Motivation The Radon Transform (RT) Hyperbolic PDEs Results Solving the Radiative Transfer Equation using the Radon Transform Megan Oeltjenbruns (Wayne State College) Collaborators: Eappen Nelluvelil (Rice University), Jacob Spainhour (Florida State University), Christine Vaughan (Iowa State University), and Dr. James Rossmanith (Iowa State University) NCUWM 2020 1
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Outline Motivation 1 The Radon Transform (RT) 2 RT Derivation in 2D Discretizing the RT Computing the IRT and Backprojection Hyperbolic PDEs 3 The Radiative Transfer Problem RTs of Derivatives Discretization of the PDE Results 4 Results 2
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Goal Imaging Problem The imaging problem is to reconstruct a distribution from several profiles taken at different angles The Radon transform is a simple way to produce profiles Certain advection problems are easier to solve profile-by-profile 3
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Goal Our Goal To efficiently solve multi-dimensional systems of hyperbolic partial differential equations using the Radon transform hi Three main steps: Forward Radon transform multi-dimensional problem 1 Solve family of 1D advection problems 2 Use inverse Radon transform to original space 3 4
Motivation The Radon Transform (RT) Hyperbolic PDEs Results RT Derivation in 2D Outline Motivation 1 The Radon Transform (RT) 2 RT Derivation in 2D Discretizing the RT Computing the IRT and Backprojection Hyperbolic PDEs 3 The Radiative Transfer Problem RTs of Derivatives Discretization of the PDE Results 4 Results 5
Motivation The Radon Transform (RT) Hyperbolic PDEs Results RT Derivation in 2D Coordinate Rotation y z s ω x 6
Motivation The Radon Transform (RT) Hyperbolic PDEs Results RT Derivation in 2D Radon Transform Suppose f : R 2 → R is a function with compact support The Radon transform of f is formally defined as follows � ∞ R ( f ) = � f ( s , ω ) := f ( x ( s , z ; ω ) , y ( s , z ; ω ) ) dz −∞ � ∞ = f ( s cos( ω ) − z sin( ω ) , s sin( ω ) + z cos( ω )) dz −∞ 7
Motivation The Radon Transform (RT) Hyperbolic PDEs Results RT Derivation in 2D Radon Transform at ( x 0 , y 0 ) z s ( x 0 , y 0 ) ω 8
Motivation The Radon Transform (RT) Hyperbolic PDEs Results RT Derivation in 2D Radon Transform z s ( s 4 , ω ) ( s 3 , ω ) ω ( s 2 , ω ) ( s 1 , ω ) 9
Motivation The Radon Transform (RT) Hyperbolic PDEs Results RT Derivation in 2D Radon Transform Example f � f 1 . 0 0 . 96 π 0 . 72 0 . 84 0 . 63 0 . 5 3 π 0 . 72 0 . 54 4 0 . 45 0 . 60 0 . 0 π 0 . 36 y 0 . 48 ω 2 0 . 27 0 . 36 0 . 18 − 0 . 5 π 0 . 24 4 0 . 09 0 . 12 0 . 00 − 1 . 0 0 . 00 0 − 1 . 0 − 0 . 5 0 . 0 0 . 5 1 . 0 − 1 . 0 − 0 . 5 0 . 0 0 . 5 1 . 0 x s 10
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Discretizing the RT Outline Motivation 1 The Radon Transform (RT) 2 RT Derivation in 2D Discretizing the RT Computing the IRT and Backprojection Hyperbolic PDEs 3 The Radiative Transfer Problem RTs of Derivatives Discretization of the PDE Results 4 Results 11
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Discretizing the RT Grid Structure Need discretized domain for inverse computation Profile-by-profile construction suggests N ω evenly-spaced diameters, with ω held constant Chose a method that allows for spectrally accurate interpolation, differentiation 12
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Discretizing the RT Grid Structure Example One diameter of the mesh 1 . 0 0 . 5 0 y − 0 . 5 − 1 . 0 − 1 . 0 − 0 . 5 0 0 . 5 1 . 0 x 13
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Discretizing the RT Grid Structure Example Full mesh 1 . 0 0 . 5 0 y − 0 . 5 − 1 . 0 − 1 . 0 − 0 . 5 0 0 . 5 1 . 0 x 14
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Discretizing the RT Computing Integrals Compact support of f means we compute line integrals across chords of the domain Compute line integrals with Clenshaw-Curtis quadrature � ∞ R ( f ) = f ( x , y ) dz −∞ N q � w i f ( z i ) ≈ i = 1 15
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Discretizing the RT Quadrature Example 1 0 y − 1 − 1 0 1 x 16
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Discretizing the RT Quadrature Example 1 ( s 0 , ω 0 ) 0 y − 1 − 1 0 1 x 17
