What I will Tell You

Vector Space ProjectionsInner Product Choiceand Applications to Signal ProcessingPerson Talking At you:Josh Carmichael 1Geophysics program, University of Washington Seismic Emissions & VSP

Vector Space ProjectionsSaying Simple Stuff about Hilbert Spacesthat is Obvious, but Interesting, which Usually isn’t Said Person Talking At you:Josh Carmichael 1Geophysics program, University of Washington Seismic Emissions & VSP

What I will Tell You • Review: inner products and basis building in Hilbert Spaces • The filtering-effect of certain Hilbert Spaces via their inner product • Examples that illustrate convolutional equivalence • Characterization of inner products as eigen-problems • Making a larger space via inner product choice • Application to multi-resolution analysis via scaling functions • Matlab code, anyone? • Questions? Seismic Emissions & VSP

What This is About ingredients • Vectors have 2 attributes: basis functions and data projectionsonto basis functions (coefficients) • Vectors are built from basis functions using an “inner product” in “Hilbert Space” • Example: re-express a given caloric intake from a meal using a sum of smaller meals with desirable properties Amount of ingredient Hilbert Space Basis Functions c1 + c2 ~ Projections Seismic Emissions & VSP

Hilbert Space Properties Lots of ways to do this  ‘distance’ between elements Fourier Expansion Energy Conservation ‘size’ of element Seismic Emissions & VSP Completeness

Making a Basis • Form a set of linearly independent elements… • Sort elements, and choose first • Pick your favorite inner product • Orthogonalize elements using chosen inner product • Cook for 15 minutes…. • Result: a orthonormal basis from which anything in your Hilbert Space can be made Seismic Emissions & VSP

Same Basis, Two Inner Products Experiment: Use the same basis with 2 distinct inner products, and project a sinusoid onto each: Hilbert Space 1 Hilbert Space 2 Zero-mean, has 1-1 boundary values Conclusion: Hilbert Space 2 is a low-pass filter Seismic Emissions & VSP

Hilbert Space 1 Hilbert Space 2 has reduced observed Gibbs Phenomenon using same of basis functions Hilbert Space 2 Seismic Emissions & VSP

Equivalence to Convolution • Last example: I characterized the inner product by its operation on a cosine basis • Inverse proportionality with the basis index, w, • Generally: apply Parseval’s theorem… • Normalize in frequency domain… • Use special transform relationship and get convolutional equivalence Seismic Emissions & VSP

Convolutional-Type Inner Products • The Hilbert Space 1 versus Hilbert Space 2 fight illustrated inner products can act like filters • Underlying concept: the inner product was characterized by projection onto basis functions Characterize the inner product through projection onto a set of basis functions Desired parameter response Seismic Emissions & VSP

Convolutional-Type Inner Products Abstract expression of the filtering effect Seismic Emissions & VSP

‘Nearby’-Orthogonality via Convolution basis functions formed by convolution with a Dirac sequence family member Convert time-domain integral expression to frequency domain via Parseval’s theorem Convolution in time domain is same as multipication in frequency domain u and v satisfy same orthogonality condition as Laguerre polynomials Seismic Emissions & VSP

Physics Behind Inner Product Choice Ground Displacement from a simple fault slip at two adjacent fault patches Observed ground displacement after being attenuated and operated upon by a recording instrument • Inner product choice not necessarily arbitrary • Typical example: Generalized Eigenvalue problems: • Elements defined by convolution or positiveness Would like these to be basis functions, since each contains independent information Would like to ‘back out’ physical basis functions using the attenuation and instrument response Seismic Emissions & VSP

Application: Haskell Waveform Modeling tR tC • ‘Haskell’ theory suggests 1-D faulting produces trapezoidal ground displacement • The ground displacement from a fault should be expressible as a superposition of waveforms produced from smaller faults • The observed displacement caused by faulting at distinct locations should be expressed independently j( t ) j( t-h ) Seismic Emissions & VSP

Requiring Orthogonality of Haskell Displacements Nonzero support overlap Similar to Sobolev Inner Product for W1 Seismic Emissions & VSP

… Vj+3 Vj+2 Vj Vj-1 … U U U U U {0} = V = Vk Vk V1 k k V2 V3 V4 Two-Scale Consequence of Sobolev Type Inner Product j Vj=span{ j( t, 2j ktC ) }, k=1,2,… Vj-1=span{ j( t, 2 j-1ktC ) }, k=1,2,… j-1 Seismic Emissions & VSP

Multi-Resolution Analysis of a Seismic Signal Seismic Emissions & VSP

Dyadic Construction of Multi-Scale Basis Functions Basis function k in level J is equal to the sum of the 2k-1 and 2kbasis functions in level J-1 Seismic Emissions & VSP

Dyadic Reconstruction via Scaling Functions Seismic Emissions & VSP

A Full-Scale Decomposition 32 Basis Functions 16 Basis Functions 8 Basis Functions Seismic Emissions & VSP

What I just Told You • Hilbert space choice, via the right inner product, may reduce artifacts present in spectral reconstruction • Filtering is energy conserving by inner product choice • Hilbert Spaces may be enlarged, a given set of independent elements • Choice natural for requirements of MRA Seismic Emissions & VSP

What I will Tell You