PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Analysis → Harmonic Analysis ↓


Wavelets


Wavelets are mathematical functions that chop up data into individual frequency components and analyze each component with a resolution that matches its scale. They have applications in many branches of science and engineering, including signal processing, data compression, and solving differential equations. Wavelets are particularly useful for processing signals that contain sharp spikes or transients.

Introduction to wavelets

The mathematical basis of wavelets lies in harmonic analysis, a branch of mathematics that studies functions or signals in terms of fundamental waves. Traditional Fourier analysis provides a way to decompose a signal into sine and cosine components. However, this method is limited in its ability to represent localized phenomena because sine and cosine are non-zero everywhere. Wavelets overcome this limitation by being localized in both the time and frequency domains.

Why use wavelets?

Wavelets provide a time-frequency representation of the signal, allowing us to zoom in on interesting regions of the data. They provide both time and frequency localization, which is particularly important for non-stationary signals. For example, in signal processing, wavelets can be used to compress data without losing essential information during transmission or storage.

Mathematical formulation

Mathematically, a wavelet transform is a representation of a function by a set of "wavelet" functions. A wavelet function, denoted as ψ(t), has zero mean and is localized in both time and frequency. The wavelet transform of a signal f(t) is given as:

Wψ[f](a, b) = ∫ f(t) * ψa,b(t) dt

Here, ψa,b(t) are the scaled and translated versions of the parent wavelet ψ(t) :

ψa,b(t) = 1/√|a| * ψ((t-b)/a)

where a is the scale parameter that stretches or compresses the wavelet, and b is the translation parameter that moves it along the t axis.

Visualization of wavelets

Let us look at some simple examples to understand the properties of wavelet functions.

Consider the Haar wavelet, which is one of the simplest wavelet functions. The Haar wavelet ψ(t) is defined as:

ψ(t) = 1, if 0 ≤ t < 0.5 = -1, if 0.5 ≤ t < 1 = 0, otherwise

The above SVG shows the Haar wavelet function. You can see that it has a piece-wise stationary structure. It is non-zero only in the interval [0, 1]. This localization property makes it particularly useful for time-domain analysis.

Wavelet transform

There are two main types of wavelet transforms: continuous wavelet transform (CWT) and discrete wavelet transform (DWT).

Continuous wavelet transform (CWT)

The continuous wavelet transform is useful for analyzing localized transient signals. It provides a way to obtain a highly detailed description of the data in both the time and frequency domains.

CWT(a, b) = ∫ f(t) ψa,b(t) dt

The CWT maps the original signal to a two-dimensional function of scale a and translation b. It is continuous in both time and scale, providing a detailed view of the signal.

Discrete wavelet transform (DWT)

The Discrete Wavelet Transform works on discrete data and is efficient for numerical calculations. It results in a finite number of coefficients that represent the original data.

DWT(n, m) = ∑ f[i] ψn,m[i]

Unlike the continuous wavelet transform, the DWT provides a non-redundant, efficient way to decompose a signal, making it useful for compression and noise reduction.

Properties of wavelets

Wavelets have many useful properties, including:

  • Localization: Wavelets are localized in both time and frequency, making accurate time-frequency analysis possible.
  • Orthogonality: Some wavelet bases are orthogonal, meaning they can represent data efficiently with non-redundancy.
  • Multiresolution: Analyzing the signal at multiple scales gives a more complete picture, which is especially useful for transient signals.

Applications of wavelets

The flexibility and localized nature of wavelets has led to their use in a variety of areas:

  • Data Compression: Wavelets are used in image and signal compression because they efficiently represent rapid changes in data.
  • Noise reduction: Wavelets can separate noise from important features in signals, facilitating clean data processing.
  • Image processing: Wavelet transform is widely used in feature recognition and enhancement tasks.
  • Medical imaging: In MRI and other imaging techniques, wavelets help improve image quality and suppress noise.

Conclusion

Wavelets provide a robust framework for analyzing a wide variety of signals and functions in both the time and frequency domains. They provide advanced tools for decomposing and reconstructing signals, ultimately expanding the reach of traditional harmonic analysis. As a result, wavelets serve as the backbone of many modern signal processing applications.


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