Applications of Wavelets and Framelets

Transcription

1 Applications of Wavelets and Framelets Bin Han Department of Mathematical and Statistical Sciences University of Alberta, Edmonton, Canada Present at 2017 International Undergraduate Summer Enrichment Program at UofA July 19, 2017 Bin Han (University of Alberta) Applications of Wavelets UofA 1 / 47

2 Outline of Tutorial Wavelets in the function setting. Some applications of wavelets and framelets Tensor product wavelets and framelets Image processing using complex tight framelets. Subdivision schemes in computer graphics. Declaration: Some figures and graphs in this talk are from various sources from Internet, or from published papers, or produced by matlab, maple, or C programming. [Details and sources of all graphs can be provided upon request of the audience.] Bin Han (University of Alberta) Applications of Wavelets UofA 2 / 47

3 What Is a Wavelet in the Function Setting? Let φ = (φ 1,...,φ r ) T and ψ = (ψ 1,...,ψ s ) T in L 2 (R). A system is derived from φ,ψ via dilates and integer shifts: AS 0 (φ;ψ) :={φ( k) : k Z} {ψ j;k := 2 j/2 ψ(2 j k) : j N {0},k Z}. {φ;ψ} is called an orthogonal wavelet in L 2 (R) if AS 0 (φ;ψ) is an orthonormal basis of L 2 (R). {φ;ψ} is a tight framelet in L 2 (R) if f 2 L 2 (R) = k Z f,φ( k) 2 l 2 + f,ψ j;k 2 l 2, f L 2 (R). Orthogonal wavelet and tight framelet representation: f = f,ψ j;k ψ j;k, f L 2 (R), k Z f,φ( k) φ( k)+ j=0 j=0 k Z k Z where f,g := R f(x)g(x)t dx is the inner product. Bin Han (University of Alberta) Applications of Wavelets UofA 3 / 47

4 Dilates of a Wavelet 1.5 ψ 2;128 1 ψ 0; ψ 2; Bin Han (University of Alberta) Applications of Wavelets UofA 4 / 47

5 Integer Shifts of a Wavelet 0.5 ψ 0;0 ψ 0; ψ 2;0 ψ 2; Bin Han (University of Alberta) Applications of Wavelets UofA 5 / 47

6 Why Wavelets? A wavelet ψ often has 1 compact support good spatial localization. 2 high smoothness/regularity good frequency localization. 3 high vanishing moments multiscale sparse representation. 4 associated filter banks fast wavelet transform to compute coefficients f,ψ j;k through filter banks. 5 singularity detecting/locating and good approximation property. 6 close relations to windowed and fast Fourier transform. Explanation: Vanishing moments: x j,ψ(x) = 0 for j = 0,...,N. suppψ j;k = 2 j k +2 j suppψ 2 j k when j. f,ψ j;k = f P,ψ j;k 0 if f a polynomial P on suppψ j;k. If f,ψ j;k is large, then the singularity is around 2 j k. Bin Han (University of Alberta) Applications of Wavelets UofA 6 / 47

7 Tight Framelets or Orthogonal Wavelets Theorem: Let φ = (φ 1,...,φ r ) T and ψ = (ψ 1,...,ψ s ) T in L 2 (R). {φ;ψ} is a tight framelet (or orthogonal wavelet) in L 2 (R) 1 lim j φ(2 j ξ) 2 l 2 = 1; 2 there exist r r matrix â and s r matrix b of 2π-periodic measurable functions in L (T) such that φ(2ξ) = â(ξ) φ(ξ), i.e., φ = 2 a(k)φ(2 k), k Z ψ(2ξ) = b(ξ) φ(ξ), i.e., ψ = 2 k Zb(k)φ(2 k), and {â; b} is a tight framelet filter bank: [â(ξ) ] T â(ξ +π) â(ξ)t b(ξ) b(ξ) b(ξ +π) â(ξ +π) T T = I 2r, a.e.ξ R. b(ξ +π) 3 s = r and {φ( k)} k Z is an orthonormal system in L 2 (R), where f(ξ) := R f(x)e iξx dx and â(ξ) := k Z a(k)e ikξ. Bin Han (University of Alberta) Applications of Wavelets UofA 7 / 47

