2.11 Fundamentals of CT Reconstruction in 2D and 3D JA Fessler, University of Michigan, Ann Arbor, MI, USA ã 2014 Elsevier B.V. All rights reserved. 2.11.1 Introduction 264 2.11.2 Radon Transform in 2D 264 2.11.2.1 Definition 264 2.11.2.2 Signed Polar Forms 265 2.11.2.3 Radon Transform Properties 266 2.11.2.4 Sinogram 267 2.11.2.5 Fourier-Slice Theorem 267 2.11.3 Back Projection 268 2.11.3.1 Image-Domain Analysis 269 2.11.3.2 Frequency-Domain Analysis 269 2.11.3.3 Summary 270 2.11.4 Radon Transform Inversion 270 2.11.4.1 Direct Fourier Reconstruction 272 2.11.4.2 The BPF Method 272 2.11.4.3 The FBP Method 273 2.11.4.4 Ramp Filters and Hilbert Transforms 274 2.11.4.5 Filtered Versus Unfiltered Back Projection 275 2.11.4.6 The CBP Method 275 2.11.4.7 PSF of the FBP Method 277 2.11.4.8 Summary 277 2.11.5 Practical Back Projection 277 2.11.5.1 Rotation-Based Back Projection 278 2.11.5.2 Ray-Driven Back Projection 278 2.11.5.3 Pixel-Driven Back Projection 278 2.11.5.4 Interpolation Effects 279 2.11.5.5 Summary 279 2.11.6 Sinogram Restoration 279 2.11.7 Sampling Considerations 279 2.11.7.1 Radial Sampling 280 2.11.7.2 Angular Sampling 280 2.11.8 Linogram Reconstruction 280 2.11.9 2D Fan-Beam Tomography 280 2.11.9.1 Fan-Parallel Rebinning Methods 282 2.11.9.2 The FBP Approach for 360 Scans 282 2.11.9.2.1 Equiangular case 284 2.11.9.2.2 Equidistant case 284 2.11.9.3 FBP for Short Scans 285 2.11.9.4 The BPF Approach 285 2.11.10 3D Cone-Beam Reconstruction 286 2.11.10.1 Equidistant Case (Flat Detector) 286 2.11.10.2 Equiangular Case (Third-Generation Multislice CT) 287 2.11.10.3 Extensions (Data Truncation and Helical Scans) 287 2.11.10.3.1 Rebinning 287 2.11.10.3.2 Offset detectors 287 2.11.10.3.3 Long object problem 287 2.11.10.3.4 Helical scans 287 2.11.10.3.5 Region of interest reconstruction 287 2.11.10.3.6 Motion compensation 287 2.11.10.3.7 Local tomography 287 2.11.11 Iterative Image Reconstruction 287 2.11.11.1 Object Model 288 2.11.11.2 Measurement Model 288 2.11.11.3 Algebraic Methods 288 2.11.11.4 Statistical Models 288 Comprehensive Biomedical Physics http://dx.doi.org/10.1016/B978-0-444-53632-7.00212-4 263
34
Embed
Fundamentals of CT Reconstruction in 2D and 3Ddownload.xuebalib.com/xuebalib.com.10982.pdf2.11.4.1 Direct Fourier Reconstruction 272 2.11.4.2 The BPF Method 272 2.11.4.3 The FBP Method
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Co
2.11 Fundamentals of CT Reconstruction in 2D and 3DJA Fessler, University of Michigan, Ann Arbor, MI, USA
ã 2014 Elsevier B.V. All rights reserved.
2.11.1 Introduction 2642.11.2 Radon Transform in 2D 2642.11.2.1 Definition 2642.11.2.2 Signed Polar Forms 2652.11.2.3 Radon Transform Properties 2662.11.2.4 Sinogram 2672.11.2.5 Fourier-Slice Theorem 2672.11.3 Back Projection 2682.11.3.1 Image-Domain Analysis 2692.11.3.2 Frequency-Domain Analysis 2692.11.3.3 Summary 2702.11.4 Radon Transform Inversion 2702.11.4.1 Direct Fourier Reconstruction 2722.11.4.2 The BPF Method 2722.11.4.3 The FBP Method 2732.11.4.4 Ramp Filters and Hilbert Transforms 2742.11.4.5 Filtered Versus Unfiltered Back Projection 2752.11.4.6 The CBP Method 2752.11.4.7 PSF of the FBP Method 2772.11.4.8 Summary 2772.11.5 Practical Back Projection 2772.11.5.1 Rotation-Based Back Projection 2782.11.5.2 Ray-Driven Back Projection 2782.11.5.3 Pixel-Driven Back Projection 2782.11.5.4 Interpolation Effects 2792.11.5.5 Summary 2792.11.6 Sinogram Restoration 2792.11.7 Sampling Considerations 2792.11.7.1 Radial Sampling 2802.11.7.2 Angular Sampling 2802.11.8 Linogram Reconstruction 2802.11.9 2D Fan-Beam Tomography 2802.11.9.1 Fan-Parallel Rebinning Methods 2822.11.9.2 The FBP Approach for 360� Scans 2822.11.9.2.1 Equiangular case 2842.11.9.2.2 Equidistant case 2842.11.9.3 FBP for Short Scans 2852.11.9.4 The BPF Approach 2852.11.10 3D Cone-Beam Reconstruction 2862.11.10.1 Equidistant Case (Flat Detector) 2862.11.10.2 Equiangular Case (Third-Generation Multislice CT) 2872.11.10.3 Extensions (Data Truncation and Helical Scans) 2872.11.10.3.1 Rebinning 2872.11.10.3.2 Offset detectors 2872.11.10.3.3 Long object problem 2872.11.10.3.4 Helical scans 2872.11.10.3.5 Region of interest reconstruction 2872.11.10.3.6 Motion compensation 2872.11.10.3.7 Local tomography 2872.11.11 Iterative Image Reconstruction 2872.11.11.1 Object Model 2882.11.11.2 Measurement Model 2882.11.11.3 Algebraic Methods 2882.11.11.4 Statistical Models 288
264 Fundamentals of CT Reconstruction in 2D and 3D
2.11.11.5 Regularized Weighted Least Squares 2892.11.11.6 Total Variation Regularization and Sparsity 2892.11.11.7 Optimization Algorithms 2892.11.11.8 Example 2902.11.12 Summary and Future Trends 290References 291
GlossaryBack projection Creating an image from a sinogram.
Sinogram Raw data measured by a tomographic imaging
system.
Tomography Forming cross-sectional images of a 3D
object.
AbbreviationsBPF Backproject-filter method for image reconstruction
FBP Filter-backproject method for image reconstruction
FT Fourier transform
PSF Point-spread function
WLS Weighted least squares
2.11.1 Introduction
Methods for tomographic image reconstruction can be cate-
gorized as either analytic methods or iterative methods.
Analytic image reconstruction methods typically are based
on idealized models for the imaging system, and these sim-
plifications lead to noniterative algorithms that are used rou-
tinely in x-ray CT clinical practice because they require only
modest computation times. (Other names are Fourier recon-
struction methods and direct reconstruction methods, a term
that emphasizes that these methods are noniterative.) Itera-
tive methods are based on more accurate models for the
imaging system, and these can lead to improved image quality
at the price of greatly increased computation. Many iterative
methods are also based on models for the measurement
statistics, and these methods can reduce image noise and
thus enable lower-dose scans.
This chapter begins with a review of classical analytic tomo-
graphic reconstruction methods. These analytic methods are
also useful for developing intuition and for initializing iterative
algorithms. See Chapter 2.03 for several x-ray CT examples.
Several books have been devoted to the subject of image
reconstruction (Deans, 1983; Helgason, 1980; Kak and Slaney,
1988; Natterer, 1986; Natterer and Wubbeling, 2001). The
treatment in this chapter slightly generalizes classical deriva-
tions by considering an angularly weighted back projection
that is described in Section 2.11.3. This weighted back-
projector is introduced here to facilitate analysis of weighted
least square (WLS) formulations of image reconstruction
methods.
There are several limitations of analytic reconstruction
methods that impair their performance. Analytic methods gen-
erally ignore measurement noise in the problem formulation
and treat noise-related problems as an ‘afterthought’ by post-
filtering operations. Analytic formulations usually assume con-
tinuous measurements and provide integral-form solutions.
Sampling issues are treated by discretizing these solutions
‘after the fact.’ Analytic methods require certain standard
geometries (e.g., parallel rays and complete sampling in radial
and angular coordinates). Statistical methods for image recon-
struction can overcome all of these limitations. The mathemat-
ical background needed to work with analytic reconstruction
methods is Fourier analysis, whereas iterative methods are
based primarily on tools from linear algebra.
2.11.2 Radon Transform in 2D
The foundation of analytic reconstruction methods is the
Radon transform that relates a 2D function f(x, y) to the col-
lection of line integrals of that function (Cormack, 1963;
Radon, 1917, 1986). (We focus initially on the 2D case.)
Emission and transmission tomography systems acquire mea-
surements that are something like blurred line integrals, so the
line-integral model represents an idealization of such systems.
Figure 1 illustrates the geometry of the line integrals associated
with the (ideal) 2D Radon transform.
