An analog of Chang inversion formula for weighted Radon transforms in multidimensions F.O. Goncharov 1 R.G. Novikov 2 1 Moscow Institute of Physics and Technology Dolgoprudny, Russian Federation 2 Ecole Polytechnique Palaiseau, France Quasilinear Equations, Inverse Problems and their Applications, 2016 F.O. Goncharov, R.G. Novikov An analog of Chang inversion formula Quasilinear Equations etc. 1 / 20
Outline Introduction 1 Weighted Radon transforms Chang inversion formula in 2D Main result 2 Novikov’s result for Chang formula in 2D Analog of Chang inversion formula for ND Possible tomographical applications in 3D Summary 3 F.O. Goncharov, R.G. Novikov An analog of Chang inversion formula Quasilinear Equations etc. 2 / 20
Outline Introduction 1 Weighted Radon transforms Chang inversion formula in 2D Main result 2 Novikov’s result for Chang formula in 2D Analog of Chang inversion formula for ND Possible tomographical applications in 3D Summary 3 F.O. Goncharov, R.G. Novikov An analog of Chang inversion formula Quasilinear Equations etc. 3 / 20
Weighted Radon transforms Let f ∈ C 0 ( R n ) , n ≥ 2 then Radon Rf and weighted Radon R W f transforms are defined correspondingly: � Rf ( s , θ ) def = f ( x ) dx H , (1) x θ = s � W ( x , θ ) f ( x ) dx H , ( s , θ ) ∈ R n × S n − 1 , R W f ( s , θ ) def = (2) x θ = s where W is complex-valued, W ∈ C ( R n × S n − 1 ) ∩ L ∞ ( R n × S n − 1 ). Typical question: Ker ( R ) , Ker ( R W ) = { 0 } ? F.O. Goncharov, R.G. Novikov An analog of Chang inversion formula Quasilinear Equations etc. 4 / 20
Classical Results on weighted Radon transforms [L.Chang, 1978] A method for attenuation correction in radionuclide computed tomography. IEEE Transactions on Nuclear Science. [J.Boman, E.T. Quinto, 1987] Support theorems for real-analytic Radon transforms. Duke Mathematical Journal. [R.G. Novikov, 2002] An inversion formula for the attenuated X-ray transformation. Arkiv f¨ ur Matematik. [L.A. Kunyansky, 1992] Generalized and attenuated Radon transforms: restorative approach to the numerical inversion. Inverse problems. [S. Gindikin, 2010] A remark on the weighted Radon transform on the plane. Inverse Problems and Imaging. ... F.O. Goncharov, R.G. Novikov An analog of Chang inversion formula Quasilinear Equations etc. 5 / 20
Outline Introduction 1 Weighted Radon transforms Chang inversion formula in 2D Main result 2 Novikov’s result for Chang formula in 2D Analog of Chang inversion formula for ND Possible tomographical applications in 3D Summary 3 F.O. Goncharov, R.G. Novikov An analog of Chang inversion formula Quasilinear Equations etc. 6 / 20
Simple example: SPECT Weighted Radon transform ( n = 2 , 3): + ∞ � , y ∈ R n , θ ∈ S n − 1 . W a ( y , θ ) = exp − a ( y + s θ ) ds (3) 0 � W a ( y , θ ) f ( y ) dy , l ∈ T S n − 1 , a ∈ S ( R n ) . R W a f ( l ) = (4) y ∈ l where T S n − 1 = { ( x , θ ) ∈ R n × S n − 1 : ( x · θ ) = 0 } – manifold of all oriented lines in R n , § ( R n ) – Schwartz class. F.O. Goncharov, R.G. Novikov An analog of Chang inversion formula Quasilinear Equations etc. 7 / 20
