1 Preliminaries As mentioned in the previous section, the matrix A - PDF document

Quincunx wavelets on T 2 Kenneth R. Hoover and Brody Dylan Johnson Abstract This article examines a notion of finite-dimensional wavelet systems on T 2 , which employ a dilation operation induced by the Quincunx matrix. A theory of multiresolution analysis (MRA) is presented which includes the characterization and construction of MRA scaling functions in terms of low-pass filters. Orthonormal wavelet systems are constructed for any given MRA. Two general examples, based upon the classical Shannon and Haar wavelets, are presented and the approximation properties of the associated systems are studied. Introduction This work examines finite-dimensional systems of functions on the torus, T 2 , which employ the basic tenets of wavelet theory: dilation and translation. The present study follows a similar analysis on the circle [2], T , where dilation of f ∈ L 2 ( T ) was accomplished by a dyadic downsampling of the Fourier transform, i.e., � Df ( k ) = ˆ f ( 2 k ) , k ∈ Z . An obvious extension to L 2 ( T 2 ) would involve downsampling of the Fourier transform by 2 I 2 ; however, this choice fails to utilize the freedom provided by the move from one to two dimensions. Instead, the dilation operation considered here will be achieved through downsampling by a 2 × 2 matrix, A , satisfying • A has integer entries; • A has eigenvalues with modulus strictly greater than 1; • A has determinant 2. The first requirement is necessary for the downsampling operation � Df ( k ) = ˆ f ( Ak ) , k ∈ Z 2 , to be well defined. The second condition ensures that repeated dilation of a function f ∈ L 2 ( T 2 ) will tend to a constant function in L 2 ( T 2 ) . Finally, the third condition specifies that A should have minimal determinant. Indeed, if λ 1 and λ 2 Kenneth R. Hoover, California State University Stanislaus, e-mail: khoover@csustan.edu · Brody Dylan Johnson, Saint Louis University, e-mail: brody@slu.edu 1

2 Kenneth R. Hoover and Brody Dylan Johnson are the eigenvalues of A , then | det A | = | λ 1 λ 2 | is an integer greater than 1. It is not difficult to see that if det A = 2, then the trace of A will also be 2 under the above assumptions. A certain amount of the analysis will be independent of a specific choice for A . Nevertheless, A will hereafter denote the Quincunx dilation matrix, � 1 − 1 � A = . 1 1 √ This dilation is the composition of a rotation by π 4 with multiplication by 2 and, consequently, facilitates a natural geometric intuition. This discussion has focused on the role that A will play in the creation of a dilation operator. In the next section, the role played by A for translation will also be discussed. 1 Preliminaries As mentioned in the previous section, the matrix A plays two roles in the proceeding theory, one dealing with dilation and another related to translation. In dilation, the Fourier transform of a function f ∈ L 2 ( T 2 ) will be downsampled over the subgroup A Z 2 of Z 2 . Translation will be considered over a discrete subgroup of T 2 formed as a quotient of A − j Z 2 by Z 2 , where j > 0 determines the scale or resolution of the translations being considered. For a fixed integer j > 0, define the lattice of order 2 j generated by A , Γ j , as the collection of 2 j distinct coset representatives of A − j Z 2 / Z 2 . It will be assumed that each element of Γ j belongs to the rectangle [ 0 , 1 ) × [ 0 , 1 ) . In the next section, a notion of shift-invariant spaces will be introduced that consists of functions in L 2 ( T 2 ) which are invariant under translation by elements of Γ j . Recall that the dilation operation induced by A downsamples the Fourier trans- form of f ∈ L 2 ( T 2 ) by A . This operation will be best understood through the quotient groups Z 2 / B j Z 2 , where B = A ∗ . Consequently, define the dual lattice of order 2 j (j > 0 ) generated by A , Γ ∗ j , as the collection of 2 j distinct coset representatives of Z 2 / B j Z 2 determined by the intersection B j R ∩ Z 2 , where R = ( − 1 2 , 1 2 ] × ( − 1 2 , 1 2 ] . Because B has integer entries it follows that B j R ⊆ B j + 1 R , so Γ ∗ j is a natural subset of Γ ∗ j + 1 . The following lemma summarizes several elementary, but useful facts about the matrices A and B as well as the lattices Γ j and Γ ∗ j . Lemma 1. Let A, B as above and let j ≥ 2 be an integer. 1. Γ j = { A − 1 α + α ′ : α ∈ Γ j − 1 , α ′ ∈ Γ 1 } . j = { B β + β ′ : β ∈ Γ ∗ j − 1 , β ′ ∈ Γ ∗ 2. Γ ∗ 1 } . � 0 1 � 3. AB − 1 = . − 1 0 4. A k B ℓ = B ℓ A k , k ,ℓ ∈ Z .