2.11.2.1 Definition
Let L r; ’ð Þ denote the line in the Euclidean plane at angle ’
counterclockwise from the y-axis and at a signed distance r
from the origin:
L r; ’ð Þ ¼ x; yð Þ 2 R2 : x cos’þ y sin’ ¼ r� �
[1]
¼ r cos’� ‘ sin’, r sin’þ ‘ cos’ð Þ : ‘ 2 Rf g [2]
Let p’(r) denote the line integral through f(x, y) along the line
L r; ’ð Þ. There are several equivalent ways to express this line
integral, each of which has its uses:
f(x, y)
x
y
r
r0
r 0
pj(
r)
2ar0
Figure 2 Projection of a centered uniform disk object, illustrated at’¼p/2.
pj(r)
j
j
Objectf(x, y)
L(r,j)
r
r
y
x
Projection
Figure 1 Geometry of the line integrals associated with the Radontransform.
Fundamentals of CT Reconstruction in 2D and 3D 265
p’ rð Þ ¼ðL r;’ð Þ
f x; yð Þd‘
¼ð1�1
f r cos’� ‘ sin’, r sin’þ ‘ cos’ð Þ d‘[3]
¼ð1�1
ð1�1
f r0cos’� ‘ sin’, r0 sin’þ ‘ cos’ð Þd r0 � rð Þ dr0 d‘
[4]
¼ð1�1
ð1�1
f x; yð Þd x cos’þ y sin’� rð Þ dx dy [5]
¼1
cos’j jð1�1
fr � t sin’
cos’; t
� �dt, cos’ 6¼ 0
1
sin’j jð1�1
f t;r � t cos’
sin’
� �dt, sin’ 6¼ 0
8>><>>: [6]
where d(�) denotes the 1D Dirac impulse. (The last form came
from Edholm and Herman, 1987). The step between [4] and
[5] uses the following change of variables:
xy
� �¼ cos’ � sin’
sin’ cos’
� �r0
‘
� �� [7]
The Radon transform of f is the complete collection of line
integrals:
f $Radon p’ rð Þ : ’ 2 0; p½ �, r 2 �1,1ð Þ� �[8]
The function p’(�) is called the projection of f at angle ’.
Sometimes, one refers to values of ’ outside of the domain
given in [8]; this is possible using the ‘periodic extension’
described in [17]. Of course, a practical system has a finite
maximum radius that defines its circular field of view.
In its most idealized form, the 2D image reconstruction
problem is to recover f(x, y) from its projections {p’(�)}. To do
this, one must somehow return the data in projection space
back to object space, as described in Section 2.11.4.
Example 1 Consider the centered uniform disk object with
radius r0:
f x; yð Þ ¼ a rectr
2r0
� �, rect tð Þ≜1 jtj�1=2f g ¼
1, jtj � 1=2
0, otherwise
([9]
Using [3], the Radon transform of this object is
p’ rð Þ ¼ð1�1
f r cos’� ‘ sin’, r sin’þ ‘cos’ð Þ d‘
¼ð1�1
a rect
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir cos’� ‘ sin’ð Þ2 þ r sin’þ ‘cos’ð Þ2
f ax;byð Þ $Radonp∠p b cos’, a sin’ð Þ rjajbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b cos’ð Þ2 þ a sin’ð Þ2q
0B@
1CA
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib cos’ð Þ2 þ a sin’ð Þ2
q[18]
for a,b 6¼0, where r� and ∠p were defined in Section 2.11.2.2.
The following two properties are special cases of the
where g tð Þ ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� t21p
jtj<1f g denotes the projection of a circle
of unity radius.
Example 3 Consider the object f(x,y)¼d2(x�x0, y�y0), the
2D Dirac impulse centered at (x0,y0). Informally, we can
think of this object as a disk function centered at (x0; y0) of
radius r0 and height 1/(pr02) (so that volume is unity) in the
limit as r0!0.
Let Cr0 rð Þ ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffir20 � r2
prect r
2r0
�denote the projection of
centered uniform disk with radius r0 as derived in [10] in
Example 1. Then, by the shift property [16], the projections
of a disk centered at (x0; y0) are
p’ rð Þ ¼ Cr 0 r � x0 cos’þ y0 sin’½ �ð Þ(See Figure 4.) Thus, the projections of the 2D Dirac impulse
are found as follows:
p’ rð Þ ¼ 1
pr20Cr 0 r � x0 cos’þ y0 sin’½ �ð Þ
! d r � x0 cos’þ y0 sin’½ �ð Þ as r0 ! 0
Sinogram for disk0 40
x
yr0
φ
0
p/2
p
Figure 3 Left: cross-section of 2D object containing three Dirac impulses. Right: the corresponding sonogram consisting of three sinusoidal impulseridges.
Fundamentals of CT Reconstruction in 2D and 3D 267
An alternative derivation uses [5]. In summary, for a 2D Dirac
impulse object located at (x0; y0), the projection at angle ’ is a
1D Dirac impulse located at r¼x0 cos’þy0 sin’ (see
Figure 3).
r
φ
−60 −40 −20 0 20 40 60p 0
Figure 4 Sinogram for a disk object of radius r0¼20 centered at(x0; y0)¼ (40; 0).
2.11.2.4 Sinogram
Because p’(r) is a function of two arguments, we can display
p’(r) as a 2D grayscale picture where usually r and ’ are the
horizontal and vertical axes, respectively. If we make such a
display of the projections p’(r) of a 2D Dirac impulse, then the
picture looks like a sinusoid corresponding to the function
r¼x0 cos’þy0 sin’. Hence, this 2D function is called a sino-
gram and (when sampled) represents the raw data available for
image reconstruction. So, the goal of tomographic reconstruc-
tion is to estimate the object f(x, y) from a measured sinogram.
Each point (x, y) in object space contributes a unique sinu-
soid to the sinogram, with the ‘amplitude’ of the sinusoid
Figure 9 Illustration of filter-backproject (FBP) method. Top row: an object f(x, y) consisting of two squares, the larger of which has severalsmall holes in it, its sinogram p’(r), its top row p0(r), and its laminogram fb(x, y). The laminogram is so severely blurred that the small holes arenot visible. Bottom row: the ramp-filtered sinogram p
^
’ rð Þ, its top row p^
0 rð Þ, and FBP image f x; yð Þ. Because of the ramp filtering described inSection 2.11.4.3, the small details are recovered.
Fb(u,v)
Projectionpj(r)
Sinogram
pj (r)
Filteredsinogram
Con
efil
ter
|r|
(con
volv
e)ra
mp
filte
r
Backprojection
1/r
**2D FT
1D FT
|n|
Ram
pfil
ter
2D FT 1D FT
Pj (n)
Pj(n)
Gridding
Slice
F(u, v)
Object
f (x, y)
fb(x, y)
Laminogram
Figure 8 Relationships between a 2D object f(x, y) and its projections and transforms. Left side of the figure is image domain; right side is projectiondomain. Inner ring is space domain; outer ring is frequency domain.
Fundamentals of CT Reconstruction in 2D and 3D 271
272 Fundamentals of CT Reconstruction in 2D and 3D
2.11.4.1 Direct Fourier Reconstruction
The direct Fourier reconstruction method is based on the
Fourier-slice theorem [22]. To reconstruct f(x, y) from {p’(r)}
by the direct Fourier method, one performs the following steps:
• Take the 1D FT of each p’(∙) to get P’(∙) for each ’.
• Create a polar representation Fo(r,F) of the 2D FT of object
F(u, v) using the Fourier-slice relationship:
Fo r; ’ð Þ ¼ P’ rð Þ• Convert from polar representation Fo(r,F) to Cartesian
coordinates F(u, v). This approach, first proposed in De
Rosier and Klug (1968), was ‘the first applicable method
for reconstructing pictures from their projections (Herman,
1972).’
For sampled data, this polar to Cartesian step, often called
gridding, requires very careful interpolation. Figure 10 illus-
trates the process. Numerous papers have considered this step
in detail (e.g., Alliney et al., 1993; Bellon and Lanzavecchia,
1997; Bracewell, 1956; Cheung and Lewitt, 1991; Choi and
Munson, 1998; Dusaussoy, 1996; Edholm and Herman, 1987;
Fourmont, 2003; Gottlieb et al., 2000; Lanzavecchia and
Bellon, 1997; Lewitt, 1983; Matej and Bajla, 1990; Mersereau,
1974, 1976; Mersereau and Oppenheim, 1974; Natterer, 1985;
O’Sullivan, 1985; Penczek et al., 2004; Potts and Steidl, 2001;
Schomberg and Timmer, 1995; Seger, 1998; Stark et al.,
1981a,b; Sweeney and Vest, 1973; Tabei and Ueda, 1992;
Walden, 2000). Of these, the nonuniform FFT methods
with good interpolation kernels are particularly appealing
(e.g., Fourmont, 2003; Matej et al., 2004; Schomberg and
Timmer, 1995).
• Take the inverse 2D FT of F(u, v) to get f(x, y).
In practice, this is implemented using the 2D inverse FFT,
which requires Cartesian samples, whereas the relationship
Fo(r,’)¼P’(r) is intrinsically polar, hence the need for
interpolation.
