1. Fourier transform and wavelet transform
Fourier transform FFT
Fourier transform is to analyze the signal completely in the frequency domain, but it cannot give the change of the signal at each time point, and a sudden change at any point on the time axis will affect the entire frequency signal.
The basis of Fourier transform is sinusoidal curves of different frequencies, so Fourier transform decomposes the signal wave into the superposition sum of sinusoidal waves of different frequencies, which cannot effectively represent sudden changes.
Wavelet transform wavelets
Wavelet transform is based on some specific functions (not just one), and expands the data signal into a series of series. It is a local transformation of time and frequency, and can perform multi-scale joint analysis of data in the time domain and frequency at the same time. , which has the function of multi-scale refinement analysis. Therefore, we can effectively analyze the signal on different decomposition levels and different wavelet basis functions.
Wavelets, loosely speaking, waveforms of finite duration, have a mean value of zero. Wavelet transform is the decomposition of a signal into a series of wavelets. Wavelets come in different sizes and shapes. The availability of multiple wavelets is a key advantage of wavelet analysis.
Comparing wavelets to sine waves, the basis of Fourier analysis, sinusoids have no limit in duration—negative infinity extends to positive infinity. While sinusoids are smooth, wavelets tend to be irregular and asymmetrical.
scaling & shifting
Scale: Stretch or shrink the signal
The wavelet has a band-pass characteristic center frequency in the frequency domain, stretching the wavelet helps to capture slowly changing changes in the signal, and compressing the wavelet helps to capture sudden changes.
shifting move
Moving the wavelet just means delaying or advancing the start of the wavelet along the length of the signal.
The two main transforms in wavelet analysis are continuous and discrete wavelet transforms.
The key applications of CWT (continuous wavelet transform) continuous wavelet transform are time-frequency analysis and time-domain frequency component filtering;
The key application of DWT (discrete wavelet transform) discrete wavelet transform is signal and image denoising and compression, because it helps to represent many naturally occurring signals and images with fewer coefficients.
2. Upsampling upsampling and downsampling subsampling
upsampling
The main purpose of enlarging an image (or called upsampling or image interpolation) is to enlarge the original image so that the image can be displayed on a higher display device. Image enlargement almost adopts the interpolation method, that is, inserts new elements between pixels using an appropriate interpolation algorithm on the basis of the original image pixels.
downsampling
There are two main purposes of shrinking images (or subsampling or downsampling):
1. Make the image conform to the size of the display area;
2. Generate a thumbnail of the corresponding image.
The zoom operation on an image does not bring more information about the image, so the quality of the image will inevitably be affected. However, there are indeed some scaling methods that can enhance the information of the image, so that the quality of the scaled image exceeds the quality of the original image.
For a scale of M*N, it is downsampled by s times, that is, an image of (M/s)*(N/s) size is obtained (s should be a common divisor of M and N). If the image is in the form of a matrix, it essentially converts the image in the original image s*s window into a pixel, and the value of the pixel is the mean value of all pixels in the window.
3. Hal Wavelet
This blogger explained it in detail, so I won’t sort it out by myself. I will guide you ⬇️:
Haar transform of wavelet transform