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# Design Method of Directional GenLOT with Trend Vanishing Moments

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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

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### Design Method of Directional GenLOT with Trend Vanishing Moments

1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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