Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Sampling Theory in R d , Cont’d We can write Poisson summation for an arbitrary lattice by a change of coordinates. Let A be an invertible d × d matrix, Λ = A Z d , and Λ ⊥ = ( A T ) − 1 Z d be the dual lattice. Then � � � ( n ) e 2 π in · A − 1 ( t ) ( f ◦ A )( A − 1 t + n ) = b f ( t + λ ) = ( f ◦ A ) λ ∈ Λ n ∈ Z d n ∈ Z d � 1 f (( A T ) − 1 ( n )) e 2 π i ( A T ) − 1 ( n ) · t . � = | det A | n ∈ Z d Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Sampling Theory in R d , Cont’d We can write Poisson summation for an arbitrary lattice by a change of coordinates. Let A be an invertible d × d matrix, Λ = A Z d , and Λ ⊥ = ( A T ) − 1 Z d be the dual lattice. Then � � � ( n ) e 2 π in · A − 1 ( t ) ( f ◦ A )( A − 1 t + n ) = b f ( t + λ ) = ( f ◦ A ) λ ∈ Λ n ∈ Z d n ∈ Z d � 1 f (( A T ) − 1 ( n )) e 2 π i ( A T ) − 1 ( n ) · t . � = | det A | n ∈ Z d Since | det A | = vol (Λ), we can write this as � � 1 f ( β ) e 2 π i β · t ( PSF ) . � f ( t + λ ) = vol (Λ) λ ∈ Λ β ∈ Λ ⊥ Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Sampling Theory in R d , Cont’d We can write Poisson summation for an arbitrary lattice by a change of coordinates. Let A be an invertible d × d matrix, Λ = A Z d , and Λ ⊥ = ( A T ) − 1 Z d be the dual lattice. Then � � � ( n ) e 2 π in · A − 1 ( t ) ( f ◦ A )( A − 1 t + n ) = b f ( t + λ ) = ( f ◦ A ) λ ∈ Λ n ∈ Z d n ∈ Z d � 1 f (( A T ) − 1 ( n )) e 2 π i ( A T ) − 1 ( n ) · t . � = | det A | n ∈ Z d Since | det A | = vol (Λ), we can write this as � � 1 f ( β ) e 2 π i β · t ( PSF ) . � f ( t + λ ) = vol (Λ) λ ∈ Λ β ∈ Λ ⊥ This extends again to the Schwartz class of distributions as � � � 1 δ λ = δ β ( PSF2 ) . vol (Λ) λ ∈ Λ β ∈ Λ ⊥ Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Sampling Theory in R d , Cont’d The dual sampling lattice can be written as Λ ⊥ = { λ ⊥ : λ ⊥ = z 1 ω 1 + z 2 ω 2 + . . . + z d ω d } . This creates a fundamental sampling parallelpiped Ω P in � R d . If the region Ω P is a hyper-rectangle, we get � sin( π sin( π ω 1 ( t − n 1 ω 1 )) ω d ( t − n d ω d )) 1 f ( t ) = f ( n 1 ω 1 , . . . ) · . . . · . vol (Λ) π ( t − n 1 ω 1 ) π ( t − n d ω d ) n ∈ Z d Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Sampling Theory in R d , Cont’d The dual sampling lattice can be written as Λ ⊥ = { λ ⊥ : λ ⊥ = z 1 ω 1 + z 2 ω 2 + . . . + z d ω d } . This creates a fundamental sampling parallelpiped Ω P in � R d . If the region Ω P is a hyper-rectangle, we get � sin( π sin( π ω 1 ( t − n 1 ω 1 )) ω d ( t − n d ω d )) 1 f ( t ) = f ( n 1 ω 1 , . . . ) · . . . · . vol (Λ) π ( t − n 1 ω 1 ) π ( t − n d ω d ) n ∈ Z d If, however, Ω P is a general parallelepiped, we first have to compute the inverse Fourier transform of χ Ω P . Let S be the generalized sinc function 1 vol (Λ)( χ Ω P ) ∨ . S = Then, the sampling formula becomes � f ( t ) = f ( λ ) S ( t − λ ) . λ ∈ Λ Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Sampling Theory in R d , Cont’d Definition (Nyquist Tiles for f ∈ PW Ω P ) Let f ∈ L 2 ( � PW Ω P = { f continuous : f ∈ L 2 ( R d ) , � R d ) , supp ( � f ) ⊂ Ω P } . Let f ∈ PW Ω P . The Nyquist Tile NT ( f ) for f is the parallelepiped of minimal area in � R d such that supp ( � f ) ⊆ NT ( f ). A Nyquist Tiling is the set of translates { NT ( f ) k } k ∈ Z d of Nyquist tiles which tile � R d . Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Sampling Theory in R d , Cont’d Definition (Nyquist Tiles for f ∈ PW Ω P ) Let f ∈ L 2 ( � PW Ω P = { f continuous : f ∈ L 2 ( R d ) , � R d ) , supp ( � f ) ⊂ Ω P } . Let f ∈ PW Ω P . The Nyquist Tile NT ( f ) for f is the parallelepiped of minimal area in � R d such that supp ( � f ) ⊆ NT ( f ). A Nyquist Tiling is the set of translates { NT ( f ) k } k ∈ Z d of Nyquist tiles which tile � R d . Definition (Sampling Group for f ∈ PW Ω P ) Let f ∈ PW Ω P with Nyquist Tile NT ( f ). The Sampling Group G is a symmetry group of translations such that NT ( f ) tiles � R d . Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Sampling Theory in R d , Cont’d We use our sampling lattices to develop Voronoi cells corresponding to the sampling lattice. These cells will be, in the Euclidean case, our Nyquist tiles. Definition (Voronoi Cells in � R d ) Let Λ = { λ k ∈ R d } be a sampling set for f ∈ PW Ω . Let Λ ⊥ be the dual lattice in frequency space. Then, the Voronoi cells { Φ k } , the Voronoi partition VP (Λ ⊥ ), and partition norm �VP (Λ ⊥ ) � corresponding to the sampling lattice are defined as follows. 1.) The Voronoi cells Φ k = { ω ∈ � R d : dist ( ω, λ ⊥ k ) ≤ inf j � = k dist ( ω, λ ⊥ j ) } , 2.) The Voronoi partition VP (Λ ⊥ ) = { Φ k ∈ � R d } k ∈ Z d , 3.) The partition norm �VP (Λ ⊥ ) � = sup k ∈ Z d sup ω,ν ∈ Φ k dist ( ω, ν ). Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Sampling Theory in R d , Cont’d Figure: 3D Nyquist Cell Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Sampling Theory in R d , Cont’d Theorem (Nyquist Tiling for Euclidean Space (C, (3)(2016)) Let f ∈ PW Ω P , and let Λ = { λ k ∈ R d } k ∈ Z d be the sampling grid which samples f exactly at Nyquist. Let Λ ⊥ be the dual lattice in frequency space. Then the Voronoi partition VP (Λ ⊥ ) = { Φ k ∈ � R d } k ∈ Z d equals the Nyquist Tiling, i.e., { Φ k ∈ � R d } k ∈ Z d = { NT ( f ) k } k ∈ Z d . Moreover, the partition norm equals the volume of Λ ⊥ , i.e., �VP (Λ ⊥ ) � = sup k ∈ Z d sup ω,ν ∈ Φ k dist ( ω, ν ) = vol (Λ ⊥ ) , and the sampling group G is exactly the group of motions that preserve Λ ⊥ . Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Why Use Voronoi Cells? Allows us to create a unified construction of Sampling in all geometries. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Why Use Voronoi Cells? Allows us to create a unified construction of Sampling in all geometries. For a fixed grid, if the geometry changes, the Voronoi Cells give the correct Nyquist Tiles for that geometry. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Why Use Voronoi Cells? Allows us to create a unified construction of Sampling in all geometries. For a fixed grid, if the geometry changes, the Voronoi Cells give the correct Nyquist Tiles for that geometry. Reduces the question of Sampling purely to the optimal sampling grid. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Why Use Voronoi Cells, Cont’d? Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST Figure: Euclidean Voronoi Diagram
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Why Use Voronoi Cells, Cont’d? Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST Figure: Manhattan Voronoi Diagram
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Geometry of Surfaces Understand the geometry by understanding the group of motions that preserve the geometry – Klein’s Erlangen Program . Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Euclidean Geometry Euclidean Geometry – Rotations and Translations. ϕ θ,α = e i θ z − α . Length – � L E (Γ) = | dz | . Γ L E ( ϕ θ,α (Γ)) = L E (Γ) . Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Spherical Geometry Spherical Geometry – M¨ obius Transformations ϕ α,β = α z − β − β z − α , where | α | 2 + | β | 2 = 1. Length – � 2 | dz | L S (Γ) = 1 + | z | 2 . Γ L S ( ϕ α,β (Γ)) = L S (Γ) . Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Spherical Geometry, Cont’d Figure: Spherical Geometry Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Hyperbolic Geometry Hyperbolic Geometry – M¨ obius-Blaschke Transformations ϕ θ,α = e i θ z − α 1 − α z , α ∈ D , where | α | < 1. Length – � 2 | dz | L H (Γ) = 1 − | z | 2 . Γ L H ( ϕ θ,α (Γ)) = L H (Γ) . Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Hyperbolic Geometry, Cont’d Figure: Hyperbolic Tesselation – SU (1 , 1) Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Hyperbolic Geometry, Cont’d Figure: Hyperbolic Upper Half Plane H – SL (2 , R ) Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Hyperbolic Geometry, Cont’d Figure: Hyperbolic Bookcase Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Curvature “in a Nutshell” Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST Figure: Curvature and Geometry