8 Example: Haar Orthogonal Wavelet {φ; ψ} Refinable function Wavelet φ = χ [0,1] ψ := χ [1/2,1] χ [0,1/2]. φ = φ(2 )+φ(2 1) ψ = φ(2 1) φ(2 ). φ and ψ have explicit expressions and φ is the B-spline of order 1. Bin Han (University of Alberta) Applications of Wavelets UofA 8 / 47

9 Example: Daubechies Orthogonal Wavelet {φ; ψ} φ = φ(2 ) φ(2 1) φ(2 2) φ(2 3). ψ = φ(2 ) φ(2 1) φ(2 2) φ(2 3). The functions φ and ψ do not have explicit expressions. Bin Han (University of Alberta) Applications of Wavelets UofA 9 / 47

10 Tensor Product (Separable) Tight Framelet Let {a;b 1,...,b s } be a 1D tight framelet filter bank. If s = 1, {a;b 1 } is called an orthonormal wavelet filter bank. Tensor product filters: [u 1 u d ](k 1,...,k d ) = u 1 (k 1 ) u d (k d ). Tensor product tight framelet filter bank: {a;b 1,...,b s } {a;b 1,...,b s }. Tensor product functions: [f 1 f d ](x 1,...,x d ) = f 1 (x 1 ) f d (x d ). Tensor product tight framelet: {φ;ψ 1,...,ψ s } {φ;ψ 1,...,ψ s }. Advantages: fast and simple algorithm. Bin Han (University of Alberta) Applications of Wavelets UofA 10 / 47

11 Tree Structure and Sparsity of Wavelet Coefficients Bin Han (University of Alberta) Applications of Wavelets UofA 11 / 47

12 Image Compression Using Orthogonal Wavelets Original Lena image and reconstructed Lena images with compression ratios 32 and 128 using SPIHT. Large coefficients are recorded with priority and tree structure is used. Bin Han (University of Alberta) Applications of Wavelets UofA 12 / 47

13 Image Denoising Using Orthogonal Wavelets Wavelet-shrinkage from statistics: small coefficients are set to 0. Bin Han (University of Alberta) Applications of Wavelets UofA 13 / 47

14 Curve Modeling: Corner Cutting Subdivision Scheme Initial control polygon v, iterated once S a v, iterated 5 times S 5 a v, where a = { 1 8, 3 8, 3 8, 1 8 } [0,3] is the B-spline filter of order 3. Bin Han (University of Alberta) Applications of Wavelets UofA 14 / 47

15 Surface Modeling by Subdivision scheme Initial mesh v, iterated once S a,m v, iterated twice S 2 a,m v. Bin Han (University of Alberta) Applications of Wavelets UofA 15 / 47

16 Subdivision Surfaces Used in Animated Movies Bin Han (University of Alberta) Applications of Wavelets UofA 16 / 47

17 Bandlimited Complex Tight Framelets TP-CTF π -π - π π 3π π 2 Tight framelet filter bank CTF 6 := {a +,a ;b 1 +,b+ 2,b 1,b 2 }: black lines for â+ and â ; dashed lines for b + 1 and b 1 ; dotted lines for b + 2 and b 2 : a := a +, b 1 := b+ 1, b 2 := b+ 2, and â + := χ [0,c];ε,ε, b+ 1 := χ [c 1,c 2 ];ε,ε, b+ 1 := χ [c 2,π];ε,ε. The tensor product tight framelet TP-CTF 6 := d CTF 6. Take advantages of wavelets and Discrete Cosine Transform. Bin Han (University of Alberta) Applications of Wavelets UofA 17 / 47

18 Two-dimensional TP-CTF 6 (14 directions) Bin Han (University of Alberta) Applications of Wavelets UofA 18 / 47

19 Denoising Comparison for Barbara Image DTCWT TP-CTF 6 UDWT TV Shearlet Redundancy N/A 49 σ = σ = σ = σ = σ = DTCWT=Dual Tree Complex Wavelet Transform. TP-CTF 6 =Han and Zhao, SIAM J. Imag. Sci. 7 (2014), UDWT=Undecimated Discrete Wavelet Transform. TV=Rudin-Osher-Fatemi (ROF) model using higher-order scheme. Shearlet=shearlet frames in W. Lim, IEEE T. Image Process., Measure of performance: PSNR = 10log MSE. The larger PSNR value the better performance. Bin Han (University of Alberta) Applications of Wavelets UofA 19 / 47