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Discretizing the RT Quadrature Example 1 0 y − 1 − 1 0 1 x 18
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Discretizing the RT Quadrature Example 1 0 y − 1 − 1 0 1 x 19
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Discretizing the RT Quadrature Example 1 0 y − 1 − 1 0 1 x 20
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Discretizing the RT Interpolation Scheme Need to sample f at arbitrary quadrature nodes, use interpolation Use a series of one-dimensional interpolation schemes: Spectrally accurate on each diameter Fourth order accurate across angles 21
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Discretizing the RT Interpolation Scheme Example 22
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Discretizing the RT Interpolation Scheme Example (cont.) 23
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Discretizing the RT Interpolation Scheme Example (cont.) 24
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Discretizing the RT Interpolation Scheme Example (cont.) 25
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Computing the IRT and Backprojection Outline Motivation 1 The Radon Transform (RT) 2 RT Derivation in 2D Discretizing the RT Computing the IRT and Backprojection Hyperbolic PDEs 3 The Radiative Transfer Problem RTs of Derivatives Discretization of the PDE Results 4 Results 26
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Computing the IRT and Backprojection Ill-posedness of the IRT Suppose f ( x , y ) : R 2 → R is a function with compact support, and R ( f ) = � f In discretized case, we only know f , � f at mesh points We can approximate R ( f ) with a matrix-vector multiplication R f = � f 27
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Computing the IRT and Backprojection Backprojection The adjoint of the Radon transform, denoted R ∗ , is known as backprojection If � f ( s , ω ) : R × [ 0 , π ] → R , the backprojection of � f is � π R ∗ � � � ( x , y ) := f ( x cos( ω ) + y sin( ω ) , ω ) d ω f 0 28
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Computing the IRT and Backprojection Graphical Representation of Backprojection 1 ( x, y ) 0 y − 1 − 1 0 1 x 29
Motivation The Radon Transform (RT) Hyperbolic PDEs Results Computing the IRT and Backprojection Using BiCGSTAB to Solve the Normal Equations We can approximate the action of R ∗ using R T We want to solve R T R f = R T � f We do not explicitly need R T R , just the matrix-vector products R f , R T � � R f We use an iterative method (BiCGSTAB) to solve R T R f = R T � f 30
Motivation The Radon Transform (RT) Hyperbolic PDEs Results The Radiative Transfer Problem Outline Motivation 1 The Radon Transform (RT) 2 RT Derivation in 2D Discretizing the RT Computing the IRT and Backprojection Hyperbolic PDEs 3 The Radiative Transfer Problem RTs of Derivatives Discretization of the PDE Results 4 Results 31
Motivation The Radon Transform (RT) Hyperbolic PDEs Results The Radiative Transfer Problem Application: Radiative Transfer We want to solve the Radiative Transfer Equation � F , t + Ω · ∇ F + σ t F = σ s S 2 F d Ω 4 π A kinetic model for subatomic particles propagating through a homogeneous medium Ω · ∇ F is a transport term � σ s S 2 F d Ω − σ t F is a collision term 4 π A linear transport equation in 1 + 5 dimensions 32
Motivation The Radon Transform (RT) Hyperbolic PDEs Results The Radiative Transfer Problem Spherical Harmonics Use spherical harmonics to create P N equations ℓ � N � F m ℓ ( t , x ) Y m F ( t , x , Ω) ≈ ℓ ( µ, φ ) ℓ = 0 m = − ℓ Already know spherical harmonics, Y m ℓ ( µ, φ ) ; removes angular dependence from F Need to solve for (most) of the F m ℓ ( t , x ) 33
Motivation The Radon Transform (RT) Hyperbolic PDEs Results RTs of Derivatives Outline Motivation 1 The Radon Transform (RT) 2 RT Derivation in 2D Discretizing the RT Computing the IRT and Backprojection Hyperbolic PDEs 3 The Radiative Transfer Problem RTs of Derivatives Discretization of the PDE Results 4 Results 34
Motivation The Radon Transform (RT) Hyperbolic PDEs Results RTs of Derivatives Solving the P N Equations We now consider the following system of M differential equations q , t + A q , x + B q , y = C q q ( t = 0 , x , y ) = q 0 ( x , y ) 35
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