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InstitutMines-Telecom
Wavelet-based imagecompression
Marco Cagnazzo
Multimedia Compression
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IntroductionDWT and MRA
Images compression with wavelets
Outline
Introduction
Discrete wavelet transform and multiresolution analysisFilter banks and DWTMultiresolution analysis
Images compression with waveletsEZWJPEG 2000
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IntroductionDWT and MRA
Images compression with wavelets
Outline
Introduction
Discrete wavelet transform and multiresolution analysis
Images compression with wavelets
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IntroductionDWT and MRA
Images compression with wavelets
Wavelets and images : Motivations
◮ Image model : trends + anomalies
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IntroductionDWT and MRA
Images compression with wavelets
Wavelets and images : Motivations
◮ Image model : trends + anomalies
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IntroductionDWT and MRA
Images compression with wavelets
Wavelets and images : Motivations
◮ Image model : trends + anomalies
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IntroductionDWT and MRA
Images compression with wavelets
Wavelets and images : Motivations
◮ Anomalies :◮ Abrupt variations of the signal◮ High frequency contributions◮ Objects’ contours◮ Good spatial resolution◮ Rough frequency resolution
◮ Trends :◮ Slow variations of the signal◮ Low frequency contributions◮ Objects’ texture◮ Rough spatial resolution◮ Good frequency resolution
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IntroductionDWT and MRA
Images compression with wavelets
Wavelets and images : Motivations
Time-frequency boxes of basis signalsTime analysis:θγ(t) = δ(t − t0)Frequency analysis:θγ(t) = e2iπf0t
Short Time Fourier Transform(STFT):
θγ(t) = g(t − t0)e2iπf0t
σt , σξ independent from γ
Wavelet Transform:
θγ(t) =1√
aψ
(
t − ba
)
σt(a,b) = aσt,(1,0) σξ(a,b) = a−1σξ,(1,0)
Time Analysis Frequency Analysis
STFT Wavelet Transform
T
T T
T
F
FF
F
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IntroductionDWT and MRA
Images compression with wavelets
Wavelets and images : Motivations
Signal model: an image row
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IntroductionDWT and MRA
Images compression with wavelets
Wavelets and images : Motivations
Signal model: an image row
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IntroductionDWT and MRA
Images compression with wavelets
Wavelets and images : Motivations
Signal model: an image row
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IntroductionDWT and MRA
Images compression with wavelets
Wavelets and Multiple resolution analysis
◮ Approximation: low resolution version◮ “Details”: zeros when the signal is polynomial
0 50 100 150 200 250 300 350 400 450 500−20
0
20
40
60
0 50 100 150 200 250 300 350 400 450 500
−20
0
20
40
60
80
c1[k ] d1[k ]
x [k ] = c0[k ]
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Outline
Introduction
Discrete wavelet transform and multiresolution analysisFilter banks and DWTMultiresolution analysis
Images compression with wavelets
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
1D filter banks
Decomposition
c[k ] c[k ]
d [k ] d [k ]
x [k ]
g
h ↓ 2
↓ 2
Analysis filter bank
2 ↓ : decimation : c[k ] = c[2k ]
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Reconstruction
c[k ]c[k ]
d [k ]d [k ]
x [k ]
g
h↑ 2
↑ 2
Synthesis filter bank
2 ↑ : interpolation operator, doubles the sample number
c[k ] =
{c[k/2] if k is even
0 if k is odd
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Filter properties
◮ Perfect reconstruction (PR)◮ FIR◮ Orthogonality◮ Vanishing moments◮ Symmetry
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Perfect reconstruction conditions
We want PR after synthesis and analysis filter banks :∀k ∈ Z,
xk = xk+ℓ ⇐⇒ X (z) = z−ℓX (z)
c[k ] c[k ]
d [k ] d [k ]
x [k ]
g
h
g
h↓ 2
↓ 2
c[k ]
d [k ]
x [k ]
↑ 2
↑ 2
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Z-domain relationships
filter C (z) =∞∑
n=−∞
cnz−n = H (z)X (z)
decimation C (z) =12
[C(
z1/2)+ C
(−z1/2
)]
interpolation C (z) = C(
z2)
output X (z) = H (z)C(
z2)+ G (z)D
(z2)
X (z) =12
[H (z)H (z) + G (z)G (z)
]X (z)
+12
[H (z)H (−z) + G (z)G (−z)
]X (−z)
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
PR conditions in Z
∀k ∈ Z,
xk = xk+ℓ ⇐⇒ X (z) = z−ℓX (z)
m
H (z)H (z) + G (z)G (z) = 2z−ℓ Non distortion
H (z)H (−z) + G (z)G (−z) = 0 Non aliasing
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Perfect reconstruction conditions
Matrix form
For simplicity, we ignore the delay, ℓ = 0If the analysis filter bank is given, the synthesis one isdetermined by:
[H (z) G (z)
H (−z) G (−z)
]·
[H (z)G (z)
]=
[20
]
We assume that the modulation matrix is invertible.