20 Remove Mixed Gaussian and Impulse Noises Gaussian and Pepper and Salt impulse noise. Cameraman: σ = 0, p = 0.3, PSNR = Lena: σ = 15, p = 0.5, PSNR = Gaussian and Random-valued impulse noises: Barbara: σ = 30, p = 0.2, PSNR = Peppers: σ = 20, p = 0.1, PSNR = Bin Han (University of Alberta) Applications of Wavelets UofA 20 / 47

21 Remove Gaussian & Pepper-and-Salt Noise AOP TP-CTF 6 AOP TP-CTF 6 AOP TP-CTF 6 σ p Cameraman House Peppers (1.88) (1.81) (0.82) (2.10) (1.85) (0.76) (1.80) (3.51) (1.43) (1.31) (2.68) (0.91) σ p Cameraman House Peppers (1.25) (5.13) (1.73) (1.59) (6.25) (2.02) (3.73) (4.49) (2.25) (2.74) (4.23) (1.72) AOP, TV-based, SIAM J. Imaging, 5 (2013), TP-CTF 6, Shen/Han/Braverman, J. Math. Imaging Vis., 54 (2016), Bin Han (University of Alberta) Applications of Wavelets UofA 21 / 47

22 Figure: Corrupted by text with σ = 20. Recovered with PSNR= Bin Han (University of Alberta) Applications of Wavelets UofA 22 / 47 Image Inpainting Using TP-CTF 6 Figure: 80% missing pixels. Recovered by our algorithm: PSNR=31.67.

23 Examples of Subdivision Curve Bin Han (University of Alberta) Applications of Wavelets UofA 23 / 47

24 Examples of Subdivision Curve Bin Han (University of Alberta) Applications of Wavelets UofA 24 / 47

25 Subdivision Schemes A dilation matrix M is a d d integer matrix such that all the eigenvalues of M are greater than one in modulus. Examples of dilation matrices: 2I d (dyadic), 3I d (ternary), M 2 = [ ], N 2 = [ ], M 3 = [ M 2 and N 2 are called the quincunx dilation matrices inducing the quincunx lattice M 2 Z2 = N 2 Z2 = {(j,k) Z 2 : j +k is even}. The subdivision operator S a,m : l(z d ) l(z d ) is [S a,m v](n) := det(m) k Z d v(k)a(n Mk), ]. where v = {v(k)} k Z d l(z d ). Bin Han (University of Alberta) Applications of Wavelets UofA 25 / 47

26 Subdivision Triplets: Symmetry is Necessary A symmetry group G is a finite set of d d integer matrices with determinants ±1 forming a group under matrix multiplication. A mask/filter a = {a(k)} k Z d : Z d R is G-symmetric with symmetry center c a if a(e(k c a )+c a ) = a(k), k Z d,e G. A dilation matrix M is compatible with G if MEM 1 G, E G. (a,m,g) is called a subdivision triplet if M is compatible with G and the mask a is G-symmetric. Bin Han (University of Alberta) Applications of Wavelets UofA 26 / 47

27 Subdivision Schemes Using Triplet (a, M, G) Subdivision scheme: calculate v n := Sa,M n v for n N and attach the value v n (k) at the point M n (k c a ), k Z d. The subdivision scheme converges if {v n } n=1 converges to a continuous function v for every bounded initial control mesh v. If the symmetry center c a = 0, it is called a primal subdivision scheme; otherwise, it is called a dual subdivision scheme. Proposition: For a subdivision triplet (a, M, G) with symmetry center c a, if â(0) = 1 with â(ξ) := k Z d a(k)e ik ξ, then φ(e( c φ )+c φ ) = φ E G with c φ := (M I d ) 1 c a, where φ is the M-refinable (or basis) function associated with the mask/filter a defined by φ(ξ) := j=1â((mt ) j ξ),ξ R d. Bin Han (University of Alberta) Applications of Wavelets UofA 27 / 47

28 Important Dilation Matrices Two important symmetry groups: { [ ] [ ] D 4 := ±,±,± { D 6 := ± [ ] 1 0,± 0 1 [ ] 0 1,± 1 1 [ ] 1 1,± 1 0 [ ] 0 1,± 1 0 [ ] 0 1,± 1 0 [ ]} 0 1, 1 0 [ ] 1 1,± 0 1 [ ]} D 4 for the quadrilateral mesh and D 6 for the triangular mesh. N is G-equivalent to M if N = EMF for some E,F G. N 2 is D 4-equivalent to M 2. Theorem: For a 2 2 real-valued matrix M, 1 if M is compatible with the symmetry group D 4, then M must be D 4 -equivalent to either ci 2 or cm 2 for some c R. 2 if M is compatible with the symmetry group D 6, then M must be D 6 -equivalent to either ci 2 or cm 3 for some c R. Bin Han (University of Alberta) Applications of Wavelets UofA 28 / 47

