wavelet-based image compression
TRANSCRIPT
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Wavelet-based imagecompression
Marco Cagnazzo
Multimedia Compression - MN907
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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
2/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
Outline
Introduction
Discrete wavelet transform and multiresolution analysis
Images compression with wavelets
3/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
Signal analysis
◮ Signal analysis: similitude to “atoms” φk (t) or φk [t] (if timeis discretized)
◮ Similitude: scalar product
c[k ] = 〈x [t], φk [t]〉
◮ Projection over a set of signals◮ Basis change◮ Linear Transform
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IntroductionDWT and MRA
Images compression with wavelets
Signal analysis
time
freq
uenc
y
Time analysis
φk (t) = δ(t − k)
5/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
Signal analysis
time
freq
uenc
y
Frequency Analysis
φk (t) = e−j2π kN t
6/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
Signal analysis
time
freq
uenc
y
Short Time Fourier Transform
φk ,τ (t) = e−j2π kN twτ (t)
Where wτ (t) is a window cen-tred in τ .
E.g., wτ (t) = 1√2πσ2
e(t−τ)2
2σ2
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IntroductionDWT and MRA
Images compression with wavelets
Signal analysis
time
freq
uenc
y
Wavelet Transform
φj ,k(t) = φ(2−j t − k)
8/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
Wavelets and images : Motivations
◮ Image model : trends + anomalies
9/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
Wavelets and images : Motivations
◮ Image model : trends + anomalies
9/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
Wavelets and images : Motivations
◮ Image model : trends + anomalies
9/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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
10/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
Wavelets and images : Motivations
Signal model: an image row
11/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
Wavelets and images : Motivations
Signal model: an image row
11/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
Wavelets and images : Motivations
Signal model: an image row
11/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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
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c1[k ] d1[k ]
x [k ] = c0[k ]
12/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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
13/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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 ]
h1
h0 ↓ 2
↓ 2
Analysis filter bank
2 ↓ : decimation : c[k ] = c[2k ]
14/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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 ]
f1
f0↑ 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
15/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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
16/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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 ]
h1
h0
f1
f0↓ 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 = H0 (z)X (z)
decimation C (z) =12
[C(
z1/2)+ C
(−z1/2
)]
interpolation C (z) = C(
z2)
output X (z) = F0 (z)C(
z2)+ F1 (z)D
(z2)
X (z) =12[F0 (z)H0 (z) + F1 (z)H1 (z)]X (z)
+12[F0 (z)H0 (−z) + F1 (z)H1 (−z)]X (−z)
18/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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
T (z) =H0 (z)F0 (z) + H1 (z)F1 (z) = 2z−ℓ Non distortion (ND)
A(z) =H0 (−z)F0 (z) + H1 (−z)F1 (z) = 0 Aliasing cancelation (AC)
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Perfect reconstruction conditions
Matrix form
If the analysis filter bank is given, the synthesis one isdetermined by:
[H0 (z) H1 (z)
H0 (−z) H1 (−z)
]·[F0 (z)F1 (z)
]=
[2z−ℓ
0
]
We need that the modulation matrix is invertible, ie.
∀z ∈ C : |z| = 1, ∆(z) = H0(z)H1(−z)− H1(z)H0(−z) 6= 0
20/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Perfect reconstruction conditions
Synthesis filter bank
If ∆(z) 6= 0 = on the unit circle, then
F0 (z) =2z−ℓ
∆(z)H1 (−z)
F1 (z) = −2z−ℓ
∆(z)H0 (−z)
Question: if the analysis FB (AFB) is made up by FIR, how toimpose that the synthesis FB (SFB) is FIR as well?
21/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Perfect reconstruction with FIR filters
In order to F0, F1 be sum of powers, ∆(z) must be a single power ofz. A sufficient condition is the alternating sign (AS) condition:
F0(z) = H1(−z)
F1(z) = −H0(−z)
Exercise: find the relationships between the analysis and synthesisFB in the time domain; justify the name “AS”.Exercise: show that AS implies AC.
22/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Analysis filters
◮ AS assures that, if AFB is FIR, SFB is FIR also, and AC issatisfied
◮ Still we need to find H0 and H1 such that ND is satisfied◮ Typically we choose some structure for H0 and H1 and we
use ND to derive the first from the second◮ QMF: H1(z) = H0(−z)◮ CQF: H1(z) = −z−(N−1)H0(−z−1)◮ Biorthogonal
23/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Orthogonality
QMF and CQF are orthogonal, which assures energyconservation:
∞∑
k=−∞(xk )
2 =
∞∑
k=−∞(ck )
2 +
∞∑
k=−∞(dk )
2
⇒ reconstruction error = quantization error on DWT coefficientsFor bi-orthogonal filters, the reconstruction errors is a weightedsum of the quantization errors on the DWT subbands, withsuitable weights ωi
24/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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?
