Wavelet-basedImage Processing

James S. Walker

Department of Mathematics

University of Wisconsin–Eau Claire

To Berlina

1

Some References

• and T.Q. Nguyen, Adaptive scanning

methods for wavelet difference reduction

in lossy image compression. Int’l Conf. on

Image Proc., Vancouver, Sept. 2000. 3, 9,

pp. 182–185.

• and T.Q. Nguyen, Wavelet-based image

compression. Chap. in Handbook of Image

Compression, CRC Press, 2000.

• , Tree-adapted wavelet shrinkage. Ad-

vances in Imaging and Electron Physics,

124, pp. 343–394, 2002.

• Software and Papers:

http://www.uwec.edu/walkerjs

2

Image Processing

• Image Compression

• Denoising

• Image Enhancement

• Image Recognition

• Feature Detection

• Texture Classification

• Image Registration

3

Transform-based Compression

Image → Transform → Quantize,Encode

→CompressedImage

Compression Process

CompressedImage

→ Decode,Estimate

→ InverseTransform

→ Image

Decompression Process

4

Level 1

Level 2

Level 3

Wavelet Transform

5

Lena Lena Histogram

Lena Transform Transform Histogram

6

Desired Features

• Progressive; Embedded

– web pages; database browsing

• Low complexity; Low memory

– narrow bandwidth

• Region of Interest (ROI)

– reconnaissance; medical diagnosis

• Operations on compressed data

– reconnaissance; denoising

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ROI Property

(a) (b) (c)

(b) A 200:1 compression of the image in (a).

(c) An update of the compression, where the

ROI (central quarter sub-image) is exact (loss-

less).

To transmit the image in (c) requires 60,746

bytes. A savings of 4.3 to 1 over the full

262,159 bytes for the original. If an exact

(lossless) compression were done, the savings

would only be 1.4 to 1.

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Compression Methods

• Zerotree methods

– EZW (Shapiro, 1992)

– SPIHT (Said & Pearlman, 1993)

• Difference Reduction methods

– WDR (Tian & Wells, 1995)

– ASWDR (Walker, 2000)

• Block-based methods

* JPG (JPEG Group, 1990)

– GenLOT + Remapping (Nguyen, et al,

1999)

– JPEG2000 (Taubman, et al, 2000)

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SPIHT Algorithm

• Embedded, progressive? Yes

• Region-of-Interest? No

– Even if a zerotree intersects the ROI at

one location, then the full zerotree must

be encoded.

• Operations on compressed data? No

• Low memory? No

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Set Partitioning in Hierarchical

Trees

1. Wavelet transform image.

2. Initialize scan order and threshold.

3. Significance pass. Encode the significance

map using code for transitions from in-

significant (zerotrees) to significant values.

4. Refinement pass. Generate refinement bits

for old significant values (bit-plane encod-

ing).

5. Divide threshold by 2, repeat Steps 3–4.

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ASWDR Algorithm

• Embedded, progressive? Yes

• Region-of-Interest? Yes

• Operations on compressed data? Yes

• Low memory? No

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Adaptively Scanned Wavelet

Difference Reduction

1. Wavelet transform image.

2. Initialize scan order and threshold.

3. Significance pass. Encode new significant

values using difference reduction.

4. Refinement pass. Generate refinement bits

for old significant values (bit-plane encod-

ing).

5. Update scan order to search through coef-

ficients that are more likely to be signifi-

cant at half-threshold.

6. Divide threshold by 2, repeat Steps 3–4.

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Difference Reduction

• Compute binary expansions of number of

steps between significant values (skipping

over old ones). Replace MSB by sign. Use

signs as delimiters between expansions.

• Example. Suppose new significant values

are

x[2] = +17, x[3] = −14, x[14] = +18.

The new values are at indices 2, 3, and

14. The steps between new values are

2 = (10)2, 1 = (1)2, and 11 = (1011)2.

Difference reduction encoding is then

0 + −011+

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Sig. Parents ≈> Sig. Children

(a) (b)

(a) Insignificant children in 1st HL subband

having significant parents, threshold 32.

