fft英文资料.docx

1、fft英文资料fft英文资料傅立叶变换在图像处理中有非常非常的作用傅立叶变换在图像处理中有非常非常的作用。因为不仅傅立叶分析涉及图像处理的很多方面，傅立叶的改进算法，比如离散余弦变换，gabor与小波在图像处理中也有重要的分量。印象中，傅立叶变换在图像处理以下几个话题都有重要作用：1.图像增强与图像去噪绝大部分噪音都是图像的高频分量，通过低通滤波器来滤除高频噪声; 边缘也是图像的高频分量，可以通过添加高频分量来增强原始图像的边缘；2.图像分割之边缘检测提取图像高频分量3.图像特征提取：形状特征：傅里叶描述子纹理特征：直接通过傅里叶系数来计算纹理特征其他特征：将提取的特征值进行傅里叶变换来使特征

2、具有平移、伸缩、旋转不变性4.图像压缩可以直接通过傅里叶系数来压缩数据；常用的离散余弦变换是傅立叶变换的实变换；傅立叶变换傅里叶变换是将时域信号分解为不同频率的正弦信号或余弦函数叠加之和。连续情况下要求原始信号在一个周期内满足绝对可积条件。离散情况下，傅里叶变换一定存在。冈萨雷斯版里面的解释非常形象：一个恰当的比喻是将傅里叶变换比作一个玻璃棱镜。棱镜是可以将光分解为不同颜色的物理仪器，每个成分的颜色由波长（或频率）来决定。傅里叶变换可以看作是数学上的棱镜，将函数基于频率分解为不同的成分。当我们考虑光时,讨论它的光谱或频率谱。同样,傅立叶变换使我们能通过频率成分来分析一个函数。傅立叶变换有很多优

3、良的性质。比如线性，对称性（可以用在计算信号的傅里叶变换里面）;时移性：函数在时域中的时移，对应于其在频率域中附加产生的相移，而幅度频谱则保持不变；频移性：函数在时域中乘以ejwt，可以使整个频谱搬移w。这个也叫调制定理，通讯里面信号的频分复用需要用到这个特性（将不同的信号调制到不同的频段上同时传输）；卷积定理：时域卷积等于频域乘积；时域乘积等于频域卷积（附加一个系数）。（图像处理里面这个是个重点）信号在频率域的表现在频域中，频率越大说明原始信号变化速度越快；频率越小说明原始信号越平缓。当频率为0时，表示直流信号，没有变化。因此，频率的大小反应了信号的变化快慢。高频分量解释信号的突变部分，而低

4、频分量决定信号的整体形象。在图像处理中，频域反应了图像在空域灰度变化剧烈程度，也就是图像灰度的变化速度，也就是图像的梯度大小。对图像而言，图像的边缘部分是突变部分，变化较快，因此反应在频域上是高频分量；图像的噪声大部分情况下是高频部分；图像平缓变化部分则为低频分量。也就是说，傅立叶变换提供另外一个角度来观察图像，可以将图像从灰度分布转化到频率分布上来观察图像的特征。书面一点说就是，傅里叶变换提供了一条从空域到频率自由转换的途径。对图像处理而言，以下概念非常的重要：图像高频分量：图像突变部分；在某些情况下指图像边缘信息，某些情况下指噪声，更多是两者的混合；低频分量：图像变化平缓的部分，也就是图像

5、轮廓信息高通滤波器：让图像使低频分量抑制，高频分量通过低通滤波器：与高通相反，让图像使高频分量抑制，低频分量通过带通滤波器：使图像在某一部分的频率信息通过，其他过低或过高都抑制还有个带阻滤波器，是带通的反。图像去噪图像去噪就是压制图像的噪音部分。因此，如果噪音是高频额，从频域的角度来看，就是需要用一个低通滤波器对图像进行处理。通过低通滤波器可以抑制图像的高频分量。但是这种情况下常常会造成边缘信息的抑制。常见的去噪模板有均值模板，高斯模板等。这两种滤波器都是在局部区域抑制图像的高频分量，模糊图像边缘的同时也抑制了噪声。还有一种非线性滤波-中值滤波器。中值滤波器对脉冲型噪声有很好的去掉。因为脉冲点

6、都是突变的点，排序以后输出中值，那么那些最大点和最小点就可以去掉了。中值滤波对高斯噪音效果较差。椒盐噪声：对于椒盐采用中值滤波可以很好的去除。用均值也可以取得一定的效果，但是会引起边缘的模糊。高斯白噪声：白噪音在整个频域的都有分布，好像比较困难。冈萨雷斯版图像处理P185：算术均值滤波器和几何均值滤波器（尤其是后者）更适合于处理高斯或者均匀的随机噪声。谐波均值滤波器更适合于处理脉冲噪声。图像增强有时候感觉图像增强与图像去噪是一对矛盾的过程，图像增强经常是需要增强图像的边缘，以获得更好的显示效果，这就需要增加图像的高频分量。而图像去噪是为了消除图像的噪音，也就是需要抑制高频分量。有时候这两个又是

7、指类似的事情。比如说，消除噪音的同时图像的显示效果显著的提升了，那么，这时候就是同样的意思了。常见的图像增强方法有对比度拉伸，直方图均衡化，图像锐化等。前面两个是在空域进行基于像素点的变换，后面一个是在频域处理。我理解的锐化就是直接在图像上加上图像高通滤波后的分量，也就是图像的边缘效果。对比度拉伸和直方图均衡化都是为了提高图像的对比度，也就是使图像看起来差异更明显一些，我想，经过这样的处理以后，图像也应该增强了图像的高频分量，使得图像的细节上差异更大。同时也引入了一些噪音。In image processing Fourier transformFourier transform very v