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Perfect reconstruction conditions
Synthesis filter bank
Modulation matrix determinant :
∆(z) = H (z)G (−z)−G (z)H (−z)
H (z) =2
∆(z)G (−z)
G (z) = −2
∆(z)H (−z)
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Perfect reconstruction with FIR filters
Finite impulse response filters:It can be shown that in this case the PR condition is equivalentto the alterning signs condition. Example:
h(k) = a b c h(k) = p -q r -s t
g(k) = p q r s t g(k) = -a b -c
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Orthogonality
Orthogonality assures energy conservation:
∞∑
k=−∞
(xk )2 =
∞∑
k=−∞
(ck )2 +
∞∑
k=−∞
(dk )2
⇒ reconstruction error = quantization error on DWT coefficientsFor non orthogonal filters, the reconstruction errors is aweighted sum of the quantization errors on the DWT subbands,with suitable weights ωi
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Vanishing moments
◮ Vanishing moments (VM) represent filter ability toreproduce polynomials: a filter with p VM can representpolynomials with degree < p
◮ The High-pass filter will not respond to a polynomial inputwith degree < p
◮ In this case all the signal information is preserved in theapproximation signal (half the samples)
◮ A filter with p VM has at least 2p taps
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Borders problem
◮ Filterbank properties such as we saw, are valid forinfinite-size signals
◮ We are interested in finite support signals◮ How to interpret the previous results for finite support
signals?
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Borders problem
◮ Filterbank properties such as we saw, are valid forinfinite-size signals
◮ We are interested in finite support signals◮ How to interpret the previous results for finite support
signals?◮ Zero padding would introduce a coefficient expansion◮ Filtering an N-size signal with an M-size produces a signal
with size N + M − 1
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Borders problem
◮ Filterbank properties such as we saw, are valid forinfinite-size signals
◮ We are interested in finite support signals◮ How to interpret the previous results for finite support
signals?◮ Zero padding would introduce a coefficient expansion◮ Filtering an N-size signal with an M-size produces a signal
with size N + M − 1◮ Periodization?◮ Symmetrization?
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Borders problem: Coefficient expansion
−10 0 10 20 30
−2
0
2
4
6
8
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12
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x
−10 0 10 20 30
−0.2
0
0.2
0.4
0.6
0.8
h
−10 0 10 20 30
−2
0
2
4
6
8
10
12
14
16
y = h ∗ x
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Borders problem: Periodization
◮ A signal x of support N is considered as a periodic signal xof period N
◮ Filtering x with h results into a periodic output y◮ y has the same period N as x◮ So we need to compute just N samples of y◮ However, periodization introduces “jumps” in a regular
signal
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Borders problem: Periodization
−10 0 10 20 30
−2
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x
−10 0 10 20 30
−2
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y
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Borders problem: Symmetry
◮ Symmetrization before periodization reduces the impact onsignal regularity
◮ But it doubles the number of coefficients...
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Borders problem: Symmetry
◮ Symmetrization before periodization reduces the impact onsignal regularity
◮ But it doubles the number of coefficients...◮ Unless the filters are symmetric, too
◮ We use x as half-period of xs◮ xs has a period of 2N samples◮ Filtering xs with h, produces ys◮ If h is symmetric, ys is periodic and symmetric, with period
2N: we only need to compute N samples
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Borders problem: Symmetry
−10 0 10 20 30
−2
0
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4
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12
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xs
−10 0 10 20 30
−2
0
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ys
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Haar filter
h(k) = 1 1 h(k) = 1 1
g(k) = 1 -1 g(k) = -1 1
◮ Symmetric◮ Orthogonal◮ VM = 1
◮ Only capable to represent piecewise constant signals
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Summary: perfect reconstruction andborders
◮ Convolution involves coefficient expansion◮ Solution: circular convolution
◮ Circular convolution allows to reconstruct an N-samplessignal with N wavelet coefficients
◮ The periodization generates borders discontinuities, i.e.spurious high frequencies coefficients that demand a lot ofcoding resources
◮ Solution: Symmetric periodization◮ No discontinuities◮ Does it double the coefficient number?◮ No, if the filter is symmetric !
