We will focus in this chapter on reconstructing two-dimensional objects fromparallel- beamprojections with a single rotation axis. The unknown object is modeled as a
two-dimensional finite and integrable function f :R2→Rwith bounded support. We
define a projectionPθof f as the line integral of f(x,y)over linelθ: Pθ(t) = Z lθ f(x,y)ds (5.1) = Z Z R2 f(x,y)δ(xcosθ+ysinθ−t)dxdy (5.2)
This integral transform is called theRadon transformof f.
Given an image f(x,y), we can model the projection geometry in parallel-beam
tomography as a number of parallel lines going through f, each rotated by a certain
projection angleθ. A point(x,y)on one such linelθobeys the equation t=xcosθ+ ysinθ. For each linelθ, a unique constanttdefines all points on that line. This model
is shown graphically in Fig. 5.1. The basic tomographic problem is to reconstruct the unknown image f(x,y)from the measured projections.
In practice, only discrete projection data is available, which consists of a matrix of measured values, one for each combination of Nθ projection angles θ ∈ Θ =
{θ0,θ1, . . . ,θNθ−1}andNddetectorsp∈ {0, 1, . . . ,Nd−1}. The position of a detectorp relative to the central detector is given byτp:
τp=d
p−Nd2−1
5.2. NOTATION AND CONCEPTS 69
whered is the width of a detector. The entire set of detector positions is given by T={τ0,τ1, . . . ,τNd−1}.
The projection data is used to reconstruct f on anN×N grid of square pixels.
Without loss of generality, we define that each pixel has a width and height of one, and that the center of the grid is positioned at the origin. In this case, the center of pixel (xi,yj)is situated in row jand columniof the pixel grid, withi∈ {0, 1, . . . ,N−1}
and j∈ {0, 1, . . . ,N−1}, andxi=yi=i−(N−1)/2.
Filtered back projection
One way of solving the reconstruction problem is to find a direct inverse of Eq. (5.2). To perform this inversion, we first convolve the projection data with a filterhθ(t):
qθ(t) =
Z ∞
−∞
hθ(τ)Pθ(t−τ)dτ (5.4)
We can also perform this operation in the Fourier domain, where ˆPandHdenote the
Fourier transforms ofPandh: qθ(t) =
Z ∞
−∞
ˆ
Pθ(u)Hθ(u)e2πıutdu (5.5)
By taking the formal adjoint of the Radon transform, it can be shown that ifHθ(u) =|u|,
we obtain a direct inverse of Eq. (5.2)[KS01]:
f(x,y) =
Z π
0
qθ(xcosθ+ysinθ)dθ (5.6)
In practice, Eq. (5.6) cannot be used directly, sincePθ(t)can only be measured for
a finite set of anglesΘand a finite set of detector positionsT. Therefore, we need to
discretize both variables to obtain a usable reconstruction algorithm. Inserting Eq. (5.4) in Eq. (5.6) and discretizing, we obtain the filtered back projection method (FBP):
f(x,y)≈FBPh(x,y) = X
θd∈Θ X
τp∈T
h(τp)Pθd(t−τp) (5.7) wheret=xcosθd+ysinθd. Because the projection data is discretized, interpolation is needed to obtain values att−τp, forτp∈T. Linear interpolation is often adequate, since projection data is usually reasonably smooth.
The convolution operation in Eq. (5.7) can be performed in Fourier space, leading to an efficient implementation of FBP: first convolve the projection data with filter
hin Fourier space inO(NθNdlogNd)time and afterwards backproject the result to
obtain the reconstruction inO(NθN2). Various discrete approximations of the ideal
filterHθ(u) =|u|are used in practice, such as the Ram-Lak (ramp), Shepp-Logan, and
Hann filters[Far+97]. The Ram-Lak filter, obtained by settingHθ(u)to 0 whenu>uc
−10 −5 0 5 10 n −0.1 0.0 0.1 0.2 0.3 d − 2h ( nd )
Figure 5.2: The widely used discrete Ram-Lak filter for the FBP algorithm (Eq. (5.7)). In this image,dis the distance between adjacent detector positions. This filter is an approximation of the ideal filter, obtained by taking the Fourier transformHθ(u) =|u|of the ideal filter, and settingHθ(u) =0 whenu>ucfor someuc.
(a) (b) (c)
Figure 5.3: Various reconstructions of the Shepp-Logan head phantom on a 512×512 pixel grid. In (a) the phantom was reconstructed by FBP using 512 projection angles∈[0,π). In (b) and (c), only 32 projection angles were used to reconstruct the phantom. FBP was used in (b), while the image (c) was obtained by using SIRT, an iterative algebraic method, with prior knowledge about the minimum and maximum possible image values.
FBP is one of the most widely used reconstruction methods in practice, because of the low computational cost compared to other methods, and good reconstruction quality if data of enough projections are available. The accuracy of the reconstructions depends on how well Eq. (5.7) approximates Eq. (5.6). If data of many projection angles are available (say, several hundreds), the approximation is often very good. When using FBP with a small number of angles, artifacts appear in the reconstructions. These artifacts can make further analysis of the reconstruction, such as segmentation, very difficult. An example of artifacts in an FBP reconstruction of limited data is shown in Fig. 5.3b. Note that a reconstruction of the same data by an algebraic method, shown in Fig. 5.3c, contains less artifacts, but takes more time to compute.