This method would work perfectly if given noiseless, con-
tinuous projections p’(r). Practical disadvantages of this
method are that it requires 2D FTs and gridding can cause
interpolation artifacts. An alternative approach uses a Hankel
transform rather than FTs (Higgins and Munson, 1988); this
method also uses interpolation.
v
u
vGridding
u
Figure 10 Illustration of polar samples of Fo(r,’)¼P’(r) that onemust interpolate onto Cartesian samples of F(u, v) for the direct Fourierreconstruction method.
Example 13 Consider the sinogram described by
p’ rð Þ ¼ rectr � x0 cos’� y0 sin’
w
� �
What is the object f(x, y) that has these projections?
First, taking the 1D FT yields
P’ nð Þ ¼ w sinc wvð Þe�i2pv x0 cos’þy0 sin’ð Þ
so by the Fourier-slice theorem, the spectrum of f(x, y) is
given by
Fo r;Fð Þ ¼ w sinc wrð Þe�i2pp x0 cosFþy0 sinFð Þ
or equivalently
F u; nð Þ ¼ w sinc wffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ n2
p �e�i2p x0uþy0nð Þ
Because (Bracewell, 2000, p. 338) w sinc wrð Þ ���!2D FT
274 Fundamentals of CT Reconstruction in 2D and 3D
where, assuming w(’)¼1 hereafter, the filtered object f∨
x; yð Þhas the following spectrum:
F∨
r;Fð Þ ¼ jrjFo r;Fð ÞOf course, in practice, we cannot filter the object before acquir-
ing its projections. However, applying the Fourier-slice theo-
rem to the scenario in the preceding text, we see that each
projection p^’ rð Þ has the following 1D FT:
p∨
’ rð Þ$FT P∨’ nð Þ ¼ F∨
r; ’ð Þ���r¼n¼ jrjFo r; ’ð Þjr¼n ¼ jnjFo n; ’ð Þ
¼ jnjP’ nð ÞThis relationship implies that we can replace the cone filter in
the preceding text with a set of 1D filters with frequency
response |n| applied to each projection p’(�). This filter is calledthe ramp filter due to its shape. The block diagram in the
preceding text becomes
This reconstruction approach is called the FBP method and is
used the most widely in tomography.
A formal derivation of the FBP method uses the Fourier-
slice theorem as follows:
f x; yð Þ ¼ðð
F u; vð Þei2p xuþyvð Þdudv
¼ðp0
ð1�1
F n cos’, n sin’ð Þei2pn x cos ’þy sin’ð Þjnjdn d’
¼ðp0
ð1�1
P’ nð Þei2pn x cos ’þy sin ’ð Þjnjdn d’
¼ðp0
p∨
’ x cos’þy sin’ð Þd’
where we define the filtered projection p∨’ rð Þ as follows:
p∨
’ rð Þ ¼ð1�1
P’ nð Þjnjei2pnrdn [39]
The steps of the FBP method are summarized as follows.
• For each projection angle ’, compute the 1D FT of the
projection p’(�) to form P’(n).
• Multiply P’(n) by |n| (ramp filtering) to get
P∨
’ nð Þ ¼ ��n��P’ nð Þ.• For each ’, compute the inverse 1D FT of P
∨
’ nð Þ to get the
filtered projection P∨
’ rð Þ in [39]. In practice, this filtering is
often done using an FFT, which yields periodic convolu-
tion. Because the space-domain kernel corresponding to |n|is not space-limited (see Figure 14), periodic convolution
can cause ‘wrap-around’ artifacts. With care, these artifacts
can be avoided by zero padding the sinogram. Sampling the
ramp filter can also cause aliasing artifacts. See Example 18
in the succeeding text for a preferable approach.
• The ramp filter nulls the DC component of each projection.
If desired, this can be restored using the volume conserva-
tion property [20]. The approach of Example 18 avoids the
need for any such DC correction. Discretizing the integrals
carefully avoids the need for empirical scale factors.
• Backproject the filtered sinogram P∨
’ rð Þn o
using [30] to get
f x; yð Þ, that is,
f x; yð Þ ¼ðp0
p∨
’ x cos’, y sin’ð Þd’ [40]
In practice, usually, the pixel-driven back projection
approach of Section 2.11.5.3 is used.
With some hindsight, the existence of such an approach
seems natural because the Fourier-slice theorem provides a
relationship between the 2D FT in object domain and the 1D
FT in projection domain.
2.11.4.4 Ramp Filters and Hilbert Transforms
It can be useful to relate the ramp filter |n| to a combination of
differentiation and a Hilbert transform.
The Hilbert transform of a 1D function f(t) is defined (using
Cauchy principal values) by Grafakos (2004, p. 248) (note that
some texts use the opposite sign, e.g., Barrett and Myers, 2003,
p. 194; Bracewell, 2000, p. 359):
fHilbert tð Þ ¼ 1
p
ð1�1
1
t � sf sð Þds ¼ 1
ptf tð Þ [41]
Note that this ‘transform’ returns another function of t. The
corresponding relationship in the frequency domain is
FHilbert nð Þ ¼ �i sgn nð ÞF nð Þ [42]
Example 16 The Hilbert transform of the rect function
rect tð Þ ¼ 1 jtj�1=2f g is (Grafakos, 2004, p. 249) 1p log tþ1=2
t�1=2��� ���.
Using the Hilbert transform frequency response [42], we
rewrite the ramp filter |n| in [39] as follows:
nj j ¼ 1
2pi2pnð Þ �isgn nð Þð Þ
The term (i2pn) corresponds to differentiation, by the dif-
ferentiation property of the FT. Therefore, another expression
for the FBP method [40] is
f x; yð Þ ¼ 1
2p
ðp0
d
drpHilbert r; ’ð Þ
����r¼x cos’þy sin’
d’ [43]
where pHilbert(r, ’) denotes the Hilbert transform of p’(r) with
respect to r. Combining [41] and [43] yields
f x; yð Þ ¼ 1
2p2
ðp0
ð1�1
@@r p’ rð Þ
x cos’þ y sin’� rdr d’ [44]
This form is closer to Radon’s inversion formula (Natterer,
1986, p. 21; Radon, 1917, 1986).
Example 17 Continuing Example 6, the spectrum of the pro-
jection at angle ’ of a rectangle object is given by [23], so its
ramp-filtered projections are given by (for sin ’ 6¼0)
P∨
’ nð Þ ¼ ��n��a sinc n acos’ð Þb sinc nb sin’ð Þ¼ 1
p cos’ sin’sin pn acos’ð Þ sgn nð Þ b sin’ð Þ sinc nb sin’ð
¼ 1
2p cos’ sin’e�ipna cos’ � eipna cos’Þð
i sgn nð Þ b sin’ð Þ sinc nb sin’ð Þ½ �
-2 -1 0 1 2-1
0
1
r
Pro
ject
ion
Projection of squareIdeal ramp-filtered projectionBand-limited ramp-filtered projection
Figure 13 Projection p’(r) of a unit square at angle ’¼p/9 and its filtered versions �p’ rð Þ for both ideal ramp filter |n| and band-limited ramp filter withcutoff frequency n0¼4.
Fundamentals of CT Reconstruction in 2D and 3D 275
Using the Hilbert transform in Example 16, the inverse 1D FT
of the bracketed term is 1pb sin’ log
x� 12 b sin’
xþ 12 b sin’
��������, so by the shift
property of the FT, the filtered projections are
�p’ rð Þ ¼ 1
2p2cos’sin’log
r2 � a cos ’þb sin ’2
�2r2 � a cos ’�b sin ’
2
�2�������
�������Compare with Servieres et al. (2004, eqn [14]). Figure 13
shows an example of the projection p’(r) of a unit square
and its filtered version p∨
’ rð Þ. The ramp filter causes singulari-
ties at each of the points of discontinuity in the projections
(cf. Figure 9).
2.11.4.5 Filtered Versus Unfiltered Back Projection
Recall that an unfiltered back projection of a sinogram gives an
image blurred by 1/|n|. This blurring is due to the fact that the
(all nonnegative) projection values ‘pile up’ in the laminogram
and there is no destructive interference. In contrast, after filter-
ing with the ramp filter, the projections have both positive and
negative values, so a destructive interference can occur, which
is desirable for the parts of the image that are supposed to be
zero, for example. Figure 9 illustrates these concepts.
2.11.4.6 The CBP Method
The ramp filter amplifies high-frequency noise; so in practice,
one must apodize it by a 1D low-pass filter A(n), in which case
[39] is replaced by
�p’ rð Þ ¼ð1�1
P’ nð ÞA nð Þjnjei2pnr dn [45]
Alternatively, one can perform this filtering operation in the
Figure 19 Illustration of pseudopolar grid in 2D Fourier spaceassociated with linogram sampling. The cross marks and circlescorrespond to the P’
EW(n) samples and the P’NS(n) samples, respectively.
Original 2 angles 4 angles
32 angles 64 angles 128 angles
Figure 18 Illustration of the effects of angular undersampling on image quality for FBP reconstruction. The image is 128�128, and the truevalues of the digital phantom are 1 in the background disk and 8 in the small disks. The grayscale display is windowed from 0.5 to 1.5 to enhancethe visibility of the artifacts.
