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arXiv:0808.3510v2 [math.NA] 29 Apr 2009
A Reconstruction Algorithm for Photoacoustic
Imaging based on the Nonuniform FFT
Markus Haltmeier, Otmar Scherzer, and Gerhard Zangerl.∗†
April 29, 2009
Abstract
Fourier reconstruction algorithms significantly outperform conventional
backprojection algorithms in terms of computation time. In photoacous
tic imaging, these methods require interpolation in the Fourier space do
main, which creates artifacts in reconstructed images. We propose a novel
reconstruction algorithm that applies the onedimensional nonuniform fast
Fourier transform to photoacoustic imaging. It is shown theoretically and
numerically that our algorithm avoids artifacts while preserving the com
putational effectiveness of Fourier reconstruction.
Key–words. Image reconstruction, photoacoustic imaging, planar mea
surement geometry, fast algorithm, nonuniform FFT.
1 Introduction
Photoacoustic imaging (PAI) is a novel promising tool for visualizing light ab
sorbing structures in an optically scattering medium, which carry valuable in
formation for medical diagnostics. It is based on the generation of acoustic
waves by illuminating an object with pulses of nonionizing electromagnetic ra
diation, and combines the high contrast of pure optical and the high resolution
of ultrasonic imaging. The method has demonstrated great promise for a vari
ety of biomedical applications, such as imaging of animals [1, 2], early cancer
diagnostics [3, 4], and imaging of vasculature [5, 6].
When an object is illuminated with short pulses of nonionizing electromag
netic radiation, it absorbs a fraction of energy and heats up. This in turn induces
acoustic (pressure) waves, that are recorded with acoustic detectors outside of
the object. Other than in conventional ultrasound imaging, where the source of
acoustic waves is an external transducer, in PAI the source is the imaged object
∗M. Haltmeier, O. Scherzer, and G. Zangerl are with the Department of Mathematics, Uni
versity Innsbruck, Technikerstr. 21a, 6020 Innsbruck, Austria, email: {markus.haltmeier,
otmar.scherzer, gerhard.zangerl}@uibk.ac.at.
†O. Scherzer is also with the Radon Institute of Computational and Applied Mathematics,
Altenberger Str. 69, 4040 Linz, Austria
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itself. The frequency bandwidth of the recorded signals is therefore generally
broad and depends on the size and the shape of illuminated structures.
1.1Planar recording geometry
Throughout this paper we assume a planar recording geometry, where the acous
tic signals are recorded with omnidirectional detectors arranged on planes (or
lines), see Figure 1. The planar geometry is of particular interest since it can
be realized most easily in practical applications. The recorded acoustic sig
nals are then used to reconstruct the initially generated acoustic pressure which
represents optically absorbing structures of the investigated object.
detector array
optical illumination
object
acoustic wave
Figure 1: Photoacoustic imaging for planar recording geometry. The
object is illuminated by a pulse of electromagnetic radiation, and reacts with
an expansion. Induced acoustic waves are measured with an array of acoustic
detectors arranged on a plane (or a line) and used to from an image of the
object.
For the planar recording geometry, two types of theoretically exact recon
struction formulas have been reported: Temporal backprojection [7, 8, 9, 10]
and Fourier domain formulas [11, 7, 9, 12, 13, 14, 15]. Numerical implemen
tations of those formulas often lead to fast and accurate image reconstruction
algorithms.
In temporal backprojection formulas the signals measured at time t are back
projected over spheres of radius vst with the detector position in the centre (vs
denotes the speed of sound). In Fourier domain formulas this backprojection is
performed by interpolation in the frequency domain. Reconstruction methods
based on Fourier domain formulas are attractive since they reconstruct an N ×
N × N image in O(N3logN) floating point operations by using of the Fast
Fourier Transform (FFT). Straightforward implementations of backprojection
type formulas, on the other hand, require O(N5) operation counts, see [16, 17].
The standard FFT algorithm assumes sampling on an equally spaced grid
and therefore, in order to implement the Fourier domain formulas, interpolation
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in the Fourier space is required. Interpolation in the Fourier domain is a critical
issue, and creates artifacts in reconstructed images, see the examples in Section
5. One obtains significantly better results by increasing the sampling density
in the Fourier space. This is achieved by either zeropadding [13] or by sym
metrizing the recorded signals around t = 0 (which is equivalent to using the
fast cosine transform instead of the FFT). In this paper, we propose an efficient
reconstruction algorithm that uses the nonuniform (or unevenly spaced) FFT
[18, 19, 20, 21, 22, 23] and further increases the quality of reconstruction.
