[DWT Note] Fourier transformation and Wavelet Transformation

 

[DWT Note] Fourier transformation and Wavelet Transformation

I. Preface

The signals we often encounter, such as sine, cosine, and even complex ECG, EEG, and seismic waves, are all signals in the time domain. We also become the original signal, but normally, the information we obtain in the original signal is limited. Therefore, in order to obtain more information, we need to perform mathematical transformation on the original signal to obtain the signal in the transformed domain, usually, the main types of transformations are Fourier transformation, Laplace transformation, z transformation, and wavelet transformation. Today we will mainly discuss Fourier transformation and wavelet transformation.

 

 

Ii. Stable and non-stable signals

Before introducing the subject, let's talk about the difference between a stable signal and a non-stable signal.

A stable signal is a signal that does not change with the distribution parameter or distribution law over time. That is to say, the statistical characteristics of a stable signal do not change with time. As shown in:

In contrast, a non-stable signal refers to a signal with the distribution parameter or distribution law changing over time. That is to say, the statistical characteristics of non-Stable Random Signals are time functions (changes with time), as shown in:

In the Time-Frequency Analysis of signals, we can simply understand that the frequency of a stable signal does not change with time, but that of a non-stable signal changes with time.

 

 

Iii. multi-resolution analysis

The concept of uncertainty has been introduced in heenburg, that is, moving particles and locations cannot be determined at the same time, because the concepts of time domain and frequency domain are inherited from physics, the time-frequency relationship also satisfies the uncertainty principle. That is to say, the signal frequency and occurrence time cannot be determined at an exact point in the time-frequency plane. At any time point, we cannot determine which spectrum component exists. What we can do is to determine which spectrum component exists within a given period of time.

This is a resolution issue.

Taking image signals as an example, we all know that the low-frequency part of an image shows the basic information of the image, while the high-frequency part is more detailed information. Just like using Google Maps, a high scale (Low Frequency) means that there is no detail, and it is an overall view. A low scale (high frequency) means more details.

In practical application, the high-frequency part of an image usually lasts for a short period of time, usually in the form of short-term abrupt changes or spikes, such as the image edge information and some noise information, in the time domain, we can understand that high frequencies correspond to large changes in images in these regions. However, low-frequency information exists in most places and is reflected in some background or content information, which is not obvious in the time domain.

Therefore, when analyzing the low-frequency part of a signal, we only need a large frequency resolution and a small time-domain resolution to reflect the low-frequency information, in the high-frequency part, a large time resolution and a small frequency resolution are required to reflect the high-frequency information.

In fact, there is no resolution problem for the basic Fourier transformation, because the resolution of the Fourier transformation in the time domain is 0, and the resolution in the time domain is also 0. Therefore, in fact, there is no resolution for Fourier transformation, while the short-term Fourier transformation analyzes signals in different time periods in the time domain by means of window addition. However, because the Window Length is fixed, the resolution is fixed, in addition, the selection of the window length is a conflict between the time domain and the resolution of the frequency domain. The wavelet transform can provide different resolutions in different frequency bands based on the scale transformation and offset, this is very useful in practice. We will introduce it in detail later.

 

 

Iv. Fourier Transformation

Intuitively, we all know that frequency means the rate at which something changes. If something (in the correct technical term it is a mathematical or physical variable) changes fast, we say it is too frequent. If it is not fast, the frequency is low. If this variable remains unchanged, we say it has a frequency of 0 or no frequency.

During Fourier transformation, the time-domain and frequency are converted. Generally, we can easily see some information that is not seen in the time-domain from the frequency-domain. For example, it is generally difficult to find these conditions in the time-domain signals of ECG. Heart disease experts generally use time-domain ECG records on tapes to analyze ECG signals. Recently, the new ECG recorder/analyzer can also provide information about the frequency domain of the ECG, through which they can determine whether the disease exists. Analysis of Frequency Domain diagrams makes it easier for them to diagnose the disease.

Fourier transform is a reversible transformation that allows conversion between the original signal and the transformed signal. However, only one type of information is available at any time. That is to say, time information is not included in the frequency domain after Fourier transformation, and time information is not included in the time domain after inverse transformation.

In fact, it is very easy to understand this formula based on Fourier variation. The formula of continuous Fourier transformation is:

The formula for Discrete Fourier transformation is:

We can clearly see that in Fourier transformation, points are valid in all time classes because they are from negative infinity to positive infinity. That is to say, no matter when the frequency component changes, the integral result will be affected globally. The same principle applies to discrete Fourier transformation. This is why we say that Fourier transform is not suitable for analyzing non-stable signals.

 

5. Short-term Fourier Transformation

Previously, we discussed that Fourier transformation is not applicable to non-stable signals. However, if we assume that the signal is stable, Fourier transformation can be applied. In fact, if we assume that the signal is stable for a very short period of time, we can observe the signal from the narrow window, and the window is narrow to the signal we see from the window is indeed stable. The mathematical approximation finally determined by the researchers, as a modified version of Fourier transformation, is called short-term Fourier transformation.

