Spatial distortion correction and crystal identification for MRIcompatible positionsensitive avalanche photodiodebased PET scanners.
ABSTRACT Positionsensitive avalanche photodiodes (PSAPDs) are gaining widespread acceptance in modern PET scanner designs, and owing to their relative insensitivity to magnetic fields, especially in those that are MRIcompatible. Flood histograms in PET scanners are used to determine the crystal of annihilation photon interaction and hence, for detector characterization and routine quality control. For PET detectors that use PSAPDs, flood histograms show a characteristic pincushion distortion when Anger logic is used for event positioning. A small rotation in the flood histogram is also observed when the detectors are placed in a magnetic field. We first present a general purpose automatic method for spatial distortion correction for flood histograms of PSAPDbased PET detectors when placed both inside and outside a MRI scanner. Analytical formulae derived for this scheme are based on a hybrid approach that combines desirable properties from two existing event positioning schemes. The rotation of the flood histogram due to the magnetic field is determined iteratively and is accounted for in the scheme. We then provide implementation details of a method for crystal identification we have previously proposed and evaluate it for cases when the PET detectors are both outside and in a magnetic field. In this scheme, Fourier analysis is used to generate a lowerorder spatial approximation of the distortioncorrected PSAPD flood histogram, which we call the 'template'. The template is then registered to the flood histogram using a diffeomorphic iterative intensitybased warping scheme. The calculated deformation field is then applied to the segmentation of the template to obtain a segmentation of the flood histogram. A manual correction tool is also developed for exceptional cases. We present a quantitative assessment of the proposed distortion correction scheme and crystal identification method against conventional methods. Our results indicate that our proposed methods lead to a large reduction in manual labor and indeed can routinely be used for calibration and characterization studies in MRIcompatible PET scanners based on PSAPDs.

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Page 1
2008 IEEE Nuclear Science Symposium Conference Record
Spatial distortion correction and crystal
identification for positionsensitive avalanche
photodiodebased PET scanners
Abhijit J. Chaudhari, Member, IEEE, Anand A. Joshi, Member, IEEE, Yibao Wu, Member, IEEE,
Richard M. Leahy, Fellow, IEEE, Simon R. Cherry, Fellow, IEEE, and Ramsey D. Badawi, Member, IEEE
M10298
Fig. 1.
PET scanner and flood histograms; (a) the PET detector module with an array
of 8 x 8 LSO crystals coupled via optical fibers to a single 14 x 14 mm2
PSAPD (reproduced from [4D, (b) flood histogram obtained for the detector
when placed outside the 7T MRI scanner's magnetic field, (c) flood histogram
obtained for the same detector when placed inside the MRI scanner's magnetic
field
A single PET detector module from the UC Davis MRIcompatible
I. INTRODUCTION
M
simultaneouslyacquired images of morphology, function and
metabolic activity are expected to have a huge positive impact
on both preclinical as well as clinical imaging fields [1][3].
At DC Davis, a preclinical MRIcompatible PET scanner has
been built and has been characterized [1], [4]. This merger
of MRI and PET became possible through the use of PET
detectors in which photomultiplier tubes (PMTs) are replaced
by magnetic fieldinsensitive positionsensitive avalanche pho
todiodes (PSAPDs). PET signals can be measured in these
detectors with minimal distortion even when they are placed
inside the bore of a MRI scanner [4]. Since positionsensitive
APDs (or PSAPDs) can read out a large number of scintillator
crystals simultaneously, they also help reduce the electronic
complexity of a PET system [5]. A photograph ofa single PET
detector module from this system is shown in Fig. l(a), where
a (8 x 8) array of polished (1.43 x 1.43 x 6) mm3Lutetium
Orthosilicate (LSO) crystals is coupled to a single (14 x 14)
mm2PSAPD via optical fibers. Sixteen such detectors make
up the system, amounting to a total of 1024 LSO crystals.
More details about this scanner are in [1], [4].
