Proc. of Proc. of Asia Pacific Signal and Information Proc. Assoc. Annual Summit and Conf. (APSIPA ASC), pp.692-701, Biopolis, Singapore, Dec. 14 – 17, 2010

1. Design Method of Directional GenLOT
with 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

2. 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

3. Background
Transforms play a crucial role in lots of applications such as
Coding/Denoising
x → [Ψ] → y →
Quantization
or Shrinkage
→ ˆy → [Ψ−1
] → ˆx
Modeling (e.g. Compressive Sensing)
x → [Φ] → v → [Estimation] → ˆy → [Ψ−1
] → ˆx
[Ψ] → y (modeled to be sparse)
Feature extraction
x → [Ψ] → y →
Classification
or Regression
→ ω
Orthogonality, i.e. ΨT Ψ = I, makes things simple
because of Perseval’s theorem x 2
2 = y 2
2.
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 3 / 18

4. Symmetric Orthogonal Transforms
Symmetric bases are preferably adopted in image processing.
Family of Symmetric Orthogonal Transforms
Haar DCT
LOT
2−D Sym. Orth. DWT w VM
M−D LPPUFB
Parameter Reduction
M−ch Regular LPPUFB
Block−wise Impl.2−D GenLOT w VM
[Tr.SP 1999]
Hadamard DST
LPPUFB/GenLOT
LBT
Contourlet
w 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

5. 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 bases
Efficient implementation
Design procedure
This work contributes to
Clarify the relation between TVM direction and overlapping factor
Generalize 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

6. 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
eo
2−(Ny+Nx)
z−1
xz−1
x
z−1
y
E0
U
{x}
1 U
{y}
Ny
U0
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

7. 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

8. 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

9. 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 VM
annhilate piece-wise polynomials
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 7 / 18

10. 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 VM
annhilate piece-wise polynomials
ny
nx
φ
uφ
Every wavelet filters with TVM
annhilate 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

11. 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 VM
annhilate piece-wise polynomials
ny
nx
φ
uφ
Every wavelet filters with TVM
annhilate 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

12. DVM vs. TVM
DVM
Every wavelet filters annihilate
piece-wise polynomials along every
directed lines.
Freq. Domain
yω
xω
0=ωuT
φ
( )T
xy uu ,=u
TVM
Every wavelet filters annihilate
directed piece-wise polynomial
surfaces.
Freq. Domain
φ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
φ
φ
φ
cos
sin
u
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

13. Trend Vanishing Moment (TVM) Condition
We say that a filter bank has P-order TVM along the direction
uφ = (sin φ, 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)p
p
q=0
p
q
sinp−q
φ cosq
φ
∂p
∂ωp−q
y ∂ωq
x
Hk ej!T
!=o
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 9 / 18

14. Trend Vanishing Moment (TVM) Condition
We say that a filter bank has P-order TVM along the direction
uφ = (sin φ, 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)p
p
q=0
p
q
sinp−q
φ cosq
φ
∂p
∂ωp−q
y ∂ωq
x
Hk ej!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
p
q
sinp−q
φ cosq
φ
∂p
∂zp−q
y ∂zq
x
E zM
d(z)
z=1
,
where cp is a constant, 1 = (1, 1, · · · , 1)T and
am = (1, 0, · · · , 0)T [PCS2010].
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 9 / 18

15. 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
x2
x3
The same approach as [Oraintara et al., IEEE
Trans. SP 2001] is applicable to obtain the design
constraint.
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 10 / 18

16. Conditions for Polyphase Order
Theorem (Necessary Condition for the Polyphase Order)
2-D GenLOT requires polyphase order [Ny, Nx] such that (Ny + Nx) > 1
to 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

17. Conditions for Polyphase Order
Theorem (Necessary Condition for the Polyphase Order)
2-D GenLOT requires polyphase order [Ny, Nx] such that (Ny + Nx) > 1
to 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 following
specified 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 overlapping
factor ≥ 2
00000000000000001111111111111111
00
00
0000
00
0000
00
11
11
1111
11
1111
11000000000000000000111111111111111111
00
00
0000
0000
00
00
00
11
11
1111
1111
11
11
11
-π
4
π
4
3π
4
Horizontal overlapping
factor ≥ 2
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 11 / 18

18. Design Procedure with Two-Order TVM
00000000000000001111111111111111
00
0000
00
0000
00
00
11
1111
11
1111
11
11000000000000000000111111111111111111
00
0000
00
0000
00
00
00
11
1111
11
1111
11
11
11
-π
4
π
4
3π
4 λ = λ1 + λ2
λ1λ2
x1
x2
x3
For φ ∈ [π/4, 3π/4] and Ny ≥ 2
1 Give a direction φ and let x3 as
given in Tab. II.
2 Impose parameter matrices
U
{y}
Ny−2 and U
{y}
Ny−1 to constitute a
triangle.
3 Optimize parameter matrices for
minimizing a given cost function
under the constraint x3 ≤ 2.
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 12 / 18

19. Design Procedure with Two-Order TVM
00000000000000001111111111111111
00
0000
00
0000
00
00
11
1111
11
1111
11
11000000000000000000111111111111111111
00
0000
00
0000
00
00
00
11
1111
11
1111
11
11
11
-π
4
π
4
3π
4 λ = λ1 + λ2
λ1λ2
x1
x2
x3
For φ ∈ [π/4, 3π/4] and Ny ≥ 2
1 Give a direction φ and let x3 as
given in Tab. II.
2 Impose parameter matrices
U
{y}
Ny−2 and U
{y}
Ny−1 to constitute a
triangle.
3 Optimize parameter matrices for
minimizing a given cost function
under the constraint x3 ≤ 2.
For φ ∈ [−π/4, π/4] and Nx ≥ 2
1 Give a direction φ and let x3 as
given 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

20. Directional Design Specification
We adopt Passband Error & Passband Energy Criteria.
π
4 α
π
π
−π
−π
π
4
π
4
ωx
ωy
An example of passband region deformation for ideal lowpass filters, and
an 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

21. Design Examples with Two-Order TVM
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
0
2
4
ωx
/πω
y
/π
Magnitude
H0 ej!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, where
Ny = Nx = 2, i.e. basis images of size 12 × 12.
A novel result
more than 2 × 2 channels.
S. Muramatsu et al. (Niigata Univ.) Design Method of DirLOT w TVMs December 16, 2010 14 / 18

22. Evaluation
QUESTION!
Do really the obtained systems satisfy the TVM condition?
Numerically verify
1 Simulation for Ramp Picture
Rotation
2 Simulation for TVM
Rotation
Sparsity ratio as a fraction of
nonzero samples and Coefs.:
Rx =
M−1
k=0 yk[m] 0
x[n] 0
,
(|x| ≤ 10−15
is regarded as zero.)
X(z) ˆX(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

23. Simulation for Ramp Picture Rotation
Direction of TVM is fixed to φ = −26.57◦.
−40 −20 0 20 40 60 80 100 120
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
φx
SparsityRatioR
x
4× 4 DirLOT with 2−Ord.TVM (φ=−26.57
°
)
2−D 4× 4 DCT
2−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

24. 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 50 100
0
0.2
0.4
0.6
0.8
1
φ [degree]
SparsityRatioRx
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

25. Conclusions
2-D GenLOT belongs to symmetric orthogonal transforms
TVM was introduced
An extention of 1-D VM to 2-D case
Different 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