Fundamentals of CT Reconstruction in 2D and 3D 281
midpoint of the detector passes through the exact center of
rotation, we allow an offset roff between that line and the center
(Gullberg et al., 1986). Let P denote the point along that line
that intersects the circle of radius roff centered at the rotation
isocenter. D0d denotes the distance from the point P to the
detector, Ds0 denotes the distance from the x-ray source to P,
and Dfs denotes the distance from the focal point of the
detector arc to the x-ray source. Define Dsd≜D0dþDs0 to be
the total distance from the x-ray source to the center of the
detector. This formulation allows the detector focal point to
differ from the x-ray source location to encompass a variety of
system configurations. For flat detectors, Dfs¼1. For third-
generation x-ray CT systems, Dfs¼0. For fourth-generation
x-ray CT systems, Dfs¼�Ds0.
In our notation, the distances D0d and Ds0 are constants,
rather than being functions of angle b. Generalizations exist toallow noncircular source trajectories (Besson, 1996).
Let s2 [�smax, smax] denote the (signed) arc length along
the detector, with s¼0 corresponding the detector center. Arc
length is a natural parameterization for detector elements that
are spaced equally along the detector. (For a flat detector
with Dfs¼1, the arc length s is simply the position along
the detector.) The various angles have the following
relationships:
a sð Þ ¼ s
Dfd, g sð Þ ¼ arctan
Dfd sin a sð ÞDfd cos a sð Þ �Dfs
� �[65]
where Dfd≜DfsþDsd. The two most important cases are
Figure 21 Left: (r, ’) coordinates for equiangular fan-beam samples based directly on [70]. The fan angle is 2gmax¼p/3. Right: after converting ’
to the range [0, p) using the periodicity property [17]. Top: for equally spaced samples in s symmetrical around s¼0. Bottom: for equally spacedsamples in s with quarter-detector offset. The samples for one particular value of b are circled for illustration.
Fundamentals of CT Reconstruction in 2D and 3D 283
360� rotation of the x-ray source and detector (Kak and Slaney,
1988, p. 77).
We start by rewriting the parallel-ray FBP formula [48] for
the case of 360� rotation:
f x; yð Þj j ¼ 1
2
ð2p0
ðp’ rð Þh∗ x cos’þ y sin’� rð Þdr d’ [78]
where h∗(�) denotes the ramp filter in [52] with 1D FT
H∗(n)¼ |n|.Now, change to fan-beam coordinates by making the trans-
formation of variables r¼ r(s), ’¼’(s, b), defined in [70]. The
Jacobian matrix is
@
@sr sð Þ @
@br sð Þ
@
@s’ s;bð Þ @
@b’ s;bð Þ
2664
3775 ¼ Ds0 cos g sð Þ � roff sin g sð Þ½ � _g sð Þ 0
_g sð Þ 1
� �
[79]
the determinant of which is
J sð Þ≜ Ds0 cos g sð Þ � roff sin g sð Þj jj _g sð Þj [80]
The reconstruction formula becomes
f x; yð Þj j ¼ 1
2
ð2p0
ðp s; bð Þh∗ x cos’ s; bð Þ þ y sin’ s; bð Þ � r sð Þð Þ J sð Þdsdb
¼ 1
2
ð2p0
ðp s; bð ÞJ sð Þh∗ x cos bþ g sð Þð Þ þ y sin bþ g sð Þð Þð
�Ds0sin g sð Þ � roff cos g sð ÞÞdsdb
Although this expression is a fan-beam reconstruction
formula, it is inconvenient for practical use; the challenge
is to manipulate it so that it has a FBP form to facilitate
implementation.
Using trigonometric identities, one can simplify the argu-
ment of h∗ in the preceding text as follows:
x cos bþ gð Þ þ y sin bþ gð Þ �Ds0sin g� roff cosg¼ Lb x; yð Þ sin gb x; yð Þ � g
Fundamentals of CT Reconstruction in 2D and 3D 291
Because of the shift invariance of (parallel-beam) projec-
tion and back projection, the primary tool for understanding
analytic image reconstruction methods is Fourier analysis. In
contrast, linear algebra, statistics, and optimization provide the
foundation for iterative methods.
Further acceleration of iterative algorithms for x-ray CT
image reconstruction will continue to be an active research
area because faster computation is essential for routine clinical
use. Moore’s law alone will not solve the computation problem
in iterative reconstruction because the problem sizes in CT
continue to grow due to higher angular sampling rates, the
use of multiple x-ray sources, and wider cone angles. More
accurate modeling of the x-ray physics, such as focal spot size
and detector fill fraction, will also increase computation
(La Riviere and Vargas, 2008). Model-based methods for
motion-compensated image reconstruction (MCIR) are also
computationally demanding (Fessler, 2010; Jacobson and
Fessler, 2003, 2006). MCIR is particularly challenging for non-
rigid motion (as expected in cardiac scans) and will remain an
important research area. Dynamic imaging problems (such as
perfusion CT) are also a significant challenge for iterative
reconstruction.
Dual-energy CT (DECT) has become available commer-
cially in recent years through various advances in technology.
There need to be more studies on iterative methods for DECT
image reconstruction to make full use of the energy informa-
tion with acceptable doses. The extension of DECT to spectral
CT measurements with more than two energy bins is also an
important future direction in image reconstruction research.
An important challenge in using iterative methods for
image reconstruction is that they typically are nonlinear,
which makes it difficult to analyze and predict the image
quality properties of the resulting images. There has been
some progress on analyzing the spatial resolution and noise
properties of nonquadratically regularized image reconstruc-
tion methods (Ahn and Leahy, 2008), but predicting perfor-
mance of detection tasks, particularly for unknown location
problems, remains a challenging problem (Zeng and
Myers, 2011).
For surveys of this active field, see De Man and Fessler
(2010), Fessler (2000), Hsieh et al. (2013), Nuyts et al.
(2013), and Peyrin and Douek (2013).
References
Abella M, Vaquero JJ, Soto-Montenegro ML, Lage E, and Desco M (2009) Sinogrambow-tie filtering in FBP PET reconstruction. Medical Physics 36(5): 1663–1671.
Abidi MA and Davis PB (1990) Radial noise filtering in positron emission tomography.Optical Engineering 29(5): 567–574.
Abramowitz M and Stegun IA (1964) Handbook of Mathematical Functions. New York:Dover.
Ahn S and Leahy RM (2008) Analysis of resolution and noise properties ofnonquadratically regularized image reconstruction methods for PET. IEEETransactions on Medical Imaging 27(3): 413–424.
Alessio A, Sauer K, and Kinahan P (2006) Analytical reconstruction of deconvolvedFourier rebinned PET sinograms. Physics in Medicine and Biology 51(1): 77–94.
Alliney S, Matej S, and Bajla I (1993) On the possibility of direct Fourier reconstructionfrom divergent-beam projections. IEEE Transactions on Medical Imaging 12(2):173–181.
Allmendiger HT, Kappler S, Bruder H, and Stierstorfer K (2012) Temporal resolution andmotion artifacts in dual-source cardiac CT and single-source CT with iterative
reconstruction. Proceedings of International Meeting on Image Formation in X-rayCT, 135–139.
Andersen AH and Kak AC (1984) Simultaneous algebraic reconstruction technique(SART): A superior implementation of the ART algorithm. Ultrasonic Imaging 6(1):81–94.
August J and Kanade T (2004) Scalable regularized tomography without repeatedprojections. Proceedings of International Parallel and Distributed ProcessingSymposium (IPDPS), 18: 3189–3196.
Averbuch A, Coifman RR, Donoho DL, Israeli M, and Shkolnisky Y (2008) A frameworkfor discrete integral transformations I – The pseudopolar Fourier transform. SIAMJournal on Scientific Computing 30(2): 764–784.
Averbuch A, Coifman RR, Donoho DL, Israeli M, Shkolnisky Y, and Sedelnikov I (2008)A framework for discrete integral transformations II – The 2D discrete Radontransform. SIAM Journal on Scientific Computing 30(2): 785–803.
Axel L, Herman GT, Roberts DA, and Dougherty L (1990) Linogram reconstructionfor magnetic resonance imaging (MRI). IEEE Transactions on Medical Imaging9(4): 447–449.
Bae KT and Whiting BR (2001) CT data storage reduction by means of compressingprojection data instead of images: Feasibility study. Radiology 219(3): 850–855.
Barrett HH and Myers KJ (2003) Foundations of Image Science. New York: Wiley.Basu S and Bresler Y (2000) O(n2log2n) filtered backprojection reconstruction
algorithm for tomography. IEEE Transactions on Image Processing 9(10):1760–1773.
Basu S and Bresler Y (2001) Error analysis and performance optimization of fasthierarchical backprojection algorithms. IEEE Transactions on Image Processing10(7): 1103–1117.
Bellon PL and Lanzavecchia S (1997) Fast direct Fourier methods, based on one- andtwo-pass coordinate transformations, yield accurate reconstructions of x-ray CTclinical images. Physics in Medicine and Biology 42(3): 443–464.
Besson G (1996) CT fan-beam parametrizations leading to shift-invariant filtering.Inverse Problems 12(6): 815–833.
Blondel C, Vaillant R, Malandain G, and Ayache N (2004) 3D tomographicreconstruction of coronary arteries using a precomputed 4D motion field. Physics inMedicine and Biology 49(11): 2197–2208.
Bracewell RN (1956) Strip integration in radio astronomy. Australian Journal of Physics9: 198–217.