Flood histograms are twodimensional probabilistic maps
RIcompatible Positron Emission Tomography (PET)
scannersthatproduceanatomicallycoregistered
AbstractPositionsensitive avalanche photodiodes (PSAPDs)
are gaining widespread acceptance in modern PET scanner
designs, and owing to their relative insensitivity to magnetic fields,
especially in those that are MRIcompatible. Flood histograms in
PET scanners are used to determine the crystal of annihilation
photon interaction and hence, for detector characterization and
routine quality control. For PET detectors that use PSAPDs, flood
histograms show a characteristic pincushion distortion when
Anger logic is used for event positioning. A small rotation in the
flood histogram is also observed when the detectors are placed
in a magnetic field. We first present a general purpose automatic
method for spatial distortion correction for flood histograms of
PSAPDbased PET detectors when placed both inside and outside
a MRI scanner. Analytical formulae derived for this scheme are
based on a hybrid approach that combines desirable properties
from two existing event positioning schemes. The rotation of
the flood histogram due to the magnetic field is determined
iteratively and is accounted for in the scheme. We then provide
implementation details of a method for crystal identification
we have previously proposed and evaluate it for cases when
the PET scanner is both outside and in a magnetic field. In
this scheme, Fourier analysis is used to generate a lowerorder
spatial approximation of the distortioncorrected PSAPD flood
histogram, which we call the 'template'. The template is then
registered to the flood histogram using a diffeomorphic iterative
intensitybased warping scheme. The calculated deformation field
is then applied to the segmentation of the template to obtain a
segmentation of the flood histogram. A manual correction tool is
also developed for exceptional cases. We present a quantitative
assessment of the proposed distortion correction scheme and
crystal identification method against conventional methods. Our
results indicate that our proposed methods lead to a large
reduction in manual labor and indeed can routinely be used
for calibration and characterization studies in MRIcompatible
PET scanners based on PSAPDs.
Index TermsPETIMRI, PSAPD, spatial distortion correction,
crystal identification
A. 1. Chaudhari, Y. Wu and S. R. Cherry are with the Department of
Biomedical Engineering, University of CaliforniaDavis, Davis, CA 95616,
USA (Email: ajchaudhari@ucdavis.edu).
A. A. Joshi was with the Signal and Image Processing Institute, University
of Southern California, Los Angeles, CA 90089, USA. He currently is with
the Laboratory of Neuro Imaging, University of CaliforniaLos Angeles, Los
Angeles, CA 90095, USA
R. M. Leahy is with the Signal and Image Processing Institute, University
of Southern California, Los Angeles, CA 90089, USA.
R. D. Badawi is with the Department of Radiology, UC Davis Medical
Center, Sacramento, CA 95817, USA.
8188rfayof
LSO Q'YStMt
141 14mm2
~
(b)
(c)
9781424427154/08/$25.00 ©2008 IEEE
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Page 2
generated using the four output signals from the PSAPD when
the PET detector is irradiated with an annihilation photon flood
source. Flood histograms obtained for a single PET detector
module placed outside and in the magnetic field of a 7T
small animal MRI scanner are shown in Fig. 1(b)(c). These
histograms were obtained using Anger logic [6]. Three spatial
effects are prominent in the flood histograms, (i) an asymmetry
about the vertical axis due to the different curvatures of the
optical fibers more apparent in Fig. 1(b), but also present in
and Fig. I(c), (ii) a pincushion distortion resulting from using
Anger logic for event positioning visible in both Fig. I(b) and
(c), and (iii) a rotation of the histogram when the detector
is placed in a magnetic field owing to the Hall effect [7],
visible in Fig. 1(c). The distortions caused due to the optical
fiber curvatures may be corrected by appropriately scaling the
corresponding signals that undergo larger attenuation. New
positioning formulae for reducing the pincushioning effect
in flood histograms of generic PSAPDbased detectors were
proposed by Zhang et al. [8]. They showed results from the ap
plication of their formulae to detectors that employed (8 x 8)
mm2PSAPDs. However, when these formulae were used for
the aforementioned detector module that uses a (14 x 14)
mm2PSAPD, pincushioning was overestimated leading to
a barreltype spatial distortion. This barrel effect may be
attributed to variability in the resistive and capacitive networks
that underlie the PSAPD chip [9]. Adequate control over
the barrel and pincushioning effects is desirable to minimize
distortions in the flood histogram.
The rotation of the flood histogram in a magnetic field, as
seen in Fig. 1(c), occurs only for those PSAPDs that have
faces at right angles to the static magnetic field direction [4].
This rotation is either clockwise or anticlockwise depending
on whether the device face forms an angle of +900or 900
with the static magnetic field vector respectively. The rotation
angle in either case was determined to be a constant and is a
function of the static magnetic field alone [4]. Different MRI
sequences have minimal impact on it. Hence, once determined,
this rotation angle may be reused for the PET detectors in a
MRI scanner with the same field strength assuming identical
geometrical placement of the PET scanner.