29 Quad and Triangular Meshes Figure: The quadrilateral mesh Z 2 Q (left) and the triangular mesh Z2 T (right). Bin Han (University of Alberta) Applications of Wavelets UofA 29 / 47

30 Definition of Linear-phase Moments Interpolation: [S a,m v](mk) = v(k) for all k Z and v l(z d ) a(0) = det(m) 1, a(mk) = 0, k Z d \{0}. Interpolation on Polynomials: [S a,m p](mk) = p(k M 1 c) for all k Z and all polynomials p with deg(p) < m a has linear-phase moments with phase c: â(ξ) = e ic ξ + O( ξ m ), ξ 0; Define lpm(a) = m with the highest possible m. a has order m sum rules: â(ξ +2πω) = O( ξ m ), ξ 0,ω Ω M \{0}, where Ω M := [0,1) d [(M T ) 1 Z d ]. Define sr(a,m) = m with the highest possible m. Note: If a has symmetry with symmetry center c a, then c = c a. Bin Han (University of Alberta) Applications of Wavelets UofA 30 / 47

31 Importance of Linear-phase Moments {a;b 1,...,b s } is called a tight M-framelet filter bank if â(ξ) 2 + b 1 (ξ) b s (ξ) 2 = 1, s â(ξ)â(ξ +2πω)+ b l (ξ) b l (ξ +2πω) = 0, ω Ω M \{0}. l=1 Called an orthogonal M-wavelet filter bank if s = det(m) 1. If det(m) = 2, then s = 1, Ω M = {0,ω}, and {a;b} is an orthogonal M-wavelet filter bank for some γ Z d \[MZ d ], â(ξ) 2 + â(ξ +2πω) 2 = 1, b(ξ) = e iγ ξâ(ξ +2πω). A filter b has n vanishing moments if b(ξ) = O( ξ n ) as ξ 0. We define vm(b) := n with the highest n. Theorem: If {a;b 1,...,b s } is a tight M-framelet filter bank and a has symmetry with symmetry center c a, then min(vm(b 1 ),...,vm(b s )) = min(sr(a), 1 2 lpm(a)). Bin Han (University of Alberta) Applications of Wavelets UofA 31 / 47

32 Tight Framelets and Wavelets A function ψ has n vanishing moments if ψ(ξ) = O( ξ n ) as ξ 0. We define vm(ψ) := n with the largest n. Theorem: If {a;b 1,...,b s } is a tight M-framelet filter bank with â(0) = 1, let φ(ξ) := j=1â((mt ) j ξ), ψ l (M T ξ) := b l (ξ) φ(ξ). Then {φ;ψ 1,...,ψ s } is a tight framelet in L 2 (R d ): f L 2 (R d ), f 2 L 2 (R d ) = f,φ( k) 2 + k Z d s f, det(m) j/2 ψ l (M j k) 2. j=0 l=1 k Z d vm(ψ l ) = vm(b l ) for all l = 1,...,s. It is a challenging problem to construct multivariate wavelets or tight framelets with symmetry and high vanishing moments. Bin Han (University of Alberta) Applications of Wavelets UofA 32 / 47

33 Fourier Transform For a function f on R d, its Fourier transform is defined to be f(ξ) := f(x)e iξ x dx, ξ R d. R d For a sequence a : Z d C, its Fourier series is â(ξ) := a(k)e ik ξ, ξ R d. k Z d Bin Han (University of Alberta) Applications of Wavelets UofA 33 / 47

34 Cascade Algorithms How to solve the refinement equation: φ = det(m) k Z d a(k)φ(m k), where the mask a : Z d R is finitely supported, equivalently, φ(ξ) = â((m T ) 1 ξ) φ((m T ) 1 ξ). Cascade algorithm: The cascade operator R is defined to be R a,m f := det(m) k Z d a(k)φ(m k). φ is a fixed point of R a,m by φ = R a,m φ. {f n := R n a,m f} n N of functions is called a cascade algorithm. The cascade algorithm converges if for every compactly supported eligible initial function f, there exists a continuous function f such that lim n f n f C(R d ) = 0. Bin Han (University of Alberta) Applications of Wavelets UofA 34 / 47