1.2 Prior work and innovations
The nonuniform FFT has been applied to a variety of medical imaging prob
lems, such as standard Xray CT, magnetic resonance imaging, and diffraction
tomography [24, 25, 26], and has also been used implicitly in gridding algo
rithms [27, 28]. All those algorithms deal with the problem of recovering a two
(or higher) dimensional object function from samples of its multidimensional
Fourier transform on a noncartesian grid.
Our approach is conceptually quite different to the above mentioned ref
erences: The special structure of our problem allows to perform several one
dimensional nonuniform FFTs instead of a single higher dimensional one. This
leads to a reduced numerical cost, compared to the above algorithms. The pro
posed algorithm is more closely related to a reconstruction algorithm for Xray
CT suggested in [29, Section 5.2], which also evaluates the Fourier transform on
irregular samples by means of the onedimensional FFT.
1.3 Outline
This article is organized as follows: In Section 2 we present the mathematical
basics of Fourier reconstruction in PAI. In Section 3 we review the nonuniform
FFT which is then used to derive the nonuniform FFT based reconstruction
algorithm in Section 4. In Section 5 we present numerical results of the proposed
algorithm and compare it with existing Fourier and back projection algorithms.
The paper concludes with a discussion of some issues related to sampling and
resolution in the Appendix.
2 Photoacoustic Imaging
Let C∞
half space H := Rd−1× (0,∞), d ≥ 2. Consider the initial value problem
?∂2
p(x,0) = f(x),
∂tp(x,0) = 0,
0(H) denote the space of smooth functions with bounded support in the
t− ∆?p(x,t) = 0,(x,t) ∈ Rd× (0,∞),
x ∈ Rd,
x ∈ Rd,
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with f ∈ C∞
with respect to t. We write x = (x,y), x ∈ Rd−1, y ∈ R, and define the operator
Q : C∞
0(H). Here ∆ denotes the Laplacian in Rdand ∂tis the derivative
0(H) → C∞(Rd) by
(Qf)(x,t) :=
?
p(x,y = 0,t),
0,
if t > 0,
otherwise.
Photoacoustic imaging for planar recording geometry is concerned with recon
structing f ∈ C∞
Of practical interest are the cases d = 2 and d = 3, see [30, 31, 32, 33].
0(H) from incomplete and possibly erroneous knowledge of Qf.
2.1Exact inversion formula
The operator Q can be inverted analytically by means of the exact inversion
formula
(Ff)(Kx,Ky) =
2Ky
?FQf??
Kx,sign(Ky)
?
Kx2+ K2
y
?
sign(Ky)
?
Kx2+ K2
y
(1)
where (Kx,Ky) ∈ Rd−1×R, and F denotes the ddimensional Fourier transform,
?
Equation (1) has been derived in [12, 13] for three spatial dimensions. It can be
proven in any dimension by using the inversion formula for the spherical mean
Radon transform of [7, 9]. A related formula using the Fourier cosine transform
instead of the Fourier transform has been obtained in [34, 15] for d = 2,3.
(Fϕ)(K) :=
Rde−iKxϕ(x) dx,
K = (Kx,Ky) ∈ Rd.
2.2Partial Fourier reconstruction
The inversion formula (1) yields an exact reconstruction of f, provided that
(Qf)(x,t) is given for all (x,t) ∈ Rd. In practical applications only a partial
(or limited view) data set is available [35, 36, 37]. In this paper we assume
that data (Qf)(x,t) are given only for (x,t) ∈ (0,X)d, see Figure 1, which are
modeled by
g(x,t) := wcut(x,t)(Qf)(x,t),(2)
where wcutis a smooth nonnegative cutoff function that vanishes outside (0,X)d.
Using data (2), the function f cannot be exactly reconstructed in a stable way
(see [38, 37]). It is therefore common to apply the exact inverse of Q to the
partial data g and to consider the result as an approximation of the object to
be reconstructed. More precisely, the function f†defined by
(Ff†)(Kx,Ky) :=
2Ky(Fg)
?