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Theorem The Uniformization Theorem – Klein, Koebe, Poincare. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Theorem The Uniformization Theorem – Klein, Koebe, Poincare. Every surface admits a Riemannian metric of constant Gaussian curvature κ . Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography Theorem The Uniformization Theorem – Klein, Koebe, Poincare. Every surface admits a Riemannian metric of constant Gaussian curvature κ . Every simply connected Riemann surface (universal covering space � S ) is conformally equivalent to one of the following: The Plane C – Euclidean Geometry – κ = 0 – D ff E e i θ z + α , ◦ The Riemann Sphere e C – Spherical Geometry – κ = 1 – D α z − β ff E , where | α | 2 + | β | 2 = 1 . , ◦ − β z − α e Disk D – Hyperbolic Geometry – κ = − 1 – The Poincar´ D ff E e i θ z − α , ◦ , where | α | < 1 . 1 − α z Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography General Surface Given connected Riemann surface S and its universal covering space S , S is isomorphic to � � S / Γ, where the group Γ is isomorphic to the fundamental group of S , π 1 ( S ). Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography General Surface Given connected Riemann surface S and its universal covering space S , S is isomorphic to � � S / Γ, where the group Γ is isomorphic to the fundamental group of S , π 1 ( S ). The corresponding covering is simply the quotient map which sends every point of � S to its orbit under Γ. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Geometry of Surfaces Sampling in Non-Euclidean Geometry Application: Network Tomography General Surface Given connected Riemann surface S and its universal covering space S , S is isomorphic to � � S / Γ, where the group Γ is isomorphic to the fundamental group of S , π 1 ( S ). The corresponding covering is simply the quotient map which sends every point of � S to its orbit under Γ. A fundamental domain is a subset of � S which contains exactly one point from each of these orbits. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Spherical Geometry The sphere is compact, and its study requires different tools. Sampling on the sphere is how to sample a band-limited function, an N th degree spherical polynomial, at a finite number of locations, such that all of the information content of the continuous function is captured. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Spherical Geometry The sphere is compact, and its study requires different tools. Sampling on the sphere is how to sample a band-limited function, an N th degree spherical polynomial, at a finite number of locations, such that all of the information content of the continuous function is captured. Since the frequency domain of a function on the sphere is discrete, the spherical harmonic coefficients describe the continuous function exactly. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Spherical Geometry The sphere is compact, and its study requires different tools. Sampling on the sphere is how to sample a band-limited function, an N th degree spherical polynomial, at a finite number of locations, such that all of the information content of the continuous function is captured. Since the frequency domain of a function on the sphere is discrete, the spherical harmonic coefficients describe the continuous function exactly. A sampling theorem thus describes how to exactly recover the spherical harmonic coefficients of the continuous function from its samples. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Spherical Geometry The sphere is compact, and its study requires different tools. Sampling on the sphere is how to sample a band-limited function, an N th degree spherical polynomial, at a finite number of locations, such that all of the information content of the continuous function is captured. Since the frequency domain of a function on the sphere is discrete, the spherical harmonic coefficients describe the continuous function exactly. A sampling theorem thus describes how to exactly recover the spherical harmonic coefficients of the continuous function from its samples. The open question is the establishment of the optimal Beurling-Landau densities. This leads to questions about sphere tiling. