wavelet-based image processing(b) a 200:1 compression of the image in (a). (c) an update of the...
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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
7
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.
8
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)
9
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
10
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.
11
ASWDR Algorithm
• Embedded, progressive? Yes
• Region-of-Interest? Yes
• Operations on compressed data? Yes
• Low memory? No
12
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.
13
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+
14
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.
15
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.
16
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.
17
Block-based Transforms
(a) (b)
(a) Subbands in a 6-level wavelet transform.
(b) Division of transform values into 64 blocks.
18
Blocked Transform Advantages
• Low memory requirements
• Localization of image statistics
– Improved handling of non-stationary
statistics for arithmetic coding
19
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
20
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
21
32:1 Compressions
Original JPG, 25.0 dB
JPEG2000, 27.2 dB ASWDR, 27.1 dB
22
32:1 Compressions
Original JPG, 27.0 dB
JPEG2000, 29.1 dB ASWDR, 28.8 dB
23
32:1 Compressions
Original SPIHT, 27.5 dB
JPEG2000, 27.2 dB ASWDR, 27.1 dB
24
32:1 Compressions
Original SPIHT, 28.1 dB
JPEG2000, 27.6 dB ASWDR, 28.8 dB
25
Original JPG, 64:1 (maximum)
JPEG2000, 64:1 ASWDR, 64:1
26
Original SPIHT, 128:1
JPEG2000, 128:1 ASWDR, 128:1
27
Edge Correlation
• Ratio of the variances (in higher spatial fre-
quencies) of decompressed image to origi-
nal
• More sensitive to image details than PSNR
28
Lena 3-Level Transform
Only high-pass values Lena’s edges
29
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
31
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