Finite frames and Sigma-Delta quantization John J. Benedetto - PowerPoint PPT Presentation

Finite frames and Sigma-Delta quantization John J. Benedetto Norbert Wiener Center, Department of Mathematics University of Maryland, College Park http://www.norbertwiener.umd.edu Finite frames and Sigma-Delta quantization – p.1/ ??

Outline and collaborators 1. Finite frames 2. Sigma-Delta quantization − theory and implementation 3. Sigma-Delta quantization − number theoretic estimates Collaborators: Matt Fickus (frame force); Alex Powell and Özgür Yilmaz ( Σ − ∆ quantization); Alex Powell, Aram Tangboondouangjit, and Özgür Yilmaz ( Σ − ∆ quantization and number theory). Finite frames and Sigma-Delta quantization – p.2/ ??

Finite Frames Frames Frames F = { e n } N n =1 for d -dimensional Hilbert space H , e.g., H = K d , where K = C or K = R . Any spanning set of vectors in K d is a frame for K d . Finite frames and Sigma-Delta quantization – p.3/ ??

Finite Frames Frames Frames F = { e n } N n =1 for d -dimensional Hilbert space H , e.g., H = K d , where K = C or K = R . Any spanning set of vectors in K d is a frame for K d . F ⊆ K d is A -tight if N � ∀ x ∈ K d , A � x � 2 = |� x, e n �| 2 n =1 Finite frames and Sigma-Delta quantization – p.3/ ??

Finite Frames Frames Frames F = { e n } N n =1 for d -dimensional Hilbert space H , e.g., H = K d , where K = C or K = R . Any spanning set of vectors in K d is a frame for K d . F ⊆ K d is A -tight if N � ∀ x ∈ K d , A � x � 2 = |� x, e n �| 2 n =1 If { e n } N n =1 is a finite unit norm tight frame (FUN-TF) for K d , with frame constant A , then A = N/d . Finite frames and Sigma-Delta quantization – p.3/ ??

Finite Frames Frames Frames F = { e n } N n =1 for d -dimensional Hilbert space H , e.g., H = K d , where K = C or K = R . Any spanning set of vectors in K d is a frame for K d . F ⊆ K d is A -tight if N � ∀ x ∈ K d , A � x � 2 = |� x, e n �| 2 n =1 If { e n } N n =1 is a finite unit norm tight frame (FUN-TF) for K d , with frame constant A , then A = N/d . Let { e n } be an A-unit norm TF for any separable Hilbert space H . A ≥ 1 , and A = 1 ⇔ { e n } is an ONB for H ( Vitali ). Finite frames and Sigma-Delta quantization – p.3/ ??

The geometry of finite tight frames The vertices of platonic solids are FUN-TFs. Finite frames and Sigma-Delta quantization – p.4/ ??

The geometry of finite tight frames The vertices of platonic solids are FUN-TFs. Points that constitute FUN-TFs do not have to be equidistributed, e.g., ONBs and Grassmanian frames. Finite frames and Sigma-Delta quantization – p.4/ ??

The geometry of finite tight frames The vertices of platonic solids are FUN-TFs. Points that constitute FUN-TFs do not have to be equidistributed, e.g., ONBs and Grassmanian frames. FUN-TFs can be characterized as minimizers of a “frame potential function” (with Fickus) analogous to Finite frames and Sigma-Delta quantization – p.4/ ??

The geometry of finite tight frames The vertices of platonic solids are FUN-TFs. Points that constitute FUN-TFs do not have to be equidistributed, e.g., ONBs and Grassmanian frames. FUN-TFs can be characterized as minimizers of a “frame potential function” (with Fickus) analogous to Coulomb’s Law . Finite frames and Sigma-Delta quantization – p.4/ ??

Frame force and potential energy F : S d − 1 × S d − 1 \ D − → R d P : S d − 1 × S d − 1 \ D − → R , where P ( a, b ) = p ( � a − b � ) , p ′ ( x ) = − xf ( x ) Coulomb force CF ( a, b ) = ( a − b ) / � a − b � 3 , f ( x ) = 1 /x 3 Finite frames and Sigma-Delta quantization – p.5/ ??

Frame force and potential energy F : S d − 1 × S d − 1 \ D − → R d P : S d − 1 × S d − 1 \ D − → R , where P ( a, b ) = p ( � a − b � ) , p ′ ( x ) = − xf ( x ) Coulomb force CF ( a, b ) = ( a − b ) / � a − b � 3 , f ( x ) = 1 /x 3 Frame force f ( x ) = 1 − x 2 / 2 FF ( a, b ) = < a, b > ( a − b ) , Finite frames and Sigma-Delta quantization – p.5/ ??

Frame force and potential energy F : S d − 1 × S d − 1 \ D − → R d P : S d − 1 × S d − 1 \ D − → R , where P ( a, b ) = p ( � a − b � ) , p ′ ( x ) = − xf ( x ) Coulomb force CF ( a, b ) = ( a − b ) / � a − b � 3 , f ( x ) = 1 /x 3 Frame force f ( x ) = 1 − x 2 / 2 FF ( a, b ) = < a, b > ( a − b ) , Total potential energy for the frame force TFP ( { x n } ) = Σ N m =1 Σ N n =1 | < x m , x n > | 2 Finite frames and Sigma-Delta quantization – p.5/ ??

Characterization of FUN-TFs For the Hilbert space H = R d and N , consider 1 ∈ S d − 1 × ... × S d − 1 and { x n } N TFP ( { x n } ) = Σ N m =1 Σ N n =1 | < x m , x n > | 2 . Theorem Let N ≤ d . The minimum value of TFP , for the frame force and N variables, is N ; and the minimizers are precisely the orthonormal sets of N elements for R d . Finite frames and Sigma-Delta quantization – p.6/ ??

