Chapter 3 Tight-frame Applications 1 Outline 1. Inpainting 1. - PDF document

Chapter 3 Tight-frame Applications 1 Outline 1. Inpainting 1. Inpainting 2. Impulse Noise Removal 2. Impulse Noise Removal 3. Inpainting in the Transformed Domain 3. Inpainting in the Transformed Domain 2 1

Inpainting 3 Notations noise set data set 4 2

Variational Approach 5 Tight Frame Algorithm 6 3

Tight Frame Algorithm 7 Numerical Test 1: 512-by-512 Lena 8 4

Numerical Results 1 1dB increase = error decreases 10% 9 Numerical Test 2: 512-by-512 Lena 10 5

Numerical Results 2 11 Numerical Test 3: 256-by-256 Pepper Text with even bigger font 12 6

Numerical Results 3 13 Results Up-close 14 7

Results Up-close 15 Advantages of Tight Frame Algorithm 16 8

An Equivalent Formulation 17 Minimization Functional 18 9

Outline 1. Inpainting 2. Impulse Noise Removal 3. Inpainting in the Transformed Domain 19 Goal Impulse noise removal 20 10

Impulse Noise Model 21 Impulse Noise Model 22 11

Salt-and-Pepper Noise 23 Salt-and-Pepper Noise 24 12

Random-Valued Impulse Noise 25 Denoising Schemes 26 13

30% Salt-and-Pepper Noise 27 Median-type Filters 28 14

Adaptive Median Filter 29 30% Salt-and-Pepper Noise 30 15

But …at 70% Salt-and-Pepper Noise 31 Characteristics of Median-type Filters 32 16

Variational Method 33 l 1 Fitting Term for Impulse Noise: (Nikolova, J. Math. Imaging & Vision , (2004)) 34 17

l 1 Fitting Term for Impulse Noise: (Nikolova, J. Math. Imaging & Vision , (2004)) 35 Two-Phase Method: (Chan, Ho, and Nikolova, IEEE TIP (2005)) 36 18

Two-Phase Method: 37 38 19

39 40 20

41 Comparison Salt-and-Pepper Lena Bridge Goldhill Cameraman Noisy 6.71 6.78 6.93 6.63 Variational Method 24.64 21.11 23.54 20.69 Adaptive 25.73 21.76 21.46 21.38 Median Filter Our Method 29.26 25.00 26.94 24.91 42 21

Two-Phase Method using Framelets 43 70% Salt-&-Pepper Noise 44 22

Numerical Results 45 90% Salt-&-Pepper Noise 46 23

Numerical Results 47 48 24

Comparison with Variational Method Noise Variational Framelet Image level 50% 30.5 31.3 Lena 256x256 70% 27.4 28.8 90% 22.9 24.2 50% 33.1 33.8 Lena 512x512 70% 29.7 31.2 90% 25.4 26.5 Cameraman 24.8 25.8 256x256 Goldhill 512x512 29.9 30.0 Boat 512x512 70% 28.0 29.1 Barbara 512x512 24.6 25.7 Bridge 512x512 24.7 24.7 49 Random-Valued Impulse Noise 50 25

51 52 26

A New Noise Detector (ROLD) 53 Outline 1. Inpainting 2. Impulse Noise Removal 3. Inpainting in the Transformed Domain 54 27

Framework for Missing Data Recovery noise set data set 55 Framework for Missing Data Recovery 56 28

Tight Frame Algorithm 57 Convergence (Cai, C., Shen, ACHA 2008) 58 29

Extension to Frequency Domain Inpainting data set  59 Tight Frame Algorithm 60 30

Convergence Results 61 Infrared Imaging 62 31

Chop-and-Nod Procedure 63 Chop and Nod 64 32

65 Minimization Properties 66 33

Reconstruction Observed Image Reconstruction by Projected from United by Framelet- Landweber’s Kingdom Infrared Based Method Iteration Telescope 67 Original Chopped & Nodded Landweber Framelet 68 34

Inpainting in Image and Frequency Domains 69 Inpainting on Cartoon and Texture 70 70 35

Inpainting on Cartoon and Texture corrupted cartoon texture 71 71 36