8、ery role in image processing. Because not only Fourier analysis involves many aspects of image processing, Fourier improved algorithmFor example, discrete cosine transform, gabor and wavelets in image processing is also an important component.Impression, Fourier transform image processing following

9、topics have an important role in:1. Image Enhancement and Image DenoisingMost of the noise image is a high-frequency component, a low-pass filter to filter out high frequency - noise; high-frequency component of the image edge is, by adding an edge to enhance the high-frequency component of the orig

10、inal image;2. The edge detection image segmentationHigh-frequency component extracted imageThe image feature extraction:Shape Feature: Fourier descriptorsTexture: Texture is calculated directly by the Fourier coefficientsOther features: the extracted eigenvalues Fourier transform to make the feature

11、 translation, scalable, rotational invariance4. Image CompressionData can be compressed by the Fourier coefficients; conventional discrete cosine transform is the Fourier transform of the real transform; Fourier transformFourier transform is a time-domain signal into different frequency sine or cosi

12、ne function signal superimposed sum. It requires continuous signal satisfies conditions of the original absolute integrability condition for one cycle. Discrete case, Fourier transform must exist. Gonzalez Edition Inside the very image interpretation: an apt analogy is the Fourier transform of as a

13、glass prism. Prisms can be broken down into different colors of light physical equipment, each color component having a wavelength (or frequency) is determined. Fourier transform can be seen as a prism in mathematics, a function based on the frequency decomposition into different components. When we

14、 consider the light, to discuss its spectrum or frequency spectrum. Similarly, the Fourier transform allows us to analyze the frequency component by a function.Fourier transform has many fine properties. Such as linear symmetry (can be used in the calculation of the Fourier transform of the signal i

15、nside);Time shifting of: function when the shift in the time domain, in the frequency domain corresponding to the generation of an additional phase shift and an amplitude spectrum remains unchanged;Frequency Shift: The function in the time domain by multiplying e jwt, can make the entire spectrum sh

16、ifting w. This is also called the modulation theorem which communication signal frequency division multiplexing need to use this feature (to a different signal modulation frequency band different simultaneous transmission);Convolution theorem: the time domain frequency domain convolution is equal to

17、 the product; the product is equal to the time domain frequency domain convolution (attach a factor). (Image processing inside this is key)Signal in the frequency domain representationIn the frequency domain, the greater the frequency of the original signal changes faster; the smaller the frequency

18、the more gentle description of the original signal. When the frequency is 0, the DC signal does not change. Therefore, the size of the frequency response of the speed of change in the signal. High-frequency component mutations explain part of the signal, and the low-frequency component determines th

19、e overall image signal.In image processing, frequency domain response of the intensity of the image gray changes in the airspace, which is the rate of change of image intensity, which is the image of the gradient magnitude. Image, the edge portion of the image is mutated portion, changes rapidly, an

20、d therefore the reaction is in the high-frequency components in the frequency domain; in most cases the image noise high frequency portion; image part is the low-frequency component changes smoothly. That is, the Fourier transform to provide a different angle observation image, the image may be tran

21、sformed from the gray distribution to the frequency distribution characteristic up observation image. Written point that is, the Fourier transform provides a way to convert from a free airspace to frequency. Image processing, the following concept is very important:High-frequency component image: mu

22、tated portion of the image; in some cases refer to edge information, in some cases refer to noise, a hybrid of the two more;Low-frequency components: part of the image changes gently, that is, the image contour informationHigh Pass Filter: Makes images so that low-frequency component suppressed by t

23、he high-frequency componentLow Pass Filter: Qualcomm Instead, let the high-frequency component of the image inhibited by low-frequency componentsBandpass filter: image in a part of the frequency information through other too high or too inhibitedThere is a band stop filter, band pass counter.Image D

24、enoisingDenoising is to suppress the noise of the image. Therefore, if the amount of high-frequency noise is, from the perspective of frequency domain point of view, is the need to use a low-pass filter for image processing. Low-pass filter can suppress high-frequency components of the image. But th

25、is situation will often result in inhibition of edge information. Common denoising template has a template mean, Gaussian templates. Both filters are suppressed high frequency components of the image in the local area, while the edges are blurred image noise is suppressed. There is also a nonlinear

26、filtering - median filter. Median filter for impulse-type noise with good removed. Because the pulse point is a point mutation, after sorting the output value, the maximum and minimum points that can be removed. Median filtering Gaussian noise less effective.Salt and pepper noise: salt and pepper fo

27、r the median filtering can be well removed. Means may also be used to achieve a certain effect, but can cause blurred edges.Gaussian white noise: white noise are distributed throughout the frequency domain, it seems more difficult.Gonzalez edition image processing P185: arithmetic and geometric mean

28、 filter mean filter (especially the latter) is more suited to handle a uniform or Gaussian random noise. Harmonic mean filter is more suited to handle the impulse noise.Image EnhancementSometimes you feel the image enhancement and denoising is a contradiction in the process, image enhancement is oft

29、en the need to enhance the edges of the image to get a better display, which requires adding high frequency components of the image. The denoising is to eliminate image noise, which is necessary to suppress high frequency components. Sometimes this means that another two similar things. For example, display an image to eliminate noise while significantly improved, then this time is the same meaning.