design method of directional genlot with trend vanishing moments
TRANSCRIPT
Design Method of Directional GenLOTwith Trend Vanishing Moments
S. Muramatsu, T. Kobayashi, D. Han and H. Kikuchi
Dept. of Electrical and Electronic Eng., Niigata University, Japan
December 16, 2010
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 1 / 18
Outline
Background
Review of Symmetric Orthogonal Transforms
Contribution of This Work
Review of GenLOT and Trend Vanishing Moment (TVM)
Two-Order TVM Condition for 2-D GenLOT
Design Procedure, Design Example and Evaluation
Conclusions
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 2 / 18
Background
Transforms play a crucial role in lots of applications such as
Coding/Denoising
x → [Ψ] → y →[
Quantizationor Shrinkage
]→ y → [Ψ−1] → x
Modeling (e.g. Compressive Sensing)
x → [Φ] → v → [Estimation] → y → [Ψ−1] → x
↘ [Ψ] → y (modeled to be sparse)
Feature extraction
x → [Ψ] → y →[
Classificationor Regression
]→ ω
Orthogonality, i.e. ΨTΨ = I, makes things simplebecause of Perseval’s theorem ‖x‖2
2 = ‖y‖22.
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 3 / 18
Symmetric Orthogonal Transforms
Symmetric bases are preferably adopted in image processing.
Family of Symmetric Orthogonal Transforms
Haar DCT
LOT2−D Sym. Orth. DWT w VM
M−D LPPUFB
Parameter ReductionM−ch Regular LPPUFB
Block−wise Impl.2−D GenLOT w VM
[Tr.SP 1999]
Hadamard DST
LPPUFB/GenLOT
LBT
Contourletw DVM
X−lets
inspire
MLT/ELT
[ICIP2009]
[ICIP2010,PC2010]
[ISPACS2009]applicable
2−D GenLOT w TVM
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 4 / 18
LPPUFB: Linear-Phase Paraunitary Filter Banks
Contribution
Experimental Results of ECSQ at 0.5bpp [ICIP2010]
Original(PSNR)
8 × 8 DCT(28.75[dB])
3-Lv. 9/7 DWT(28.85[dB])
3-Lv. GenLOT w VM2(28.68[dB])
3-Lv. GenLOT w TVM(29.82[dB])
Current issues on 2-D non-separable GenLOT
Adaptive control of basesEfficient implementationDesign procedure
This work contributes to
Clarify the relation between TVM direction and overlapping factorGeneralize the design procedure in terms of overlapping factor
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 5 / 18
What is GenLOT?
Generalized Lapped Orthogonal Transform
Orthogonality, symmetry and variability of basis
Compatibility w block DCT
Constructed by a lattice structure
-
- -
- -
- -
- -
- -
-
ee
oo
oe
eo2−(Ny+Nx)
z−1xz−1
xz−1y
E0
U{x}1 U{y}
NyU0
W0
↓M↓M↓M↓M
d(z)
Lattice structure of a 2-D non-separable GenLOT (forward transform)
E(z) =
Ny∏ny=1
{R{y}ny Q(zy)
}·
Nx∏nx=1
{R{x}nx Q(zx)
}· R0E0,
where W0, U0 and U{d}nd are parameter matrices.
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 6 / 18
What is TVM?
Trend vanishing moment is an extention of 1-D VM to 2-D case.
0 = μ(0)k =
∑n∈Z
hk [n],
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 7 / 18
What is TVM?
Trend vanishing moment is an extention of 1-D VM to 2-D case.
0 = μ(0)k =
∑n∈Z
hk [n],
0 = μ(1)k =
∑n∈Z
hk [n]n, · · ·
n
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 7 / 18
What is TVM?
Trend vanishing moment is an extention of 1-D VM to 2-D case.
0 = μ(0)k =
∑n∈Z
hk [n],
0 = μ(1)k =
∑n∈Z
hk [n]n, · · ·
n
Every wavelet filters with VMannhilate piece-wise polynomials
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 7 / 18
What is TVM?
Trend vanishing moment is an extention of 1-D VM to 2-D case.
0 = μ(0)k =
∑n∈Z
hk [n],
0 = μ(1)k =
∑n∈Z
hk [n]n, · · ·
n
Every wavelet filters with VMannhilate piece-wise polynomials
ny
nxφuφ
Every wavelet filters with TVMannhilate piece-wise polynomial
surfaces in the direction φ
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 7 / 18
What is TVM?