Bad news: the only orthogonal symmetric FIR filter is Haar!
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Biorthogonal filters
Cohen-Daubechies-Fauveau filters
With biorthogonal (i.e. PR) filters, if h has p VM and h has pVM, the filter has at least p + p − 1 taps.The CDF filters have the following properties:
◮ They are symmetric (linear phase)◮ They maximize the VM for a given filter length◮ They are close to orthogonality (weights ωi are close to
one)
They are by far the most popular in image compression
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
9/7 biorthogonal filters
Filter coefficients:
n 0 ±1 ±2 ±3 ±4
h[l] 0.852699 0.377403 −0.110624 −0.023849 0.037828h[l] 0.788486 0.418092 −0.040689 −0.064539
Impulse response of low-pass filters
For high pass filters, we have (alterning sign condition):
g[l] = (−1)l+1 h [l − 1] and g[l] = (−1)l−1 h [l + 1] .
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
1D Multiresolution analysis
Decomposition
c0[k ]
c1[k ]
c2[k ]
c3[k ]
d1[k ]
d2[k ]
d3[k ]g
g
g
h
h
h
2 ↓
2 ↓
2 ↓
2 ↓
2 ↓
2 ↓
Three level wavelet decomposition structure
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Reconstruction
c0[k ]
c1[k ]
c2[k ]
c3[k ]c
d1[k ]
d2[k ]
d3[k ]
g
g
g
h
h
h
2 ↑
2 ↑
2 ↑
2 ↑
2 ↑
2 ↑
Reconstruction from wavelet coefficients
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
2D AMR
2D Filter banks for separable transform
One decomposition level
aj [n,m]
aj+1[n,m]
dH
j+1[n,m]
dV
j+1[n,m]
dD
j+1[n,m]
h[k ]
g[k ]
h[ℓ]
h[ℓ]
g[ℓ]
g[ℓ]
(2,1) ↓
(2,1) ↓
(1,2) ↓
(1,2) ↓
(1,2) ↓
(1,2) ↓
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
2D-DWT subbands: orientations
fver
fhor(A) (V)
(H) (D)
(A), (H), (V) and (D) respectively correspond to approximationcoefficients, horizontal, vertical and diagonal detail coefficients.
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
2D AMR: multiple levels
a0
a1
dH
1dV
1dD
1
a2
dH
2dV
2dD
2
a3
dH
3dV
3dD
3
Three levels of separable 2D-AMR.
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
2D-DWT subbands: orientations
fver
fhor
(A)
(V1)
(H1) (D1)
(V2)
(H2) (D2)
(V3)
(H3) (D3)
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Example
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Example
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Example
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Example
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Example
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Example
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Example
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Outline
Introduction
Discrete wavelet transform and multiresolution analysis
Images compression with waveletsEZWJPEG 2000
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Compression with DWT
◮ Methods based on inter-scale dependencies:
◮ EZW (Embedded Zerotrees of Wavelet coefficients),◮ SPIHT (Set Partitioning in Hierarchical Trees)◮ Tree-based representation of dependencies◮ Advantages: good exploitation of inter-scale dependencies,
low complexity◮ Disadvantage: no resolution scalability
◮ Methods not based on inter-scale dependencies
◮ Explicit bit-rate allocation among subbands◮ Entropy coding of coefficients◮ Advantages: Good exploitation of intra-scale dependencies,
random access, resolution scalability◮ Disadvantage: no exploitation of inter-scale dependencies
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Embedded Zerotrees of Wavelet coefficients
Main characteristics
◮ Quality scalability (i.e. progressive representation)
◮ Lossy-to-lossless coding
◮ Small complexity
◮ Rate-distortion performance much better than JPEG above allat small rates
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Progressive representation of DWTcoefficients
◮ Each new coding bit must convey the maximum ofinformation
m
◮ Each new coding bit must reduce as much as possibledistortion
◮ We first send the largest coefficients◮ Problem: localization overhead
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Example: an image and its waveletcoefficients
LL HL
LH HH
ApproximationApproximation
Horizontal detailsHorizontal details
Vertical detailsVertical details
Diagonal detailsDiagonal details
DWT 2 further levels
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Progressive representation: subband order
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
EZW Algorithm
◮ The subband scan order alone is not enough to assure thatlargest coefficients are sent first
◮ We need to localize the largest coefficients◮ Without having to send explicit localization information◮ Idea: to exploit the inter-band correlation to predict the
position of non-significant coefficients◮ If the prediction is correct we save many coding bits (for all
the predicted coefficients)
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Zero-tree of wavelet coefficients
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
EZW idea
