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Wavelets

Learn about wavelets, a powerful tool in scientific computing using hierarchical basis functions with compact support. Discover their history, applications, and advantages over Fourier analysis.

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Wavelets

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  1. Wavelets Paul Heckbert Computer Science Department Carnegie Mellon University 15-859B - Introduction to Scientific Computing

  2. Function Representations • sequence of samples (time domain) • finite difference method • pyramid (hierarchical) • polynomial • sinusoids of various frequency (frequency domain) • Fourier series • piecewise polynomials (finite support) • finite element method, splines • wavelet (hierarchical, finite support) • (time/frequency domain) 15-859B - Introduction to Scientific Computing

  3. What Are Wavelets? In general, a family of representations using: • hierarchical (nested) basis functions • finite (“compact”) support • basis functions often orthogonal • fast transforms, often linear-time 15-859B - Introduction to Scientific Computing

  4. Simple Example: Haar Wavelet • Consider piecewise-constant functions over 2j equal sub-intervals of [0,1]: • j=2: four intervals • A basis 15-859B - Introduction to Scientific Computing

  5. Nested Function Spaces for Haar Basis • Let Vj denote the space of all piecewise-constant functions represented over 2j equal sub-intervals of [0,1] • Vj has basis 15-859B - Introduction to Scientific Computing

  6. Function Representations –Desirable Properties • generality – approximate anything well • discontinuities, nonperiodicity, ... • adaptable to application • audio, pictures, flow field, terrain data, ... • compact – approximate function with few coefficients • facilitates compression, storage, transmission • fast to compute with • differential/integral operators are sparse in this basis • Convert n-sample function to representation in O(nlogn) or O(n) time 15-859B - Introduction to Scientific Computing

  7. Wavelet History, Part 1 • 1805 Fourier analysis developed • 1965 Fast Fourier Transform (FFT) algorithm … • 1980’s beginnings of wavelets in physics, vision, speech processing (ad hoc) • … little theory … why/when do wavelets work? • 1986 Mallat unified the above work • 1985 Morlet & Grossman continuous wavelet transform … asking: how can you get perfect reconstruction without redundancy? 15-859B - Introduction to Scientific Computing

  8. Wavelet History, Part 2 • 1985 Meyer tried to prove that no orthogonal wavelet other than Haar exists, found one by trial and error! • 1987 Mallat developed multiresolution theory, DWT, wavelet construction techniques (but still noncompact) • 1988 Daubechies added theory: found compact, orthogonal wavelets with arbitrary number of vanishing moments! • 1990’s: wavelets took off, attracting both theoreticians and engineers 15-859B - Introduction to Scientific Computing

  9. Time-Frequency Analysis • For many applications, you want to analyze a function in both time and frequency • Analogous to a musical score • Fourier transforms give you frequency information, smearing time. • Samples of a function give you temporal information, smearing frequency. • Note: substitute “space” for “time” for pictures. 15-859B - Introduction to Scientific Computing

  10. Comparison to Fourier Analysis • Fourier analysis • Basis is global • Sinusoids with frequencies in arithmetic progression • Short-time Fourier Transform (& Gabor filters) • Basis is local • Sinusoid times Gaussian • Fixed-width Gaussian “window” • Wavelet • Basis is local • Frequencies in geometric progression • Basis has constant shape independent of scale 15-859B - Introduction to Scientific Computing

  11. Wavelet Applications • Medical imaging • Pictures less corrupted by patient motion than with Fourier methods • Astrophysics • Analyze clumping of galaxies to analyze structure at various scales, determine past & future of universe • Analyze fractals, chaos, turbulence 15-859B - Introduction to Scientific Computing

  12. Wavelets for Denoising • White noise is independent random fluctuations at each sample of a function • White noise distributes itself uniformly across all coefficients of a wavelet transform • Wavelet transforms tend to concentrate most of the “energy” in a small number of coefficients • Throw out the small coefficients and you’ve removed (most of) the noise • Little knowledge about noise character required! 15-859B - Introduction to Scientific Computing

  13. Wavelets, Vision, and Hearing Human vision & hearing: The retina and brain have receptive fields (filters) sensitive to spots and edges at a variety of scales and translations Somewhat similar to wavelet multiresolution analysis Human hearing also uses approximately constant shape filters Computer vision: Pyramid techniques popular and powerful for matching, tracking, recognition Gaussian & Laplacian pyramids – wavelet precursors Speech processing: Quadrature Mirror Filters – wavelet precursors 15-859B - Introduction to Scientific Computing

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