Construction Of Generalized Multivariate Interpolating Refinable Function Vectors And Wavelets Applications In Grayscale Image Hiding Techniques

In this paper, we discuss a novel type of multivariate refinable functionsvectors with interpolation property and investigate their properties. Thecorresponding construction method is also presented. As a complementary,a new grayscale image hiding method is exhibited later.The main works of this dissertation he in the following three parts:.1. Based on the present results of interpolating refinable functions andfunction vectors, by introducing scaling dilation matrix and multiplicity dilationmatrix, we provide a generalized notion of multivariate interpolatingrefinable function vectors, that is, interpolating M-refinable function vectorsof type (M,R). With the theories of multivariate refinable functionvectors and vector cascade algorithms, a necessary and sufficient conditionfor the existence of generalized interpolating refinable function vectors isgiven strictly.2. We characterize the generalized interpolation property, orthogonalityand symmetry property in terms of their matrix masks. The sum rulestructure of the interpolatory masks of type (M, R) is also studied. As acomplementory, some special characters of the interpolatory masks are alsostudied. Some substantial examples is presented later. 3. In the last part of this paper, combining with wavelets processing techniques, we present a novel and effective grayscale image hiding method. Thenew method exhibits excellent perceptual transparency and anti-attackingability.This paper is organized as follows.In the introduction, we have briefly introduced the history of wavelet analysisand its applications, also including the background of this dissertation.In chapter 1, we have reviewed some necessary definitions, glossaries andsome important theories. In addition, some necessary notations and assumptionswere also introduced.In chapter 2, we generalized the notion of the existing interpolating refinablefunction vectors to compactly supported (M, R)-interpolating Mrefinablefunction vectors, that is, interpolating refinable function vectorsof type (M, R), which is characterized in terms of their masks. Consideringsome important and necessary properties of wavelets analysis, we also dicussedorthogonality and symmetry property of the interpolating refinablefunction vectors of type (M, R).In chapter 3, we studied the sum rule structure of the interpolatory masksof type (M, R), which plays a central role in our construction of interpolatorymasks of type (M, R) with increasing orders of sum rules. Additionally,some special properties of the interpolatory masks were investigated in thisthis section. As an important complementary, some examples of symmetricand orthogonal interpolating refinable function vectors were presented inthis section.In chapter 4, a novel and effective grayscale image hiding technique wasprovided detailedly. By introducing multi-resolution transform of an embeddedimage, the new image embedding method utilizes spatial-frequencyquantization and Peano transposition for embedded information generation.Experimental results show that our watermarking method is feasible, ratio- nal and robust to common image signal processing attacks. In addition, itexhibits excellent perceptual transparency.Finally, we have briefly summarized the research work in the dissertationand pointed out further problems to resolve.Our main results covers chapter 2 to chapter 4. The main results fromthe dissertation can be listed as follow.1. IntroductionThroughout this dissertation, R^s, Z^s, N₀^s denote the real plane, the set ofall integers and the set of all non-negative integers, respectively.Forμ=(μ₁,…,μ_s)∈N₀^s, we denote |μ|:=|μ₁|+…+|μ_s|,μ!:=μ₁!…μ_s! andξ^μ?