From SPECT to Chang inversion formula in 2D Let f ∈ C 0 ( R 2 ) , W ∈ C ( R 2 × S 1 ) ∩ L ∞ ( R 2 × S 1 ), then 1 � W ( x θ ⊥ , θ ) d θ, h ′ = d def S 1 h ′ f appr = ds h ( s , θ ) , (5) 4 π w 0 ( x ) h W ( s , θ ) = 1 � R W ( t , θ ) dt , ( s , θ ) ∈ T S 1 ≃ R × S 1 , π p . v . (6) s − t R w 0 ( x ) = 1 � S 1 W ( x , θ ) d θ, w 0 ( x ) � = 0 . (7) 2 π [L.Chang, 1978] A method for attenuation correction in radionuclide computed tomography. IEEE Transactions on Nuclear Science. [R.G. Novikov, 2002] An inversion formula for the attenuated X-ray transformation. Arkiv f¨ ur Matematik. – exists exact inversion formula for R W a . F.O. Goncharov, R.G. Novikov An analog of Chang inversion formula Quasilinear Equations etc. 8 / 20
Outline Introduction 1 Weighted Radon transforms Chang inversion formula in 2D Main result 2 Novikov’s result for Chang formula in 2D Analog of Chang inversion formula for ND Possible tomographical applications in 3D Summary 3 F.O. Goncharov, R.G. Novikov An analog of Chang inversion formula Quasilinear Equations etc. 9 / 20
Inversion with Chang formula in 2D Let W ( x , θ ) , ( x , θ ) ∈ R 2 × S 1 is complex valued, W ∈ C ( R 2 × S 1 ) ∩ L ∞ ( R 2 × S 1 ) and 1 � w 0 ( x ) def = S 1 W ( x , θ ) d θ, w 0 ( x ) � = 0 . (8) 2 π Theorem (R.G. Novikov, 2011) Let W satisfies conditions above, f ∈ C 0 ( R 2 ) and f appr is defined by the Chang inversion formula. Then f = f appr (in terms of distributions) if and only if W ( x , θ ) − w 0 ( x ) ≡ w 0 ( x ) − W ( x , − θ ) . (9) R.G. Novikov, Weighted Radon transforms for which Chang’s approximate inversion formula is exact. Russian Mathematical Surveys , 66(2): 442-443, 2011. F.O. Goncharov, R.G. Novikov An analog of Chang inversion formula Quasilinear Equations etc. 10 / 20
Sketch of the proof in 2D Let f ∈ C 0 ( R 2 ), recall: 1 � W ( x θ ⊥ , θ ) d θ, h ′ = d def S 1 h ′ f appr = ds h ( s , θ ) , 4 π w 0 ( x ) h W ( s , θ ) = 1 � R W f ( t , θ ) dt , ( s , θ ) ∈ T S 1 ≃ R × S 1 . π p . v . s − t R Main idea of the proof – “symmetrization” of W: = 1 W s ( x , θ ) def 2( W ( x , θ ) + W ( x , − θ )) , (10) R W s f ( s , θ ) = 1 2( R W f ( s , θ ) + R W f ( − s , − θ )) , (11) h W s ( s , θ ) = 1 2( h W ( s , θ ) − h W ( − s , − θ )) , (12) W s ( s , θ ) = 1 h ′ 2( h ′ W ( s , θ ) + h ′ W ( − s , − θ )) , ( s , θ ) ∈ R × S 1 . (13) F.O. Goncharov, R.G. Novikov An analog of Chang inversion formula Quasilinear Equations etc. 11 / 20
Sketch of the proof in 2D From identities (10)-(13) and definition of f appr it follows: 1 � W s ( x θ ⊥ , θ ) d θ, h ′ = d S 1 h ′ f appr ≡ ds h ( s , θ ) , 4 π w 0 ( x ) h W s ( s , θ ) = 1 � R W s f ( t , θ ) dt , ( s , θ ) ∈ T S 1 ≃ R × S 1 . π p . v . s − t R Sufficiency: W s ( x , θ ) ≡ w 0 ( x ) . Necessity: From Radon inversion formula and definition f appr it follows: � S 1 ( h ′ w 0 ( x θ ⊥ , θ ) − h ′ W s ( x θ ⊥ , θ )) d θ = 0 . (14) From 2D-Fourier transform of (14) it follows: h w 0 ≡ h W s ⇒ R w 0 f = R W s f ( ∀ f ∈ C 0 ( R 2 )) ⇒ w 0 ≡ W s (15) F.O. Goncharov, R.G. Novikov An analog of Chang inversion formula Quasilinear Equations etc. 12 / 20