Bracewell R (2000) The Fourier Transform and its Applications, 3rd edn. New York:McGraw-Hill.
Bracewell RN and Riddle AC (1967) Inversion of fan-beam scans in radio astronomy.The Astrophysical Journal 150: 427–434.
Brady SD (1993) New Mathematical Programming Approaches to the Problem of ImageReconstruction from Projections. PhD Thesis, Department of Operations Research,Stanford.
Brasse D, Kinahan PE, Clackdoyle R, Defrise M, Comtat C, and Townsend DW (2004)Fast fully 3-D image reconstruction in PET using planograms. IEEE Transactions onMedical Imaging 23(4): 413–425.
Censor Y (1983) Finite series expansion reconstruction methods. Proceedings of theIEEE 71(3): 409–419.
Chen G-H (2003) A new framework of image reconstruction from fan beam projections.Medical Physics 30(6): 1151–1161.
Chen G-H, Leng S, and Mistretta CA (2005) A novel extension of the parallel-beamprojection-slice theorem to divergent fan-beam and cone-beam projections.Medical Physics 32(3): 654–665.
Chen G-H, Tang J, and Leng S (2008) Prior image constrained compressed sensing(PICCS): A method to accurately reconstruct dynamic CT images from highlyundersampled projection data sets. Medical Physics 35(2): 660–663.
Cheung WK and Lewitt RM (1991) Modified Fourier reconstruction method usingshifted transform samples. Physics in Medicine and Biology 36(2): 269–277.
Choi H and Munson DC (1998) Direct-Fourier reconstruction in tomography andsynthetic aperture radar. International Journal of Imaging Systems and Technology9(1): 1–13.
Chu G and Tam KC (1977) Three-dimensional imaging in the positron camera usingFourier techniques. Physics in Medicine and Biology 22(2): 245–265.
Chun SY and Fessler JA (2013) Noise properties of motion-compensatedtomographic image reconstruction methods. IEEE Transactions on Medical Imaging32(2): 141–152.
Clackdoyle R and Noo F (2004) A large class of inversion formulae for the 2DRadon transform of functions of compact support. Inverse Problems 20(4):1281–1292.
Colsher JG (1980) Fully three dimensional positron emission tomography. Physics inMedicine and Biology 25(1): 103–115.
Cormack AM (1963) Representation of a function by its line integrals, with someradiological applications. Journal of Applied Physics 34(9): 2722–2727.
292 Fundamentals of CT Reconstruction in 2D and 3D
Courdurier M, Noo F, Defrise M, and Kudo H (2008) Solving the interior problem ofcomputed tomography using a priori knowledge. Inverse Problems 24(6): 065001.
Davison ME and Grunbaum FA (1981) Tomographic reconstruction with arbitrarydirections. Communications on Pure and Applied Mathematics 34(1): 77–119.
De Man B and Basu S (2002) A high-speed, low-artifact approach to projection andbackprojection. Proc. IEEE Nuclear Science Symp. and Medical Imaging Conf.,vol. 3, 1477–1480. http://dx.doi.org/10.1109/NSSMIC.2002.1239600.
De Man B and Basu S (2004) Distance-driven projection and backprojection in threedimensions. Physics in Medicine and Biology 49(11): 2463–2475.
De Man B and Fessler JA (2010) Statistical iterative reconstruction for X-raycomputed tomography. In: Censor MJ Yair and Wang G (eds.) BiomedicalMathematics: Promising Directions in Imaging, Therapy Planning and InverseProblems, pp. 113–140. Madison, WI: Medical Physics Publishing, ISBN:9781930524484.
De Rosier D and Klug A (1968) Reconstruction of three-dimensional structures fromelectron micrographs. Nature 217(5124): 130–138.
Deans SR (1983) The Radon Transform and Some of its Applications. New York: Wiley1993 2nd edn. by Krieger Publishing Co., Malabar, FL.
Defrise M, Noo F, Clackdoyle R, and Kudo H (2006) Truncated Hilbert transform andimage reconstruction from limited tomographic data. Inverse Problems 22(3):1037–1054.
Donohue KD and Saniie J (1989) A scanning and sampling scheme for computationallyefficient algorithms of computer tomography. IEEE Transactions on AcousticsSpeech and Signal Processing 37(3): 402–414.
Duerinckx AJ, Zatz LM, and Macovski A (1978) Non-linear smoothing filters andnoise structure in computed tomography (CT) scanning: a preliminary report.Recent & Future Developments in Medical Imaging, Proceedings of SPIE,vol. 152, 19–25.
Dusaussoy NJ (1996) VOIR: A volumetric image reconstruction algorithm based onFourier techniques for inversion of the 3-D Radon transform. IEEE Transactions onImage Processing 5(1): 121–131.
Edholm PR and Herman GT (1987) Linograms in image reconstruction fromprojections. IEEE Transactions on Medical Imaging 6(4): 301–307.
Edholm P, Herman GT, and Roberts DA (1988) Image reconstruction from linograms:Implementation and evaluation. IEEE Transactions onMedical Imaging 7(3): 239–246.
Elbakri IA and Fessler JA (2003) Efficient and accurate likelihood for iterative imagereconstruction in X-ray computed tomography. Medical Imaging 2003: ImageProcessing, Proceedings of SPIE, vol. 5032.
Erdogan H and Fessler JA (1999a) Monotonic algorithms for transmission tomography.IEEE Transactions on Medical Imaging 18(9): 801–814.
Erdogan H and Fessler JA (1999b) Ordered subsets algorithms for transmissiontomography. Physics in Medicine and Biology 44(11): 2835–2851.
Faridani A, Finch DV, Ritman EL, and Smith KT (1997) Local tomography II. SIAMJournal on Applied Mathematics 57(4): 1095–1127.
Faridani A and Ritman EL (2000) High-resolution computed tomography from efficientsampling. Inverse Problems 16(3): 635–650.
Faridani A, Ritman EL, and Smith KT (1992) Local tomography. SIAM Journal onApplied Mathematics 52(2): 459–484.
Feldkamp LA, Davis LC, and Kress JW (1984) Practical cone beam algorithm. Journal ofthe Optical Society of America A 1(6): 612–619.
Fessler JA (1993) Tomographic reconstruction using information weighted smoothingsplines. In: Barrett HH and Gmitro AF (eds.) Information Processing in MedicalImaging. Lecture Notes in Computer Science, vol. 687, pp. 372–386. Berlin:Springer.
Fessler JA (1994) Penalized weighted least-squares image reconstruction for positronemission tomography. IEEE Transactions on Medical Imaging 13(2): 290–300.
Fessler JA (1995) Resolution properties of regularized image reconstruction methods.Technical Report 297, Comm. and Sign. Proc. Lab., Dept. of EECS, Univ. ofMichigan, Ann Arbor, MI, 48109-2122.
Fessler JA (1996) Mean and variance of implicitly defined biased estimators (such aspenalized maximum likelihood): Applications to tomography. IEEE Transactions onImage Processing 5(3): 493–506.
Fessler JA (2000) Statistical image reconstruction methods for transmission tomography.In: Sonka M and Michael Fitzpatrick J (eds.) Medical Image Processing and Analysis.Handbook of Medical Imaging, vol. 2, pp. 1–70. Bellingham: SPIE.
Fessler JA (2010) Optimization transfer approach to joint registration/reconstruction formotion-compensated image reconstruction. Proceedings of IEEE InternationalSymposium on Biomedical Imaging, 596–599.
Fessler JA and Booth SD (1999) Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction. IEEE Transactions on Image Processing 8(5):688–699.
Fessler JA and Rogers WL (1996) Spatial resolution properties of penalized-likelihoodimage reconstruction methods: Space-invariant tomographs. IEEE Transactions onImage Processing 5(9): 1346–1358.
Fourmont K (2003) Non-equispaced fast Fourier transforms with applications totomography. Journal of Fourier Analysis and Applications 9(5): 431–450.
Gelfand IM and Graev MI (1991) Crofton function and inversion formulas in real integralgeometry. Functional Analysis and Its Applications 25(1): 1–5.
Gelfand IM and Shilov GE (1977) Generalized Functions. Properties and Operations,vol. 1. New York: Academic Press.
Glick SJ, Penney BC, King MA, and Byrne CL (1994) Noniterative compensation for thedistance-dependent detector response and photon attenuation in SPECT. IEEETransactions on Medical Imaging 13(2): 363–374.
Gordon R, Bender R, and Herman GT (1970) Algebraic reconstruction techniques (ART)for the three-dimensional electron microscopy and X-ray photography. Journal ofTheoretical Biology 29(3): 471–481.
Gottlieb D, Gustafsson B, and Forssen P (2000) On the direct Fouriermethod for computer tomography. IEEE Transactions on Medical Imaging 19(3):223–232.
Grafakos L (2004) Classical and Modern Fourier Analysis. New Jersey: Pearson.Grangeat P, Koenig A, Rodet T, and Bonnet S (2002) Theoretical framework for a
dynamic cone-beam reconstruction algorithm based on a dynamic particle model.Physics in Medicine and Biology 47(15): 2611–2626.
Grass M, Kohler T, and Proksa R (2000) 3D cone-beam CT reconstruction for circulartrajectories. Physics in Medicine and Biology 45(2): 329–348.
