[PDF] - Chapter 3 Tight-frame Applications 1 Outline 1. Inpainting 1. PDF Document

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Tight-frame Applications Chapter 3

1. Inpainting
2. Impulse Noise Removal
3. Inpainting in the Transformed Domain

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1. Inpainting
2. Impulse Noise Removal
3. Inpainting in the Transformed Domain

Outline

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Inpainting

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Notations

noise set data set

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Variational Approach

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Tight Frame Algorithm

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Tight Frame Algorithm

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Numerical Test 1: 512-by-512 Lena

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1dB increase = error decreases 10%

Numerical Results 1

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Numerical Test 2: 512-by-512 Lena

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Numerical Results 2

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Numerical Test 3: 256-by-256 Pepper

Text with even bigger font

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Numerical Results 3

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Results Up-close

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Results Up-close

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Advantages of Tight Frame Algorithm

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An Equivalent Formulation

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Minimization Functional

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1. Inpainting
2. Impulse Noise Removal
3. Inpainting in the Transformed Domain

Outline

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Impulse noise removal

Goal

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Impulse Noise Model

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Impulse Noise Model

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Salt-and-Pepper Noise

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Salt-and-Pepper Noise

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Random-Valued Impulse Noise

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Denoising Schemes

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30% Salt-and-Pepper Noise

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Median-type Filters

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Adaptive Median Filter

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30% Salt-and-Pepper Noise

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But …at 70% Salt-and-Pepper Noise

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Characteristics of Median-type Filters

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Variational Method

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l1 Fitting Term for Impulse Noise:

(Nikolova, J. Math. Imaging & Vision, (2004))

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l1 Fitting Term for Impulse Noise:

(Nikolova, J. Math. Imaging & Vision, (2004))

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Two-Phase Method:

(Chan, Ho, and Nikolova, IEEE TIP (2005))

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Two-Phase Method:

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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 Median Filter

25.73 21.76 21.46 21.38

Our Method

29.26 25.00 26.94 24.91

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Comparison

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Two-Phase Method using Framelets

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70% Salt-&-Pepper Noise

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Numerical Results

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90% Salt-&-Pepper Noise

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Numerical Results

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25 Image Noise level Variational Framelet 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 256x256 24.8 25.8 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

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Comparison with Variational Method

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Random-Valued Impulse Noise

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A New Noise Detector (ROLD)

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1. Inpainting
2. Impulse Noise Removal
3. Inpainting in the Transformed Domain

Outline

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noise set data set

Framework for Missing Data Recovery

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Framework for Missing Data Recovery

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Tight Frame Algorithm

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Convergence

(Cai, C., Shen, ACHA 2008)

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Extension to Frequency Domain Inpainting

 data set

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Tight Frame Algorithm

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Convergence Results

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Infrared Imaging

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Chop-and-Nod Procedure

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Chop and Nod

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Minimization Properties

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Observed Image from United Kingdom Infrared Telescope Reconstruction by Projected Landweber’s Iteration Reconstruction by Framelet- Based Method

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Original Chopped & Nodded Landweber Framelet

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Inpainting in Image and Frequency Domains

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Inpainting on Cartoon and Texture