35 Cascade Algorithm and Subdivision Schemes Cascade algorithm: the iterative sequence {f n := R n a,m f} n N of functions. Subdivision scheme: calculate v n := Sa,M n v for n N and attach the value v n (k) at the point M n (k c a ), k Z d. Relation: f n = R n a,m f = k Z d [S n a,m δ](k)f(mn k), where δ is the Dirac sequence such that δ(0) = 1 and δ(k) = 0 for all k 0. Let h be the hat function (in 1d, h = max(1 x,0)). Then connecting points of v n be flat pieces to form a function g n is equivalent to (assume c a = 0) g n = R n a,mf with f := k Z d v(k)h( k). Bin Han (University of Alberta) Applications of Wavelets UofA 35 / 47

36 Role of a Dilation Matrix Figure: represents vertices in the coarse mesh Z 2 and represents new vertices in the refinement mesh M 1 Z 2. The M-refinement of the reference mesh Z 2, from left to right, are for subdivision triplets (a,2i 2,D 4 ), (a,m 2,D 4), (a,2i 2,D 6 ), and (a,m 3,D 6), where M 2 and M 3. Bin Han (University of Alberta) Applications of Wavelets UofA 36 / 47

37 Implemented by Convolution Subdivision scheme: calculate v n := Sa,M n v for n N and attach the value v n (k) at the point M n (k c a ), k Z d. For β,γ Z d, [S a,m v](γ +Mβ) = det(m) k Z d v(k)a(γ +Mβ Mk) = det(m) [v a [γ:m] ](β), where the coset mask a [γ:m] of the mask a is defined to be a [γ:m] (k) := a(γ +Mk), k,γ Z d. Local averaging: det(m) k Z d a [γ:m] (k) = 1 for all γ Z d. The value [S a,m v](γ +Mβ) = v(β+ ), det(m) a [γ:m] ( ), is put at β +M 1 γ M 1 c a. M 1 γ-stencil of the mask a: { det(m) a(γ Mk)} k Z d. Bin Han (University of Alberta) Applications of Wavelets UofA 37 / 47

38 1D Subdivision Triplets For a finitely supported sequence a : Z R, we define a(z) := k Z a(k)z k, z C\{0}. Let M be an integer greater than one. Subdivision operator: [S a,m v](z) = Mv(z 2 )a(z). a has order n sum rules if and only if a(z) = (1+z + +z M 1 ) n b(z) for some Laurent polynomial b. a has order n linear-phase moments if and only if a(z) = z c + O( z 1 n ), z 1. a is interpolatory with respect to M if a(0) = 1, a(mk) = 0, k Z\{0}. M Bin Han (University of Alberta) Applications of Wavelets UofA 38 / 47

39 1D Subdivision Triplet The triplet (a,2,{ 1,1}) is a primal subdivision triplet with where a = 1 2 {w 3,w 2,w 1,w 0,w 1,w 2,w 3 } [ 3,3], w 0 = 3+t 4, w 1 = 8+t 16, w 2 = 1 t 8, w 3 = t 16 with t R. If t = 1, then a = 2 ab 6 ( 3) and sr(a,2) = 6, lpm(a) = 2 and sm p (a,2) = 5+1/p for all 1 p. If t 1/2, then sr(a,2) = 4. sm (a,2) = 3 log 2 (1+t) provided t > 1/2. We only have sm (a,2) 3 log 2 t for t 1/2. When t = 0, a = a4( 2) B is the centered B-spline filter of order 4 with sr(a,2) = 4 and lpm(a) = 2. When t = 1, a is an interpolatory 2-wavelet filter with sr(a,2) = 4 and lpm(a) = 4. Bin Han (University of Alberta) Applications of Wavelets UofA 39 / 47

40 Subdivision Stencils w 2 w 0 w 2 w 3 w 1 w 1 w 3 Figure: The 0-stencil (left) and the 1 2-stencil (right) of the primal subdivision scheme. It is an interpolatory 2-wavelet filter if w 2 = 1 t 8 = 0 (i.e. t = 1). Since M = 2, each line segment (with endpoints ) in the coarse mesh Z is equally split into two line segments with one new vertex ( ) in the middle. Bin Han (University of Alberta) Applications of Wavelets UofA 40 / 47