Reduction of spatial distortion is desirable for automated
crystal identification, which in tum, is necessary for detailed
characterization ofPET detectors, as well as for routine quality
control of PET scanners. Crystal identification can be posed as
a segmentation problem where one requires a segmentation of
the flood histogram into regions equal to the total number of
scintillator crystals in the detector array, such that each region
has one peak. Existing segmentation schemes are derived from
a broad range of image processing and pattern recognition
techniques. The relatively straightforward but the most time
consuming scheme is to manually click on peak locations
on a computer screen and then use a watershed method for
segmenting the individual regions [10]. This method is labor
intensive and hence, impractical for modem PET scanners that
typically have thousands of crystals. A semiautomatic scheme
involving thresholding the flood histogram to automatically
identifying peaks followed by watershedbased segmentation
has been proposed [11]. However, because of its dependence
on intensities of individual crystals in the flood histogram, this
method produces inaccurate results especially in cases where
crystals in the same scintillator array have large efficiency vari
ations. Sophisticated methods based on selforganizing maps,
multilevel neural networks, wavelets, and Weiner filtering
have been developed [12][15]. However, these methods are
primarily designed for specific scanners the investigators are
developing.
We previously have developed a distortion correction
scheme for PET detectors based on PSAPDs [16]. In this
scheme, adaptive event positioning formulae were derived
using those proposed by Anger [6] and Zhang [8]. These
proposed formulae result in reduced pincushion or barrel
distortions. However, additional compensation is needed in the
case of our PETIMRI detectors to account for the asymmetry
in the flood histograms caused due to the curvatures of the
optical fibers and for the flood histogram rotation in mag
netic field. For the distortioncorrected flood histogram, we
previously have developed a general purpose semiautomatic
segmentation scheme based on Fourier space analysis [16].
In this scheme, we first obtain a template image that exploits
the spatial frequency information in the given flood histogram.
This template image can be segmented simply with horizontal
and vertical lines drawn midway between adjacent peaks in the
histogram. A diffeomorphic polynomialbased scheme that is
capable of iteratively minimizing intensity differences is then
used to register the template to the given flood histogram.
The estimated warping field is applied to the segmentation of
the template resulting in the segmentation of the given flood
histogram.
In this paper, we first present modified adaptive formulae
for event positioning that provide a corrective mechanism for
asymmetry and rotation of the flood histogram in addition to
accounting for pincushion or barrel distortions. The optimal
parameters that lead to the least distortion in each case are
determined iteratively and automatically. Further, to facilitate
implementation by others, we provide specific implementation
details of our previously proposed segmentation scheme. We
then evaluate quantitatively the performance of both the spatial
distortion correction scheme and the segmentation method
for PET detectors from the DC Davis MRIcompatible PET
scanner. We show results for cases when the PET detector is
both outside and in the magnetic field of a 7T MRI scanner.
II. MATERIALS AND METHODS
A. PETdata measurement inside and outside the MRI scanner
The PET scanner was first locked in position after inserting
it into the bore of a Biospec 7T MRI scanner (Broker BioSpin
Corporation, Billerica, MA, USA). A cylindrical phantom
(internal diameter = 5 cm, length =4 cm) was filled with
14.8 MBq (400 /LCi) of 18FDG solution and was placed
into the field of view of the PET scanner. Five sets of data
were acquired in singles mode when the static magnetic field
was turned off. Each measurement lasted 5 min. The static
magnetic field was then switched on and the data acquisition
process was repeated. The data acquisition system consisted
of NIM electronics and PowerDAQ PD2MFS boards (United
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Page 3
(a)
(b)
(c)
TABLE I
TABLE OF FLOOD HISTOGRAM FORMULAE
Zhang's method:
Xb = ~ + g
Yb = ~ + g
cos (7r/4) + ~ + g
sin (7r/4),
sin (7r/4) 
~ + g
cos (7r/4).
Adaptive method:
Xc = aXa + (1  a)Xb' where a E [0,1],
Yc = aYa + (1 Q)Yb•
In magnetic field:
Xd = Xc cos (B) + Ycsin (0),
Yd= Xc sin (0) + Yc cos (B).
2) Formulae for generating flood histograms: Anger's and
Zhang's event positioning formulae as a function of output
signals A, B, C, and D are shown in row 1 and row 2 of
Fig. 2.
