wavelet for view

7/31/2019 Wavelet for View

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ee.sharif.edu/~dip

E. Fatemizadeh, Sharif University of Technology, 2011

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Wavelets and Multi Resolution Processing

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Ifyou painted a picture with a sky,

clouds, trees, and flowers, you would usea different size brush depending on the

size of the features. Wavelets are like those

brushes.

Ingrid Daubechies


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Image: A non-stationary Phenomenon


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Image Pyramid


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Gaussian (up) and Laplacian (down) Pyramid


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Subband Coding (1D)


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Subband Coding (2D)


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Four-band Split:

A(LL): Approximation

H(HL): Horizontal

V(LH): Vertical D (HH): Diagonal

A H

V D


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Multi-Level Decomposition:

Haar Basis function


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Two Stage FWT* Analysis:

*: Fast Wavelet Transform


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Two Stage FWT Synthesis:


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2D FWT:

Analysis

Decomposition

Synthesis


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Three Scale FWT:

Approximation

Horizontal Edge

Vertical Edge Details


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Modifying DWT*:

Edge Detection

*: Discrete Wavelet Transform


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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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Wavelet Packet Analysis:

3 scale full analysis three


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Example (1):

Full wavelet packet decomposition


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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 (.,): Thresholding operator ( being the threshold)

X t S t N t

1,

Y W X

Z D Y

S W Z


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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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Hard and Soft Thresholding:

Hard:

Soft:

,0

u uD u

u

, 0

u u

D u u

u u


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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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VisuShrink (Universal Thresholding):

N: Sample (Signal/Image) size (# of pixels in image)

: Noise variance

Thresholding sub-band:

All

Details (HL,LH, HH) Noise Estimation:

2N Ln N

2N

2

2

1 ,

0.6745

ij

N ij

median YY HH


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SureShrink (Adaptive Thresholding):

Sub-band adaptive thresholding (each detail sub-band)

Based on Steins Unbiased Estimator for Risk (SURE), amethod for estimating the loss in an unbiased fashion.

:Wavelet coefficients in thejth sub-band

For the soft threshold estimator:

We have:

Optimal threshold:

1

d

i iY

,i iY D Y

2

1

( ; ) 2# : min ,d

i i

i

SURE Y d i Y Y

* arg min ( ; )SURE Y


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NormalShrink:

For maximumJ scale and for scale k:

2

2

2

1

2

,0.6745

: Standard deviation of the considered subband

: Length of subband k

Nk

Y

k

ij

N ij

Y

LLn

J

median YY HH

Lk


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Challenges:

Wavelet base.

Threshold Selection.

Threshold function.

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