Design of High-Performance Filter Banks for Image Coding

Transcription

1 Design of High-Performance Filter Banks for Image Coding Di Xu Michael D. Adams Dept. of Elec. and Comp. Engineering University of Victoria, Canada IEEE Symposium on Signal Processing and Information Technology, 2006 Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 1 / 21

2 Outline 1 Background Information 2 Design Method 3 Experimental Results 4 Summary Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 2 / 21

3 Outline 1 Background Information 2 Design Method 3 Experimental Results 4 Summary Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 3 / 21

4 Desirable Characteristics perfect reconstruction linear phase high coding gain (two models) good frequency selectivity certain vanishing moment Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 4 / 21

5 Desirable Characteristics perfect reconstruction linear phase high coding gain (two models) good frequency selectivity certain vanishing moment Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 4 / 21

6 Desirable Characteristics perfect reconstruction linear phase high coding gain (two models) G SBC = L 1 k=0 ( α k A k B k ) α k, where A k = m Z h hk (m) n Z h vk (n) p Z h hk (p) h vk (q)r(m p, n q), q Z B k = α k m Z g hk 2 (m) g vk 2 (n), n Z α kis sampling factor, good frequency selectivity certain vanishing moment { ρ x + y for separable model r(x, y) = x ρ 2 +y 2 for isotropic model, Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 4 / 21

7 Desirable Characteristics perfect reconstruction linear phase high coding gain (two models) good frequency selectivity certain vanishing moment Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 4 / 21

8 Desirable Characteristics perfect reconstruction linear phase high coding gain (two models) good frequency selectivity certain vanishing moment Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 4 / 21

9 Desirable Characteristics perfect reconstruction linear phase high coding gain (two models) good frequency selectivity certain vanishing moment Problems? Difficult to obtain all properties Difficult to obtain a good tradeoff Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 4 / 21

10 Outline 1 Background Information 2 Design Method 3 Experimental Results 4 Summary Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 5 / 21

11 Lifting Scheme (a) analysis side (b) synthesis side Figure: The lifting realization of a 1-D two-channel filter bank. perfect reconstruction linear phase easily imposed by lifting realization high coding gain good frequency selectivity certain vanishing moment properties left to design Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 6 / 21

12 Lifting Scheme (a) analysis side (b) synthesis side Figure: The lifting realization of a 1-D two-channel filter bank. perfect reconstruction linear phase easily imposed by lifting realization high coding gain good frequency selectivity certain vanishing moment properties left to design Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 6 / 21

13 Lifting Scheme (a) analysis side (b) synthesis side Figure: The lifting realization of a 1-D two-channel filter bank. Canonical Form: (a) analysis side (b) synthesis side Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 6 / 21

14 Lifting Scheme (a) analysis side (b) synthesis side Figure: The lifting realization of a 1-D two-channel filter bank. H 0 (z) = H 0,0 (z 2 ) + zh 0,1 (z 2 ) and H 1 (z) = H 1,0 (z 2 ) + zh 1,1 (z 2 ), ] λ 1 ([ ] [ ] ) 1 0 where = F 2k (z) 1 [ H0,0 (z) H 0,1 (z) H 1,0 (z) H 1,1 (z) k=0 1 F 2k+1 (z) 0 1 Canonical Form: (a) analysis side (b) synthesis side Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 6 / 21

15 Constrained Optimization Objective function: G sep (x) G(x) = G iso (x) min{g sep (x), G iso (x)} separable only isotropic only joint. Stopband energy: b k (x) S k ĥk(ω, x) 2 dω, k {0, 1}. Moment functions: c k (x) m k (x), k {1, 2,..., n}. Abstract Optimization Problem maximize G(x) subject to: b k (x) ε k, k {0, 1} and c k (x) γ k, k {1, 2,..., n}. Details for Different Objective Functions Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 7 / 21