Kx,sign(Ky)
?
Kx2+ K2
y
?
sign(Ky)
?
Kx2+ K2
y
,(3)
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is considered an approximation of f. The function f†is called partial Fourier
reconstruction.
Fourier reconstruction algorithms in PAI name numerical implementations
of (3). In this paper we apply the onedimensional nonuniform FFT to derive a
fast and accurate algorithm for implementing (3).
3The Nonuniform Fast Fourier Transform
The discrete Fourier transform of a vector g = (gn)N−1
the nodes ω = (ωk)N/2−1
n=0∈ CNwith respect to
k=−N/2(with N even) is defined by
T[g](ωk) :=
N−1
?
n=0
e−iωkn2π/Ngn,k = −N/2,...,N/2 − 1. (4)
Direct evaluation of the N sums in (4) requires O(N2) operations. Using the
classical fast Fourier transform (FFT) this effort can be reduced to O(N logN)
operations. However, application of the classical FFT is restricted to the case
of evenly spaced nodes ωk= k, k = −N/2,...,N/2 − 1.
The onedimensional nonuniform FFT (see [18, 19, 20, 29, 21, 22, 23]) is an
approximate but highly accurate method for evaluating (4) at arbitrary nodes
ωk, k = −N/2,...,N/2 − 1, in O(N logN) operations.
3.1Derivation of the nonuniform FFT
To derive the nonuniform FFT we closely follow the presentation of [29], which
is based on the following lemma:
Lemma 1 ([29, Proposition 1]). Let c > 1 and α < π(2c − 1). Assume that
Ψ : R → R is continuous in [−α,α], vanishing outside [−α,α], and positive in
[−π,π]. Then
e−iωθ=
2πΨ(θ)
j∈Z
c
?
ˆΨ(ω − j/c)e−ijθ/c,ω ∈ R,θ ≤ π .(5)
HereˆΨ(ω) :=
of Ψ.
?
Re−iωθΨ(θ)dθ denotes the onedimensional Fourier transform
Proposition 2. Let c, α, Ψ, andˆΨ be as in Lemma 1. Then, for every g =
(gn)N−1
n=0∈ CNand ω ∈ R we have
N−1
?
with
n=0
e−iωn2π/Ngn=
?
j∈Z
e−iπ(ω−j/c)ˆΨ(ω − j/c)ˆGj,(6)
ˆGj:=
c
2π
?N−1
n=0
?
gne−ijn2π/(Nc)
Ψ(n2π/N − π)
?
,j ∈ Z. (7)
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Proof. Taking θ = n2π/N − π ∈ [−π,π] in (5), gives
c
2πΨ(n2π/N − π)
e−iωn2π/N=
?
j∈Z
ˆΨ(ω − j/c)e−ijn2π/(cN)e−iπ(ω−j/c),
and therefore
N−1
?
n=0
e−iωn2π/Ngn=
c
2π
N−1
?
n=0
?
j∈Z
e−iπ(ω−j/c)ˆΨ(ω − j/c)gne−ijn2π/(cN)
Ψ(n2π/N − π).
Interchanging the order of summation in the right hand side of the above equa
tion shows (6), (7) and concludes the proof.
In the following we assume that cN is an even number. Then
ˆGj=
c
2π
?cN−1
n=0
?
gn
Ψ(n2π/N − π)e−ijn2π/(Nc)
?
,j ∈ Z,(8)
where gn:= 0 for n ≥ N, is an oversampled discrete Fourier transform with the
oversampling factor c. Moreover we assume thatˆΨ is concentrated around zero
and decays rapidly away from zero. The nonuniform FFT uses the formulas (6),
(8) to evaluate T[g] at the nodes ωk. The basic steps of the algorithm are as
follows:
(i) Append (c − 1)N zeros to the vector g = (gn)N−1
−Nc/2,...,Nc/2 − 1, in (8) with the FFT algorithm.
(ii) Evaluate the sums in (6) approximately by using only the terms with
ωk− j/c ≤ K, where the interpolation length K is a small positive
parameter.
n=0and evaluateˆGj, j =
SinceˆΨ is assumed to decay rapidly, the truncation error in Step (ii) is small.
