Wavelet Transforms in Image Processing (II)
“There is no greater education than one that is self-driven”-Neil deGrasse Tyson.
Before we proceed further, let us be aware of some important notions, such as the “uncertainty principle’. The uncertainty principle implies that predicting a quantity’s value with arbitrary certainty is generally impossible.
We don’t know what frequencies exist at what time, but we can understand what frequency bands exist at what time interval.
From Fourier to Wavelet
In the Fourier transform, time localization is achieved by windowing the data. This phenomenon or transform where we window the data is called the Short-Term Fourier Transform (STFT). The major backdrop of the STFT is that the window function is finite, so frequency resolution decreases. The fixed length of the window means time and frequency resolutions are fixed for the entire length of the signal. If one decides on a narrow window, it gives good time resolution but poor frequency resolution, and vice versa if a wide window is considered.
So later, engineers, physicists, and mathematicians devised wavelet transformations and its theory. The wavelet transform overcomes the limitations of the STFT by using wavelet functions that can be scaled and translated, allowing for variable time and frequency resolutions.
So, wavelet transformation analyzes a signal into different frequencies at different resolutions, known as multiresolution analysis. The wavelet is our new basis function and acts as a window function. We can change the width of the wavelet and its central frequency as we move it across our signal by changing s. “s” denotes scaling. Wavelet theory gives a better localization of time-frequency.
An expanded wavelet is better at revolving low-frequency components of the signal with bad time resolution.
Shrunken wavelet is better at resolving high-frequency components of the signal with good time resolution.
All the windows used are the dilated (or compressed) and shifted versions of the mother wavelet.
Wavelet Packet: The wavelet transform is actually a subset of a far more versatile transform. Wavelet packets are perpendicular linear combinations of wavelets. They form bases that retain many of their parent wavelets' orthogonality, smoothness, and localization properties.
Low-pass filter And its wavelet analysis
A low-pass filter is a type of filter that allows low-frequency components of a signal to pass through while attenuating or removing high-frequency components. In wavelet analysis, a low-pass filter decomposes and analyzes the signal by separating it into different frequency bands. The low-pass filter in wavelet analysis is applied to the signal at each level of the decomposition process. At each level, the low-pass filter removes high-frequency components from the signal and retains the low-frequency components (Mallat, 1999). This allows for progressive filtering out of high-frequency details while preserving low-frequency information, resulting in efficient signal compression. In image compression, wavelet transforms are commonly used to reduce the size of an image file while preserving important visual information. By applying wavelet analysis with a low-pass filter in image compression, the high-frequency details of the image can be progressively filtered out while preserving the important structural features and overall visual quality of the image. This technique allows for more efficient storage and transmission of images without significant loss of quality. Mathematically, it can be denoted as a convolution of the input signal with a low-pass filter coefficient sequence.
Convolution is a mathematical operation that combines two functions to produce a third function. In wavelet analysis, convolution refers to the mathematical operation of combining the input signal with a sequence of coefficients from a low-pass filter. The result of this convolution is a filtered version of the input signal, where the high-frequency components are attenuated or removed. Another term we often encounter in wavelet analysis of image compression is "thresholding.”.
Thresholding is a technique used in wavelet analysis for image compression to enhance the compression process's efficiency further. Thresholding involves setting small values in the wavelet coefficients to zero, effectively removing noise or irrelevant details from the signal (Osborne et al., 2002). This helps to reduce the overall size of the compressed image file and improve its visual quality.
Deconvolution is essential in image processing applications such as image restoration or image enhancement. It is used to reverse the effects of convolution by estimating the original signal or image from its degraded or blurred version. Mathematically, deconvolution can be performed by dividing the Fourier transform of the degraded image by the Fourier transform of the degradation filter, thereby removing the effects of the convolution operation.
High-pass filter and its wavelet analysis
A high-pass filter is a type of filter that allows high-frequency components of a signal to pass through while attenuating or removing low-frequency components. In wavelet analysis, a high-pass filter decomposes and analyzes the signal by separating it into different frequency bands. The high-pass filter in wavelet analysis is applied to the signal at each level of the decomposition process. It extracts the high-frequency details and edges of the signal while suppressing the low-frequency components. The high-pass filter in wavelet analysis helps to capture sharp transitions, fine textures, and other high-frequency information in an image.
Did you know?
Dr. Ingrid Daubechies is a Belgian mathematician and physicist and one of the pioneers of the wavelet theory revolution. She is known for her work with wavelets in image compression.
The Daubechies wavelets, named after Dr. Daubechies, are a family of orthogonal wavelets that define a discrete wavelet transform. They are characterized by a maximal number of vanishing moments for a given support. Each wavelet type in this class has a scaling function, also known as the father wavelet, which generates an orthogonal multiresolution analysis.
That’s it for today!! See you next week with a very new topic!! In the meantime, you can suggest topics you want discussions on- I would be happy to read them!! Btw I was thinking of constructing a Cosmo-Math online journal club for girls (preferably college and Uni women), but I need your opinions, ideas, and help! Do suggest me if you feel it’s a nice plan worth executing-email me at gsreya99@gmail.com or hit me up on my Instagram https://www.instagram.com/bloggers_perspective/?hl=en