=ξ₁^μ₁…ξ_s^μ_s forξ= (ξ₁,…,ξ_s)∈R^s. Let O_k be theordered set {μ∈N₀^s:|μ|= k} under the lexicographic order. That is,v = (v₁,…,v_s) is less thanμ=(μ₁,…,μ_s) in the lexicographic order ifv_j =μ_j for j=1,…, i-1 and v_i <μ_i. By #O_k we denote the cardinalityof the set O_k. Throughout the paper, for a smooth function f, the partialderivative of a differentiable function f with respect to the jth coordinateis denoted by D_jf, j=1,…, s and forμ=(μ₁,…,μ_s)∈N₀^s, D^μdenotesthe differential operator D₁^μ₁…D_s^μ_s.For 1≤p≤∞, L_p(R^s) denotes the set of all Lebesgue measurable functionsf such that ||f||_Lp(R^p)^s:=∫_R^s|f(x)|^p dx <∞. And ((?)₀(Z^s))^r×s denotesthe linear space of all finitely supported sequences of r×s matrices onZ^s, and ((?)_p(Z^s))^r×s for 1≤p≤∞denotes the linear space of all sequencesv of r×s matrices on Z^s such that ||v||_(?)p(Z^s)^r×s:=(∑_{β∈Z^s}|v|^p)^1/p<∞for 1≤p <∞and ||v||_{(?)∞(Z^s)}^r×s:=sup_{β∈Z^s}|v(∞) ||, where ||·|| denotesany matrix norm on r×s matrices.Definition 1.1.1 An s×s integer matrix M is called a dilation matrix ifall its eigenvalues are greater than one in modulusDefinition 1.1.2 A dilation matrix M is isotropic if M is similar to adiagonal matrix diag(σ₁,…,σ_s) such that |σ₁|=…= |σ_s|=|det M |^1/s. Definition 1.1.3 An MRA of L²(R^s) consists of a set of closed subspaces{V_j}_j∈Z(V_j (?) V_j+1) that satisfies following conditions:·There exists (?)(x)∈V₀ such that {(?)(x-k)}_k∈Z^s form the Riesz basesof V₀。From the definition of MRA , the following equation holds.Definition 1.1.4 We calledφ=(φ₁,…,φ_r)^T:R^S(?)C^r×1 is a M-refinablefunction vector ifφsatisfies the following vector refinement equationa:Z^s(?) C^r×r is a finitely supported sequence of r×r matrices on Z^s, calledthe matrix mask for the M-refinable function vectorφ.In the frequency domain, the vector refinement equation in (1) can berewritten aswhereDefinition 1.1.5 The following iteration scheme is called a (vector) cascadealgorithm associated with mask a and dilation matrix Mwhere Q_a,M is the cascade operator on (L_p(R^s))^r×1. Definition 1.1.6 A finite set G(?){B∈Z^s×s:|detM|=1} of matricesis a symmetry group with respect to a scaling dilation matrix M on Z^s if Gforms a group under matrix multiplication and for all B∈G, MBM^-1∈G.Definition 1.1.7 A sequence a on Z^s is G-symmetric with a center c_a∈Z^sifIt is easy to see that (5) is equivalent toDefinition 1.1.8 A function f on R^s is G-symmetric with a center c_f∈R^sifDefinition 1.1.9 For two s×s matrices N and M, we say that N is Gequivalentto M if there exists A, B∈G such that N = AMB.For discussion conveniently, let us introduce the following assumptionsinto this dissertation.Assumption 1.2.1 Let M and R be two s×s dilation matrix with m:=| detM | and r:= |detR|.·LetΓ_M:={(?)₀,…,(?)_m-1} and (?)_M:= {(?)₀,…,(?)_m-1} be the completesets of representatives of different cosets of (M^T)^-1Z^s\Z^s and Z^s\MZ^srespectively.·LetΓ_R:={ρ₀,…,ρ_r-1} and (?)_R:={(?)₀,…,(?)_r-1} be the complete setsof representatives of different cosets of (R^T)^-1Z^s\Z^s and R^-1Z^s\Z^s.Without loss of generality, we set (?)₀ = (?)₀=ρ₀=(?)₀= 0.Definition 1.3.1 For an r×1 compactly supported function vector f on R^s,we say that the shifts of f are stable if span{(?)(ξ+ 2πβ):β∈Z^s}= C^r×1 for allξ∈R^s, and the shifts of f are linear independent if span{(?)(ξ+ 2πβ) :β∈Z^s}=C^r×1 for allξ∈C^r×1.Definition 1.3.3 For 0 < v≤1 and 1≤p≤∞, we say that f∈Lip(v, L_p(R^s)) if there is a positive constant C such thatThe L_p smoothness of a function f∈L_p(R^s) is measured by its L_p criticalexponent v_p(f) defined byParticularly with multiplicity r, for a function vector f = [f₁,…,f_r]^T,v_p(f):=min{v_p(f_e):e=1,…,r}.Definition 1.3.4 For a matrix mask a∈(e₀(Z^s))^r×r, we say that a satisfiesthe sum rules of order k + 1 with respect to MZ^s if there exists a sequence(?)∈(e₀(Z^s))^1×r such that (?)