26/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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
26/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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?
26/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Borders problem: Coefficient expansion
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y = h0 ∗ 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 h0 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
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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 h0, produces ys◮ If h0 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
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Haar filter
h0(k) = 1 1 f0(k) = 1 1
h1(k) = 1 -1 f1(k) = -1 1
◮ Symmetric◮ Orthogonal (Haar is both QMF and CQF)◮ 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 filters, if h0 has p VM and f0 has p VM, thefilter 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
DWT and MRA
A multiresolution analysis (MRA) is a set of subspaces Vj in L2(R)such that:
1. ∀j ∈ Z,Vj ⊂ Vj+1
2.⋂
j Vj = {0}
3.⋃
j Vj = L2(R)
4. f (t) ∈ Vj ⇔ f (2t) ∈ Vj+1
5. f (t) ∈ V0 ⇔ ∀k ∈ Z, f (t − k) ∈ V0
6. ∃φ ∈ V0 : {φ(t − k)}k∈Z is a basis of V0
φ is called father wavelet.
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
MRA and DWT
◮ The projection fj of f into Vj is an approximation of f◮ Increasing j , we increase the resolution◮ The projection fj of f into Vj converges to f :
fj →j f
◮ MRA implies φj ,k = 2j/2φ(2j t − k) generates a basis of Vj
◮ ∆f = fj+1 − fj are the details at level j◮ The space of details is Wj
◮ Vj ⊕Wj = Vj+1
◮ Vj = V0 ⊕W0 ⊕W1 ⊕ . . . ⊕Wj−1
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
MRA and DWT
◮ Projection of f over V0 : approximation
◮ Projection of f over Wj : details allowing to increase theresolution (pass from Vj−1 to Vj
◮ The basis of Wj is generated by ψj,k (t) = 2jψ(2j t − k)
◮ ψ(t) is the mother wavelet
◮ ψ(t) =√
2∑
k b(k)φ(2t − k) (wavelet equation)
◮ φ(t) =√
2∑
k a(k)φ(2t − k) (dilation equation)
◮ c and d are related as CQF: B(z) = −z−(N−1)A(−z−1)
◮ Thus ψj,k (t) are a basis for L2(R) and the coefficients of f can begenerated through a FB
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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 ]h1
h1
h1
h0
h0
h0
2 ↓
2 ↓
2 ↓
2 ↓
2 ↓
2 ↓
Three levels wavelet decomposition structure
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Multiresolution Analysis 1D
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
-200
20406080
0 50 100 150 200 250 300 350 400 450 500
-200
20406080
0 50 100 150 200 250 300 350 400 450 500
-50
0
50
100
150
c1[k ]
c2[k ]
d2[k ]
d2[k ]
d1[k ]
d1[k ]
d1[k ]
c3[k ] d3[k ]
x [k ] = c0[k ]
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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 ]
d1[k ]
d2[k ]
d3[k ]
f1
f1
f1f0
f0
f0
2 ↑
2 ↑
2 ↑
2 ↑
2 ↑
2 ↑
Reconstruction from wavelet coefficients
40/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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
cj [n,m]
cj+1[n,m]
dH
j+1[n,m]
dV
j+1[n,m]
dD
j+1[n,m]
h0[k ]
h1[k ]
h0[ℓ]
h0[ℓ]
h1[ℓ]
h1[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
c0
c1
dH
1dV
1dD
1
c2
dH
2dV
2dD
2
c3
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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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
48/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Example
49/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Example
50/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
Filter banks and DWTMultiresolution analysis
Example
51/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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
52/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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 possible
distortion◮ 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
56/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Progressive representation: subband order
57/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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 SN SP SP 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 HH1LH177/98 07.12.20 Une école de l’IMT 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
83/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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
84/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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
85/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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
86/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
JPEGComparison JPEG / JPEG2000
Image Originale, 24 bpp
87/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Comparison JPEG / JPEG2000
Rate: 1bpp
88/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Comparison JPEG / JPEG2000
Rate: 0.75bpp
89/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Comparison JPEG / JPEG2000
Rate: 0.5bpp
90/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Comparison JPEG / JPEG2000
Rate: 0.3bpp
91/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Comparison JPEG / JPEG2000
Rate: 0.2bpp
92/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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
93/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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IntroductionDWT and MRA
Images compression with wavelets
EZWJPEG 2000
Error effect: JPEG
JPEG, pE = 10−4 JPEG, pE = 10−4
94/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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
95/98 07.12.20 Une école de l’IMT Wavelet-based image compression
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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
98/98 07.12.20 Une école de l’IMT Wavelet-based image compression