frequencies, while the reflectance with the high frequencies.
  • By specify the filter function affecting the low- and high-frequency components in different ways, we can get a good deal of control over the illumination and reflectance components.

image.png

例子

image.png

傅里叶变换性质

平移：
image.png
image.png
image.png
image.png
image.png
image.png
image.png
由于傅里叶变换是默认输入序列为周期的，所以在卷积操作时，原序列和原序列的周期序列进行卷积操作之后得到的结果是不同的 => 卷绕错误。
解决办法：增大周期，补零。=> padding

The Padding

image.png
image.png
image.png
image.png

The Correlation Theorems

The principal use of correlation is for matching.
If we want to determine whether an image f contains a particular object or region in which we are interested, we let h be that object or region (normally call this image a template). Then, if there is a match, the correlation of the two functions will be maximum at the location where h finds a correspondence in f.
image.png
（现代信号处理里面的相关 => 考虑相关性）
image.png
image.png

  • The major objective of this chapter was the development for the Fourier transform, and the combination with the spatial domain enhancement.
  • The area of image enhancement really is a collection of tools that have been found in practice to produce acceptable results in given applications.
  • Although most of the tools themselves are well grounded in mathematical and statistical concepts, their use is strictly problem oriented.Image enhancement is more an art than a science.