Grimmer R, Oelhafen M, Elstrom U, and Kachelriess M (2009) Cone-beam CT imagereconstruction with extended z range. Medical Physics 36(7): 3363–3370.
Guedon JV and Normand N (1997) Mojette transform: Applications for imageanalysis and coding. Visual Communications and Image Processing,Proceedings of SPIE, vol. 3024, 873–884.
Guedon JP and Normand N (2005) The Mojette transform: The first ten years.In: Andres E, et al. (ed.) Discrete Geometry for Computer Imagery. LNCS, vol. 3429,p. 79. Berlin, Germany: Springer.
Guedon J, Servieres M, Beaumont S, and Normand N (2004) Medical software controlquality using the 3D Mojette projector. Proceedings of IEEE InternationalSymposium on Biomedical Imaging 1: 836–839.
Gullberg GT (1979) The reconstruction of fan-beam data by filtering the back-projection. Computer Graphics and Image Processing 10(1): 30–47.
Gullberg GT, Crawford CR, and Tsui BMW (1986) Reconstruction algorithm for fanbeam with a displaced center-of-rotation. IEEE Transactions on MedicalImaging 5(1): 23–29.
Gullberg GT and Zeng GL (1995) Backprojection filtering for variable orbit fan-beamtomography. IEEE Transactions on Nuclear Science 42(4–1): 1257–1266.
Hebert TJ (1992) A union of deterministic and stochastic methods for imagereconstruction. Proceedings of IEEE Nuclear Science Symposium and MedicalImaging Conference 2: 1117–1119.
Hebert TJ and Gopal SS (1991) An improved filtered back-projection algorithm usingpre-processing. Proceedings of IEEE Nuclear Science Symposium and MedicalImaging Conference 3: 2068–2072.
Helgason S (1980) The Radon transform. Progress in Mathematics, vol. 5. Boston:Birkhauser.
Herman GT (1972) Two direct methods for reconstructing pictures from theirprojections: A comparative study. Computer Graphics and Image Processing 1(2):123–144.
Herman GT and Lent A (1976) Iterative reconstruction algorithms. Computers in Biologyand Medicine 6(4): 273–294.
Herman GT, Lent A, and Lutz PH (1978) Relaxation methods for image reconstruction.Communications of the ACM 21(2): 152–158.
Herman GT and Naparstek A (1977) Fast image reconstruction based on a Radoninversion formula appropriate for rapidly collected data. SIAM Journal on AppliedMathematics 33(3): 511–533.
Higgins WE and Munson DC (1988) A Hankel transform approach to tomographicimage reconstruction. IEEE Transactions on Medical Imaging 7(1): 59–72.
Hinkle J, Szegedi M, Wang B, Salter B, and Joshi S (2012) 4D CT imagereconstruction with diffeomorphic motion model. Medical Image Analysis 16(6):1307–1316.
Horbelt S, Liebling M, and Unser M (2002a) Discretization of the Radon transformand of its inverse by spline convolutions. IEEE Transactions on Medical Imaging21(4): 363–376.
Horbelt S, Liebling M, and Unser M (2002b) Filter design for filtered back-projectionguided by the interpolation model. Medical Imaging 2002: Image Processing,Proceedings of SPIE, vol. 4684, 806–813.
Hsieh J (1998) Adaptive streak artifact reduction in computed tomography resultingfrom excessive x-ray photon noise. Medical Physics 25(11): 2139–2147.
Fundamentals of CT Reconstruction in 2D and 3D 293
Hsieh J, Nett B, Yu Z, Sauer K, Thibault J-B, and Bouman CA (2013) Recent advances inCT image reconstruction. Current Radiology Reports 1(1): 39–51.
Hsieh J and Tang X (2006) Tilted cone-beam reconstruction with row-wise fan-to-parallel rebinning. Physics in Medicine and Biology 51(20): 5259–5276.
Huesman RH, Salmeron EM, and Baker JR (1989) Compensation for crystal penetrationin high resolution positron tomography. IEEE Transactions on Nuclear Science36(1): 1100–1107.
Hutchins GD, Rogers WL, Chiao P, Raylman R, and Murphy BW (1990) Constrainedleast-squares projection filtering in high resolution PET and SPECT imaging. IEEETransactions on Nuclear Science 37(2): 647–651.
Hutchins GD, Rogers WL, Clinthorne NH, Koeppe RA, and Hichwa RD (1987)Constrained least-squares projection filtering: A new method for the reconstructionof emission computed tomographic images. IEEE Transactions on Nuclear Science34(1): 379–383.
Isola AA, Grass M, and Niessen WJ (2010) Fully automatic nonrigid registration-basedlocal motion estimation for motion-corrected iterative cardiac CT reconstruction.Medical Physics 37(3): 1093–1109.
Jacobson MW and Fessler JA (2003) Joint estimation of image and deformationparameters in tomographic image reconstruction. IEEE Workshop on StatisticalSignal Processing 149–152.
Jacobson MW and Fessler JA (2006) Joint estimation of respiratory motion and activityin 4D PET using CT side information. Proceedings of IEEE International Symposiumon Biomedical Imaging, 275–278.
Joseph PM and Spital RD (1981) The exponential edge-gradient effect in x-raycomputed tomography. Physics in Medicine and Biology 26(3): 473–487.
Kachelrieß M, Watzke O, and Kalender WA (2001) Generalized multi-dimensionaladaptive filtering for conventional and spiral single-slice, multi-slice, and cone-beam CT. Medical Physics 28(4): 475–490.
Kak AC and Slaney M (1988) Principles of Computerized Tomographic Imaging.New York: IEEE Press.
Kao C-M and Pan X (2000) Non-iterative methods incorporating a priori sourcedistribution and data information for suppression of image noise and artefacts in 3DSPECT. Physics in Medicine and Biology 45(10): 2801–2819.
Kao C-M, Pan X, and Chen C-T (2000) Accurate image reconstruction using DOIinformation and its implications for the development of compact PET systems. IEEETransactions on Nuclear Science 47(4–2): 1551–1560.
Kao C-M, Wernick MN, and Chen C-T (1998) Kalman sinogram restoration for fast andaccurate PET image reconstruction. IEEE Transactions on Nuclear Science 45(6):3022–3029.
Karp JS, Muehllehner G, and Lewitt RM (1988) Constrained Fourier space method forcompensation of missing data in emission computed tomography. IEEETransactions on Medical Imaging 7(1): 21–25.
Karuta B and Lecomte R (1992) Effect of detector weighting functions on the pointspread function of high-resolution PET tomographs. IEEE Transactions on MedicalImaging 11(3): 379–385.
Katsevich AI (1997) Local tomography for the generalized Radon transform. SIAMJournal on Applied Mathematics 57(4): 1128–1162.
Katsevich A (2002) Theoretically exact filtered backprojection-type inversion algorithmfor spiral CT. SIAM Journal on Applied Mathematics 62(6): 2012–2026.
Katsevich A (2006) Improved cone beam local tomography. Inverse Problems 22(2):627–644.
Katsevich A (2008) Motion compensated local tomography. Inverse Problems 24(4):045012.
Katsevich A, Silver M, and Zamyatin A (2011) Local tomography and the motionestimation problem. SIAM Journal on Imaging Sciences 4(1): 200–219.
Katz M (1978) Questions of uniqueness and resolution in reconstruction fromprojections. Lecture Notes in Biomathematics, vol. 26. Springer.
Kazantsev IG (1998) Tomographic reconstruction from arbitrary directions using ridgefunctions. Inverse Problems 14(3): 635–646.
Kazantsev IG, Matej S, and Lewitt RM (2004) System and gram matrices of 3-Dplanogram data. IEEE Transactions on Nuclear Science 51(5): 2579–2587.
Kazantsev IG, Matej S, and Lewitt RM (2006) Inversion of 2D planogramdata for finite-length detectors. IEEE Transactions on Nuclear Science 53(1):160–166.
Kim D, Pal D, Thibault J-B, and Fessler JA (2013) Accelerating ordered subsets imagereconstruction for X-ray CT using spatially non-uniform optimization transfer. IEEETransactions on Medical Imaging 32(11): 1965–1978.
Kim D, Ramani S, and Fessler JA (2013) Ordered subsets with momentum foraccelerated X-ray CT image reconstruction. Proceedings of IEEE Transactions onAcoustics Speech and Signal Processing, 920–923.
Kinahan P, Clackdoyle R, and Townsend D (1999) Fully-3D image reconstruction usingplanograms. Radiology and Nuclear Medicine, Proceedings of International Meetingon Fully 3D Image Reconstruction, 329–332.
King MA, Doherty PW, Schwinger RB, and Penney BC (1983) A Wiener filter for nuclearmedicine images. Medical Physics 10(6): 876–880.
Kudo H, Courdurier M, Noo F, and Defrise M (2008) Tiny a priori knowledge solves theinterior problem in computed tomography. Physics in Medicine & Biology 53(9):2207–2232.
La Riviere PJ, Bian J, and Vargas PA (2006) Penalized-likelihood sinogram restorationfor computed tomography. IEEE Transactions on Medical Imaging 25(8):1022–1036.
La Riviere PJ and Billmire DM (2005) Reduction of noise-induced streakartifacts in X-ray computed tomography through spline-basedpenalized-likelihood sinogram smoothing. IEEE Transactions on Medical Imaging24(1): 105–111.