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Fourier Analysis in Hyperbolic Geometry Let dz denote the area measure on the unit disc D = { z | | z | < 1, and let the measure dv be given by the SU (1 , 1)-invariant measure on D , given by dv ( z ) = dz / (1 − | z | 2 ) 2 . Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Fourier Analysis in Hyperbolic Geometry Let dz denote the area measure on the unit disc D = { z | | z | < 1, and let the measure dv be given by the SU (1 , 1)-invariant measure on D , given by dv ( z ) = dz / (1 − | z | 2 ) 2 . For f ∈ L 1 ( D , dv ) the Fourier-Helgason transform (FHT) – � f ( z ) e ( − i λ +1) � z , b � dv ( z ) � f ( λ, b ) = D for λ > 0 and b ∈ T . Here � z , b � denotes the hyperbolic distance from z to a point b on the boundary of D . Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Fourier Analysis in Hyperbolic Geometry Let dz denote the area measure on the unit disc D = { z | | z | < 1, and let the measure dv be given by the SU (1 , 1)-invariant measure on D , given by dv ( z ) = dz / (1 − | z | 2 ) 2 . For f ∈ L 1 ( D , dv ) the Fourier-Helgason transform (FHT) – � f ( z ) e ( − i λ +1) � z , b � dv ( z ) � f ( λ, b ) = D for λ > 0 and b ∈ T . Here � z , b � denotes the hyperbolic distance from z to a point b on the boundary of D . FHT Inversion � � f ( z ) = 1 � f ( λ, b ) e ( i λ +1) � z , b � λ tanh( λπ/ 2) d λ db . 2 π R + T Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry → L 2 ( R + × T , FHT: L 2 ( D ) − 1 2 π λ tanh( λπ/ 2) d λ db ) Let dist denote the distance in R + × T , weighted by 1 2 π λ tanh( λπ/ 2). Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry → L 2 ( R + × T , FHT: L 2 ( D ) − 1 2 π λ tanh( λπ/ 2) d λ db ) Let dist denote the distance in R + × T , weighted by 1 2 π λ tanh( λπ/ 2). A function f ∈ L 2 ( D , dv ) is called band-limited if its Fourier-Helgason transform � f is supported inside a bounded subset [0 , Ω] of R + . The collection of band-limited functions with band-limit inside a set [0 , Ω] will be denoted PW Ω = PW Ω ( D ). Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry → L 2 ( R + × T , FHT: L 2 ( D ) − 1 2 π λ tanh( λπ/ 2) d λ db ) Let dist denote the distance in R + × T , weighted by 1 2 π λ tanh( λπ/ 2). A function f ∈ L 2 ( D , dv ) is called band-limited if its Fourier-Helgason transform � f is supported inside a bounded subset [0 , Ω] of R + . The collection of band-limited functions with band-limit inside a set [0 , Ω] will be denoted PW Ω = PW Ω ( D ). One approach to sampling (Feichtinger-Pesenson) proceeds as follows. To sample, tile R + × T with Ω bands. Then, since we don’t know Nyquist, we cover the bands with disks. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry → L 2 ( R + × T , FHT: L 2 ( D ) − 1 2 π λ tanh( λπ/ 2) d λ db ) Let dist denote the distance in R + × T , weighted by 1 2 π λ tanh( λπ/ 2). A function f ∈ L 2 ( D , dv ) is called band-limited if its Fourier-Helgason transform � f is supported inside a bounded subset [0 , Ω] of R + . The collection of band-limited functions with band-limit inside a set [0 , Ω] will be denoted PW Ω = PW Ω ( D ). One approach to sampling (Feichtinger-Pesenson) proceeds as follows. To sample, tile R + × T with Ω bands. Then, since we don’t know Nyquist, we cover the bands with disks. There is a natural number N such that for any sufficiently small r there are points x j ∈ D for which B ( x j , r / 4) are disjoint, B ( x j , r / 2) cover D and 1 ≤ � j χ B ( x j , r ) ≤ N . Such a collection of { x j } will be called an ( r , N )-lattice. Let φ j be smooth non-negative functions which are supported in B ( x j , r / 2) and satisfy that � j φ j = 1 D Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d Let φ j be smooth non-negative functions which are supported in B ( x j , r / 2) and satisfy that � j φ j = 1 D . Define the operator � , Tf ( x ) = P Ω f ( x j ) φ j ( x ) j where P Ω is the orthogonal projection from L 2 ( D , dv ) onto PW Ω ( D ). Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d Let φ j be smooth non-negative functions which are supported in B ( x j , r / 2) and satisfy that � j φ j = 1 D . Define the operator � , Tf ( x ) = P Ω f ( x j ) φ j ( x ) j where P Ω is the orthogonal projection from L 2 ( D , dv ) onto PW Ω ( D ). By decreasing r one can obtain the inequality � I − T � < 1, in which case T can be inverted by ∞ � T − 1 f = ( I − T ) k f . k =0 Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d Since � I − T � < 1, T can be inverted by ∞ � T − 1 f = ( I − T ) k f . k =0 Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d Since � I − T � < 1, T can be inverted by ∞ � T − 1 f = ( I − T ) k f . k =0 For given samples, we can calculate Tf and the Neumann series, which provides the recursion formula f n +1 = f n + Tf − Tf n . Then f n +1 → f as n → ∞ in norm. The rate of convergence – � f n − f � ≤ � I − T � n +1 � f � . Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d Theorem (Irregular Sampling by Iteration (C, (2016)) Let S be a Riemann surface whose universal covering space � S is hyperbolic. Then there exists an ( r , N ) -lattice on S such that given f ∈ PW Ω = PW Ω ( S ) , f can be reconstructed from its samples on the lattice via the recursion formula f n +1 = f n + Tf − Tf n . We have f n +1 → f as n → ∞ in norm. The rate of convergence is � f n − f � ≤ � I − T � n +1 � f � . This, however, leaves open questions about densities. We address this in the next few frames. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d Equip the unit disc D with normalized area measure d σ ( z ), let O be the set of holomorphic functions, and let 1 ≤ p < ∞ be given. Definition (Bergman Space) A p ( D ) = L p ( D , d σ ) ∩ O ( D ) . This is a reproducing kernel Banach space with reproducing kernel 1 K ( z , w ) = (1 − wz ) 2 . Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d Equip the unit disc D with normalized area measure d σ ( z ), let O be the set of holomorphic functions, and let 1 ≤ p < ∞ be given. Definition (Bergman Space) A p ( D ) = L p ( D , d σ ) ∩ O ( D ) . This is a reproducing kernel Banach space with reproducing kernel 1 K ( z , w ) = (1 − wz ) 2 . Lower and upper Beurling-Landau densities on Bergman spaces A p ( D ) on the unit disc by Seip and Schuster. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d Equip the unit disc D with normalized area measure d σ ( z ), let O be the set of holomorphic functions, and let 1 ≤ p < ∞ be given. Definition (Bergman Space) A p ( D ) = L p ( D , d σ ) ∩ O ( D ) . This is a reproducing kernel Banach space with reproducing kernel 1 K ( z , w ) = (1 − wz ) 2 . Lower and upper Beurling-Landau densities on Bergman spaces A p ( D ) on the unit disc by Seip and Schuster. Define � � � � z − ζ � � ρ ( z , ζ ) = � � 1 − ¯ z ζ Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d Equip the unit disc D with normalized area measure d σ ( z ), let O be the set of holomorphic functions, and let 1 ≤ p < ∞ be given. Definition (Bergman Space) A p ( D ) = L p ( D , d σ ) ∩ O ( D ) . This is a reproducing kernel Banach space with reproducing kernel 1 K ( z , w ) = (1 − wz ) 2 . Lower and upper Beurling-Landau densities on Bergman spaces A p ( D ) on the unit disc by Seip and Schuster. Define � � � � z − ζ � � ρ ( z , ζ ) = � � 1 − ¯ z ζ Let Γ k = { z k } be a set of uniformly discrete points, that is inf j � = k ρ ( z j , z k ) > 0 Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d For each z let n z ( r ) be the number of points from Γ k in the disk | ζ | < r , and define � r N z ( r ) = n z ( τ ) d τ 0 Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d For each z let n z ( r ) be the number of points from Γ k in the disk | ζ | < r , and