16 Optimization Scheme Highly Nonlinear Iterative SOCP Algorithm 1 reduce order at an operating point x by using Taylor series approximation functions of δ 2 impose an additional constraint, s.t. δ is small 3 solve the second-order cone programming (SOCP) problem 4 update operating point x = x + δ 5 go to step 1, unless algorithm converges maximize subject to: T G(x)δ Q 1/2 k (x)δ + q k (x) ε k b k (x) +q T k (x)q k (x), k {0, 1}, T m k (x)δ + m k (x) γ k, δ β, k {1, 2,..., n}, and where Q k (x) = xĥk(ω, x) T x ĥk(ω, x)dω, S k q k (x) = Q 1/2 k (x) ĥ k (ω, x) S k T x ĥk(ω, x)dω. Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 8 / 21

17 Optimization Scheme Highly Nonlinear Iterative SOCP Algorithm 1 reduce order at an operating point x by using Taylor series approximation functions of δ 2 impose an additional constraint, s.t. δ is small 3 solve the second-order cone programming (SOCP) problem 4 update operating point x = x + δ 5 go to step 1, unless algorithm converges maximize subject to: T G(x)δ Q 1/2 k (x)δ + q k (x) ε k b k (x) +q T k (x)q k (x), k {0, 1}, T m k (x)δ + m k (x) γ k, δ β, k {1, 2,..., n}, and where Q k (x) = xĥk(ω, x) T x ĥk(ω, x)dω, S k q k (x) = Q 1/2 k (x) ĥ k (ω, x) S k T x ĥk(ω, x)dω. Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 8 / 21

18 Optimization Scheme Highly Nonlinear Iterative SOCP Algorithm 1 reduce order at an operating point x by using Taylor series approximation functions of δ 2 impose an additional constraint, s.t. δ is small 3 solve the second-order cone programming (SOCP) problem 4 update operating point x = x + δ 5 go to step 1, unless algorithm converges maximize subject to: T G(x)δ Q 1/2 k (x)δ + q k (x) ε k b k (x) +q T k (x)q k (x), k {0, 1}, T m k (x)δ + m k (x) γ k, δ β, k {1, 2,..., n}, and where Q k (x) = xĥk(ω, x) T x ĥk(ω, x)dω, S k q k (x) = Q 1/2 k (x) ĥ k (ω, x) S k T x ĥk(ω, x)dω. Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 8 / 21

19 Optimization Scheme Highly Nonlinear Iterative SOCP Algorithm 1 reduce order at an operating point x by using Taylor series approximation functions of δ 2 impose an additional constraint, s.t. δ is small 3 solve the second-order cone programming (SOCP) problem 4 update operating point x = x + δ 5 go to step 1, unless algorithm converges maximize subject to: T G(x)δ Q 1/2 k (x)δ + q k (x) ε k b k (x) +q T k (x)q k (x), k {0, 1}, T m k (x)δ + m k (x) γ k, δ β, k {1, 2,..., n}, and where Q k (x) = xĥk(ω, x) T x ĥk(ω, x)dω, S k q k (x) = Q 1/2 k (x) ĥ k (ω, x) S k T x ĥk(ω, x)dω. Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 8 / 21

20 Optimization Scheme Highly Nonlinear Iterative SOCP Algorithm 1 reduce order at an operating point x by using Taylor series approximation functions of δ 2 impose an additional constraint, s.t. δ is small 3 solve the second-order cone programming (SOCP) problem 4 update operating point x = x + δ 5 go to step 1, unless algorithm converges maximize subject to: T G(x)δ Q 1/2 k (x)δ + q k (x) ε k b k (x) +q T k (x)q k (x), k {0, 1}, T m k (x)δ + m k (x) γ k, δ β, k {1, 2,..., n}, and where Q k (x) = xĥk(ω, x) T x ĥk(ω, x)dω, S k q k (x) = Q 1/2 k (x) ĥ k (ω, x) S k T x ĥk(ω, x)dω. Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 8 / 21

21 Outline 1 Background Information 2 Design Method 3 Experimental Results 4 Summary Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT 06 9 / 21

