Wavelets & Time-Frequency Analysis
Wavelets & Time-Frequency Analysis
The Fourier transform gives perfect frequency resolution but zero time resolution — a single sinusoid extends over all time. The Short-Time Fourier Transform (STFT) localizes in time with a sliding window, but is limited by the uncertainty principle. Wavelets provide simultaneous time-frequency localization by dilating and translating a mother wavelet $\psi$. The Haar, Daubechies, and Mexican hat wavelets are used in signal compression, edge detection, and multi-resolution analysis.
Wavelet Transform
The continuous wavelet transform (CWT) of $f\in L^2(\mathbb{R})$ is $W_f(a,b)=\frac{1}{\sqrt{|a|}}\int f(t)\overline{\psi\left(\frac{t-b}{a}\right)}\,dt$ where $a\neq 0$ is the scale (dilation) and $b$ is the translation. The mother wavelet $\psi$ must satisfy the admissibility condition $C_\psi=\int\frac{|\hat{\psi}(\xi)|^2}{|\xi|}\,d\xi<\infty$, implying $\hat{\psi}(0)=0$ (zero mean). The inverse CWT: $f(t)=\frac{1}{C_\psi}\int\int W_f(a,b)\psi\left(\frac{t-b}{a}\right)\frac{da\,db}{a^2}$.
Multi-Resolution Analysis (MRA)
An MRA is a sequence of subspaces $\cdots V_{-1}\subset V_0\subset V_1\cdots$ of $L^2(\mathbb{R})$ with: $\overline{\bigcup V_j}=L^2$; $\bigcap V_j=\{0\}$; $f\in V_j\iff f(2\cdot)\in V_{j+1}$ (scaling); there exists a scaling function $\phi$ with $\{\phi(x-k)\}$ an orthonormal basis for $V_0$. The wavelet $\psi$ spans the orthogonal complement $W_j=V_{j+1}\ominus V_j$. Discrete wavelet transform (DWT): apply low-pass (scaling) and high-pass (wavelet) filter banks, then downsample — iterated to get multi-scale decomposition.
Example 1
Compute the Haar wavelet transform of $x=[4,6,10,2]$.
Solution: Haar mother wavelet: $\psi=[1,-1]/\sqrt{2}$, scaling: $\phi=[1,1]/\sqrt{2}$. Level 1: averages $[(4+6)/2,(10+2)/2]=[5,6]$; differences $[(4-6)/2,(10-2)/2]=[-1,4]$. Level 2: averages $[(5+6)/2]=[5.5]$; differences $[(5-6)/2]=[-0.5]$. Wavelet coefficients: $[-0.5,-1,4]$; scaling $[5.5]$. Perfect reconstruction: invert the filters. Large coefficients indicate edges/discontinuities; small coefficients indicate smooth regions — basis of wavelet compression.
Example 2
Explain why the Mexican hat wavelet $\psi(x)=(1-x^2)e^{-x^2/2}$ is used for edge detection.
Solution: $\psi(x)$ is the negative second derivative of a Gaussian: $\psi=-\frac{d^2}{dx^2}e^{-x^2/2}$. At large scales $a$, $W_f(a,b)\approx a^2\cdot f''(b)$ (up to constants). Thus zero crossings of $W_f$ at scale $a$ correspond to inflection points of $f$ smoothed at scale $a$ — these are edges. The Mexican hat wavelet detects blobs and ridges; at multiple scales, it implements a scale-space edge detector (Marr-Hildreth algorithm).
Practice
- Prove the admissibility condition implies $\int\psi(t)\,dt=0$ (zero mean), giving wavelets their oscillatory character.
- Show the Haar basis $\{\psi_{j,k}(t)=2^{j/2}\psi(2^j t-k):j,k\in\mathbb{Z}\}$ is orthonormal in $L^2(\mathbb{R})$.
- Compare the time-frequency resolution of the STFT and CWT using the Heisenberg uncertainty principle.
- Explain why JPEG 2000 uses wavelets instead of the DCT used in JPEG, in terms of ringing artifacts and compression efficiency.
Show Answer Key
1. Admissibility: $C_\psi=\int_0^\infty\frac{|\hat{\psi}(\xi)|^2}{\xi}d\xi<\infty$ requires $\hat{\psi}(0)=0$, i.e., $\int\psi(t)dt=0$. So wavelets must oscillate (have zero mean).
2. Orthonormality: $\langle\psi_{j,k},\psi_{j',k'}\rangle=\delta_{jj'}\delta_{kk'}$. For Haar: $\psi(t)=\mathbf{1}_{[0,1/2)}-\mathbf{1}_{[1/2,1)}$. Translates are supported on disjoint intervals (orthogonal), and dilates are orthogonal by the multiresolution structure.
3. STFT: fixed window $\Rightarrow$ fixed time-frequency resolution $\Delta t\cdot\Delta f\ge1/(4\pi)$. CWT: variable window (dilated wavelet) $\Rightarrow$ good time resolution at high frequency, good frequency resolution at low frequency. CWT adapts the resolution to the signal's local frequency content.
4. DCT produces block artifacts at $8\times8$ boundaries. Wavelets use overlapping basis functions, eliminating blocking artifacts. Wavelets also provide multiresolution decomposition, better preserving edges. JPEG 2000 achieves 10-20% better compression at the same quality compared to JPEG.