41 1D Subdivision Triplet The triplet (a,2,{ 1,1}) is a dual subdivision triplet with where a = 1 2 {w 2,w 1,w 0,w 0,w 1,w 2 } [ 2,3], w 0 = 12+3t 16, w 1 = 8 3t 32, w 2 = 3t 32 with t R. If t = 2, then a = 3 ab 5 ( 2) and sr(a,2) = 5,lpm(a) = 2 and sm p (a,2) = 4+1/p for all 1 p. sr(a,2) = 3 and sm (a,2) = 4 log 2 (4+3t) provided t > 2/3. We only have sm (a,2) 1 log 2 (3 t ) for t 2/3. When t = 0, a = a3 B( 1) is the shifted B-spline filter of order 3 with sr(a,2) = 3 and lpm(a) = 2. When t = 1, sr(a,2) = 3 and lpm(a) = 4. Bin Han (University of Alberta) Applications of Wavelets UofA 41 / 47

42 Subdivision Stencils w 1 w 0 w 2 w 2 w 0 w 1 Figure: The 0-stencil (left) and the 1 2-stencil (right) of the dual subdivision scheme. The 1 2-stencil is the same as the 0-stencil. The value [S a,2 v](k) for k Z is attached to the center k 1 2 of the line segment [k 1,k] instead of the vertex k 2. Since M = 2, each line segment is equally split into two. Bin Han (University of Alberta) Applications of Wavelets UofA 42 / 47

43 1D Subdivision Triplet The triplet (a,3,{ 1,1}) is a primal subdivision triplet with where a = 1 3 {w 5,w 4,w 3,w 2,w 1,w 0,w 1,w 2,w 3,w 4,w 5 } [ 5,5], w 0 = 7 2t 1 8t 2, w 9 1 = 6 2t 1 5t 2, w 9 2 = 3+t 1+t 2, 9 w 3 = 1+t 1+4t 2, w 9 4 = t 1+3t 2 with t, w 9 5 = t 2 1,t 2 R. 9 If t 1 = 2/9 and t 2 = 1/9, then sr(a,3) = 5 and sm p (a,3) = 4+1/p for all 1 p whose 3-refinable function is the B-spline of order 5. sm (a,2) 2 log 3 max( 1 2t 1 2t 2, 2t 1, 2t 2 ). If t 1 = 7/9 and t 2 = 4/9, then a is an interpolatory 3-wavelet filter with sr(a,3) = 4 = lpm(a) and sm (a,3) log If t 1 = 5/11 and t 2 = 4/11, then a is an interpolatory 3-wavelet filter with sr(a,3) = 3 = lpm(a) and sm (a,3) 2+log 3 (11/10) > 2. Bin Han (University of Alberta) Applications of Wavelets UofA 43 / 47

44 1D Subdivision Triplet w 3 w 0 w 3 w 4 w 1 w 2 w 5 w 5 w 2 w 1 w 4 Figure: The 0-stencil (left), the 1 3 -stencil (middle), and 2 3-stencil of the subdivision scheme. Due to symmetry, 2 3-stencil is the same as the 1 3 -stencil. It is an interpolatory 3-wavelet filter if w 3 = 1+t 1+4t 2 9 = 0. Since M = 3, each line segment (with endpoints ) is equally split into three line segments with two new inserted vertices ( ) at 1 3 +Z and 2 3 +Z. Bin Han (University of Alberta) Applications of Wavelets UofA 44 / 47

45 Examples of Subdivision Curve Bin Han (University of Alberta) Applications of Wavelets UofA 45 / 47

46 Examples of Subdivision Curve Bin Han (University of Alberta) Applications of Wavelets UofA 46 / 47

47 Masks Used Subdivision curves at levels 1, 2, 3 with the initial control polygons at the first row. (1) uses the subdivision triplet (a,2,{ 1,1}) with a = a B 4 ( 2) (2) uses interpolatory subdivision triplet (a, 2,{ 1, 1}). (3) uses (a,2,{ 1,1}) with a = a B 3( 1). (4) the corner cutting scheme (5) uses (a,3,{ 1,1}). (6) uses interpolatory (a, 3,{ 1, 1}). Bin Han (University of Alberta) Applications of Wavelets UofA 47 / 47