1. L ECTURE 5 Wavelet Frames: theory, construction and fast algorithms. B. Dong and Z. Shen, MRA-based wavelet frames and applications, IAS Lecture Notes Series, in Mathematics in Image Processing (H. Zhao Eds), 2013.

2. O UTLINE Frames in finite dimensional spaces Multiresolution analysis (MRA) Unitary extension principle (UEP): construction of wavelet frames Fast wavelet frame transforms Multivariate wavelet frame systems

3. F RAMES IN FINITE DIMENSIONAL SPACES Characterization frames and importance of redundancy

4. T IGHT F RAMES IN Orthonormal basis Riesz basis Tight frame: Mercedes-Benz frame Expansions: Unique Not unique

5. T IGHT F RAMES IN F INITE D IMENSIONAL V ECTOR S PACES Operators: Analysis operator Synthesis operator Frame operator

6. T IGHT F RAMES IN F INITE D IMENSIONAL V ECTOR S PACES When the frame is a tight frame, then. Due to redundancy, the representation of elements in is not unique.

7. W HY R EDUNDANCY ? Conditions for frames is more flexible than basis, which makes it possible to construct systems satisfy prescribed properties. More robust to errors: e.g. signal transmission Given, we are sending Received: and Recall The operator has nontrivial null space, and the dimension of the null space increases when redundancy increases.

8. W HY R EDUNDANCY ? How to quantify? Define mean square error (MSE): Redundancy is good, so are tight frames

9. M ULTIRESOLUTION A NALYSIS (MRA) Tight frames in L2 space, MRA, refinable functions, approximation of operators

10. T IGHT F RAMES OF General tight frame system: Tight wavelet frames: given When elements of are orthogonal to each other, or equivalently each element has norm 1, it is called an orthonormal wavelet bases. They are redundant systems satisfying Parseval’s identity Or equivalently Where or Wavelet/Affine system Quasi-Affine system

11. G ENERAL C HARACTERIZATION Beautiful but not useful…

12. M ULTIRESOLUTION A NALYSIS (MRA)

13. Refinable function and its refinement mask : Equivalently in Fourier domain If is refinable, then (1.1)-(i) and (iii) are satisfied If is compactly supported and, then (1.1)-(ii) is satisfied where

14. M ULTIRESOLUTION A NALYSIS (MRA) Properties of generators of MRA o Stable: o Orthonormal: Bessel property

15. M ULTIRESOLUTION A NALYSIS (MRA) Approximate a function from : quasi-interpolator operator:

16. M ULTIRESOLUTION A NALYSIS (MRA) Fourier domain:

17. M ULTIRESOLUTION A NALYSIS (MRA) Linear (m=2) and Cubic (m=4)

18. MRA-B ASED C ONSTRUCTION OF T IGHT W AVELET F RAMES Unitary extension principle for affine system and fast algorithms

19. MRA- BASED T IGHT W AVELET F RAMES Define wavelets by wavelet masks: Question: Given a refinable function and its associated refinement mask, how to construct a tight wavelet frame system ? Answer: Unitary extension principle (UEP) Or

20. UEP FOR A FFINE S YSTEMS

30. V ANISHING M OMENTS An equivalent characterization: Then, we have Therefore, short support + high order of vanishing moment means sparse approximation. However, you cannot have both in general.

31. A PPROXIMATION OF T IGHT F RAME S YSTEM o o o o Key observation:

32. E XAMPLES B-spline tight wavelet frames:

33. E XAMPLES Pseudo-spline tight wavelet frames: In particular when, we have Daubechies’ orthonormal wavelets where Square root by Fejer-Riesz lemma

34. O RTHONORMAL W AVELETS UEP Theorem continued: Daubechies’ construction: Additional condition to ensure orthogonality: stability

35. O RTHONORMAL W AVELETS Cohen’s criteria continued Example of K Meaning of the criteria  

36. F AST T RANSFORMS FOR A FFINE S YSTEMS Transition from continuum to discrete. o Recall: o We under stand discrete data takes the form o Underlying function is approximated by

37. F AST T RANSFORMS FOR A FFINE S YSTEMS Transition from continuum to discrete. o Define operators (infinite matrix) on sequence space o Perfect reconstruction:

38. F AST T RANSFORMS FOR A FFINE S YSTEMS    

41. E XAMPLES Multi-level decomposition of a 1D signal

42. E XAMPLES Analyzing singularities [S. Mallat, A wavelet tour of signal processing, 2009]

43. E XAMPLES Wavelet-version: Localized singularity analysis:

44. E XAMPLES Localized singularity analysis

45. E XAMPLES Self-similar function analysis

46. MRA-B ASED C ONSTRUCTION OF T IGHT W AVELET F RAMES Unitary extension principle for quasi-affine system and fast algorithms

47. UEP FOR Q UASI -A FFINE S YSTEMS

54. F AST T RANSFORMS FOR Q UASI -A FFINE S YSTEMS Transition from continuum to discrete. o Recall: o Perfect reconstruction

55. F AST T RANSFORMS FOR Q UASI -A FFINE S YSTEMS Transition from continuum to discrete. o Define operators (infinite matrix) on sequence space o Perfect reconstruction:

56.  F AST T RANSFORMS FOR Q UASI -A FFINE S YSTEMS

57. A DVANTAGE OF Q UASI -A FFINE S YSTEMS Translation-invariant thresholding Original Noisy

58. A DVANTAGE OF Q UASI -A FFINE S YSTEMS Translation-invariant thresholding AffineQuasi-Affine

59. G ENERALIZATION TO M ULTIVARIATE T IGHT W AVELET F RAME S YSTEMS Tensor product

60. H IGHER D IMENSIONS : T ENSOR P RODUCT Given univariate masks, define Then we have tensor product wavelet system with Then is a tight wavelet frame whenever the univariate counterpart is. Or equivalently and

61. 2D F AST T RANSFORMS

64. S PARSE A PPROXIMATION Sparsity and sparse approximation

65. S PARSITY AND S PARSE A PPROXIMATION What is sparsity? DCT I II

66. S PARSITY AND S PARSE A PPROXIMATION Sparse approximation IDCT

67. S PARSITY AND S PARSE A PPROXIMATION DCT Sparse approximation

68. S PARSITY AND S PARSE A PPROXIMATION IDCT Sparse approximation Different signals may have rather different sparse approximation

69. S PARSITY AND S PARSE A PPROXIMATION Wavelet Frame Transform Keeping 1% largest coefficients Apply inverse wavelet transform Sparse approximation of images

70. O THER S PARSE R EPRESENTATIONS FOR I MAGES Short-Time Fourier Transform: Curvelets: Bandlets, shearlets, contourlets, complex wavelets, etc… Where and