Spatial domain and frequency domain image compression of images

What is image spatial domain( Space domain ) and frequency domain( frequency domain )？

Graphic spatial domain Refers to the color space of the image , Based on the direct processing of image pixels . In the space domain (x,y) It is considered to be a point in two-dimensional space , Count Word image f(x,y)=color It is a discrete function defined on a rectangular region in two-dimensional space .

Graphic frequency domain It refers to describing the characteristics of an image with frequency as an independent variable , The spatial variation of the pixel value of an image can be decomposed into different amplitudes 、 Linear superposition of simple vibration functions of spatial frequency and phase . The composition and distribution of various spatial frequency components in an image is called Image spectrum .

It can be transformed by some means, such as （ The Fourier transform , Discrete cosine transform , Wavelet transform, etc ） Process the image in the frequency domain . Because some characteristics are outstanding in the frequency domain , Easy to handle .

The relationship between the two ：

Spatial domain and frequency domain can be converted to each other ; Apply Two dimensional discrete Fourier transform or Wavelet transform , The image can be converted from spatial domain to frequency domain ; adopt Corresponding inverse transformation It can also be converted back to spatial domain images , That is, images that people can recognize directly .

Image frequency domain filtering

The filtering of two-dimensional digital images is mainly divided into Spatial domain filtering and spectrum shaping

Spatial domain filtering ： Use various templates to directly interact with images Convolution operation

spectrum shaping ： When implementing some image processing , Frequency domain processing is simpler than spatial domain ; For digital images in spatial domain , according to Convolution theorem Can pass Fourier transformation take Spatial convolution filtering Transformation for spectrum shaping , however Then the frequency domain filtered image Reverse transformation Back to space domain

The frequency of the image ： An indicator of the intensity of gray value change , It's the gradient of gray level in plane space .

Low frequency of image

Low frequency That is, the color changes slowly , That is, ash Degrees change slowly , It means that it is An area of continuous gradient , This part is low frequency . For an image , Except for the high frequency, it is the low frequency , That is, the content within the edge is low frequency , The content in the edge is most of the information of the image , That is, the general outline and outline of the image , Is the approximate information of the image .

  Low frequency of image

High frequency means that the frequency changes quickly . When does the gray change quickly in the image ? That is, there is a great difference in gray between adjacent areas , This is fast change . Image , One Edges of images and backgrounds , There are usually obvious differences , That is to say, change the sideline , Gray changes quickly , That is, the parts with high change frequency . therefore , The gray value of the image edge changes quickly , It corresponds to the high frequency , That is, the edge of the high-frequency display image . The details of the image also belong to the area where the gray value changes sharply , Because of the sharp change of gray value , The details will appear .

in addition noise （ Noise ） In the same way , In the position of a pixel , The reason is noise , It's because it's different from the normal dot color , In other words, the gray value of the pixel is obviously different , That is, the gray level changes rapidly , So it's the high-frequency part , So there is noise at high frequency .

The frequency domain components of the results obtained from the two-dimensional discrete Fourier transform of the digital image are shown in the figure below , The upper left corner is the DC component , The four corners of the transformation result correspond to low-frequency components , The central part corresponds to the high-frequency part

For the convenience of observation , Often take transposition Method make the DC component appear in the center of the window （ Centralization ）, After transformation, the center is low frequency , Outward is high frequency .

In the frequency domain , It can be easily realized Sharpening and blurring of the image ：

Intercept the low-frequency component of the frequency , Make an inverse Fourier transform , What you get is the blurred image , namely Low pass filtering

Intercept the high-frequency component of the frequency , Make an inverse Fourier transform , What you get is the sharpened image , namely High pass filter

Human eyes are not sensitive to high-frequency information .

Image compression

Lossy compression

  We start from the image rgb To begin , Then use the compression algorithm to encode it, which is what we store in memory , It's more compact but With our original rgb It means completely different , Therefore, part of the compression scheme also needs to define a decoding component , Transform the stored representation of our data into something that the computer can render as an image rgb Format .jpeg Part of the standard defines the code And decoding .jpeg A key point in is that the final decoded image will not be the same as the original uncompressed image , This is why we call it lossy compression in the compression part of the pipeline . We will deliberately lose information to obtain compression At the 5% level .