Characterization of FUN-TFs For the Hilbert space H = R d and N , consider 1 ∈ S d − 1 × ... × S d − 1 and { x n } N TFP ( { x n } ) = Σ N m =1 Σ N n =1 | < x m , x n > | 2 . Theorem Let N ≤ d . The minimum value of TFP , for the frame force and N variables, is N ; and the minimizers are precisely the orthonormal sets of N elements for R d . Theorem Let N ≥ d . The minimum value of TFP , for the frame force and N variables, is N 2 /d ; and the minimizers are precisely the FUN-TFs of N elements for R d . Finite frames and Sigma-Delta quantization – p.6/ ??

Characterization of FUN-TFs For the Hilbert space H = R d and N , consider 1 ∈ S d − 1 × ... × S d − 1 and { x n } N TFP ( { x n } ) = Σ N m =1 Σ N n =1 | < x m , x n > | 2 . Theorem Let N ≤ d . The minimum value of TFP , for the frame force and N variables, is N ; and the minimizers are precisely the orthonormal sets of N elements for R d . Theorem Let N ≥ d . The minimum value of TFP , for the frame force and N variables, is N 2 /d ; and the minimizers are precisely the FUN-TFs of N elements for R d . Problem Find FUN-TFs analytically, effectively, computationally. Finite frames and Sigma-Delta quantization – p.6/ ??

Sigma-Delta quantization − theory and implementation Given u 0 and {x n } n=1 u n = u n-1 + x n -q n q n = Q(u n-1 + x n ) u n = u n-1 + x n -q n x n q n + + D + Q - First Order Σ∆ Finite frames and Sigma-Delta quantization – p.7/ ??

A quantization problem Qualitative Problem Obtain digital representations for class X , suitable for storage, transmission, recovery. Quantitative Problem Find dictionary { e n } ⊆ X : 1. Sampling [continuous range K is not digital] � x n e n , x n ∈ K ( R or C ) . ∀ x ∈ X, x = 2. Quantization. Construct finite alphabet A and � Q : X → { q n e n : q n ∈ A ⊆ K } such that | x n − q n | and/or � x − Qx � small. Methods Fine quantization, e.g., PCM. Take q n ∈ A close to given x n . Reasonable in 16-bit (65,536 levels) digital audio. Coarse quantization, e.g., Σ∆ . Use fewer bits to exploit redundancy. Finite frames and Sigma-Delta quantization – p.8/ ??

Quantization A δ K = { ( − K + 1 / 2) δ, ( − K + 3 / 2) δ, . . . , ( − 1 / 2) δ, (1 / 2) δ, . . ., ( K − 1 / 2) δ } 6 (K−1/2) δ 4 δ { q u δ { 2 3 δ /2 δ { δ /2 0 u−axis u −2 −4 f(u)=u (−K+1/2) δ −6 −6 −4 −2 0 2 4 6 Q ( u ) = arg min {| u − q | : q ∈ A δ K } = q u Finite frames and Sigma -Delta quantization – p.9/ ??

Setting n =1 is a FUN -TF for R d . Thus, Let x ∈ R d , � x � ≤ 1 . Suppose F = { e n } N we have N � x = d x n e n N n =1 with x n = � x, e n � . Note: A = N/d , and | x n | ≤ 1 . Goal Find a “good” quantizer, given K = { ( − K + 1 2) δ, ( − K + 3 2) δ, . . . , ( K − 1 A δ 2) δ } . Example Consider the alphabet A 2 1 = {− 1 , 1 } , and E 7 = { e n } 7 n =1 , with e n = (cos( 2 nπ 7 ) , sin( 2 nπ 7 )) . Finite frames and Sigma-Delta quantization – p.10/ ??

A 2 1 = {− 1 , 1 } and E 7 1.5 1 0.5 0 −0.5 −1 −1.5 −1.5 −1 −0.5 0 0.5 1 1.5 � 7 1 ( E 7 ) = { 2 n =1 q n e n : q n ∈ A 2 Γ A 2 1 } 7 Finite frames and Sigma -Delta quantization – p.11/ ??

PCM N � x = d Replace x n ↔ q n = arg { min | x n − q | : q ∈ A δ K } . Then ˜ q n e n N n =1 satisfies N N � � x � ≤ d ( x n − q n ) e n � ≤ d δ � e n � = d � x − ˜ N � 2 δ. N 2 n =1 n =1 Not good! Bennett’s “white noise assumption” Assume that ( η n ) = ( x n − q n ) is a sequence of independent, identically distributed random variables with mean 0 and variance δ 2 12 . Then the mean square error (MSE) satisfies 12 A δ 2 = ( dδ ) 2 d x � 2 ≤ MSE = E � x − ˜ 12 N Finite frames and Sigma -Delta quantization – p.12/ ??

Remarks 1. Bennett’s “white noise assumption” is not rigorous, and not true in certain cases. 2. The MSE behaves like C/A . In the case of Σ∆ quantization of bandlimited functions, the MSE is O ( A − 3 ) (Gray, Güntürk and Thao, Bin Han and Chen). PCM does not utilize redundancy efficiently. 3. The MSE only tells us about the average performance of a quantizer. Finite frames and Sigma -Delta quantization – p.13/ ??

A 2 1 = {− 1 , 1 } and E 7 Let x = ( 1 3 , 1 2 ) , E 7 = { (cos( 2 nπ 7 ) , sin( 2 nπ 7 )) } 7 n =1 . Consider quantizers with A = {− 1 , 1 } . Finite frames and Sigma -Delta quantization – p.14/ ??