Apparatus and method for smoothing random event data obtained from a Positron Emission Tomography scanner
View Patent ↗A method and apparatus for smoothing random event data obtained from a Positron Emission Tomography (PET) scanner. The method includes obtaining initial random event data u(s, φ, t=0)=u 0 (s, φ), corresponding to t=0, calculating second-order central differences u ss , u φφ with respect to s, φ, calculating a gradient u t , using u t =2(u ss +u φφ )−λ(u−u 0 ), where λ is a constant parameter, and updating the random event data using u(s, φ, t 2 )=u(s, φ, t 1 )+Δt u t , where Δt=t 2 −t 1 , t 1 =0 in a first iteration, and Δt is greater than 0. The method repeats the steps of calculating the second-order central differences, calculating the gradient, and updating the random event data until a change in u(s, φ, t) from a previous iteration is less than a predetermined threshold value.
1. A method for smoothing random event data obtained from a Positron Emission Tomography (PET) scanner, the method comprising:
obtaining initial random event data u(s, φ, t=0)=u 0 (s, φ), corresponding to t=0, the initial random event data corresponding to line-of-responses of the PET scanner;
calculating second-order central differences u ss , u φφ with respect to s, φ;
calculating a gradient u t , using u t =2(u ss +u φφ )−λ(u−u 0 ), where λ is a constant parameter;
updating the random event data using u(s, φ, t 2 )=u(s, φ, t 1 )+Δt u t , where Δt=t 2 −t 1 , t 1 =0 in a first iteration, and Δt is greater than 0; and
repeating the steps of calculating the second-order central differences, calculating the gradient, and updating the random event data until a change in u(s, φ, t) from a previous iteration is less than a predetermined threshold value.
2. The method according to claim 1 , wherein λ has a value between 0.05 and 2.0.
3. The method according to claim 1 , wherein λ has a value between 0 and 0.01.
4. The method according to claim 1 , wherein Δt has a value between 0.05 and 0.1.
5. The method according to claim 1 , wherein u(s, φ, t) converges when
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6. An apparatus for smoothing random event data obtained from a Positron Emission Tomography (PET) scanner, the apparatus comprising:
a processing circuit configured to
obtain initial random event data u(s, φ, t=0)=u 0 (s, φ), corresponding to t=0, the initial random event data corresponding to lines-of-responses of the PET scanner,
calculate second-order central differences u ss , u φφ with respect to s, φ;
calculate a gradient u t , using u t =2(u ss +u φφ )−λ(u−u 0 ), where λ is a constant parameter,
update the random event data using u(s, φ, t 2 )=u(s, φ, t 1 )+Δt u t , where Δt=t 2 −t 1 , t 1 =0 in a first iteration, and Δt is greater than 0, and
repeat the calculating the second-order central differences, calculating the gradient, and updating the random event data until a change in u(s, φ, t) from a previous iteration is less than a predetermined threshold value.
7. The apparatus according to claim 6 , wherein λ has a value between 0.05 and 2.0.
8. The apparatus according to claim 6 , wherein λ has a value between 0 and 0.01.
9. The apparatus according to claim 6 , wherein Δt has a value between 0.05 and 0.1.
10. The apparatus according to claim 6 , wherein u(s, φ, t) converges when
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11. A method for smoothing random event data obtained from a Positron Emission Tomography (PET) scanner, the method comprising:
obtaining initial random event data u(s, φ, z, θ, t=0)=u 0 (s, φ, z, θ), corresponding to t=0, the initial random event data corresponding to lines-of-responses of the PET scanner;
calculating second-order central differences u ss , u φφ , u zz , u θθ , with respect to s, φ, z, θ;
calculating a gradient u t , using u t =2(u ss +u φφ +u zz +u θθ )−λ(u−u 0 ), where λ is a constant parameter;
updating the random event data using u(s, φ, z, θ, t 2 )=u(s, φ, z, θ, t 1 )+Δt u t , where Δt=t 2 −t 1 , t 1 =0 in a first iteration, and Δt is greater than 0; and
repeating the steps of calculating the second-order central differences, calculating the gradient, and updating the random event data until a change in u(s, φ, z, θ, t) from a previous iteration is less than a predetermined threshold value.
12. The method according to claim 11 , wherein λ has a value between 0.05 and 2.0.
13. The method according to claim 11 , wherein λ has a value between 0 and 0.01.
14. The method according to claim 11 , wherein Δt has a value between 0.05 and 0.1.
15. The method according to claim 11 , wherein u(s, φ, z, θ, t) converges when
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is less than the predetermined threshold value.
16. The method according to claim 11 , further comprising:
applying a mask so that the obtained initial random event data is restricted to a region containing all measurable line-of-responses of the PET scanner.
17. An apparatus for smoothing random event data obtained from a Positron Emission Tomography (PET) scanner, the method comprising:
a processing circuit configured to
obtain initial random event data u(s, φ, z, θ, t=0)=u 0 (s, φ, z, θ), corresponding to t=0, the initial random event data corresponding to lines-of-responses of the PET scanner;
calculate second-order central differences u ss , u φφ , u zz , u θθ , with respect to s, φ, z, θ;
calculate a gradient u t , using u t =2(u ss +u φφ +u zz +u θθ )−λ(u−u 0 ), where λ is a constant parameter;
update the random event data using u(s, φ, z, θ, t 2 )=u(s, φ, z, θ, t 1 )+Δt u t , where Δt=t 2 −t 1 , t 1 =0 in a first iteration, and Δt is greater than 0; and
repeat the calculating the second-order central differences, calculating the gradient, and updating the random event data until a change in u(s, φ, z, θ, t) from a previous iteration is less than a predetermined threshold value.
18. The apparatus according to claim 17 , wherein λ has a value between 0.05 and 2.0.
19. The apparatus according to claim 17 , wherein λ has a value between 0 and 0.01.
20. The apparatus according to claim 17 , wherein Δt has a value between 0.05 and 0.1.
21. The apparatus according to claim 17 , wherein u(s, φ, z, θ, t) converges when
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is less than the predetermined threshold value.
22. The apparatus according to claim 17 , wherein the processing circuit is further configured to apply a mask so that the obtained initial random event data is restricted to a region containing all measurable line-of-responses of the PET scanner.