Quincunx wavelets on T 2 3 3 4 5 4 3 2 3 2 2 1 1 1 0 0 0 −1 −1 −1 −2 −2 −3 −2 −3 −4 −3 −4 −5 −3 −2 −1 0 1 2 3 −4 −3 −2 −1 0 1 2 3 4 −5 −4 −3 −2 −1 0 1 2 3 4 5 Γ ∗ Γ ∗ Γ ∗ 3 4 5 Fig. 1 The dual lattices Γ ∗ 3 , Γ ∗ 4 , and Γ ∗ 5 . Another important feature of the lattices, Γ ∗ j , is their behavior under multipli- cation by A . Hence, let d j : Γ ∗ j → Γ ∗ j be the mapping defined by α �→ A α . For 1 ≤ k ≤ j − 1, let B k denote the kernel of d k j , i.e., B k = { β ∈ Γ ∗ j ( β ) = 0 } . j : d k The following proposition provides two useful characterizations of B k . Proposition 1. Let j ∈ N , j ≥ 2 and 1 ≤ k ≤ j − 1 . 1. B k = B j − k Γ ∗ j , i.e., d k j is a 2 k -to- 1 mapping. � � k ∑ b ℓ A j − ℓ β 1 : b ℓ ∈ { 0 , 1 } , where β 1 is the nonzero element of Γ ∗ 2. B k = 1 . ℓ = 1 Proof. To demonstrate the first claim, let β ∈ Γ ∗ j ( β ) = 0. Hence, j and assume that d k A k β ∈ B j Z 2 or A k B − k B k β ∈ B j Z 2 . Now, since powers of A and B commute and AB − 1 is a rotation, it follows that B k β ∈ B j Z 2 and thus β ∈ B j − k Z 2 . Likewise if β ∈ B j − k Z 2 , then A k β ∈ B j Z 2 and, thus, d k j ( β ) = 0. To prove the second claim observe first that k k ∑ ∑ b ℓ A j − ℓ β 1 = b ℓ A k − ℓ A j β 1 = 0 , A k ℓ = 1 ℓ = 1 because A k − ℓ has integer entries and A j β 1 ∈ B j Z 2 . To see that the 2 k elements are unique, assume that k k ∑ ∑ b ℓ A j − ℓ β 1 = ℓ A j − ℓ β 1 , b ′ ℓ = 1 ℓ = 1 or, equivalently, that k − 1 ∑ ℓ ) A j − ℓ β 1 = ( b ′ k − b k ) A j − k β 1 . ( b ℓ − b ′ ℓ = 1

4 Kenneth R. Hoover and Brody Dylan Johnson Thus the left- and right-hand quantities lie in the intersection of A j − 1 Γ ∗ j and A j − k Γ ∗ j . However, A j − k β 1 / ∈ A j − 1 Γ ∗ j , so it follows each expression must equal zero, i.e., b k = b ′ k . This argument may be repeated to show that b ℓ = b ′ ℓ , 1 ≤ ℓ ≤ k . ⊓ ⊔ 2 Shift-invariant spaces This section introduces a notion of shift-invariant spaces for L 2 ( T 2 ) which make use of the lattices Γ j , j > 0. The translation operator generated by α ∈ Γ j will be denoted T α : L 2 ( T 2 ) → L 2 ( T 2 ) and is defined by T α f ( x ) = f ( x − α ) , x ∈ T 2 . A shift-invariant space in this context will consist of a closed subspace V of L 2 ( T 2 ) with the property that f ∈ V if and only if T α f ∈ V for all α ∈ Γ j . Of course, if V is invariant under shifts in Γ j , then V is also invariant under shifts in Γ k , 1 ≤ k ≤ j . This work shall focus attention on shift-invariant spaces generated by the Γ j -translates of a single function. Definition 1. Let φ ∈ L 2 ( T 2 ) . The principal A-shift-invariant space of order 2 j gen- erated by φ , denoted V j ( φ ) , is the finite-dimensional subspace of L 2 ( T 2 ) spanned by the collection X j ( φ ) = { T α φ : α ∈ Γ j } . (1) A function in V j ( φ ) is simply a linear combination of the Γ j -translates of φ , which motivates the following definition. Let ℓ ( Γ j ) denote the space of complex valued functions on Γ j , with an analogous meaning for ℓ ( Γ ∗ j ) . Define e j , α ∈ ℓ ( Γ ∗ j ) , j > 0, α ∈ Γ j , by e j , α ( β ) = exp ( 2 π i � α , β � ) , β ∈ Γ ∗ j . Lemma 2. The collection { 2 − j 2 e j , α } α ∈ Γ j is an orthonormal basis for ℓ ( Γ ∗ j ) . Proof. Given α ′ , α ′′ ∈ Γ j , the inner product of e j , α ′ with e j , α ′′ can be expressed as � e j , α ′ , e j , α ′′ � = ∑ exp ( 2 π i � α , β � ) , β ∈ Γ ∗ j where α = α ′ − α ′′ . If α = 0 then the inner product is 2 j . However, if α � = 0, then there exists β ′ ∈ Γ ∗ j such that � α , β ′ � / ∈ Z , in which case exp ( 2 π i � α , β ′ � ) � = 1. Since j + β ′ ≡ Γ ∗ Γ ∗ j , this leads to exp ( 2 π i � α , β ′ � ) ∑ exp ( 2 π i � α , β � ) = ∑ exp ( 2 π i � α , β � ) , β ∈ Γ ∗ β ∈ Γ ∗ j j ⊓ ⊔ which forces the sum, and hence the inner product, to be zero.