Chapter 2 Tight-frames An Introduction 1 Outline 1. Tight-frame - PDF document

Chapter 2 Tight-frames An Introduction 1 Outline 1. Tight-frame 1. Tight-frame 2. Matrix Representation 2. Matrix Representation 2 1

Tight Frames 3 Construction of Haar Wavelet = + 4 2

Construction of Haar Wavelet 5 Construction of Haar Wavelet 6 3

Haar Function ... 7 Haar Wavelet Filters 8 4

Piecewise Linear Tight Frame Piecewise Linear Refinable Function φ φ (x)= φ (2x+1)/2+ φ (2x)+ φ (2x−1)/2 1 1 0.8 0.8 0.6 0.6 φ (2x) 0.4 0.4 0.2 0.2 φ (2x−1) /2 φ (2x+1) /2 0 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 9 Piecewise Linear Tight Frame 10 5

Piecewise Linear Tight Frame 0.8 ψ 1 (x)=0.707 φ (2x+1)−0.707 φ (2x−1) Piecewise Linear Refinable Function φ 0.6 1 0.4 0.8 0.707 φ (2x+1) 0.2 0.6 0 −0.707 φ (2x−1) −0.2 0.4 −0.4 0.2 −0.6 0 −0.8 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 11 Piecewise Linear Tight Frame ψ 2 (x)=− φ (2x+1)/2+ φ (2x)− φ (2x−1)/2 Piecewise Linear Refinable Function φ 1 1 0.8 0.8 0.6 φ (2x) 0.4 0.6 0.2 0.4 0 −0.2 − φ (2x+1)/2 0.2 − φ (2x−1) /2 −0.4 0 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 12 6

Piecewise Linear Tight Frame 13 Unitary Extension Principle 14 7

Unitary Extension Principle 15 1D Piecewise Linear Tight Frame 16 8

1D Piecewise Linear Tight Frame 17 Outline 1. Tight-frame 2. Matrix Representation 18 9

Matrix Representation 19 Matrix Representation 20 10

Analysis and Synthesis Operators 21 Important Observation Tight Frames = Redundant Bases 22 11

Perfect Reconstruction Formula 23 Multi-level Decomposition without Down-sampling 24 12

Multi-level Decomposition without Down-sampling 25 Multi-level Decomposition 26 13

2D Tight Frame 27 2D Piecewise Linear Framelets 28 14

1D Piecewise Cubic Tight Frame 29 1D Piecewise Cubic Tight Frame 30 15

Tight Frames 31 Spline Framelet Systems 32 16