The nonuniform Fourier transform is summarized in Algorithm 1. All evalu
ations of Ψ andˆΨ are precomputed and stored. Moreover, the classical FFT is
applied to a vector of length cN. Therefore the numerical complexity of Algo
rithm 1 is O(cN logN). Typically c = 2, in which case the numerical effort of
the nonuniform FFT is essentially twice the effort of the onedimensional clas
sical FFT applied to an input vector of the same length. See [29, Section 3] for
an exact operation count, and a comparison between actual computation times
of the classical and the nonuniform FFT.
3.2Kaiser Bessel window
In our implementation we choose for Ψ the Kaiser Bessel window,
Ψα,K
KB(θ) :=
1
I0(αK)
?
I0(K√α2− θ2),
0,
if θ ≤ α,
if θ > α.
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Algorithm 1 Nonuniform FFT with respect to the nodes ω = (ωk)N/2−1
using input vector g = (gn)N−1
and window function Ψ.
k=−N/2,
n=0, oversampling c > 1, interpolation length K,
1: Ψ ←?Ψ(2πn/N − π)?
3:
4: function nufft(g,ω,c,K,Ψ,ˆΨ)
5:
g ← g/Ψ · c/(2π)
6:
g ←?g,zeros(1,(c − 1)N?
8:
for k = −N/2,...,N/2 − 1 do
9:
ˆ gk←?
11:
return (ˆ gk)k
12: end function
n
⊲ precomputations
2:ˆΨ ←?e−i(ωk−j/c)π/cˆΨ(ωk− j/c)?
k,j
⊲ zeropadding
7:
g ← fft(g)⊲ onedimensional FFT
j−cωk≤cKˆΨk,jgj
⊲ interpolation
10:
end for
Here I0 is the modified Bessel function of order zero. The onedimensional
Fourier transform of Ψα,K
KBis
ˆΨα,K
KB(ω) = 2sinh(α
?
K2− ω2)/?I0(αK)
?
K2− ω2?,
if ω ∈ R \ {−K,K} and 2α/(I0(αK)) otherwise.
The Kaiser Bessel window is a good and often used candidate for Ψ, since
ˆΨα,K
KB(ω) becomes extremely small for ω ≥ K. For example, with the parame
ters K = 3, and α = 3π, we have for ω ≥ K,
ˆΨα,K
KB(ω)/ˆΨα,K
KB(0) ≤ ˆΨα,K
KB(K)/ˆΨα,K
KB(0) ≃ 3 ∗ 10−11.
Remark 3. Take c = 1 and let Ψ be the characteristic function of the interval
[−π,π]. ThenˆΨ(ω) = 2π sinc(πω) and (6), (7) reduce to the sinc series
N−1
?
which is a discretized version of Shannon’s sampling formula [39, 40]
n=0
e−iωn2π/Ngn=
?
j∈Z
e−iπ(ω−j)sinc(ω − j)
?N−1
n=0
?
gne−ijn2π/N
?
,
ˆ g(ω) =
?
j∈Z
e−iπ(ω−j)sinc(ω − j)ˆ g(j),ω ∈ R,
applied to the Fourier transform of a function g : R → R that vanishes outside
[0,2π].
See Figure 2 for a comparison of sinc andˆΨα,K
One realizes thatˆΨα,K
KBdecays much faster than sinc and is therefore much better
suited for truncated interpolation. In fact, ˆΨα,K
ω ≥ 3, whereas sinc(ω) < 0.01 only for ω ≥ 100/π.
KB, with K = 3 and α = 3π.
KB(ω)/ˆΨα,K
KB(0) < 3 ∗ 10−11for
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64 2
spatial?variable θ
0246
0
0.2
0.4
0.6
0.8
1
amplitude
ΨKB
α,?K(θ)
χ[ π, π](θ)
10 505 10
160
140
120
100
80
60
40
20
0
frequency?variable ω
amplitude?(dB)
ΨKB
^
α,?K(ω)
sinc?(ω)
Figure 2: Left: KaiserBessel window Ψα,K
the interval [−π,π]. Right: Fourier transformsˆΨα,K
(decibel). Here dB denotes the logarithmic decay 10log10(φ(ω)/φ(0)) of some
quantity φ(ω).
KB(θ) and characteristic function of
KB(ω) and 2πsinc(ω) in dB
An error estimate for the nonuniform FFT using the Kaiser Bessel window
is given in [29]. The result is
???e−iωθ−
c
2πΨ(θ)
?
