Resolution and Reconstruction for a Helical CT-Scanner

Frank Natterer

December 1994

A helical CT-scanner is a fan-beam scanner in which the patient is moved along the axis of the scanner with a constant speed as the x-ray source spins around the patient, allowing for 3D imaging [2]. We give the conditions on the scanning parameters (i.e. detector width, angular speed ω, translation speed v) for reliable reconstruction of a function of essential bandwidth Ω (i.e. spatial resolution 2π/Ω in all directions), and we describe an algorithm which actually achieves this resolution.

Assume that the patient is lying in a cylinder C_{ρ} of radius ρ around the axes (the x_{1} axes) of the scanner, and that the source sits on a concentric circle of radius r ≥ ρ. Let f be a function which is supported in C_{ρ}. We define

g(x_{1}, α, β) = ∫ f(x_{1}e_{1} + rθ(α) − sθ(β))ds  (1)

were e_{1} is the first unit vector, and

θ(α) = ( \begin{matrix}0 \\ cos α \\ sin α \end{matrix} ) = ( \begin{matrix}0 \\ θ′(α) \end{matrix} ) .

In helical scanning the function g is sampled at

x_{1} = kΔx , Δx = vΔt  ,
α = kΔα , Δα = \frac{2π}{p} = ωΔt , k = 0, 1, …
β = ℓΔβ , ℓ = −q, …, q , Δβq = arcsin(ρ/r) .

(2)

Here, Δβ is the detector spacing and Δα the angle between two source positions. The condition that Δα = 2π/p with an integer p makes the sampling lattice periodic in α. We sketch the sampling points in the x_{1}-α-plane:

Thus the sampling points (2) are precisely the lattice

W Z^{3} ,  W = ( \begin{matrix}pΔx & Δx & 0 \\ 0 & Δα & 0 \\ 0 & 0 & Δβ \end{matrix} ) .  (3)

The reciprocal lattice is generated by the matrix

2πW^{−T} = ( \begin{matrix}2π/pΔx & 0 & 0 \\ −1 & 2π/Δα & 0 \\ 0 & 0 & 2π/Δβ \end{matrix} ) .

Determining the essential support of the Fourier transform of g and using the Petersen-Middleton sampling theorem would possibly lead to an efficient sampling scheme. In this note we pursue a less ambitious goal. We use the inversion formula for the 2D fan-beam transform ([1], p. 368)

V_{Ω} ⋆ f(x) = r ∫_{0}^{2π} ∫_{−π/2}^{π/2} v_{Ω}(x′ · θ′(α + β) − r sin β)g(x_{1}, α, β) cos βdβdα . (4)

Here, v is defined by

v̂_{Ω}(σ) = \frac{1}{2}(2π)^{−3/2}|σ|ϕ̂(\frac{|σ|}{Ω})

where ϕ̂ is a filter factor vanishing outside [0, 1], and the convolution in (4) with the function V_{Ω} defined by

V̂_{Ω}(ξ) = \frac{1}{2π} ϕ̂(\frac{|ξ|}{Ω}) , ξ ∈ IR^{2}

is with respect to the 2D variable x′ = ( \begin{matrix}x_{2} \\ x_{3} \end{matrix} ). Convolving (4) in the variable

x_{1} with the filter

δ_{Ω}(x_{1}) = (2π)^{−1/2} ∫_{−Ω}^{Ω} e^{ix_{1}ψ_{1}}dψ_{1}

(or any other filter with bandwidth Ω) gives a reconstruction formula for helical CT, namely

f_{Ω}(x) = r ∫_{−∞}^{+∞}∫_{0}^{2π} ∫_{−π/2}^{π/2} δ_{Ω}(x_{1} − y_{1})v_{Ω}(x′ · θ′(α + β) − r sin β)
g(y_{1}, α, β) cos βdβdαdy_{1} .  (5)

f_{Ω} deviates from the true f only in details of size ≤ \frac{2π}{Ω}. Applying the Petersen-Middleton sampling theorem to (5) as in [1] one obtains

f_{Ω}(x) = rdet(W) ∑_{k,ℓ} δ_{Ω}(x_{1} − kΔx)v_{Ω}(x′ · θ′(kΔα + ℓΔβ) − r sin(ℓΔβ))
g(kΔx, kΔα, ℓΔβ) cos(ℓΔβ) .  (6)

Arranging the k-sum into complete cycles of length p and putting x′ = dθ′(jΔα) we obtain

f_{Ω}(x_{1}, dθ′(jΔα)) = rdet(W) ∑_{ℓ=−q}^{q} ∑_{L=0}^{M} ∑_{k=Lp}^{(L+1)p−1} δ_{Ω}(x_{1} − kΔx)

(7)

v_{Ω}(d cos((k − j)Δα + ℓΔβ) − r sin(ℓΔβ))g(kΔx, kΔα, ℓΔβ) .

For each x_{1}, ℓ, d, L the k-sum is now a cyclic convolution of length p, making the evaluation amenable to FFT techniques. For each x_{1}, only those values of k for which kΔx is close to x_{1} contribute significantly. This means that only a few values of L have to be taken into account. Thus the evaluation of (7) for a fixed x_{1} takes about the same number of operations as in 2D fan-beam tomography.

From [1] we know that the error in (6) is negligible provided that

Δα ≤ \frac{r + ρ}{ρ} \frac{π}{Ωr} , Δβ ≤ \frac{π}{Ωr} , Δx ≤ \frac{π}{Ωp} ,  (8)

the last condition being obvious. In view of (2) this translates into

ωΔt ≤ \frac{r + ρ}{ρ} \frac{π}{Ωr} , Δβ ≤ \frac{π}{Ωr} , v ≤ \frac{ω}{2Ω} .  (9)

If these conditions are met, no aliasing occurs in (6).

If one wants a different resolution in the x-direction, characterized by a bandwidth Ω_{1}, one simply has to replace Ω by Ω_{1} in the last inequalities of (8) and (9).

References

[1] Natterer, F.: Sampling in fan-beam tomography, SIAM J. Appl. Math. 53, 358-380 (1993).

[2] Kalender, W.A. - Seissler, W. - Klotz, E. - Vock, P.: Spiral Volumetric CT with Single-Breath-Hold Technique, Continuous Transport and Continuous Scanner Rotation, Radiology 176, 181-183 (1990).

0LaTeX: berg f berger paperfne HelicalCT.tex