La Riviere PJ and Pan X (2000) Nonparametric regression sinogram smoothing using aroughness-penalized Poisson likelihood objective function. IEEE Transactions onMedical Imaging 19(8): 773–786.
La Riviere PJ, Pan X, and Penney BC (1998) Ideal-observer analysis of lesiondetectability in planar, conventional SPECT, and dedicated SPECTscintimammography using effective multi-dimensional smoothing. IEEETransactions on Nuclear Science 45(3): 1273–1279.
La Riviere PJ and Vargas P (2008) Correction for resolution nonuniformities caused byanode angulation in computed tomography. IEEE Transactions on Medical Imaging27(9): 1333–1341.
Lakshminarayanan AV (1975) Reconstruction from divergent ray data. Technical Report92. Buffalo, NY: Department of Computer Science, State University of New York.
Lange K and Carson R (1984) EM reconstruction algorithms for emission andtransmission tomography. Journal of Computer Assisted Tomography 8(2):306–316.
Lanzavecchia S and Bellon PL (1997) The moving window Shannon reconstruction indirect and Fourier domain: Application in tomography. Scanning Microscopy11: 155–170.
Lewitt RM (1990) Multidimensional digital image representations using generalizedKaiser-Bessel window functions. Journal of the Optical Society of America A 7(10):1834–1846.
Lewitt RM, Edholm PR, and Xia W (1989) Fourier method for correction of depth-dependent collimator blurring. Medical Imaging III: Image Processing, Proceedingsof SPIE, vol. 1092, 232–243.
Liang Z (1994) Detector response restoration in image reconstruction of high resolutionpositron emission tomography. IEEE Transactions on Medical Imaging 13(2):314–321.
Logan BF and Shepp LA (1975) Optimal reconstruction of a function from itsprojections. Duke Mathematical Journal 42(4): 645–659.
Long Y, Fessler JA, and Balter JM (2010) 3D forward and back-projection for X-ray CTusing separable footprints. IEEE Transactions on Medical Imaging 29(11):1839–1850.
Louis AK and Rieder A (1989) Incomplete data problems in X-ray computerizedtomography II. Truncated projections and region-of-interest tomography.Numerische Mathematik 56(4): 371–383.
Ma J, Liang Z, Fan Y, et al. (2012) Variance analysis of x-ray CT sinogramsin the presence of electronic noise background. Medical Physics 39(7):4051–4065.
Maass C, Knaup M, Lapp R, Karolczak M, Kalender WA, and Kachelriess M (2008)A new weighting function to achieve high temporal resolution in circular cone-beamCT with shifted detectors. Medical Physics 35(12): 5898–5909.
Macovski A (1983) Medical Imaging Systems. New Jersey: Prentice-Hall.Magnusson M (1993) Linogram and Other Direct Fourier Methods for Tomographic
Reconstruction. PhD Thesis, Linkoping, Sweden.Marzetta TL and Shepp LA (1999) A surprising Radon transform result and its
application to motion detection. IEEE Transactions on Image Processing 8(8):1039–1049.
Matej S and Bajla I (1990) A high-speed reconstruction from projections using directFourier method with optimized parameters-an experimental analysis. IEEETransactions on Medical Imaging 9(4): 421–429.
Matej S, Fessler JA, and Kazantsev IG (2004) Iterative tomographic imagereconstruction using Fourier-based forward and back-projectors. IEEE Transactionson Medical Imaging 23(4): 401–412.
McGaffin MG, Ramani S, and Fessler JA (2012) Reduced memory augmentedLagrangian algorithm for 3D iterative X-ray CT image reconstruction. MedicalImaging 2012: Physics of Medical Imaging, Proceedings of SPIE, vol. 8313,p. 831327.
Mersereau RM (1974) Recovering multidimensional signals from their projections.Computer Graphics and Image Processing 1(2): 179–185.
294 Fundamentals of CT Reconstruction in 2D and 3D
Mersereau RM (1976) Direct Fourier transform techniques in 3-D image reconstruction.Computers in Biology and Medicine 6(4): 247–258.
Mersereau RM and Oppenheim AV (1974) Digital reconstruction ofmultidimensional signals from their projections. Proceedings of the IEEE62(10): 1319–1338.
Natterer F (1985) Fourier reconstruction in tomography. Numerische Mathematik 47(3):343–353.
Natterer F (1986) The Mathematics of Computerized Tomography. Stuttgart: Teubner-Wiley.
Natterer F and Wubbeling F (2001) Mathematical Methods in Image Reconstruction.Philadelphia: Society for Industrial and Applied Mathematics.
Nien H and Fessler JA (2013) Splitting-based statistical X-ray CT image reconstructionwith blind gain correction. Medical Imaging 2013: Physics of Medical Imaging,Proceedings of SPIE, vol. 8668, p. 86681J.
Noo F, Clackdoyle R, and Pack JD (2004) A two-step Hilbert transformmethod for 2D image reconstruction. Physics in Medicine and Biology 49(17):3903–3924.
Noo F, Defrise M, and Clackdoyle R (1999) Single-slice rebinning method for helicalcone-beam CT. Physics in Medicine and Biology 44(2): 561–570.
Noo F, Defrise M, Clackdoyle R, and Kudo H (2002) Image reconstruction fromfan-beam projections on less than a short scan. Physics in Medicine and Biology47(14): 2525–2546.
Nowak RD and Baraniuk RG (1999) Wavelet-domain filtering for photon imagingsystems. IEEE Transactions on Image Processing 8(5): 666–678.
Nuyts J, De Man B, Fessler JA, Zbijewski W, and Beekman FJ (2013) Modelling thephysics in iterative reconstruction for transmission computed tomography. Physicsin Medicine and Biology 58(12): R63–R96.
Nuyts J, Dupoint P, Schieper C, and Mortelmans L (1994) Efficient storage of thedetection probability matrix for reconstruction in PET. Journal of Nuclear Medicine(Abs. Book) 35(5): 187.
O’Sullivan JD (1985) A fast sinc function gridding algorithm for Fourierinversion in computer tomography. IEEE Transactions on Medical Imaging 4(4):200–207.
Olson T and DeStefano J (1994) Wavelet localization of the Radon transform. IEEETransactions on Signal Processing 42(8): 2055–2067.
Oppenheim BE (1974) More accurate algorithms for iterative 3-dimensionalreconstruction. IEEE Transactions on Nuclear Science 21(3): 72–77.
Pan X (1999) Optimal noise control in and fast reconstruction of fan-beam computedtomography image. Medical Physics 26(5): 689–697.
Pan X, Metz CE, and Chen C-T (1996) A class of analytical methods that compensate forattenuation and spatially-variant resolution in 2D SPECT. IEEE Transactions onNuclear Science 43(4–1): 2244–2254.
Pan X, Xia D, Zou Y, and Yu L (2004) A unified analysis of FBP-based algorithms inhelical cone-beam and circular cone- and fan-beam scans. Physics in Medicine andBiology 49(18): 4349–4370.
Parker DL (1982) Optimal short scan convolution reconstruction for fan beam CT.Medical Physics 9(2): 254–257.
Pawitan Y, Bettinardi V, and Teras M (2005) Non-gaussian smoothing of low-counttransmission scans for PET whole-body studies. IEEE Transactions on MedicalImaging 24(1): 122–129.
Penczek PA, Renka R, and Schomberg H (2004) Gridding-based direct Fourierinversion of the three-dimensional ray transform. Journal of the Optical Society ofAmerica A 21(4): 499–509.
Penney BC, King MA, Schwinger RB, Baker SP, and Doherty PW (1988) Modifyingconstrained least-squares restoration for application to SPECT projection images.Medical Physics 15(3): 334–342.
Peyrin F and Douek P (2013) X-ray tomography. In: Fanet H (ed.) Photon-BasedMedical Imagery. Hoboken, NJ: Wiley.
Potts D and Steidl G (2001) A new linogram algorithm for computerized tomography.IMA Journal of Numerical Analysis 21(3): 769–782.
Qu G, Wang C, and Jiang M (2009) Necessary and sufficient convergence conditions foralgebraic image reconstruction algorithms. IEEE Transactions on Image Processing18(2): 435–440.
Radon J (1917) On the determination of functions from their integrals along certainmanifold. Berichte Sachs. Akad. Wiss. (Leipzig) 69: 262–278 Uber dieBestimmung von Funktionen durch ihre Intergralwerte Langs gewisserManningfultigkeiten.
Radon J (1986) On the determination of functions from their integral values alongcertain manifolds. IEEE Transactions on Medical Imaging 5(4): 170–176.
Ramachandran GN and Lakshminarayanan AV (1971) Three-dimensional reconstructionfrom radiographs and electron micrographs: Application of convolutions instead ofFourier transforms. Proceedings of the National Academy of Sciences 68(9):2236–2240.
Ramani S and Fessler JA (2012) A splitting-based iterative algorithm for acceleratedstatistical X-ray CT reconstruction. IEEE Transactions on Medical Imaging 31(3):677–688.
Ramm AG and Katsevich AI (1996) The Radon Transform and Local Tomography. BocaRaton: CRC Press.
