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Wavelets

Wavelets. Paul Heckbert Computer Science Department Carnegie Mellon University. Function Representations. sequence of samples (time domain) finite difference method pyramid (hierarchical) polynomial sinusoids of various frequency (frequency domain) Fourier series

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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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