design method of directional genlot with trend vanishing moments

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


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.


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


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


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.


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],


What is TVM?


0 = μ(0)k =

∑n∈Z

hk [n],

0 = μ(1)k =

∑n∈Z

hk [n]n, · · ·

n


What is TVM?


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


What is TVM?


0 = μ(0)k =

∑n∈Z

hk [n],

0 = μ(1)k =

∑n∈Z

hk [n]n, · · ·

n


ny

nxφuφ

Every wavelet filters with TVMannhilate piece-wise polynomial

surfaces in the direction φ


What is TVM?


0 = μ(0)k =

∑n∈Z

hk [n],

0 = μ(1)k =

∑n∈Z

hk [n]n, · · ·

n


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]


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.


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


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


∂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


∂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].


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.


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.


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

��

��

��

��

��

��

-π4

π4

3π4

Horizontal overlappingfactor ≥ 2


Design Procedure with Two-Order TVM

��

��

��

��

��

��

-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.


Design Procedure with Two-Order TVM

��

��

��

��

��

��

-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.


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.


U{x}Nx−2 and U

{x}Nx−1 to constitute a

triangle.

3 (← same)


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]


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.


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◦


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


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 .


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


design method of directional genlot with trend vanishing moments

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