(0)=0where k = 0,…, m-1 andδis the Dirac sequence such thatδ₀ = 1 andδ_β=0 for allβ∈Z^s\{0}.Equivalently, the definition of sum rules can be given in time domain asfollows.Definition 1.3.5 a satisfies the sum rules of order k + 1 if there exists aset of r×1 vectors {y_μ∈R^s:μ∈N₀^s, |μ|≤k} with y₀≠0 such thatholds for allμ∈N₀^s with |μ|≤k and all (?)_k∈(?)_M where and the numbers m(μ, v) are uniquely determined byLet us recall an important quantity v_p(a;M) from [54, 57], which characterizesconvergence of a vector cascade algorithm in Sobolev space and L_psmoothness exponent of a refinable function vector.For y∈(e₀(Z^s))^1×r and a positive integer k, as in [57], we define the spaceBy convention, V_0,y:= (e₀(Z^s))^r×1.Let us definewhere (?)_n:=Π_j=1ⁿ(?)((M^T)^n-jξ) and V_k,y is defined in (11). DefineAs in [57], we define the following quantityParticularly, up to a scalar multiplicative constant, the vectors D^μ(?)(0),μ∈N₀^s are quite often uniquely determined.The quantity v_p(a;M) will play an important role in our investigation ofmultivariate interpolating refinable function vectors. It is showed in [57,Theorem 4.3] and [54, Theorem 3.1] that for a nonnegative integer k, thevector cascade algorithm converges in the Sobolev Space W_p^k(R^s):= {f∈L_p(R^s):D^μf∈L_p(R^s),(?)_μ∈N₀^s, |μ|≤k} if and only if v_p(a;M)>k.In general, v_p(a;M)≤v_p(φ) always holds. In addition by recalling [54, Theorem 4.1], if the shifts of the refinable function vectorφassociatedwith mask a and dilation matrix M are stable in L_p(R^s), then v_p(a;M)characterizes the L_p smoothness exponent of a refinable function vectorφwith mask a and dilation matrix M, that is, v_p(a;M) = v_p(φ) in this case.2. Characterization of Interpolating Refinable Function VectorsIn this section, we concentrated on interpolating refinable function vectorsof type (M, R) on R^s and investigate their properties.Definition 2.1.2 Let M be an s×s dilation matrix and let N be an s×sdialtion matrix such thatRMR^-1 is an integer matrix. (15)hold. A refinable function vectorφ= (φ₁,…,φ_r)^T:R^s(?)C^r×1 is (M, R)-interpolating with respect to an s×s dilation matrix R ifφis continuousandφsatisfies the following interpolation conditionwhere e=1,…, r, p = 0,…, r-1. Here,we callφas interpolatingrefinable function vector of type(M, R). And M is a scaling dilation matrixand R is a multiplicity dilation matrix.If (15) holds, then for all (?)_p∈(?)_R, there exists k_p∈{0,…,r-1} suchthat (?)_kp∈(?)_R satisfiesIt is easy to verify that the shifts ofφin (16) are linearly independent.The following theorem characterizes the compactly supported interpolatingrefinable function vector of type (M, R) in R^s.Theorem 2.2.5 Let M be an s×s isotropic dilation matrix and R be an s×sdilation matrix such that (15) holds. Letφ=[φ₁,…,φ_r]^T be a compactly supported M-refinable function vector such that (?)(M^Tξ) =(?)(ξ)(?)(ξ). Thenφis (M, R-interpolating with respect to an s×s dilation matrix R, that is,φis a continuous function vector and (16) holds if and only if the followingstatements holds:(i) [1,…,1](?)(0)=1;(ii) a is an interpolatory mask of type (M, R): [1,…,1](?)(0)=[1,…,1]andwhereγ_p∈Z^s, p∈{0,…, r-1} and k_p∈{0,…, r-1} are uniquelydetermined by the relation (17).(iii)v_∞(a;M)>0.As a consequence of Theorem 2.2.5, the following result characterizesorthogonal (M, R)-interpolating M-refinable function vectors with compactsupport.Theorem 2.3.1 Let M be an s×s isotropic dilation matrix and R be ans×s dilation matrix such that (15) holds. Let a : Z^s (?)C^r×r be a finitelysupported sequence of r×r matrices on Z^s. Suppose thatφ=[φ₁,…,φ_r]^Tbe a compactly supported M-refinable function vector such that (?)(M^Tξ)=(?)(ξ)(?)