In image compression, what information can be discarded ？

         The human eye is more sensitive to the change of brightness than to the change of color , therefore JPGE Take advantage of this .rgb Color space ： From the origin to the color on the diagonal 255 255 255 You will gradually get brighter colors , actually , Indeed, tangents between these points define all possible grayscale colors , They are A direct measure of brightness .

 ycbcr: Y brightness , cb and cr Components will encode colors .

One way to compress the original image is , Reduce to cbcr Sampling of components , And keep all brightness components . This technique is called chrominance subsampling or more commonly chrominance subsampling .

By way of cb and cr On the aisle 2x2 The blocks are merged into one color , It is usually difficult to see any changes after secondary sampling , We leave a quarter of the original data in each color channel , Reduce the total file size 50%, But and jpeg Image compression 5% It's a long way off .

Deleting pixels is down sampling

Look at the image from a different perspective

        One way to think about images is to think of them as signals , If I slice a specific line of an image , I have a row of pixels , A certain value of each pixel is 0 To 255 Between . If we plot these values , We can get an approximate value to visualize the image as a signal . Allow us to discuss the frequency components in the image .

High frequency components correspond to rapid changes between pixels , Low frequency components correspond to smoother changes between pixels .

Two key points

1、 If I choose random parts of real images , Then the lower frequency component is likely to be the pixels in the region .

2、 The human visual system is usually less sensitive to high-frequency details in images .

How to obtain frequency components from images

Discrete cosine transform (DCT), Suppose we only have 8 A little bit , but dct The clever and absolutely inconspicuous idea of this is 8 Points are represented as coming from Sum of sample points of cosine wave .dct Get the input of the sampling point from our original signal , And provide us with an output coefficient of the same size as that we will refer to , These coefficients represent the weights of cosine waves of different frequencies that contribute to the original signal . A good analogy is to decompose complex signals into weighted sums of simple cosine waves . The first output coefficient It seems that the frequency is 1 The cosine wave of corresponds well , And the second one. coefficient And the frequency is 2 Cosine wave correlation of .

Frequency cosine wave What is it? , It is just a constant signal in terms of image , This means that it provides us with a measure of the overall brightness of a set of pixels , Brighter images will have a larger zero coefficient Compared with darker images .

 

jpeg how Use it exclusively

jpeg Take the image and split it 8x8 block .

And then by subtracting 128 Concentrate their values on 0 near . Then we get the block and put dct Apply to each line of the block , Give us eight groups dct coefficient , Then apply dct To each column of the block, this process defines two dimensions dct.

  So in the end we have 64 A coefficient of , Each coefficient is in a specific 8x8 pattern On .

  Note that the first row and the first column correspond to the early one-dimensional pattern , Other elements are combinations of these patterns , Just like in the case of one dimension , The main idea here is We can build any 8x8 block Use this 64 Images in three basic modes . But there is an interesting phenomenon , When merging a small number of coefficients , Our signals and images have looked Very close to the original image , Add dimensions again , There is no visual improvement effect .

Slowly build the image after a coefficient , We basically finally get the blur of the original image , When we add dct When the coefficient is , Slowly notice how fast the image starts to look like the original image , When we get less than 25% Of dct When the coefficient is , You can hardly tell the difference between the two images , This confirms jpeg Why it applies to this particular The key aspect of fixed image is that almost all blocks are composed of the lowest frequency component , We are usually insensitive to changes in high-frequency details .

but The next natural question is how we actually do this , Remove high frequency

 

  eliminate jpeg The process of higher frequency components in is called quantization . Quantification is a simple idea , Given from dct Frequency coefficient of 8x8 matrix , What we need to do is basically take the element of each scalar value and round it to an integer Keep low frequency , Remove high frequency , Then use   The combination of length coding and Huffman coding reduces storage .

 

 

 

https://www.youtube.com/watch?v=0me3guauqOU

Digital image spatial domain frequency domain , Deeply understand the principle and process formula of image transformation - You know