( nonlinear constraints)

Transcription

1 Wavelet Design & Applications Basic requirements: Admissibility (single constraint) Orthogonality ( nonlinear constraints)

2 Sparse Representation Smooth functions well approx. by Fourier High-frequency coefficients are roughly zero Piecewise smooth less All wavelengths respond to discontinuities & shocks Wavelets localization overcomes that Design Principles: Compact support Vanishes on smoothness (vanishing moments)

3 Vanishing Moments An analytic notion of smooth functions: polynomials The following are equivalent: i. The wavelet ψ has p vanishing moments ii. The filter h has p vanishing moments iii. and its p-1 derivatives are zero at ω=0 iv. and its p-1 derivatives are zero at ω=π v. For any polynomials of degree k Smooth funcs. locally a polynomial (Taylor) low d j [n]

4 Compact Support The scaling func. support is equal to the length of h. If h in [N 1,N 2 ] then ψ in [(N 1 -N 2 +1)/2, (N 2 -N 1 +1)/2] Sparsity at the cost of other attributes (less dofs) For example: Wavelets with p vanishing moments are at least 2p-1

5 Applications of Sparse Rep. Compression: - Between 1924 and today, the US Federal Bureau of Investigation has collected over 200 million cards of fingerprints - The FBI is digitizing the nation's fingerprint database at 500 dots per inch with 8 bits of grayscale resolution. At this rate, a single fingerprint card turns into about 10 MB of data! Multiplied by 200 million cards about 2,000 terabytes!

6 Compression vs. JPEG Wavelets-based, file size JPEG i fil i b t compression ratio bytes, compression ratio JPEG image, file size bytes,

7 Original image (512x512) bytes Compressed 20: bytes

8

9

10

11 Applications of Sparse Rep. Random, spatially uncorrelated noise, is equally distributed over the spectrum But the piecewise smooth signal doesn t! Solution: drop high-freq components (limited it ringing), i or threshold h (and keep edges sharp / avoid ringing).

12

13 Example Wavelets (finite filters) Haar The only on of length2, the anti-symmetric one Length 4: 3 conditions: 1d parameterization: Daubechies-4 by α=π/3 with 2 V.M. (maximal) Daubechies 4 by α π/3 with 2 V.M. (maximal) Length 6:

14 Example Wavelets (infinite filters) Shannon Wavelet d h i fi it VM zero around zero, hence infinite V.M. has compact support, thus is

15 Example Wavelets (infinite filters) Spline Wavelets (Battle-Lamarie) If m even ε=1 t is centered around ½, odd m then symmetric about 0 m+1 vanishing moments (scaling func. is polynomial of degree m) Truncation in practice, exponential decay. Almost orthogonal

16

17 Spatially Dependent Models Many spatial-lattice models consists of a (smoothness) linear Hamiltonian e.g., L=D T D is a spatially-invariant Laplacian. These models can be easily fit/sampled once diagonalized using Fourier: L=C -2 where C ij = c i-j Invert using Fourier, or Map each sample to Fourier and measure the variance. But what if the smoothness modulus changes in space? (C is no longer a convolution matrix)

18 Spatially Dependent Models Very smooth, large A Less smooth, weaker A Wavelets Decorrelate since they are relatively localized in Fourier space. They are also localized in space, thus are good coordinates for these models.

19 Preconditioning Iterative linear solvers for Ax=b (A PSD) are usually of the like: x n+1 =x n -(Ax n -b), that is x n+1 =x n +A(x * -x n ), where x * solution. The right direction, just filtered by A. Preconditioning is about undoing this filtration Again, for translation-invariant equations (Poisson), FFT is a good preconditioning (even a solver), It is inadequate for translationdependent ones (inhomogeneous Poisson). And again, search D such that

20 Interpolation Given N points, we can produce more by interpreting them as a j [n] and viewing them at a 0 [n]. Inverse DWT with no data Linear in the number of output nodes. Example: Lifted splines bi-orth. wavelets

21 Biorthgonal Wavelets Orthogonal Wavelets are design limited due to too many constraints/dofs Can t be finite & symmetric, relatively large Relax the construction by using of dual scaling func. and dual wavelet: Analysis: Synthesis:

22 Biorthgonal Wavelets Rather than: where W * =W -1 (orthogonality) Use arbitrary matrix and its inverse (constructed by dilations and translations of course..)

23 Biorthgonal Wavelets The dual story: Scaling eqns Admissibility ibili Birothogonality conds: linear constaints

24 Design Compact Support if and then and Vanishing Moments The number of vanishing moments equals the number of zeros of at ω=π The number of vanishing moments equals the number of zeros of at ω=π Symmetry It is possible to construct finite symmetric and anti-symmetric Odd lengths are also possible The filters and dual filters can be exchanged

25 Example B-Wavelets Splines i th b f VM (f ) it Th i i is the number of V.M. (free param.) same parity as. The minimum length dual filter is given by

26

27 The Spectral Picture As we know is 2π-periodic For real wavelets, And we know: Power Spectrum

28 Lifting scheme Let s review the Haar WT (non-normalized) Three steps can be identified: Split: reordering into even + odd Predict: pred. odd based on even and storing the difference. Update: (just) make sure that

29 Lifting scheme Inverse is just: Undo update: Undo predict: Merge: Properties: In-place No Fourier Analysis! (boundary conditions, non-regular domains, other spatial dependencies)

30 Example: Splines Splines and the (subdivision methods) obtained using polynomial prediction. Linear case: Zero on linear functions Gives the biorthogonal (2,2) of Cohen-Daubechies- Feauveau Linear time + avoids fitting polynomials.

31 Second Generation Wavelets Vary in space no Fourier analysis Interval boundary conditions Irregular samples Weighted measure Still maintain Stable basis (orthogonal / biorthogonal) Locality in space / time & frequency MRA, cascade construction

32 Second Generation Wavelets Concept of V.M. regularity remains

33 Operator Notation Convolution matrices in case of 1 st Generation Wavelets

34 Lifting Scheme S j parameterizes the lifted wavelets j Use S j to locally design the wavelets Begin with simple wavelets, e.g., the Lazy wavelets

35 Unbalanced Haar a[n] no longer at points, but on intervals a[n] a[n] Constants give zero detail