(ξ). Thenφis an orthogonal (M, R)-interpolating function vector,that is,φis a continuous function vector and (16) holds andif and only if, (i)-(iii) of Theorem 2.2.5 hold and a is an orthogonal mask: The following result characterizes compact supported orthogonal interpolatingrefinable function vectors of type (M, R) with their norms.Corollary 2.3.2 Let M be an s×s isotropic dilation matrix and R be ans×s dilation matrix such that (15) holds. Let a andφbe as in Theorem2.2.5. Ifφis an orthogonal interpolating refinable function vector of type(M, R), then (i)-(iii) of Theorem 2.2.5 hold andφsatisfiesAs an important supplementary, we shall investigate the symmetry propertyof multivariate interpolating refinable function vectors of type (M, R)in this section.Corollary 2.4.1 Let M be an s×s scaling dilation matrix and G be asymmetric group with respect to M. Let a : Z^s (?) C^r×r be a finitelysupported sequence of r×r matrices on Z^s and satisfies that 1 is a simpleeigenvalue of the matrix (?)(0) and all the other eigenvalues of (?)(0) are lessthanρ(M)^-k in modulus. Suppose thatφ=[φ₁,…,φ_r]^T be a compactlysupported M-refinable function vector such that [1,…,1](?)(0)=1 and(?)(M^Tξ)=(?)(ξ)(?)(ξ).If for all B∈G and k∈{1,…,r}, (I_s- B)c_k∈Z^s.Then for k∈{1,…,r}, each component functionφ_k ofφis G-symmetricwith a center c_k∈R^s if and only if for allβ∈Z^s and j∈{1,…, r},[a(β)]_k,j is G-symmetric with a center M_ck-c_j.In a general way, if taking c_e=(?)_p, then as to an interpolating refinablefunction vectorφof type (M, R), a complete freedom for selecting B∈Gcan be achieved.Theorem 2.4.2 Let M and R be two s×s dilation matrices such that(15) holds and G be a symmetric group with respect to M. Let a : Z^s (?)C^r×r be a finitely supported sequence of r×r matrices on Z^s. Supposethatφ=[φ₁,…,φ_r]^T be a compactly supported (M,R)-interpolating M-refinable function vector such that (?)(M^Tξ)=(?)(ξ)(?)(ξ).If for all B∈G and e∈{1,…,r}, (I_s-B)(?)_e-1∈Z^s. Then for e∈{1,…, r}, each componentfunctionφ_e ofφis G-symmetric with a center (?)_e-1∈(?)^R if and only if forallβ∈Z^s and j∈{0,…,r-1}, [a(β)]_e,j is G-symmetric with a center3. Construction of Interpolatory Masks of Type (M,R)In this section, we studied the structure of vector (?) of the family of interpolatorymasks of type (M, R), and provided a family of interpolatorymasks of type (M, R) with increasing orders of sum rules.Lemma 3.1.1 Let M and R be two s×s dilation matrices such that (15)holds for p= 0,…, r-1. Let a be a finitely supported sequence ofr×rmatrices on Z^s. Suppose that a is an interpolatory mask of type (M,R),that is, [1,…,1](?)(0)=[1,…,1] and (18) holds. If a satisfies the sum rulesof order k + 1 with a sequence (?)∈(e₀(Z^s))^1×r and (?)(0)=[1,…,1], thenThe following lemma is about the y_μin the time domain as defined inDefinition 1.3.5.Lemma 3.1.2 Let M, R and a be as in Lemma 3.1.1. Suppose that a isan interpolatory mask of type (M, R). If a satisfies the sum rules of orderk + 1, then the r×1 vectors y_μ,|μ|≤k satisfiesFor discussion conveniently, we denote [A]_k,j as the (k,j)-entry of thematrix A, and [A]_:,j as the j-th column of the matrix A.Let us denoteLemma 3.1.3 Let M, R and a be as in Lemma 3.1.1. Suppose that a isan interpolatory mask of type (M, R). If for a given k∈N₀^s, a satisfies the sum rules of order k+1, then for each ((?)_k,(?)_j)∈(?)*_M,R, the following relationalways holdswhereγ_p = Mβ_p+ (?)_k*_p, p∈{0,…, r - 1}.In this way, let us denoteTheorem 3.1.5 Let M and R be two s×s dilation matrices such that (15)holds for p=0,…, r-1. Suppose that for every ((?)_k, (?)_j)∈(?)_M,R,E(k,j) is asubset of Z^s such that [a(Mβ+(?)_k)]_:,j+1=0 for allβ∈Z^s\E(k,j). Supposethat a∈(e₀(Z^s))^r×r is an interpolatory mask of type (M,R). Then asatisfies k+1 orders of sum rule if and only if for all ((?)_k,(?)_j)∈(?)