◮ Auto-similarity : When a coefficient is small (below a threshold)it is probable that its descendants are small as well
◮ In this case we use a single coding symbol to represent thecoefficient and all its descendants. If c and all its descendantsare smaller than the threshold, c is called a zero-tree root
◮ With just one symbol, (ZT) we code (1 + 4 + 42 + . . .+ 4N−n )coefficients
◮ The localization information is implicit in the significanceinformation
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
EZW algorithm
1. k = 0
2. n = ⌊log2 (|c|max)⌋
3. Tk = 2n
4. while (rate < available rate)◮ Dominant pass◮ Refining pass◮ Tk+1 ← Tk/2◮ k ← k + 1
5. end while
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Dominant pass
◮ For each coefficient c (in the scan order)
◮ If |c| ≥ Tn, the coefficient is significant
◮ If c > 0 we encode SP (Significant Positive)◮ If c < 0 we encode SN (Significant Negative)
◮ If |c| < Tn, we compare all its descendants with the threshold
◮ If no descendant is significant, c is coded as a zero-treeroot (ZT)
◮ Otherwise the coefficient is coded as Isolated Zero (IZ)
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Refining pass
◮ We encode a further bit for all significant coefficients◮ This is equivalent to halve the quantization step
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Iteration and termination
◮ The k-th dominant pass allows to encode the k-th bit-plane◮ A significant coefficient c is such that 2k < |c| < 2k+1
◮ For the next step we halve the threshold: it is equivalent topass to the next bitplane
◮ Algorithm stops when◮ the bit budget is exhausted; or when◮ all the bitplanes have been coded
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
EZW Algorithm: summary
◮ Bitplane coding: at the k-th pass, we encode the bitplanelog2 Tk
◮ Progressive coding: each new bitplane allows refining thecoefficients quantization
◮ Lossless coding of significance symbols◮ Lossless-to-lossy coding: When an integer transform is
used, and all the bitplanes are coded, the original imagecan be restored with zero distortion
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
EZW Algorithm: Example
26 6 13 10-7 7 6 44 -4 4 -32 -2 -2 0
T0 = 2⌊log2 26⌋ = 16
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
EZW Algorithm: Example
26 6 13 10-7 7 6 44 -4 4 -32 -2 -2 0
T0 = 2⌊log2 26⌋ = 16
Bitstream:SP ZR ZR ZR 1IZ ZR ZR ZR SP SP IZ IZ 0 1 0SP IZ SP SI SP SP SN IZ IZ SP IZ IZ IZ
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
JPEG2000
◮ JPEG2000 aims at challenges unresolved by previous standards:
◮ Low bit-rate coding: JPEG has low quality for R < 0.25 bpp
◮ Synthetic images compression
◮ Random access to image parts
◮ Quality and resolution scalability
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
New functionalities
◮ Region-of-interest (ROI) coding◮ Quality and resolution scalability◮ Tiling◮ Exact coding rate◮ Lossy-to-lossless coding
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Algorithm
JPEG2000 is made up of two tiers
◮ First tier◮ DWT and quantization◮ Lossless coding of codeblocks
◮ Second tier◮ EBCOT: embedded block coding with optimized truncation◮ Scalability (quality, resolution) and ROI management
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Quantization in JPEG2000
◮ DWT coefficients are encoded with a very fine quantizationstep
◮ For the lossless coding case, DWT coefficients areintegers, and they are not quantified
◮ In summary, it is not in the quantization step that the reallylossy operations are performed
◮ The lossy coding is performed by the bitstream truncationof Tier 2
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
EBCOT
Embedded Block Coding with Optimized Truncation◮ Each subband is split in equally sized blocks of
coefficients, called codeblocks◮ The codeblocks are losslessly and independently coded
with an arithmetic coder◮ We generate as much bitstreams as codeblocks in the
image
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Bitplane coding
Most significant bitplane
LL2 HL2
LH2 HH2
HL1
LH1 HH1
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Bitplane coding
Second bitplane
LL2 HL2
LH2 HH2
HL1
LH1 HH1
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Bitplane coding
Third bitplane
LL2 HL2
LH2 HH2
HL1
LH1 HH1
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Bitplane coding
Fourth bitplane
LL2 HL2
LH2 HH2
HL1
LH1 HH1
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Bitplane coding
Fifth bitplane
LL2 HL2
LH2 HH2
HL1
LH1 HH1
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Example of bitstreams associated tocodeblocks
BP2
BP3
BP4
BP5
BP6
BP7
BP8
BP1
LL2 HL1LH2 HL2 HH2 HH1LH169/90 12.11.15 Institut Mines-Telecom Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
EBCOTOptimization
◮ If we keep all the bitstreams of all the codeblocks, we endup with a huge bitrate
◮ We have to truncate the bitstream to attain the targetbit-rate
◮ Problem: how to truncate the bitstreams with a minimumresulting distortion?