Trend vanishing moment is an extention of 1-D VM to 2-D case.
0 = μ(0)k =
∑n∈Z
hk [n],
0 = μ(1)k =
∑n∈Z
hk [n]n, · · ·
n
Every wavelet filters with VMannhilate piece-wise polynomials
ny
nxφuφ
Every wavelet filters with TVMannhilate piece-wise polynomial
surfaces in the direction φ
2-D VM – [Stanhill et al., IEEE Trans. on SP 1996]
Directional VM (DVM) – [Do et al., IEEE Trans. on IP 2005]
Trend VM (TVM) – [ICIP2010, PCS2010]
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 7 / 18
DVM vs. TVM
DVMEvery wavelet filters annihilatepiece-wise polynomials along everydirected lines.
Freq. Domain
yω
xω
0=ωuT
φ
( )Txy uu ,=u
TVMEvery wavelet filters annihilatedirected piece-wise polynomialsurfaces.
Freq. Domain
φ
⎟⎟⎠
⎞⎜⎜⎝
⎛=
φφ
φ cos
sinu
yω
xω
NOTE: u must be INTEGER,while uφ is STEERABLE FLEXIBLY.
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 8 / 18
Trend Vanishing Moment (TVM) Condition
We say that a filter bank has P-order TVM along the directionuφ = (sinφ, cos φ)T if and only if the following condition holds:
For wavelet filters (k = 1, 2, · · · ,M − 1, p = 0, 1, 2, · · · ,P − 1)
0 = (−j)pp∑
q=0
(pq
)sinp−q φ cosq φ
∂p
∂ωp−qy ∂ωq
x
Hk
(e j!T
)∣∣∣∣∣∣!=o
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 9 / 18
Trend Vanishing Moment (TVM) Condition
We say that a filter bank has P-order TVM along the directionuφ = (sinφ, cos φ)T if and only if the following condition holds:
For wavelet filters (k = 1, 2, · · · ,M − 1, p = 0, 1, 2, · · · ,P − 1)
0 = (−j)pp∑
q=0
(pq
)sinp−q φ cosq φ
∂p
∂ωp−qy ∂ωq
x
Hk
(e j!T
)∣∣∣∣∣∣!=o
An equivalent condition is derived for FIR PU systems
For a polyphase matrix (p = 0, 1, 2, · · · ,P − 1)
cpaM =
p∑q=0
(pq
)sinp−q φ cosq φ
∂p
∂zp−qy ∂zq
x
E(zM
)d(z)
∣∣∣∣∣∣z=1
,
where cp is a constant, 1 = (1, 1, · · · , 1)T andam = (1, 0, · · · , 0)T [PCS2010].
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 9 / 18
Two-Order TVM Conditions for Lattice Parameters
o = My sin φ
Ny∑ky=1
Ny∏ny=ky
U{y}ny · aM
2
+ Mx cos φ
Ny∏ny=1
U{y}ny ·
Nx∑kx=1
Nx∏nx=kx
U{x}nx · aM
2
+
Ny∏ny=1
U{y}ny ·
Nx∏nx=1
U{x}nx · U0bφ,
λ = λ1 + λ2
λ1λ2
x1
x2x3
The same approach as [Oraintara et al., IEEETrans. SP 2001] is applicable to obtain the designconstraint.
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 10 / 18
Conditions for Polyphase Order
Theorem (Necessary Condition for the Polyphase Order)
2-D GenLOT requires polyphase order [Ny,Nx] such that (Ny + Nx) > 1to hold the two-order TVM except for some singular angles.
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 11 / 18
Conditions for Polyphase Order
Theorem (Necessary Condition for the Polyphase Order)
2-D GenLOT requires polyphase order [Ny,Nx] such that (Ny + Nx) > 1to hold the two-order TVM except for some singular angles.