_M,R, thefollowing relation holdsAs a direct consequence of Theorem 3.1.5, we may have the followingremark.Theorem 3.1.7 Let M and R be two s×s dilation matrices such that (15)holds for p = 0,…, r-1. Suppose that a∈(e₀(Z^s))^r×r is an interpolatorymask of type (M, R). Then a satisfies k + 1 orders of sum rule if and onlyif for all j=0,…, r-1 and |μ|≤k,μ∈N₀^s, the following relation holdsThe following lemma shows that the dilation matrix M can be replacedby G-equivalent dilation matrices. Lemma 3.2.1 Let M be an s×s dilation matrix and G be a symmetricgroup with respect to M. Suppose that N is an s×s integer matrix. ThenN is G-equivalent to M, that is, for A, B∈G, N=AMB if and only ifMN^-1∈G, or equivalently, M^-1N∈G.Lemma 3.2.2 Let M be an s×s dilation matrix and G be a symmetricgroup with respect to M. If an integer matrix N is G-equivalent to M, thenTV is an s×s dilation matrix and G is also a symmetric group with respectto N.Lemma 3.2.3 Let M be an s×s isotropic dilation matrix and G be asymmetric group with respect to M. For an s×s integer matrix N, if thereexists an s×s integer matrix (?)∈G such that M =(?)N(?)^-1, then N is ans×s isotropic dilation matrix and G is also a symmetric group with respectto N.Remark 3.2.4 Let M be an s×s isotropic dilation matrix and G be asymmetric group with respect to M. Let R be an s×s isotropic dilationmatrix such that (15) holds for p = 0,…, r-1. Let N and (?)∈G be as inthe Lemma 3.2.3. Then R'NR'^-1 is an integer matrix, where R' := R(?).Theorem 3.2.5 Let M be an s×s isotropic dilation matrix and G be a symmetricgroup with respect to M. Let a∈(e₀(Z^s))^r×r and satisfies that 1 is asimple eigenvalue of the matrix (?)(0) and all the other eigenvalues of (?)(0) areless thanρ(M)^-k in modulus. Suppose thatφ=[φ₁,…,φ_r]^T is a compactlysupported M-refinable function vector that satisfies (?)(M^Tξ)=(?)(ξ)(?)(ξ) and[1,…,1](?)(0)=1. If an s×s integer matrix N is G-equivalent to M, andfor e∈{1,…, r}, each component functionφ_e ofφis G-symmetric with acenter c_e, then (?)(N^Tξ)=(?)(ξ)(?)(ξ),whereMoreover the following relation always holds Theorem 3.2.6 Let M and R be two s×s dilation matrices such that(15) holds for p = 0,…, r-1. Let G be a symmetric group with respectto M. Suppose that a∈(e₀(Z^s))^r×r andφ=[φ₁,…,φ_r]^T is a compactlysupported interpolating refinable function vector of type (M, R), and fore∈{1,…,r}, each component functionφ_e ofφis G-symmetric with acenter (?)_e-1∈(?)_R. Let N and (?)∈G be as in the Lemma 3.2.3. Thenφis a compactly supported interpolating refinable function vector of type(N,R(?)) such that (?)(N^Tξ) =(?)(ξ)(?)(ξ), where (?)(ξ) can be given by (27).In many applications, balancing is a very important property for discretewavelet transform. A generalized notion of the balancing as follows can beconsulted in [18, 19].Definition 3.3.1 An orthogonal r-scaling vectorφ=[φ₁,…,φ_r]^T is kbalancedrelative to {ξ₁,…,ξ_r} (?) R^s if and only iffor all a∈N₀^s with a < k.Theorem 3.3.2 Letφbe a compactly supported interpolating orthogonalM-refinable function vector of type (M, R), and a∈(e₀(Z^s))^r×r satisfies thesum rules of order k. Thenφis k-balanced relative to (?)_R.3. Grayscale Image Hiding Technique Based on MRA SchemesIn this part, we presented a novel approach to the generation of watermarksfor grayscale image embedding by using MRA transform and a newprocess, which we call the Peano transposition. The Peano transpositionis applied to the wavelet transformed coefficients to ensure that the coefficientsin all the frequency domains are uniformly redistributed in the wholetime domain. A special spatial-frequency quantization method is used toreduce the energy in the low frequency resulting in a uniform signal amplitudein the watermark. In addition, corresponding to this multi-resolutionembedded informations, an embedding method with multiple strengths is also given. Experimental results show that the proposed watermark generationand embedding method has excellent perceptual transparency, and isrobust to common image signal processing attacks.