Construction Of Multivariate Biorthogonal Interpolating Refinable Function And Wavelets And Wavelets Application

In this paper, we mainly discuss the constructions of bivariate Box spline periodic interpolating biorthogonal wavelets in 3-directional coordinate and the corresponding rapid decomposition and reconstruction algorithm, the construction of multivariate biorthogonal (M, R) interpolating G symmetric refinable function vectors, and the application of wavelets in finger-vein image enhancement.This dissertation is consisted of five parts that are the introduction, body and conclusion.definition 1 Let M be s×s isotropic dilation matrix, R is s×s dilation matrix, RMR^-1 is a integer matrix,Φ= [Φ₁,…,Φ_r]^T is (M,R) interpolating refinable function vector,(?) is compactly M refinable function vector,α(β), (?) are their corresponging finite supported mask matrix sequence,β∈Z^s. we callΦ= [Φ₁,…,Φ_r]^T, (?) are (M,R) biorthogonal interpolating refinable function vector,if(?), (1.1)Next we will give the necessary condition of biorthogonal refinable function vector.proposition 2 LetΦ= [Φ₁,…,Φ_r]^T and (?) are biotthogonal M-refinable function vector.α,(?) are their corresponging finite supported mask matrix sequence , thenΦ= [Φ₁,...,Φ_r}^Tå’Œ(?) are M-refinable vector ifα,(?) contend(?). (1.2)The biorthogonal condition will be simple if we introduce the interpolating condition.proposition 3 LetΦ= [Φ₁,...,Φ_r}^T and (?) are biothogonal compactly (M, R) interpolating refinable function vector.α,(?) are their corresponging mask matrix function. If M = R,then (1.2) if and only if(?). (1.3)Han bin give a theorem on (?) in biorthogonal condition, accroding to the theorem, we can get several equations on (?). Then we a new proof.Theorem 4 Let M is s×s isotropic dilation matrix, R is s×s dilation matrix,RMR^-1 is a integer matrix , LetΦ= [Φ₁,...,Φ_r}^T and (?) are M biorthogonal refinable vector,α,(?) are their corresponging mask matrix function ,α(β) , (?) are their corresponging finite supported mask matrix sequence, (?) condend k order sum rules,k is a integer, (?) is not equal to 0. Let O_n isαorder set under the lexicographic order. {μ₁,...,μ_N} = (?), 0 < n < k, which cardinality (?) equals N,Letthen(1.4) Han bin does not give the multivariate exam, we give in the condition (M, R) interpolating, which algorithm is as follow:Algorithm 5 (1) Select two dilation matrix M and R which contend (1.23). then we may get the cosets of Z²\MZ² and R^-1Z²\Z². thenγ_iå’Œ(?) , i = 1,…,r.(2) Select E (?) Z²(3) According to (1.29) in 1.2.7, we may get {α(β),β∈E} which contends (1.29).(4) Select G, by 1.3.5, according to (?) ofΦ= {Φ₁,…,Φ_r }, we mayobtain (?) symmetry of [α(β)]. where l,j∈{1,…,r},then we may obtain the symmetric points {[α(β)]_{l, j},l, j = 1,…, r,β∈E } by (1.30).(5) Select appropriate sum rules order,by 1.1.13, give the linear equations on (?) and get the solutions.(6) Like 4, compute the symmetric points of (?).(7) Select E' (?)Z² of (?).(8) Apply biorthogonal condition to [α(β)] and (?), obtain the corresponding equations.(9) Add the equations on (?) to above equations.(10) By 1.1.13, give the linear equations on (?).