digital image processing ee.sharif.edu/~dip
TRANSCRIPT
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20111
Digital Image Processing
Wavelets and Multi Resolution Processing
1
“If you painted a picture with a sky,clouds, trees, and flowers, you would usea different size brush depending on thesize of the features. Wavelets are like thosebrushes.”
Ingrid Daubechies
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20112
Digital Image Processing
Wavelets and Multi Resolution Processing
2
• Image: A non‐stationary Phenomenon
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20113
Digital Image Processing
Wavelets and Multi Resolution Processing
3
• Image Pyramid
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20114
Digital Image Processing
Wavelets and Multi Resolution Processing
4
• Gaussian (up) and Laplacian (down) Pyramid
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20115
Digital Image Processing
Wavelets and Multi Resolution Processing
5
• Subband Coding (1D)
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20116
Digital Image Processing
Wavelets and Multi Resolution Processing
6
• Subband Coding (2D)
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20117
Digital Image Processing
Wavelets and Multi Resolution Processing
7
• Four‐band Split:– A(LL): Approximation– H(HL): Horizontal– V(LH): Vertical– D (HH): Diagonal
A H
V D
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20118
Digital Image Processing
Wavelets and Multi Resolution Processing
8
• Multi‐Level Decomposition:– Haar Basis function
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 20119
Digital Image Processing
Wavelets and Multi Resolution Processing
9
• Two Stage FWT* Analysis:
*: Fast Wavelet Transform
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201110
Digital Image Processing
Wavelets and Multi Resolution Processing
10
• Two Stage FWT Synthesis:
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201111
Digital Image Processing
Wavelets and Multi Resolution Processing
11
• 2D FWT:– Analysis– Decomposition– Synthesis
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201112
Digital Image Processing
Wavelets and Multi Resolution Processing
12
• Three Scale FWT:– Approximation– Horizontal Edge– Vertical Edge– Details
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201113
Digital Image Processing
Wavelets and Multi Resolution Processing
13
• Modifying DWT*:– Edge Detection
*: Discrete Wavelet Transform
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201114
Digital Image Processing
Wavelets and Multi Resolution Processing
14
• Modifying DWT*:– Noise Reduction, Denoising:
• Compute DWT of Noisy Image• Thresholding the details!• Compute IDWT of alerted coefficients
– We will discuss more, later.
*: Discrete Wavelet Transform
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201115
Digital Image Processing
Wavelets and Multi Resolution Processing
15
• Wavelet Packet Analysis:– 3 scale full analysis three
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201116
Digital Image Processing
Wavelets and Multi Resolution Processing
16
• Example (1):– Full wavelet packet decomposition
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201117
Digital Image Processing
Wavelets and Multi Resolution Processing
17
• Image/Signal Denoising:– Noisy image/signal model:
• X(t): Corrupted Signal• S(t): Uncorrupted Signal• N(t): Additive Noise
– Wavelet Denoising Scheme
• W, W‐1: Forward and Inverse wavelet transform• D (.,λ): Thresholding operator (λ being the threshold)
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201118
Digital Image Processing
Wavelets and Multi Resolution Processing
18
• Motivation for Thresholding:– Small coefficients: Dominated by noise.– Large coefficients: Dominated by signal.– Then replacing small coefficients with zero!
• Some Assumption:– Wavelet de‐correlating property generate a sparse signal.– Noise spreads out equally along all coefficients.– The noise level is NOT too high.
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201119
Digital Image Processing
Wavelets and Multi Resolution Processing
19
• Hard and Soft Thresholding:– Hard:
– Soft:
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201120
Digital Image Processing
Wavelets and Multi Resolution Processing
20
• Threshold Selection:– The most important question.– Very Low threshold: Noisy‐Like result– Very High Threshold: Too smooth result.– Several methods proposed:
• VisuShrink• SureShrink• …
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201121
Digital Image Processing
Wavelets and Multi Resolution Processing
21
• VisuShrink (Universal Thresholding):
– N: Sample (Signal/Image) size (# of pixels in image)– : Noise variance
• Thresholding sub‐band:– All– Details (HL,LH, HH)
• Noise Estimation:
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201122
Digital Image Processing
Wavelets and Multi Resolution Processing
22
• SureShrink (Adaptive Thresholding):– Sub‐band adaptive thresholding (each detail sub‐band)– Based on Stein’s Unbiased Estimator for Risk (SURE), a method for estimating the loss in an unbiased fashion.
– :Wavelet coefficients in the jth sub‐band– For the soft threshold estimator:
– We have:
– Optimal threshold:
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201123
Digital Image Processing
Wavelets and Multi Resolution Processing
23
• NormalShrink:– For maximum J scale and for scale k:
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ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 201124
Digital Image Processing
Wavelets and Multi Resolution Processing
24
• Challenges:– Wavelet base.– Threshold Selection.– Threshold function.