Theorem (Sufficient Condition for the Polyphase Order)
2-D GenLOT can hold the two-order TVM for any angle in the followingspecified range when the corresponding condition is satisfied:
1 For φ ∈ [−π/4, π/4], the horizontal polyphase order Nx ≥ 2.
2 For φ ∈ [π/4, 3π/4], the vertical polyphase order Ny ≥ 2.
Vertical overlappingfactor ≥ 2
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-π4
π4
3π4
Horizontal overlappingfactor ≥ 2
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 11 / 18
Design Procedure with Two-Order TVM
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-π4
π4
3π4 λ = λ1 + λ2
λ1λ2
x1
x2x3
For φ ∈ [π/4, 3π/4] and Ny ≥ 2
1 Give a direction φ and let x3 asgiven in Tab. II.
2 Impose parameter matrices
U{y}Ny−2 and U
{y}Ny−1 to constitute a
triangle.
3 Optimize parameter matrices forminimizing a given cost functionunder the constraint ‖x3‖ ≤ 2.
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 12 / 18
Design Procedure with Two-Order TVM
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-π4
π4
3π4 λ = λ1 + λ2
λ1λ2
x1
x2x3
For φ ∈ [π/4, 3π/4] and Ny ≥ 2
1 Give a direction φ and let x3 asgiven in Tab. II.
2 Impose parameter matrices
U{y}Ny−2 and U
{y}Ny−1 to constitute a
triangle.
3 Optimize parameter matrices forminimizing a given cost functionunder the constraint ‖x3‖ ≤ 2.
For φ ∈ [−π/4, π/4] and Nx ≥ 2
1 Give a direction φ and let x3 asgiven in Tab. I.
2 Impose parameter matrices
U{x}Nx−2 and U
{x}Nx−1 to constitute a
triangle.
3 (← same)
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 12 / 18
Directional Design Specification
We adopt Passband Error & Passband Energy Criteria.
π4 α
π
π
−π
−π
π4
π4
ωx
ωy
An example of passband region deformation for ideal lowpass filters, andan example set of magnitude response specification for My = Mx = 4,d = x and α = −2.0 [ICIP2009]
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 13 / 18
Design Examples with Two-Order TVM
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
10
2
4
ωx/πω
y/π
Ma
gn
itud
e
∣∣∣H0
(e j!T
)∣∣∣ Basis images
A design example with two-order TVMs of φ = cot−1 α ∼ −26.57◦
optimized for the specification given in the previous slide, whereNy = Nx = 2, i.e. basis images of size 12 × 12.
A novel resultmore than 2 × 2 channels.
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 14 / 18
Evaluation
QUESTION!Do really the obtained systems satisfy the TVM condition?
Numerically verify1 Simulation for Ramp Picture
Rotation2 Simulation for TVM
Rotation
Sparsity ratio as a fraction ofnonzero samples and Coefs.:
Rx =
∑M−1k=0 ‖yk [m]‖0
‖x [n]‖0
,
(|x | ≤ 10−15 is regarded as zero.)
X(z) X(z)
Yk(z)
↓M
↓M
↓M
↓M
↑M
↑M
↑M
↑MH00(z)
H01(z)
H10(z)
H11(z)
F00(z)
F01(z)
F10(z)
F11(z)
Analysis bank Synthesis bank
A ramp picture of size 128 × 128,where φx = 30.00◦
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 15 / 18
Simulation for Ramp Picture Rotation
Direction of TVM is fixed to φ = −26.57◦.
−40 −20 0 20 40 60 80 100 1200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
φx
Spa
rsity
Ra
tio R
x
4× 4 DirLOT with 2−Ord.TVM (φ=−26.57°)2−D 4× 4 DCT2−D 2−Level 9/7−Transform
Sparsity ratio Rx against for directions of trend surface in ramp pictures
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 16 / 18
Simulation for TVM Rotation
Direction of input picture is fixed to φx = 30.00◦
2-D GenLOT w TVM of minimum order is designed for every φ
3-lv. DWT structure of 2 × 2-ch 2-D GenLOT is adopted.
0 50 1000
0.2
0.4
0.6
0.8
1
φ [degree]
Spa
rsity
Ra
tio R
x
Sparsity ratio Rx against for directions of two-order TVMs.
Spiky drop can be seen when φ = φx .
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 17 / 18
Conclusions
2-D GenLOT belongs to symmetric orthogonal transforms
TVM was introduced
An extention of 1-D VM to 2-D caseDifferent from classical VM and DVM
A novel generalized design procedure was given
Capability of trend surface annihilation property was shown
Future works
Design parameter reduction
Adaptive control of local basis
Fast hardware-friendly implementation
Killer application
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 18 / 18