Sampling Theory Taylor Baudry, Kaitland Brannon, and Jerome Weston - PowerPoint PPT Presentation

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography Sampling Theory Taylor Baudry, Kaitland Brannon, and Jerome Weston July 5, 2012 Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography Table of Contents Introduction 1 Background 2 Framework 3 Discovery 4 Conclusion 5 Acknowledgements 6 Bibliography 7 Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography What is Sampling Theory? Sampling theory: Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography What is Sampling Theory? Sampling theory: the study of the reconstruction of a function from its values (samples) on some subset of the domain of the function. Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography What is Sampling Theory? Sampling theory: the study of the reconstruction of a function from its values (samples) on some subset of the domain of the function. uses a vector space, V , of functions over some domain X for which it is possible to evaluate functions. Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography What is Sampling Theory? Sampling theory: the study of the reconstruction of a function from its values (samples) on some subset of the domain of the function. uses a vector space, V , of functions over some domain X for which it is possible to evaluate functions. has applications in image reconstruction and cd storage. Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography What is Sampling Theory? Sampling theory: the study of the reconstruction of a function from its values (samples) on some subset of the domain of the function. uses a vector space, V , of functions over some domain X for which it is possible to evaluate functions. has applications in image reconstruction and cd storage. We will use V = P N ( R ), the set of polynomials of degree N or less. Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography Background Information Lemma (The Alpha Lemma) Let V be a finite dimensional vector space of dimension n. If { v i } n i =1 is a set vectors that span V and, for all i, v i ∈ V , then { v i } n i =1 is a basis. Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography Necessary Info Definition If we let { v 1 . . . v k } be any set of vectors, it is then classified as a frame if there are numbers A , B > 0 such that ∀ v ǫ V the following inequality stands true k A � v � 2 ≤ � | ( v , v i ) | 2 ≤ B � v � 2 . i =1 Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography Necessary Info For simplistic purposes, the following lemma will be what is used to define a frame. Lemma The set { v 1 , . . . , v k } is said to be a frame if and only if it is a spanning set of V . Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography Necessary Info For every frame there exist what is known as a dual frame . Theorem (The Dual Theorem) Suppose { v i } k i =1 is a frame, then the dual frame { w i } k i =1 of V there exist for all v ǫ V k k � � v = ( v | v i ) w i = ( v | w i ) v i . i =1 i =1 Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography Necessary Info It is necessary to introduce what is known as a analysis operator . Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography Necessary Info It is necessary to introduce what is known as a analysis operator . Definition The analysis operator, denoted as Θ, is a linear map from the vector space V to R k such that for a given v ∈ V and frame { v i } k i =1   ( v | v 1 ) ( v | v 2 )   Θ( v ) = .   .   .   ( v | v k ) Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography Necessary Info The frame operator, denoted S, has the following properties Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography Necessary Info The frame operator, denoted S, has the following properties S = Θ ∗ Θ , where Θ ∗ is the adjoint of Θ . Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography Necessary Info The frame operator, denoted S, has the following properties S = Θ ∗ Θ , where Θ ∗ is the adjoint of Θ . S is invertible and is equal to its own adjoint and thus is self-adjoint, Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography Necessary Info The frame operator, denoted S, has the following properties S = Θ ∗ Θ , where Θ ∗ is the adjoint of Θ . S is invertible and is equal to its own adjoint and thus is self-adjoint, S − 1 is self-adjoint. Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography Necessary Info The frame operator, denoted S, has the following properties S = Θ ∗ Θ , where Θ ∗ is the adjoint of Θ . S is invertible and is equal to its own adjoint and thus is self-adjoint, S − 1 is self-adjoint. The following proposition and theorem, along with the properties of S − 1 , allow for S − 1 to be computed explicitly over a set X . Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography Necessary Info The frame operator, denoted S, has the following properties S = Θ ∗ Θ , where Θ ∗ is the adjoint of Θ . S is invertible and is equal to its own adjoint and thus is self-adjoint, S − 1 is self-adjoint. The following proposition and theorem, along with the properties of S − 1 , allow for S − 1 to be computed explicitly over a set X . Definition We say the set X can be a set of uniqueness for P N if for every f , g ∈ P N , f | X = g | X ⇒ f = g . Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography Necessary Info With every dual frame there also exist what is known as the canonical dual frame , where w i = S − 1 v i . Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography Necessary Info With every dual frame there also exist what is known as the canonical dual frame , where w i = S − 1 v i . Proposition If we define P N := { f : R → R : f ( x ) = � N n =0 a n x n , a n ǫ R } , which is a set of polynomials of degree N or less, then set X can be a set of uniqueness for P N if and only if Θ X is injective. Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography As a consequence, Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography As a consequence, Theorem Let X = { x 0 , x 1 , . . . , x N } ⊂ R ; then X is a set of uniqueness for P N . Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory

Introduction Background Framework Discovery Conclusion Acknowledgements Bibliography Necessary Info By the definiton of S , the matrix representation of S can be derived using the matrix previously defined. Thus  � N � N � N  i =0 x 2 i =0 x N N + 1 i =0 x i . . . i i � N � N � N � N i =0 x N +1 i =0 x 2 i =0 x 3 i =0 x i . . .   i i i   � N � N � N � N i =0 x 2 i =0 x 3 i =0 x 4 i =0 x N +2  . . .  [ S ] =  i i i i  . . . . ...   . . . . . . . .     � N � N i =0 x N +1 � N i =0 x N +2 � N i =0 x N i =0 x 2 N . . . i i i i Taylor Baudry, Kaitland Brannon, and Jerome Weston Sampling Theory