(11) Solve above linear equations and select the appropriate solution.In this chapter, starting from the Box-spline, we make full use of the generalized Fourier analysis method on three-directional hexagonal partition and construct the orthogonal periodic Box-spline wavelet.By generalized Fast Fourier transform on three-directional hexagonal partition we give a kind of a concrete realization of the fast algorithm which greatly reduces the computation. Let m₁, m₂ is positive integer, m₃ is nonnegative integer, denote (?)= [m₁, m₂, m₃]^T, (?) = m₁ + m₂ + m₃. letThe bivariate Box spline function (?)(x) (denote B(t)) which is corresponding to A :Let(?), (1.5)obviously (?)(ω) is trigonometric polynomial, which Fourier series :Let(?). (1.7)For any j≥0, k∈A, let B_k^j(t) come from (1.7). let(?). (1.8)thenTheorem 6 {V_j}_j≥0 in (1.8) forms a periodic multiresolution analysis in L_*²(Ω){B_k^j(t); k∈△_j} forms a basis of V_j, but not orthogonal, next we construct orhonormal basis of V_j. for j≥0, k∈△_j, let(?), (1.9) where(?) (1.10)(?). and for any p∈A, s∈(?), (?)(?),thenTheorem 7 for (?), k∈△_j, then (?) formsαorhonormalbasis of V_j. and(1) (?).(2) (?)(3)Φ_k^j(t) contends(?) (1.11)where(?) (1.12)Where C(ω) in (1.6),(?)Next we construct corresponding period orthogonal wavelets.For j≥0, k∈△_j,let(?) (1.13)B(j,k) contends (?) (1.14)let(?) (1.15)For j≥0,let W_j^ν=span{φ_k^j,ν;k∈△_j}ν=1,2,3.let W_j is orthogonalcomplement of V_j in V_j+1, then L_*² (Ω)=V₀ (?)W_j. and then Theorem 8 For j≥0, W_j^ν(ν=1,2,3) is is orthogonal subspace decomposition of W_j. when 1≤μ≠ν≤3, W_j =(?)W_j^ν, (?). and {φ_k^j,ν(x); k∈△_j,ν= 1,2,3} forms a orhonormal basis of W_j. Furthermore {φ_k⁰(x), k∈△₀}∪{φ_k^j,ν(x); k∈△_j,ν=1,2,3} _j≥0 forms a orhonormal basis of L₂(Ω).Define(?) , (1.16)(?) , (1.17)whereα_k,p^j =α_P^jg_p N(-2^-Jk),thenproposition 9 for j≥0, {D_k^j(x); k∈△_j} forms a basis ofV_j,and(?). (1.18)proposition 10 For all j≥0, k∈△_j, let (?)(x) is defined by (1.17), then(?). (1.19)furthomore, {(?)(x) ,k∈△_j} forms a basis ofV_j.proposition 11 For j≥0, k, l∈△_j, let D_k^j(x),(?)(x) are defined by (1.16) and (1.17),thenThen we obtain biorthogonal scaling funtions D_k^j(x) and (?)(x)。For j≥0, k∈△_j,ν=1,2,3,let Define(?), (1.20)whereλ_p^j,ν=(?).proposition 12 For j≥0, k,l∈△_j,ν= 1,2,3, let G_k^j,ν{x), (?)(x) are defined by (1.20) and (1.21),thenThen j≥0,ν= 1,2,3, {G_k^j,ν(x),k∈△_j} and {(?)(x),k∈△_j} forms a dual orhonormal basis of W_j^ν,which called biorthogonal period wavelets.proposition 13 for j≥0, k∈△_j, (ν= 1,2,3), D_k^j(x) and G_k^j,ν{x) are real value,and contend(?). (1.22)proposition 14 For j≥0, k∈△_j,ν= 1,2,3,(?)(x)and (?)(x) are real value,and contend(?). (1.23)For j≥0, k∈△_j,ν= 1,2,3,we give the two scale equation of D_k^j(x), (?),G_k^j,ν (x), (?)(x)as follow:(?),(1. 24) where(?),(1.25)(?),(1.26)(?),(1.27)Decomposition :(?) (1.28)Reconstruction , (?) (1.29)Next we consider the fast implementation of the Aalgorithms,First, we consider the initialization of the algorithms.If f∈V_j,then f(x) can be written the following formCombine with the cardinal interpolatory property of D_p^J(x), we havehis shows that the projection coefficients for a given scale are the uniform samples of the signal without prefiltering.Let E_j^pK=g_p^N(2^-jk,then where(?) (1.30)(?) (1.31)Furthermore, (1.31) can