Rashid-Farrokhi F, Liu KJR, Berenstein CA, and Walnut D (1997) Wavelet-basedmultiresolution local tomography. IEEE Transactions on Image Processing 6(10):1412–1430.
Riviere PJL and Pan X (1999) Few-view tomography using roughness-penalizednonparametric regression and periodic spline interpolation. IEEE Transactions onNuclear Science 46(4–2): 1121–1128.
Sauer K and Bouman C (1993) A local update strategy for iterative reconstruction fromprojections. IEEE Transactions on Signal Processing 41(2): 534–548.
Sauer K and Liu B (1991) Nonstationary filtering of transmission tomograms in highphoton counting noise. IEEE Transactions on Medical Imaging 10(3): 445–452.
Schmidlin P (1994) Improved iterative image reconstruction using variable projectionbinning and abbreviated convolution. European Journal of Nuclear Medicine 21(9):930–936.
Schomberg H and Timmer J (1995) The gridding method for image reconstruction byFourier transformation. IEEE Transactions on Medical Imaging 14(3): 596–607.
Seger MM (1998) Three-dimensional reconstruction from cone-beam data using anefficient Fourier technique combined with a special interpolation filter. Physics inMedicine and Biology 43(4): 951–960.
Servieres MCJ, Normand N, Subirats P, and Guedon J (2004) Some links betweencontinuous and discrete Radon transform. Medical Imaging 2004: ImageProcessing, Proceedings of SPIE, vol. 5370, 1961–1971.
Shepp LA and Logan BF (1974) The Fourier reconstruction of a head section. IEEETransactions on Nuclear Science 21(3): 21–43.
Smith KT and Keinert F (1985) Mathematical foundations of computed tomography.Applied Optics 24(23): 3950–3957.
Smith PR, Peters TM, and Bates RHT (1973) Image reconstruction from finite numbersof projections. Journal of Physics A: Mathematical, Nuclear and General 6(3):361–382.
Snyder DL, Hammoud AM, and White RL (1993) Image recovery from data acquiredwith a charge-coupled-device camera. Journal of the Optical Society of America A10(5): 1014–1023.
Snyder DL, Helstrom CW, Lanterman AD, Faisal M, and White RL (1995) Compensationfor readout noise in CCD images. Journal of the Optical Society of America A 12(2):272–283.
Soumekh M (1986) Image reconstruction techniques in tomographic imaging systems.IEEE Transactions on Acoustics Speech and Signal Processing 34(4): 952–962.
Spyra WJT, Faridani A, Smith KT, and Ritman EL (1990) Computed tomographicimaging of the coronary arterial tree-use of local tomography. IEEE Transactions onMedical Imaging 9(1): 1–4.
Stark H, Woods JW, Paul I, and Hingorani R (1981a) An investigation of computerizedtomography by direct Fourier inversion and optimum interpolation. IEEETransactions on Biomedical Engineering 28(7): 496–505.
Stark H, Woods JW, Paul I, and Hingorani R (1981b) Direct Fourier reconstruction incomputer tomography. IEEE Transactions on Acoustics Speech and SignalProcessing 29(2): 237–244.
Stierstorfer K, Rauscher A, Boese J, Bruder H, Schaller S, and Flohr T (2004) WeightedFBP-A simple approximate 3D FBP algorithm for multislice spiral CT with gooddose usage for arbitrary pitch. Physics in Medicine and Biology 49(11):2209–2218.
Subirats P, Servieres M, Normand N, and Guedon J (2004) Angular assessment of theMojette filtered back projection. Medical Imaging 2004: Image Processing,Proceedings of SPIE, vol. 5370, 1951–1960.
Sweeney DW and Vest CM (1973) Reconstruction of three-dimensional refractive indexfields from multidirectional interferometric data. Applied Optics 12(11): 2649.
Tabei M and Ueda M (1992) Backprojection by upsampled Fourier series expansion andinterpolated FFT. IEEE Transactions on Image Processing 1(1): 77–87.
Thibault J-B, Bouman CA, Sauer KD, and Hsieh J (2006) A recursive filter for noisereduction in statistical iterative tomographic imaging. Computational Imaging IV,Proceedings of SPIE, vol. 6065, p. 60650X.
Thibault J-B, Sauer K, Bouman C, and Hsieh J (2007) A three-dimensional statisticalapproach to improved image quality for multi-slice helical CT. Medical Physics34(11): 4526–4544.
Thibault J-B, Yu Z, Sauer K, Bouman C, and Hsieh J (2007) Correction of gainfluctuations in iterative tomographic image reconstruction. Radiology and NuclearMedicine, Proceedings of International Meeting on Fully 3D Image Reconstruction,112–115.
Tsui ET and Budinger TF (1979) A stochastic filter for transverse section reconstruction.IEEE Transactions on Nuclear Science 26(2): 2687–2690.
Fundamentals of CT Reconstruction in 2D and 3D 295
Tuy HK (1983) An inversion formula for cone-beam reconstruction. SIAM Journal onApplied Mathematics 43(3): 546–552.
Unser M, Thevenaz P, and Yaroslavsky L (1995) Convolution-based interpolation forfast, high quality rotation of images. IEEE Transactions on Image Processing 4(10):1371–1381.
Vainberg E, Kazak E, and Kurczaev V (1981) Reconstruction of the internal three-dimensional structure of objects based on real-time integral projections. SovietJournal of Nondestructive Testing 17: 415–423.
van Stevendaal U, von Berg J, Lorenz C, and Grass M (2008) A motion-compensatedscheme for helical cone-beam reconstruction in cardiac CT angiography. MedicalPhysics 35(7): 3239–3251.
Villain N, Goussard Y, Idier J, and Allain M (2003) Three-dimensional edge-preservingimage enhancement for computed tomography. IEEE Transactions on MedicalImaging 22(10): 1275–1287.
Walden J (2000) Analysis of the direct Fourier method for computer tomography. IEEETransactions on Medical Imaging 19(3): 211–222.
Wang G, Yu H, and Ye Y (2009) A scheme for multisource interior tomography. MedicalPhysics 36(8): 3575–3581.
Wernick MN and Chen CT (1992) Superresolved tomography by convex projectionsand detector motion. Journal of the Optical Society of America A 9(9): 1547–1553.
Whiting BR (2002) Signal statistics in x-ray computed tomography. Medical Imaging2002: Medical Physics, Proceedings of SPIE, vol. 4682, 53–60.
Whiting BR, Massoumzadeh P, Earl OA, O’Sullivan JA, Snyder DL, and Williamson JF(2006) Properties of preprocessed sinogram data in x-ray computed tomography.Medical Physics 33(9): 3290–3303.
Xia W, Lewitt RM, and Edholm PR (1995) Fourier correction for spatially variantcollimator blurring in SPECT. IEEE Transactions on Medical Imaging 14(1):100–115.
Xia D, Zou Y, and Pan X (2006) Reconstructions from parallel- and fan-beam data withtruncations. Medical Imaging 2006: Physics of Medical Imaging, Proceedings ofSPIE, vol. 6142, p. 614222.
Xu Q, Mou X, Wang G, Sieren J, Hoffman EA, and Yu H (2011) Statistical interiortomography. IEEE Transactions on Medical Imaging 30(5): 1116–1128.
Xu B, Pan X, and Chen C-T (1998) An innovative method to compensate for distance-dependent blurring in 2D SPECT. IEEE Transactions on Nuclear Science 45(4):2245–2251.
Xu J and Tsui BMW (2007) A compound Poisson maximum-likelihood iterativereconstruction algorithm for X-ray CT. Radiology and Nuclear Medicine,Proceedings of International Meeting on Fully 3D Image Reconstruction,108–111.
Xu J and Tsui BMW (2009) Electronic noise modeling in statistical iterativereconstruction. IEEE Transactions on Image Processing 18(6): 1228–1238.
Yang J, Yu H, Jiang M, and Wang G (2010) High-order total variation minimization forinterior tomography. Inverse Problems 26(3): 035013.
You J and Zeng GL (2007) Hilbert transform based FBP algorithm for fan-beam CT fulland partial scans. IEEE Transactions on Medical Imaging 26(2): 190–199.
Yu DF and Fessler JA (2000) Mean and variance of singles photon counting withdeadtime. Physics in Medicine and Biology 45(7): 2043–2056.
Yu DF and Fessler JA (2002) Mean and variance of coincidence photon counting withdeadtime. Nuclear Instruments and Methods in Physics Research Section A488(1–2): 362–374.
Yu H and Wang G (2009) Compressed sensing based interior tomography. Physics inMedicine and Biology 54(9): 2791–2806.
Zamyatin AA and Nakanishi S (2007) Extension of the reconstruction field of view andtruncation correction using sinogram decomposition. Medical Physics 34(5):1593–1604.
Zeng GL (2004) Nonuniform noise propagation by using the ramp filter in fan-beamcomputed tomography. IEEE Transactions on Medical Imaging 23(6): 690–695.
Zeng R and Myers KJ (2011) Task-based comparative study of iterative imagereconstruction methods for limited-angle x-ray tomography. Medical Imaging 2011:Physics of Medical Imaging, Proceedings of SPIE, vol. 7961, p. 796137.
Zhang F, Bi G, and Chen YQ (2002) Tomography time-frequency transform. IEEETransactions on Signal Processing 50(6): 1289–1297.