ω−j/c<K
ˆΨα,K
KB(ω − j/c)e−ijθ/c???
≤
30
πI0
?Kπ√α2− 1/c2? .
For example, taking c = 2, α = 3π and K = 3, the above error is as small as
3 ∗ 10−8.
4 A Fourier reconstruction Algorithm based on
the nonuniform FFT
In this section we apply the onedimensional nonuniform FFT to photoacoustic
imaging. Throughout the following we restrict our attention to two dimensions,
noting that the general case d ≥ 2 can be treated in an analogous manner.
Assume that f is a smooth function that vanishes outside (0,X)2, and set
g := wcutQf, where wcutis as in (2). Fourier reconstruction names an imple
mentation of (3), that uses discrete data
gm,n:= g(m∆samp,n∆samp), (9)
with (m,n) ∈ {0,...,N − 1}2and reconstructs an approximation
fm,n≃ f†(m∆samp,n∆samp),
with (m,n) ∈ {0,...,N − 1}2. Here f†is defined by (3), N is an even number
and ∆samp:= X/N. In the Appendix we show that the sampling in (10), (9) is
(10)
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sufficiently fine, provided that ∆samp≤ π/Ω, where Ω is the essential bandwidth
of f.
Discretizing (3) with the trapezoidal rule gives
N−1
?
n=0
?N−1
m=0
?
e−i(ln+km)2π/Nfm,n
?
=
2k
ωk,l
N−1
?
n=0
e−iωk,ln2π/N
?N−1
m=0
?
e−ikm2π/Ngm,n
?
, (11)
where
ωk,l:= sign(l)
?
k2+ l2,(k,l) ∈ {−N/2,...,N/2 − 1}2.
One notices that the inner sums in (11),
˜ gk,n:=
N−1
?
m=0
e−ikm2π/Ngm,n
(12)
can be exactly evaluated with N onedimensional FFTs, and the outer sums
ˆ gk(ωk,l) :=
N−1
?
n=0
e−iωk,ln2π/N˜ gk,n
(13)
can be approximately evaluated with N onedimensional nonuniform FFTs. De
noting the resulting approximation by ˆ gk,l≃ ˆ gk(ωk,l) and setting
ˆfk,l:=2k ˆ gk,l
ωk,l
,(k,l) ∈ {−N/2,...,N/2 − 1}2,(14)
we finally find
fn,m:=
1
N2
N/2−1
?
k,l=N/2
ei(km+ln)2π/Nˆfk,l
(15)
with the inverse twodimensional FFT algorithm.
The nonuniform FFT based reconstruction algorithm is summarized in Al
gorithm 2. Its numerical complexity can easily be estimated. Evaluating (12)
requires NO(N logN) operations (N onedimensional FFTs), evaluating (13)
requires NO(N logN) operations (N nonuniform FFTs), and (15) is evaluated
with the inverse twodimensional FFT in O(N2logN) operations. Therefore
the overall complexity of Algorithm 1 is O(N2logN).
In the next section we numerically compare Algorithm 2 with standard
Fourier algorithms presented in the literature [41, 13], which all differ in the
way how the sums in (13) are evaluated:
1. Direct Fourier algorithm. Equation (13) cannot be evaluated with
the classical FFT algorithm because the nodes ωk,l are nonequispaced.
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Algorithm 2 Nonuniform FFT based algorithm for calculating f = (fm,n)N−1
using data g = (gm,n)N−1
window function Ψ.
n,m=0
m,n=0, oversampling factor c, interpolation length K, and
1: Ψ ←?Ψ(2πn/N − π)?
3:
4: function FouRecNufft(g,c,K,Ψ,ˆΨ)
5:
for n = 0,...,N − 1 do
6:
h ← (gm,n)m
7:
(˜ gk,n)k← fft(h)
8:
end for
l ← (−N/2,...,N/2 − 1)
10:
for k = −N/2,...,N/2 − 1 do
11:
h ← nufft(h,ω,c,K,Ψ,ˆΨ)
13:
(fk,l)l← 2kh/ω
14:
end for
f ← (fk,l)k,l
16:
f ← ifft2(f)
17:
return f
18: end function
n
⊲ precomputations
2:ˆΨ ←?e−i(ωk−j/c)π/cˆΨ(ωk− j/c)?
k,j
⊲ onedimensional FFT
9:
ω ← sign(l)√k2+ l2
12:
⊲ nonuniform FFT
15:
⊲ twodimensional inverse FFT
The most simple way to evaluate (13) is with direct summation. Because
there are N2such sums, direct Fourier reconstruction requires O(N3)
operations. Consequently it does not lead to a fast algorithm. However,
since (13) is evaluated exactly, it is optimally suited to evaluate the image
quality of reconstructions with fast Fourier algorithms.