define � r N z ( r ) = n z ( τ ) d τ 0 The hyperbolic area of | ζ | < r is a ( r ) = 2 r 2 (1 − r 2 ) − 1 , and define � r A ( r ) = a ( ρ ) d ρ 0 Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d Now we can define the lower density and upper density of points in D of the sequence Γ k Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d Now we can define the lower density and upper density of points in D of the sequence Γ k Define the lower density and upper density, respectively � � N z ( r ) D − (Γ k ) = lim inf inf A ( r ) r → 1 z ∈ Γ k Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d Now we can define the lower density and upper density of points in D of the sequence Γ k Define the lower density and upper density, respectively � � N z ( r ) D − (Γ k ) = lim inf inf A ( r ) r → 1 z ∈ Γ k � � N z ( r ) D + (Γ k ) = lim sup sup A ( r ) r → 1 z ∈ Γ k Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d Theorem (Seip and Schuster) Let Λ be a set of distinct points in D . 1.) A sequence Λ is a set of sampling for A p if and only if it is a finite union of uniformly discrete sets and it contains a uniformly discrete subsequence Λ ′ for which D − (Λ ′ ) > 1 / p. 2.) A sequence Λ is a set of interpolation for A p if and only if it is uniformly discrete and D + (Λ) < 1 / p. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d Recall: (FHT) – � f ( z ) e ( − i λ +1) � z , b � dv ( z ) � f ( λ, b ) = D for λ > 0 and b ∈ T . Here � z , b � denotes the hyperbolic distance from z to a point b on the boundary of D . Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d Recall: (FHT) – � f ( z ) e ( − i λ +1) � z , b � dv ( z ) � f ( λ, b ) = D for λ > 0 and b ∈ T . Here � z , b � denotes the hyperbolic distance from z to a point b on the boundary of D . Because of � z , b � , � f ( λ, b ) is not analytic. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d Recall: (FHT) – � f ( z ) e ( − i λ +1) � z , b � dv ( z ) � f ( λ, b ) = D for λ > 0 and b ∈ T . Here � z , b � denotes the hyperbolic distance from z to a point b on the boundary of D . Because of � z , b � , � f ( λ, b ) is not analytic. Seip and Schuster results do not apply. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling in Hyperbolic Geometry, Cont’d Let dist denote the weighted distance in R + × T , weighted by 1 2 π λ tanh( λπ/ 2). Using this distance, we can define the following. Definition (Voronoi Cells in � D ) Let Λ = { λ k ∈ D : k ∈ N } be a sampling set on D . Let Λ ⊥ ⊂ R + × T be the dual lattice in frequency space. Then, the Voronoi cells { Φ k } , the Voronoi partition VP (Λ ⊥ ), and partition norm �VP (Λ ⊥ ) � corresponding to the sampling lattice are defined as follows. 1.) The Voronoi cells Φ k = { ω ∈ � D : dist ( ω, λ ⊥ k ) ≤ inf j � = k dist ( ω, λ ⊥ j ) } , 2.) The Voronoi partition VP (Λ ⊥ ) = { Φ k ∈ � D } k ∈ Z d , 3.) The partition norm �VP (Λ ⊥ ) � = sup k ∈ Z d sup ω,ν ∈ Φ k dist ( ω, ν ). Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface Recall the following. Given connected Riemann surface S and its universal covering space S , S is isomorphic to � � S / Γ, where the group Γ is isomorphic to the fundamental group of S , π 1 ( S ). Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface Recall the following. Given connected Riemann surface S and its universal covering space S , S is isomorphic to � � S / Γ, where the group Γ is isomorphic to the fundamental group of S , π 1 ( S ). The corresponding covering is simply the quotient map which sends every point of � S to its orbit under Γ. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface Recall the following. Given connected Riemann surface S and its universal covering space S , S is isomorphic to � � S / Γ, where the group Γ is isomorphic to the fundamental group of S , π 1 ( S ). The corresponding covering is simply the quotient map which sends every point of � S to its orbit under Γ. A fundamental domain is a subset of � S which contains exactly one point from each of these orbits. