Wavelet Transforms in Image Processing (III)
“What happens when you slap someone with a high frequency?-it hertz!”
Wavelet transforms are mathematical functions. They cut up the data into different frequency components and then studied each with a resolution matching its scale. Wavelet Transforms in Image Processing (III)
Mathematical transformations are applied to signals to obtain further information from that signal that is not readily available in the raw motion. Several transformations can be used, among which the Fourier transforms are the most popular. Most of the signals, in practice, are time-domain signals in their raw format. That is, whatever that signal is measuring is a function of time.
In other words, when we plot the signal, one of the axes is time (independent variable), and the other (dependent variable) is usually amplitude. When we plot time-domain signals, we obtain a time-amplitude representation of the signal. This representation is not always the best signal representation for most signal processing-related applications.
In many cases, the most distinguished information is hidden in the signal’s frequency content. The frequency spectrum of a signal is the frequency component (spectral component) of that signal. The frequency spectrum of a signal shows what frequencies exist in the signal. If the Fourier transform of a signal in the time domain is taken, the frequency-amplitude representation of that signal is obtained.
The WT was developed as an alternative to the STFT. In STFT, the signal is divided into small enough segments, and these segments (portions) of the signal can be assumed to be stationary. For this purpose, the window function “w” is chosen. The width of this window must be equal to the signal segment where its stationarity is valid.
x(t) is the signal itself, w(t) is the window function, and * is the complex conjugate. As you can see from the equation, the STFT of the signal is nothing but the Fourier transform of the signal multiplied by a window function.
The problem with STFT is that its roots go back to what is known as the Heisenberg Uncertainty Principle. This principle, initially applied to the momentum and location of moving particles, can be applied to the time-frequency information of a signal. , this principle states that one cannot know the exact time-frequency representation of a signal, i.e., one cannot know what spectral components exist at what times. What one can know are the time intervals in which certain bands of frequencies exist, which is a resolution problem.
The problem with the STFT has something to do with the width of the used window function. To be technically correct, this width of the window function is known as the support of the window. If the window function is narrow, it is compactly supported.
The wavelet transform is a transformation that gives a time-frequency representation of a signal. (Other transforms give this information, too, such as short-time Fourier transforms, Wigner distributions, etc.). Knowing the time intervals in which these particular spectral components occur may be very beneficial. For example, in EEGs, the latency of an event-related potential is of particular interest (Event-related potential is the brain’s response to a specific stimulus like a flashlight; the latency of this response is the amount of time elapsed between the onset of the stimulus and the response).
The wavelet transformations are of two types:
- Continuous Wavelet Transformations
- Discrete wavelet transformations.
Continuous Wavelet Transforms were developed as an alternative approach to the short-time Fourier
transform to overcome the resolution problem.
The continuous wavelet transform is defined as follows:
As the above equation shows, the transformed signal is a function of two variables, $\tau$ and s, the translation and scale parameters, respectively. $\psi(t)$ is the transforming function, and it is called the
mother wavelet. The term mother wavelet gets its name due to two essential properties of the wavelet.
Analysis, as explained below:
The term wavelet means a small wave. The smallness refers to the condition that this (window) function is
of finite length (compactly supported). The term wave refers to the condition that this function is oscillatory.
The term mother implies that the functions have different regions of support used in the
The transformation process is derived from one main function, the mother wavelet. In other words, the mother wavelet is a prototype for generating the other window functions.
Discrete wavelet transforms are mathematical instruments for analyzing and processing signals to offer time and frequency information.
Let the image function be S($n_1$,$n_2$) with size $N_1$X$N_2$.