2. Interpolation based algorithm. A fast and simple alternative to direct
Fourier reconstruction is as follows: Choose an oversampling factor c ≥ 1
and exactly evaluate
ˆ gk(ω) := ∆samp
N−1
?
n=0
e−iωn2π/N˜ gk,n,
at the uniformly spaced nodes ω = ∆sampj/c, j ∈ {0,...,Nc − 1}, with
the onedimensional FFT algorithm. In a next step, linear interpolation
is used to find approximate values ˆ gk,l ≃ ˆ gk(ωk,l), see [13]. Evaluating
ˆ gk,lwith linear interpolation requires O(N2) operation and therefore the
overall numerical effort of linear interpolation based Fourier reconstruction
is O(N2logN).
Algorithms using nearest neighbor interpolation instead of the linear one
have the same numerical complexity and have also been applied to PAI
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(see, e.g., [42]). Higher order polynomial interpolation has been applied
in [43] for a cubic recording geometry.
3. Truncated sinc reconstruction. If the function Ψ in Algorithm 2 is
chosen as the characteristic function of the interval [−cπ,cπ], c ≥ 1, then
the nonuniform fast Fourier transform reduces to the truncated sinc in
terpolation considered in [41]. However, due to the slow decay of sinc(ω),
truncation will introduce a nonnegligible error in the reconstructed image
(see Remark 3).
The Fourier algorithms are also be compared with a numerical implementa
tion of the backprojection formula
f(x,y) = −2y
π
?
R
??∞
r
(∂tt−1Qf)(x′,t)
√t2− r2
dt
?
dx′, (16)
where r =
(x′,0) and the reconstruction point (x,y). Equation (16) has been obtained
in [8] by applying the method of descent to the threedimensional universal
backprojection formula discovered by Xu and Wang [10]. Again (16) gives an
exact reconstruction only if it is applied to complete data (Qf)(x,t), (x,t) ∈
R2. In the numerical experiments the backprojection formula is applied to the
partial data wcutQf, see (2), and implemented with O(N3) operation counts
as described in [8, Section 3.3].
?(x − x′)2+ y2denotes the distance between the detector location
5Numerical Results
In the following we numerically compare the proposed nonuniform FFT based
algorithm with standard Fourier algorithm and the back projection algorithm
based on (16).
The cutoff function wcutis constructed by convolution of
ϕǫ(x,t) =
?
Cǫexp?−1/(ǫ − x2− t2)4?,
0,
if x2+ t2< ǫ,
otherwise ,
with the characteristic function of [0,X]2, where ǫ is a small parameter and Cǫ
is chosen in such a way that
?
In all numerical experiments we take X = 1, and N = 512. The win
dow width α is chosen to be slightly smaller than π(2c − 1), where c is the
oversampling factor that determines the accuracy of the Fourier reconstruction
algorithms.
R2ϕǫ(x,t)dxdt = 1. Typically, ǫ is chosen as a
“small” multiple of the sampling step size ∆samp= X/N.
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fcirc
Qfcirc
nearest, c = 1linear, c = 1
nearest, c = 2linear, c = 2
Figure 3: Reconstruction with interpolation based Fourier algorithms.
White corresponds to function value 1, black to function value 0.4. Top Line:
Phantom and analytic data. Middle line: Reconstruction without oversampling
(c = 1). Bottom line: Reconstruction with oversampling (c = 2).
5.1Circular shaped object
As first case example we use a circular shaped object
fcirc(x) =2
a
??a2− x − x02?1/2, if x − x0 < a,
otherwise,0,
centered at x0:= (x0,y0) (see top left image in Figure 3). For such a simple
object reconstruction artifacts can be identified very clearly. Moreover, the data
Qfcirccan be evaluated analytically (see [8, Equation (B.1)]) as
(Qfcirc)(x,t) =1
aRe
?