min∑
i
Di subject to∑
i
Ri ≤ Rtot
◮ Solution : Lagrange multiplier
J =∑
i
Di + λ
(∑
i
Ri −R
)
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
EBCOTRate-distortion curve per each codeblock
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
EBCOTRate-distortion curve per each codeblock
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
EBCOT
Embedded block coding with optimized truncation
◮ Optimal truncation point:
∂Di
∂Ri= −λ
◮ The value of the Lagrange multiplier can be find by aniterative algorithm.
◮ We can have several truncations for several target rates(quality scalability)
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Example of bit-rate allocation with EBCOTAllocation for maximal quality and minimal resolution
BP2
BP3
BP4
BP5
BP6
BP7
BP8
BP1
LL2 HL1LH2 HL2 HH2 HH1LH1
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Example of bit-rate allocation with EBCOTAllocation for maximal quality and medium resolution
BP2
BP3
BP4
BP5
BP6
BP7
BP8
BP1
LL2 HL1LH2 HL2 HH2 HH1LH1
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Example of bit-rate allocation with EBCOTAllocation for maximal quality and maximal resolution
BP2
BP3
BP4
BP5
BP6
BP7
BP8
BP1
LL2 HL1LH2 HL2 HH2 HH1LH1
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Example of bit-rate allocation with EBCOTAllocation for perceptual quality and maximal resolution
BP2
BP3
BP4
BP5
BP6
BP7
BP8
BP1
LL2 HL1LH2 HL2 HH2 HH1LH1
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Example of bit-rate allocation with EBCOTAllocation for a given bit-rate, maximal quality and resolution
BP2
BP3
BP4
BP5
BP6
BP7
BP8
BP1
LL2 HL1LH2 HL2 HH2 HH1LH1
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Example of bit-rate allocation with EBCOTAllocation pour several layers and maximal resolution
BP2
BP3
BP4
BP5
BP6
BP7
BP8
BP1
LL2 HL1LH2 HL2 HH2 HH1LH1
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
JPEGComparison JPEG / JPEG2000
Image Originale, 24 bpp
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Comparison JPEG / JPEG2000
Rate: 1bpp
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Comparison JPEG / JPEG2000
Rate: 0.75bpp
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Comparison JPEG / JPEG2000
Rate: 0.5bpp
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Comparison JPEG / JPEG2000
Rate: 0.3bpp
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Comparison JPEG / JPEG2000
Rate: 0.2bpp
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Comparison JPEG / JPEG2000
Rate: 0.2bpp pour JPEG, 0.1 pour JPEG2000
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Error effect: JPEG
JPEG, pE = 10−4 JPEG, pE = 10−4
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Error effect: JPEG and JPEG 2000
JPEG, pE = 10−4 JPEG 2000, pE = 10−4
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Error effect: JPEG and JPEG 2000
JPEG, pE = 10−3 JPEG 2000, pE = 10−3
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Image coding and robustness
◮ Markers insertion◮ Markers period◮ Marker emulation prevention◮ Trade-off between robustness and rate
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Error robustness in JPEG2000
◮ Data priorization is possible◮ No dependency among codeblocks
◮ No error propagation◮ No block-based transform
◮ No blocking artifacts
90/90 12.11.15 Institut Mines-Telecom Wavelet-based image compression