It is defined as follows:\frac{1}{\sqrt{N_1N_2}}\sum_{n_1=0}^{N_1–1}\sum_{n_2=0}^{N_2–1}S(n_1,n_2)\phi_{j_0,k_1,k_2}(n_1,n_2)
It denotes the low-pass filter.
$$\omega_\Psi^i(j_0,k_1,k_2)=\frac{1}{\sqrt{N_1N_2}}\sum_{n_1=0}^{N_1–1}\sum_{n_2=0}^{N_2–1}S(n_1,n_2)\Psi_{j_0,k_1,k_2}^i(n_1,n_2)$$
denotes the high pass filter, i={H,V,D}.\\
It is a highly effective approach in signal processing, picture compression, and data analysis. The DWT decomposes a signal into wavelet coefficients that reflect the signal’s multiple frequency components at various scales or resolutions. In contrast to the Fourier transform, which only offers frequency information, the DWT simultaneously delivers time and frequency information. The DWT functions by running the signal through a succession of wavelet filters. These filters are intended to capture various frequency components of a signal. The DWT is a hierarchical transform, meaning the signal is periodically divided into lower frequency bands by cascading the filters. The name given to this breakdown process is the decomposition tree, or pyramid. Several benefits distinguish the DWT from other transform algorithms. First, it gives a time-frequency signal representation, which is important for analyzing non-stationary, time-varying signals. Second, it offers multi-resolution analysis, which means that the signal may be divided into several frequency bands with varying levels of information. This enables efficient signal compression and denoising. Finally, the DWT has high localization qualities, so it can properly represent transient and steady-state signal components. \\
The discrete wavelet transform is mathematically represented using wavelet functions, which are tiny wave-like functions localized in both the time and frequency domains. These wavelet functions analyze and decompose data into various scales or frequency bands. The DWT is based on the scaling and wavelet functions concepts, in which the signal is first processed through a low-pass filter to extract the low-frequency components and then through a high-pass filter to retrieve the high-frequency details.
A signal’s DWT representation consists of a collection of coefficients characterizing the signal at various scales. These coefficients give information about the signal’s amplitude and phase at multiple frequencies. The DWT may be represented as a tree structure, with the original signal at the top and the decomposition filters applied iteratively to yield wavelet coefficients at various scales. This mathematical description of the DWT enables efficient and compact signal encoding, making it helpful in multiple applications such as picture and audio compression, denoising, and feature extraction.
It is required to analyze discontinuities—sharp spikes in the signal.
Wavelet-based image compression is a popular approach for reducing the size of an image file without affecting image quality. This method is based on wavelet transformations and mathematical procedures to transfer signals into a new format. Wavelet transforms, as opposed to typical Fourier transforms, which utilize sine and cosine waves to describe signals, employ a sequence of tiny waves known as wavelets to represent signals. Because it provides for more efficient encoding of picture data, wavelet-based image compression is an excellent approach to minimizing the size of an image file.
Wavelet transformations are mathematical functions used to analyze data and detect patterns. In contrast to Fourier transforms, which divide signals into sine and cosine waves, wavelet transforms depict information utilizing a succession of tiny waves known as wavelets. These wavelets are generally of short duration and are temporal and frequency-localized. As a result, wavelet transformations are beneficial for analyzing signals with both high- and low-frequency components.
Despite the benefits of image compression, these approaches have limits. Lossy compression, for example, can result in a picture quality reduction that is visible to the human eye. This is due to the compression method removing information that is less significant to the human eye, but it may also retain critical features required for particular applications. Lossless compression, on the other hand, may not be as successful as lossy compression in terms of picture file size reduction. This method may only yield a compression ratio of 2:1, which may not be sufficient in some situations.
Wavelet-based image compression solves some of the shortcomings of standard image compression approaches. This method decomposes a picture into its frequency components, which are subsequently compressed using a set of coding principles. This approach offers the advantage of compressing photos without sacrificing crucial features, resulting in a higher-quality image. Furthermore, wavelet-based compression yields excellent compression ratios, making it more efficient in lowering picture file size. This approach, however, can be more computationally demanding than typical compression algorithms, necessitating more processing power and time.
The discrete wavelet transform (DWT) is a mathematical method used in signal processing and data analysis to separate a signal’s frequency components. It is useful in image compression, denoising, and feature extraction applications. The implementation of the DWT is a crucial feature since it includes numerous steps to transform the input signal into its wavelet representation. The first step in implementing the DWT is to select an appropriate wavelet basis. Wavelet bases have diverse features and are best suited to different signals. The input signal is decomposed when a wavelet basis is chosen.