22 Choice of Objective Function G sep (x)? G(x) = G iso (x) min{g sep (x), G iso (x)} separable only isotropic only joint. Test Environment Table: Statistical results over all 26 test images and 5 bit rates Transform Mean (%) Median (%) Outperform (%) 9/7-sep /7-iso/jnt /14-sep /14-iso/jnt Conclusion: better jointly optimizing both of the G sep (x) and G iso (x) Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT / 21

23 Choice of Objective Function G sep (x) G(x) = G iso (x) min{g sep (x), G iso (x)} separable only isotropic only joint. Test Environment Table: Statistical results over all 26 test images and 5 bit rates Transform Mean (%) Median (%) Outperform (%) 9/7-sep /7-iso/jnt /14-sep /14-iso/jnt Conclusion: better jointly optimizing both of the G sep (x) and G iso (x) Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT / 21

24 Design Examples Table: Characteristics of various filter banks Transform {L k } G sep G iso b 0 b 1 Van. Mom. 9/7-J {2,2,2,2} , /7 {2,2,2,2} , /11 {4,2,2} , /11 {4,2,2,2} , /11 {2,2,4,4} , /15 {6,2,2} , Table: Statistical results over all 26 test images and 5 bit rates (compared with 9/7-J from JPEG-2000 standard) Transform Mean (%) Median (%) Outperform (%) 9/ / / / / Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT / 21

25 Design Examples (Cont d) Table: Specific results for three representative images Image PSNR (db) (model) CR 9/7-J 9/7 9/11 13/11 17/11 13/ gold (sep) target ( ) sar (iso) Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT / 21

26 Subjective Image Quality (compression ratio: 32) (a) original image (b) 9/7-J from JPEG 2000 (c) 9/7 design Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT / 21

27 Outline 1 Background Information 2 Design Method 3 Experimental Results 4 Summary Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT / 21

28 Summary Designed filter banks with: perfect reconstruction linear phase high coding gain good frequency selectivity certain prescribed moment properties Outperformed the well-known 9/7-J filter bank (from the JPEG-2000 standard) Proposed 9/7 design has same computational complexity as 9/7-J filter bank Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT / 21

29 Questions? Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT / 21

30 Design Parameter Selection Frequency Response L2-norm Error: for a stopband width of 3π 8, tolerances within [0.02, 0.14] is highly effective Moment Constraint: the norm of the vector of zeroth dual and primal moments is less than finding multiple solutions from many different initial points; effective when consider lifting-filter coefficients within [ 2, 2] Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT / 21

31 Impulse Responses of the Lifting Filters 9/7-J from JPEG 2000: proposed 9/7 design: Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT / 21

32 Constrained Optimization separable only or isotropic only case maximize G sep (x) or G iso (x) subject to: b k (x) ε k, k {0, 1} and c k (x) γ k, k {1, 2,..., n}. joint case maximize t subject to: G sep (x) t, G iso (x) t, b k (x) ε k, k {0, 1}, and c k (x) γ k, k {1, 2,..., n}. Return Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT / 21

33 Choice of Objective Function Test data: all of the 26 reasonably-sized continuous-tone grayscale images from the JPEG-2000 test set Table: Characteristics of a subset of the test images Image Size, Precision Model Description gold , 8 separable houses and countryside target , 8 patterns and textures sar , 12 isotropic synthetic aperture radar Codecs: EZW, SPIHT, and MIC Return Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT / 21

34 References Y. Chen, M. D. Adams, and W.-S. Lu Design of Optimal Quincunx Filter Banks for Image Coding. Proc. of IEEE International Symposium on Circuits and Systems, May J. Katto and Y. Yasuda Performance evaluation of subband coding and optimization of its filter coefficients. Proc. of SPIE Visual Communications and Image Processing, vol 1605, pp , Nov M. S. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret Applications of second-order cone programming. Linear Algebra and its Applications, vol 248, pp , Nov Di Xu, Michael Adams (UVic) Filter Bank Design for Image Coding ISSPIT / 21