be written the following formwhere(?) (1.32)then(?) (1.33)Forμ= 1,2,3, similarly(?), (1.34)(?),(1.35)With respect to the reconstruction algorithm, let(?),(1.36)where and k∈△_j+1, there exists 0≤ν≤3 and l∈△_j, make k = 2l+ e_ν.andFurthermore(?) (1.37)where(?) , (1.38)(?) , (1.39)Similarly(?) (1.40) where(?) (1.41)(?) (1.42)(?) (1.43) The above conduct shows that we can implement the decomposition and reconstruction algorithms by the following scheme : ·Decomposition algorithms : - Split {c_k^j+1}_{k∈△j+1} to four subbands {c_2p+eμ^j+1}_{p∈△j;μ=0,1,2,3}.- Obtain B_j,μ^D(μ=( 0,1,2,3) from (1.32)by using FFT- Obtain {S_j,μ^D(q);S_j,μ^ν,G(q)}_{q∈△j;μ=0,1,2,3;ν=1,2,3} from (1.30)and (1.35)- Compute{c_p^j;d_p^j,μ}_{p∈△j;μ=1,2,3} as in (1.33) and (1.34)by IFFT .·Reconstruction algorithms :- Compute {F_j^D(q); F_j^μ,G} (q∈△_j,μ= 1,2,3), From {c_p^j;d_p^j,μ}_{p∈△j;μ=1,2,3} by FFT as in (1.39) and (1.43).- For all k∈△_j+1, deter 0≤ν≤3 and l∈△_j such as k = 2l + e_ν- From (1.38) and (1.41) and (1.42), Obtain {R_J,ν^D(q);Γ_J,0^μ,G(q);Γ_J,l,ν^μ,G(q)} (q∈△_j,μ= 1,2,3) from (1.37) and (1.40) by IFFT .- Compute {c_2l+eν^(j+1),D;c_{μ,(2l+eν)}^(j+1),G}- Obtain {c_k^j+1}_{k∈△j+1}from (1.36).An image of finger-vein captured under infrared light is not clear because of light and electrical noise added to the CCD camera image.In this system, an infrared light is transmitted from the backside of the hand. A finger is placed between the infrared light source and camera. As hemoglobin in the blood absorbs the infrared light, the pattern of veins in the finger side of the hand is captured as a pattern of shadows. Unfortunately, the captured images contain not only vein patterns but also irregular shading and noise. The shading is produced by the varying thickness of finger bones and muscles. Therefore, we have to enhance the contrast and obtain the binary image of finger-vein for late pattern identification.In this paper, we use wavelet-transform method. As we know, Using discrete wavelet transform, we can transform the signal into its different frequency bands, and for each band, we design different method of denoising. First we perform stationary wavelet decomposition, and transform the image into four frequency bands. Because this frequency band contains the main information of the image, we use soft-threshold method to maintain the smoothness of the image. Because it consists of isolate white noise, so we simply apply hard-threshold method to eliminate noise. Because middle frequency bands LH and HL contain the edge information of the vein-image, and most of them are local extrema in row or column. So we only save these extrema in row or column of the middle frequency band, and other wavelet coefficients are set to be zero. Classical poly-threshold and multi-threshold segmentation methods are not adapt to finger-vein segmentation because different part may has different thresholds. In practice, we find that the effect of multi-threshold is better than that of poly-threshold method. With the increasing number of threshold, the effect of segmentation turn well. Therefore, we increase the number to 1/4 of the number of all the effective image pixels. Experiment shows the favorable segmentation effect.