This entails passing the signal through high-pass and low-pass filters and downsampling to create a reduced-resolution version. This procedure is iterated until the appropriate level of breakdown is achieved.
The generated wavelet coefficients indicate the multiple frequency components of the original signal at different scales after the decomposition stage. These coefficients can then be used in a variety of applications. For example, the wavelet coefficients can be quantized and encoded in picture compression to provide adequate compression. The wavelet coefficients can be thresholded in denoising to reduce noise while maintaining significant signal properties. The implementation of DWT necessitates a thorough comprehension of the underlying theory as well as practical knowledge of signal-processing techniques.
Overall, the Discrete Wavelet Transform is implemented by selecting a wavelet basis, decomposing the input signal using a sequence of filters, and processing the output wavelet coefficients for specific applications. It is a valuable instrument in signal processing and data analysis, with multiple applications in various industries. Mastering the DWT implementation necessitates theoretical and practical expertise, making it an interesting topic for students interested in signal processing and data analysis.
Advantages:
Wavelet-based image compression is a popular approach for lowering picture file size while maintaining image quality. This approach is based on wavelets, mathematical functions that may divide a picture into smaller sub-images. These sub-images, known as wavelet coefficients, can be reduced using various compression techniques to shrink the total picture size. Wavelet-based compression provides advantages over other compression algorithms. However, there are certain drawbacks to be aware of.
The ability to retain image quality is one advantage of wavelet-based compression. Wavelet-based compression can maintain more detail in an image than other compression approaches since it divides it into smaller sub-images. This implies that the image stays crisp and sharp even after compression. Furthermore, wavelet-based compression allows for the selective reconstruction of distinct areas of an image, which can be valuable in applications such as medical imaging, where specific parts of an image require closer examination.
Another benefit of wavelet-based compression is its adaptability. Wavelet-based compression may be employed on a variety of picture types and with a variety of compression methods. As a result, it may be modified to fit the unique demands of many applications, making it a flexible technology. Furthermore, wavelet-based compression may be employed with lossless and lossy compression, allowing users to choose between different levels of compression and picture quality.
Limitations:
Despite its benefits, wavelet-based compression has several drawbacks. Its computational complexity is one drawback. Because wavelet-based compression involves dividing a picture into smaller sub-images, it can be computationally demanding, necessitating a large amount of computing power. Furthermore, wavelet-based compression may be ineffective for images with poor contrast or detail since there is little to no information to compress. Finally, wavelet-based compression may not be appropriate for all picture types, as specific images may require more specialized compression algorithms to maintain their quality.
Future scopes of applying Wavelet Transforms in Image Compression:
Wavelet-based image compression is a common approach for compressing digital photographs without sacrificing quality. This approach is based on the wavelet notion, a mathematical function that may describe signals and pictures at various sizes and resolutions. Wavelet-based compression is chosen over other compression methods because it can eliminate unnecessary data from a picture while maintaining critical information.
Wavelet-based compression offers various applications, from medical imaging to satellite photography. In medical imaging, for example, wavelet-based compression is used to compress enormous volumes of medical data so that it may be conveniently communicated and stored. Wavelet-based compression is used in satellite photography to compress big pictures acquired by satellites to deliver them to the ground station with minimal bandwidth needs. Furthermore, wavelet-based compression is employed in video streaming applications, where compressed video data may be provided with little latency via the Internet.
Wavelet-based image compression can completely transform the field of image and video compression in the future. With advancements in hardware and software technologies, wavelet-based compression may now be used in real-time applications such as video conferencing and gaming. Furthermore, the rise of deep learning and artificial intelligence approaches may be used with wavelet-based compression to improve compression efficiency while decreasing the computing complexity of the compression algorithms. As a result, wavelet-based compression is predicted to continue to be an important technology for image and video compression in the future.
That’s it for today! Have a great day ahead!!