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface By the Uniformization Theorem – The only covering surface of Riemann sphere � C is itself, with the covering map being the identity. The plane C is the universal covering space of itself, the once punctured plane C \ { z 0 } (with covering map exp( z − z 0 )), and all tori C / Γ, where Γ is a parallelogram generated by → z + n γ 1 + m γ 2 , n , m ∈ Z and γ 1 , γ 2 are two fixed complex z �− numbers linearly independent over R . Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface By the Uniformization Theorem – The only covering surface of Riemann sphere � C is itself, with the covering map being the identity. The plane C is the universal covering space of itself, the once punctured plane C \ { z 0 } (with covering map exp( z − z 0 )), and all tori C / Γ, where Γ is a parallelogram generated by → z + n γ 1 + m γ 2 , n , m ∈ Z and γ 1 , γ 2 are two fixed complex z �− numbers linearly independent over R . The universal covering space of every other Riemann surface is the hyperbolic disk D . Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface By the Uniformization Theorem – The only covering surface of Riemann sphere � C is itself, with the covering map being the identity. The plane C is the universal covering space of itself, the once punctured plane C \ { z 0 } (with covering map exp( z − z 0 )), and all tori C / Γ, where Γ is a parallelogram generated by → z + n γ 1 + m γ 2 , n , m ∈ Z and γ 1 , γ 2 are two fixed complex z �− numbers linearly independent over R . The universal covering space of every other Riemann surface is the hyperbolic disk D . Therefore, the establishment of exact the Beurling-Landau densities for functions in Paley-Wiener spaces in spherical and especially hyperbolic geometries will allow the development of sampling schemes on arbitrary Riemann surfaces. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, cont’d This will split into Compact vs. Non-Compact. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, cont’d This will split into Compact vs. Non-Compact. Sampling on a compact surface is how to sample a band-limited function, an N th degree polynomial, at a finite number of locations, such that all of the information content of the continuous function is captured. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, cont’d This will split into Compact vs. Non-Compact. Sampling on a compact surface is how to sample a band-limited function, an N th degree polynomial, at a finite number of locations, such that all of the information content of the continuous function is captured. Since the frequency domain of a function on a compact surface is discrete, the coefficients describe the continuous function exactly. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, cont’d This will split into Compact vs. Non-Compact. Sampling on a compact surface is how to sample a band-limited function, an N th degree polynomial, at a finite number of locations, such that all of the information content of the continuous function is captured. Since the frequency domain of a function on a compact surface is discrete, the coefficients describe the continuous function exactly. A sampling theorem thus describes how to exactly recover the coefficients of the continuous function from its samples. The underlying geometry for sampling is inherited from the universal cover. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, cont’d Figure: Torus Fundamental Domain Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, cont’d Figure: Two Torus Fundamental Domain Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, cont’d Figure: Two Torus Fundamental Domain – A Second Look Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, cont’d Compact vs. Non-Compact, cont’d Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, cont’d Compact vs. Non-Compact, cont’d Sampling on a non-compact surface is how to sample a band-limited function at an infinite number of locations, such that all of the information content of the continuous function is captured. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, cont’d Compact vs. Non-Compact, cont’d Sampling on a non-compact surface is how to sample a band-limited function at an infinite number of locations, such that all of the information content of the continuous function is captured. Since the frequency domain of a function on a non-compact surface is a continuum, we need a Sampling Group and Nyquist Tile to reconstruct. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, cont’d Compact vs. Non-Compact, cont’d Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, cont’d Compact vs. Non-Compact, cont’d For a non-compact surface, given a discrete subgroup Γ of SL (2 , R ) acting on H , Γ contains a parabolic element. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, cont’d Compact vs. Non-Compact, cont’d For a non-compact surface, given a discrete subgroup Γ of SL (2 , R ) acting on H , Γ contains a parabolic element. The conjugacy class of the parabolic element corresponds to a cusp in the quotient manifold. When you “unfold” the surface in the universal cover, the cusp corresponds to a set of ideal vertices of your fundamental region. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, cont’d Compact vs. Non-Compact, cont’d For a non-compact surface, given a discrete subgroup Γ of SL (2 , R ) acting on H , Γ contains a parabolic element. The conjugacy class of the parabolic element corresponds to a cusp in the quotient manifold. When you “unfold” the surface in the universal cover, the cusp corresponds to a set of ideal vertices of your fundamental region. Since the frequency domain of a function on a non-compact surface is a continuum, we need a Sampling Group and Nyquist Tile to reconstruct. Here, the Sampling Group is Γ ⊥ ◦ G , where Γ ⊥ lives in frequency space. The Nyquist Tile is a subregion of the transform of the fundamental domain. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, cont’d Compact vs. Non-Compact, cont’d For a non-compact surface, given a discrete subgroup Γ of SL (2 , R ) acting on H , Γ contains a parabolic element. The conjugacy class of the parabolic element corresponds to a cusp in the quotient manifold. When you “unfold” the surface in the universal cover, the cusp corresponds to a set of ideal vertices of your fundamental region. Since the frequency domain of a function on a non-compact surface is a continuum, we need a Sampling Group and Nyquist Tile to reconstruct. Here, the Sampling Group is Γ ⊥ ◦ G , where Γ ⊥ lives in frequency space. The Nyquist Tile is a subregion of the transform of the fundamental domain. A sampling theorem thus describes how to exactly recover the coefficients of the continuous function from its samples. The underlying geometry for sampling is inherited from the universal cover. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, cont’d Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, Cont’d We have the machinery to develop a sampling theory for general analytic orientable surfaces. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, Cont’d We have the machinery to develop a sampling theory for general analytic orientable surfaces. Using covering space theory, we can develop sampling on the space of bandlimited functions on the fundamental domain of a given surface. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
Sampling Theory in Euclidean Geometry Spherical Geometry Geometry of Surfaces Hyperbolic Geometry Sampling in Non-Euclidean Geometry General Surfaces Application: Network Tomography Sampling on a General Surface, Cont’d We have the machinery to develop a sampling theory for general analytic orientable surfaces. Using covering space theory, we can develop sampling on the space of bandlimited functions on the fundamental domain of a given surface. This breaks down into compact vs. non-compact surfaces. Stephen Casey Sampling in Euclidean and Non-Euclidean Domains: A Unified Approach NIST
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