There is only one small difference between short-term Fourier transformation and Fourier transformation. In short-term Fourier transformation, signals are divided into small enough segments, and the signals of these segments can be regarded as stable signals. For this reason, a window function is required. The width of the window must be equal to the width of the signal segment, so that its stability is valid.

We will only discuss the basic theory here, so we will not go into the specific process of pushing to the window. Previously, we said that the resolution of short-term Fourier transformation is fixed because the Window Length we add is fixed. If we have an infinite window, then perform Fourier transformation to obtain the perfect frequency resolution, but the result does not contain the time information, which is the basic Fourier transformation. However, in order to obtain the smoothness of the signal, we must have a window function with a short width. In this very short period of time, the signal is stable. The shorter the window, the higher the time resolution, the higher the signal stability, but the lower the frequency differentiation rate. The following is a detailed graphic analysis:

The time-frequency resolution diagram for the shortest Window Length is shown as follows:

We can clearly see that the time domain resolution is relatively high, and the frequency domain contains many overlapping parts, so the frequency resolution is poor. The following shows the window length, as shown in:

Obviously, at this time, some overlapping operations have already occurred in the time domain, and the resolution in the frequency domain is much better than before. Let's continue to look at the display chart with the smallest Window Length, as shown in:

At this time, the resolution in the time domain is very poor, and the resolution in the frequency domain is quite high, so in short: narrow window => high-time resolution, low-frequency resolution; wide window => high-frequency resolution, low time resolution. Therefore, the selection of Window Function length is a key step during short-term Fourier transformation. We need to make a lot of trade-offs here.

 

Vi. Wavelet Transformation

A wavelet is a waveform with a small area, a limited length, and a mean value of 0. Wavelet transform is to select the appropriate basic wavelet or mother waveletPSI (t)Through the translation and scaling of the basic wavelet, a series of wavelet are formed. These clusters of wavelet can form a series of nested (signal) subspaces, and then the signals to be analyzed (sample images) project to different (signal) subspaces to observe the corresponding characteristics. In this way, we can observe an object with different focal lengths, from macro to micro, from overview to detail. Therefore, wavelet transformation is also called a mathematical microscope ".

This kind of translation and scaling is a feature of wavelet transformation. Therefore, the signal can be analyzed in different frequency ranges and in different time (Space) locations. Through this multi-resolution analysis, obtain a good time resolution and a poor frequency resolution in the high-frequency signal, a good frequency resolution and a high time resolution in the low-frequency signal, it obviously solves the disadvantages of Fourier transformation application and non-smooth signal. Wavelet transform provides a mixture of time-frequency and time-frequency Representation of signals, which is very efficient in many fields, such as image denoising, edge detection, compression coding, and image fusion.

6. 1. Discrete Wavelet Transformation

The frequently used wavelet transformations are mainly used in image processing. Here, continuous wavelet transformations are not summarized.

Discrete Wavelet Transform (DWT) is not a simple continuous transform sample, but must provide good redundancy to implement completely reversible transformation. Such redundancy requires more computer resources and increases the computing workload. Discrete wavelet transform can provide sufficient information for signal analysis and synthesis, and reduce computer resource consumption and computing workload. Compared with continuous wavelet transformation, discrete transformation is easier to implement.

Similar to continuous wavelet transformation, discrete wavelet transformation also requires the use of digital filter technology to obtain the time-range representation of digital signals. In addition, continuous wavelet transformation is completed by constantly changing the size of the window: Moving the window function in the time domain, and then performing convolution with the signal. In discrete wavelet transform, the filter truncates certain frequency components of the signal at different scales: the signal is obtained by using different high-pass filters to obtain a series of high-frequency components of the signal, different low-pass filters are used to obtain a series of low-frequency components, so that different signal frequencies can be analyzed.

The signal resolution can be used to measure the signal details. After the signal is filtered, the scale signal changes due to the filter's upper and lower Sampling operations on the signal. Subsampling of a signal can be achieved by reducing the sampling frequency or removing some components from the signal. Signal Sampling increases the sampling frequency by adding a new sampling point to the signal, and the added sampling point can be 0 or an intermediate value.

The expression of a digital signal is usually a positive integer. Discrete Wavelet Transform first needs to pass the digital signal through the digital low-pass filter to obtain the further sample value of the signal. Low-pass filtering is usually used for convolution. Its Expression is as follows:

The low-pass filter removes all signal components higher than the cutoff frequency. For example, if the maximum signal frequency is 1000Hz, the low-pass filter removes the signal components higher than Hz.