(b) New significant values in 1st HL subband

when threshold is halved to 16.

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Sig. Parents ≈> Sig. Children

Thresholds

Parent Level σ 128 64 32 16

Lena, 4th 37 0.46 0.57 0.66 0.68

Lena, 3rd 15 0.31 0.50 0.56 0.55

Lena, 2nd 19 0.95 0.51 0.54 0.49

Barbara, 4th 38 0.54 0.60 0.63 0.68

Barbara, 3rd 22 0.09 0.26 0.38 0.51

Barbara, 2nd 12 0.03 0.21 0.37 0.51

Airfield, 4th 66 0.46 0.56 0.61 0.76

Airfield, 3rd 28 0.39 0.46 0.50 0.76

Airfield, 2nd 10 0.30 0.43 0.43 0.38

Noise, 4th 42 0.01 0.18 0.50 0.74

Noise, 3rd 44 0.01 0.19 0.52 0.74

Noise, 2nd 43 0.01 0.21 0.54 0.76

Fraction of new significant values captured by

first part of the new scan order created by

ASWDR. The standard deviations σ are for the

child subbands.

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Create New Scan Order

The scan order is created for each level in the

wavelet transform as follows:

• The first part of the scan order at level j−1

are the insignificant children of significant

parents in level j.

• The second part of the scan order at level

j − 1 are the insignificant children of in-

significant parents, at least one of whose

siblings is significant.

• The third part of the scan order at level

j − 1 are the insignificant children of in-

significant parents, none of whose siblings

are significant.

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Block-based Transforms

(a) (b)

(a) Subbands in a 6-level wavelet transform.

(b) Division of transform values into 64 blocks.

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Blocked Transform Advantages

• Low memory requirements

• Localization of image statistics

– Improved handling of non-stationary

statistics for arithmetic coding

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JPEG 2000

• EBCOT algorithm

– Embedded, Block Coding, Optimal

Truncation

• Optimal Truncation:

– variational problem: encode refinement

bits in order of their decrease of MSE

• Embedded, progressive? Yes

• Region-of-Interest? Yes

• Operations on compressed data? Yes

• Low Memory? Yes

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PSNR

• Peak Signal to Noise Ratio

• Standard Error Measure in Image Process-

ing

PSNR = 10 log10

2552

1N

∑

i,j |f(i, j) − g(i, j)|2

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32:1 Compressions

Original JPG, 25.0 dB

JPEG2000, 27.2 dB ASWDR, 27.1 dB

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32:1 Compressions

Original JPG, 27.0 dB

JPEG2000, 29.1 dB ASWDR, 28.8 dB

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32:1 Compressions

Original SPIHT, 27.5 dB

JPEG2000, 27.2 dB ASWDR, 27.1 dB

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32:1 Compressions

Original SPIHT, 28.1 dB

JPEG2000, 27.6 dB ASWDR, 28.8 dB

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Original JPG, 64:1 (maximum)

JPEG2000, 64:1 ASWDR, 64:1

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Original SPIHT, 128:1

JPEG2000, 128:1 ASWDR, 128:1

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Edge Correlation

• Ratio of the variances (in higher spatial fre-

quencies) of decompressed image to origi-

nal

• More sensitive to image details than PSNR

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Lena 3-Level Transform

Only high-pass values Lena’s edges

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Edge correlation: γ =σ2

c

σ2o

Barb’s edges SPIHT, γ = 0.74

JPG2000, γ = 0.80 ASWDR, γ = 0.81

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Average PSNR Values(Airfield, Barbara, Goldhill, Lena)

CR\Method SPIHT JPG2000 ASWDR

16:1 32.51 31.91 32.22

32:1 29.45 28.87 29.21

64:1 26.92 26.41 26.74

Average Edge Correlations(Airfield, Barbara, Goldhill, Lena)

CR\Method SPIHT JPG2000 ASWDR

16:1 0.88 0.90 0.92

32:1 0.76 0.80 0.81

64:1 0.61 0.62 0.67

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