(s+− s−) − tlog
?s++ (t + ai)
s−+ (t − ai)
??
.
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back projectiondirect
truncated sinc, c = 2nonuniform FFT, c = 2
truncated sinc, c = 2 nonuniform FFT, c = 2
Figure 4: Improved reconstructions. Top Line: Back projection (left) and
direct reconstruction (right). Middle Line: Truncated sinc (left) and nonuni
form FFT based Fourier algorithm (right). Here white corresponds to function
value 1, black to function value 0.4. Bottom Line: Difference images between
direct and truncated sinc reconstruction (left), and direct and nonuniform FFT
based reconstruction (right). Here white (resp. black) corresponds to function
value 0.04 (resp. −0.04).
Here s± := ((t ± a)2+ (x,0) − x02)1/2, log(·) is the principal branch of the
complex logarithm, and Re[z] denotes the real part of complex number z. The
reconstruction results are depicted in Figures 3 and 4. Table 1 and Figure 5
compare run times with the relative ℓ2error
?f − f†?ℓ2
?f†?ℓ2
=
??
m,n(fm,n− f†
??
13
m,n)2?1/2
m,n)2?1/2
m,n(f†
,
Page 14
02468
−2.5
−2
−1.5
−1
−0.5
0
linear
nearest
Kaiser Bessel
reconstruction time
log10
??f − f†?ℓ2??f†?ℓ2?
Figure 5: Reconstruction time versus error. The points on the graphs
belong to runtimes and errors for reconstruction with oversampling factors c ∈
{1,2,4,8,16,32}.
were f†= (f†
struction. Run times were measured for Matlab implementations on a personal
computer with 2.4 GHz Athlon processor.
In order to demonstrate the stability of the Fourier algorithms, we also
performed reconstructions from noisy data, where Gaussian noise was added
with a variance equal to 20% of the maximal data value. The reconstruction
results are depicted in Figure 7.
m,n) denotes the discrete image obtained by direct Fourier recon
cℓ2error runtime (sec)
back projection
direct reconstruction
nearest neighbor
nearest neighbor
linear
linear
Truncates sinc
Kaiser Bessel


1
2
1
2
2
2


88.9
54.1
0.65
0.85
0.8
0.95
1.6
1.6
0.75
0.40
0.65
0.21
0.04
0.006
Table 1: Run times and error of different reconstruction methods.
14
Page 15
Qfcirc
0 100200300400500
−0.2
−0.1
0
0.1
0.2
0.3
exact
noisy
Figure 6: Noisy data used in Figure 7
.
5.2Shepp–Logan phantom
In the next example we consider the Shepp–Logan phantom fphant, which is
shown in top left image in Figure 8. The data were calculated numerically by
implementing d’Alemberts formula [44],
(Qfphant)(x,0,t) = ∂t
?t
0
r(Mfphant)(x,0,r)
√t2− r2
dr
with
(Mfphant)(x,0,r) :=
1
2π
?
σ=1
fphant
?(x,0) + rσ?dσ
denoting the spherical mean Radon transform of fcirc. The reconstruction re
sults from simulated data are depicted in Figure 8.
5.3Discussion
We emphasize that none of the above Fourier algorithms are designed to cal
culate an approximation of f but an approximation to the partial Fourier re
construction f†defined in (3). Therefore even in the direct reconstruction (top
right image in Figure 4) and in the back projection reconstruction one can see
some blurred boundaries in the reconstruction. Such artifacts are expected using
limited view data (2), see [38, 37].
The results of interpolation based reconstruction without oversampling (c =
1) are quite useless. The reconstructions are significantly improved by using a
larger oversampling factor c. However, even then, the results never reach the
quality of the nonuniform FFT based reconstruction. Moreover, the numerical
effort of linear interpolation based reconstruction is proportional to the over
sampling factor, which prohibits the use of “very large” values for c (see Figure
5). In the reconstruction with c = 2 (bottom line in Figure 3 and middle line
in Figure 8) artifacts are still clearly visible.
The images in the middle line of Figure 4 suggest that truncated sinc and
nonuniform FFT based reconstruction seem to perform quite similar. However,
15