After low-pass filtering, half of the signal is removed. However, according to the nyhowever theorem, the highest frequency of the signal is not. In this way, the signals after simple low-pass filtering will not meet the nyquest theorem. Therefore, the signal scale factor will be doubled. At this time, low-pass filtering will remove high-frequency components, but it can maintain recoverable low-frequency sampling. In this case, the resolution is related to the total amount of signal information. Therefore, the scale factor affects the filtering operation of the signal. From the above discussion, we can know that although the sampling obtained by simple low-pass filtering removes the high-frequency component, the signal information is lost, so the signal resolution is halved. In order to ensure the reversible wavelet transformation, the scale factor will be doubled after subsampling, so the low-pass filtering of signals can be completed by the following equations:

After filtering the signal, the computation of wavelet transformation can begin. First, the signal must be decomposed into the initial estimation approximation of the signal and the detailed information of the signal through the filter, and then the signal is analyzed and processed using different scale factors on different frequency bands. After these steps, the discrete wavelet transformation is completed. Discrete transform uses two sets of functions: the scale function and the wavelet function, which correspond to the low-pass filter and the high-pass filter respectively. The signal is decomposed on different frequency bands-through simple low-pass and high-pass filtering of the obtained signal in the time domain: first, the original signal is filtered through the high-pass filter and low-pass filter, complete the final filtering according to the constraints of the scale factor described in the previous section, as shown below:

WhereYhighAndYlowThey are output by Qualcomm filter and low-pass filter respectively.

. Image Wavelet Transformation

The previous things are theoretically strong, but I think they still have a lot of theoretical support for applications. Of course, they are not really very familiar with it, we can also easily use wavelet transform in MATLAB. Here, we have to be amazed at the fact that the contributions made by many of our predecessors have really brought us great help in our current study and research. Let's talk about some specific things.

Image Wavelet transformation is the basis for applying wavelet to image processing and is based on two-dimensional discrete wavelet transformation. An image can be viewed as a two-dimensional matrix. Generally, assume that the size of the image matrix is and there are (non-negative integers ). After each wavelet transform, the image is decomposed into four sub-blocks of the original size of 1/4, including the wavelet coefficients of the corresponding band, it is equivalent to sampling intervals in the horizontal direction and in the straight direction. When performing the next layer of wavelet transformation, the transformed dataset is in the band.

The spatial distribution of wavelet coefficients corresponds to the spatial distribution of original images.

LlThe frequency band is the thumbnail of the image content,It is the frequency band in which image data is concentrated.

After regularization of wavelet coefficients, we can see the image content from the display of coefficients. However, the detailed information of the image is stored with the band. The specific relationship is shown below:

HlThe frequency band stores high-frequency information in the horizontal direction of the image.It reflects the changes and edge information in the horizontal direction of the image;

LHThe frequency band stores high-frequency information in the vertical direction of the image.It reflects the gray-scale changes and edge information of the image in the vertical direction;

HHThe frequency band stores the high-frequency information of the image in the diagonal line.It reflects the comprehensive gray-scale changes of the horizontal and vertical direction images, and contains a small amount of edge information.

You can use graphs to display the specific information, which is the frequency distribution chart of a wavelet transform:

Below is the frequency distribution of quadratic Wavelet Transformation:

It is not hard to see that wavelet transform establishes a very good correlation between the original image and the transform coefficient. Therefore, in the design of filters (this involves specific implementations, but there are ready-made functions that can be called), we can design different bandwidths separately. For example, to weaken the glitch or high-frequency information in the horizontal direction of the imageHlThe wavelet coefficient of the band without affecting the edge information in other directions. In addition, for multi-layer wavelet transform, the wavelet coefficients at different resolution levels can be processed separately to achieve the expected filtering effect. If it is a low-pass filter, You can retainLlHowever, the high-frequency wavelet coefficients are effectively reduced.

The frequencies of the wavelet image correspond to the details of the original image at different scales and resolutions, as well as the optimal approximation of the original image at the minimum scale and resolution determined by the decomposition level of the wavelet transform. From the perspective of multi-resolution analysis, when we consider the frequency bands of wavelet images, these frequencies are not purely irrelevant, especially for each high-frequency band, because they are the descriptions of the same edge, contour, and texture of an image in different directions, at different scales, and at different resolutions, there must be a relationship between them, obviously, the corresponding edges in these frequencies also correspond to those contained in the high-frequency subbands at the same scale. Because the edge and contour of the image have a great impact on the subjective quality of the image observed by the human eye, this mechanism will undoubtedly improve the subjective quality of the encoded image.

 

VII. Summary

This section briefly summarizes the wavelet transform in the Fourier transform domain. This blog post is relatively theoretical and boring and involves many specific implementations. You only need to understand this part, in practical application, we will not be asked to design filters and so on. Thanks to the work of our predecessors, we only need one function in MATLAB to implement wavelet transformation.

We will summarize the application of wavelet transform in Image Noise Reduction, compression, feature extraction, image enhancement, image fusion, and other fields.

 

[DWT Note] Fourier transformation and Wavelet Transformation