Outline Introduction 1 Weighted Radon transforms Chang inversion formula in 2D Main result 2 Novikov’s result for Chang formula in 2D Analog of Chang inversion formula for ND Possible tomographical applications in 3D Summary 3 F.O. Goncharov, R.G. Novikov An analog of Chang inversion formula Quasilinear Equations etc. 13 / 20
Analog of Chang formula in ND Let f – test function, W – weight, than the following formulas are defined: n is odd ( − 1) ( n − 1) / 2 � [ R W f ] ( n − 1) ( x θ, θ ) d θ. f appr ( x ) def = (16) 2(2 π ) n − 1 w 0 ( x ) S n − 1 n is even ( − 1) ( n − 2) / 2 � H [ R W f ] ( n − 1) ( x θ, θ ) d θ f appr ( x ) def = (17) 2(2 π ) n − 1 w 0 ( x ) S n − 1 where [ R W f ] ( n − 1) ( s , θ ) = d n − 1 ds n − 1 R W f ( s , θ ) , s ∈ R , θ ∈ S n − 1 , (18) = 1 φ ( t ) � H φ ( s ) def π p . v . s − t dt , s ∈ R . (19) R [F.Natterer, 1986] The mathematics of computerized tomography, vol.32, SIAM . F.O. Goncharov, R.G. Novikov An analog of Chang inversion formula Quasilinear Equations etc. 14 / 20
Analog of Chang inversion formula for ND Let W ( x , θ ) , ( x , θ ) ∈ R n × S n − 1 is complex valued, W ∈ C ( R n × S n − 1 ) ∩ L ∞ ( R n × S n − 1 ) and 1 � w 0 ( x ) def = S n − 1 W ( x , θ ) d θ, w 0 ( x ) � = 0 . (20) | S n − 1 | Theorem (F.O. Goncharov, R.G. Novikov, 2016) Let W satisfies conditions above, f ∈ C 0 ( R n ) and f appr is defined by the analog Chang inversion formula in multidimensions. Then f = f appr (in terms of distributions) if and only if W ( x , θ ) − w 0 ( x ) ≡ w 0 ( x ) − W ( x , − θ ) . (21) F.O. Goncharov, R.G. Novikov, An analog of Chang inversion formula for weighted Radon transforms in multidimensions. EJMCA , 4(2): 23-32, 2016. F.O. Goncharov, R.G. Novikov An analog of Chang inversion formula Quasilinear Equations etc. 15 / 20
Sketch of the proof in multidimensions 1 Symmetrization: = 1 W s ( x , θ ) def 2( W ( x , θ ) + W ( x , − θ )) , (22) R W s f ( s , θ ) = 1 2( R W f ( s , θ ) + R W f ( − s , − θ )) , (23) then ( − 1) ( n − 1) / 2 � [ R W s f ] ( n − 1) ( x θ, θ ) d θ, f appr ( x ) ≡ (24) 2(2 π ) n − 1 w 0 ( x ) S n − 1 ( − 1) ( n − 2) / 2 � H [ R W s f ] ( n − 1) ( x θ, θ ) d θ f appr ( x ) ≡ (25) 2(2 π ) n − 1 w 0 ( x ) S n − 1 F.O. Goncharov, R.G. Novikov An analog of Chang inversion formula Quasilinear Equations etc. 16 / 20
Sketch of the proof in multidimensions Sufficiency: W s ≡ w 0 , then f appr coincides with exact Radon inversion formulas. Neccesity: Same idea as in 2D case ND Fourier transform → R W s f = R w 0 f ( H [ R W s f ] ≡ H [ R w 0 f ]) for all f ∈ C 0 ( R n ) → W s ≡ w 0 . F.O. Goncharov, R.G. Novikov An analog of Chang inversion formula Quasilinear Equations etc. 17 / 20
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