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

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

ee.sharif.edu/~dip

E. Fatemizadeh, Sharif University of Technology, 20113

Digital Image Processing

Wavelets and Multi Resolution Processing

3

• Image Pyramid

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

ee.sharif.edu/~dip

E. Fatemizadeh, Sharif University of Technology, 20115

Digital Image Processing

Wavelets and Multi Resolution Processing

5

• Subband Coding (1D)

ee.sharif.edu/~dip

E. Fatemizadeh, Sharif University of Technology, 20116

Digital Image Processing

Wavelets and Multi Resolution Processing

6

• Subband Coding (2D)

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

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

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

ee.sharif.edu/~dip

E. Fatemizadeh, Sharif University of Technology, 201110

Digital Image Processing

Wavelets and Multi Resolution Processing

10

• Two Stage FWT Synthesis:

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

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

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

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

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 

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

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)

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.

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:

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• …

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:

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:

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:

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.