(a), (b) and (c) are for the PET detector when placed outside the 7T magnetic
field, and (d), (e) and (f) are when the detector is placed in the magnetic field.
(a) and (d) use Anger's equations, (b) and (e) use Zhang's method, and (c)
and (e) use the proposed adaptive formulae with a = 0.7 and B = 0°. All
formulae are listed in in Table I
Flood histograms after correction for effects of curved optical fibers;
Industries Inc., Boston, MA, USA) synchronized with an Intel
Pentium4 Multiprocessor PC [17]. Since the static magnetic
field alone was known to be the major contributor to signal
distortion [4], no MRI sequence was used. The temperature
of the PSAPDs was maintained at 1DoC throughout the
experiment. The energy window used was 350650 keV.
B. Flood histogram generation.
Flood histograms for the detector module are generated in
two steps; (i) the output signals A, B, C, and D from the
PSAPDs are preprocessed to compensate for effects due to
optical fiber bending, and (ii) event positioning formulae are
used to generate the probabilistic maps for measured events.
c. Segmentation method
We perform the segmentation of the distortion corrected
flood histogram in three steps; (i) intensity compensation in
the flood histogram, (ii) generation of a template image and
its segmentation, and (iii) registration of the template to the
flood histogram. Using the warping field computed in step
(iii), the segmentation of the template can be transformed to
the coordinates of the flood histogram, hence segmenting the
flood histogram. The detailed procedure is outlined below. We
demonstrate the procedure on the flood histogram shown in
Fig. 2(c).
For our discussion, let h(x, y) represent the flood histogram
with 0 ~
x ~
M  1 and 0 ~
spatial frequencies corresponding to x and y by I x and I y
where 0 ~ Ix ~
Discrete Fourier Transform (DFT) pair h(x, y) ~
y ~
N  1. We denote the
M  1 and 0 ~ Iy ~
N  1. Thus, the 2D
H(lx, Iy)
table I respectively. The corresponding flood histograms in
the absence of the static magnetic field are shown in Fig.
2(a) and (b) respectively. When the detectors are placed in
the magnetic field, the flood histograms obtained are shown in
Fig. 2(d) and (e). The previously proposed adaptive formulae
are given in table I, row 3 [16]. The parameter
adaptive formulae is chosen iteratively to maintain a balance
between the pincushioning and barrel effects for each device.
This flexibility in the choice of Q allows for compensating
effects due to small changes in the resistive and capacitive
networks in PSAPDs. However, once Q is chosen for a single
PSAPD or PSAPDs manufactured using identical processes,
it may not require modification throughout the lifetime of the
devices assuming stable operation. Details about choosing ()'
are in Section IIE. With Q = 0.7 in the adaptive formulae, we
obtain the flood histograms shown in Fig. 2(c) and (f) in the
absence and presence of magnetic field respectively. To further
minimize spatial distortion in the flood histograms of detectors
in magnetic field, the rotation angle needs to be estimated. An
iterative scheme to automatically estimate () is described in
Section IIE. In row 4 of table I, formulae that compensate
for the rotation of the flood histogram due to magnetic field
are presented. Results after this compensation is applied will
be shown in Section IIIA.
Q in the
(f)
(e)(d)
1) Preprocessing: The preprocessing step primarily com
pensates for the different curvatures of the optical fibers used
in the PET detector. This effect causes asymmetry along the
central vertical axis in the flood histogram clearly visible in
Fig. 1(b). Since this is a systematic effect, we correct it by
appropriately scaling the four output signals ofthe PSAPD. We
empirically found that a scaling of 1.3 for signals Band D and
1 for signals A and D using the Anger's formulae produced
a flood histogram that looks approximately symmetrical about
the central horizontal and vertical axis. The resultant flood
histograms outside and inside the MRI scanner after this
compensation are shown in Fig. 2(a) and (d). The proposed
scaling factors were found to be consistent for all modules of
the scanner and were applied only for the purpose of flood
histogram generation and crystal lookup and have no impact
on energy computations.
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Page 4
(a)
(b)
(c)
(a)
(b)
(c)
Fig. 3.
for comparison, (b) the smoothed image Ik(x, y)I, and (c) the intensity
compensated image p(x, y)
Intensity compensation; (a) the flood histogram from Fig. 2(c)
Fig. 4.
Fourier analysis, (b) the template image t(x, y), (d) segmentation w(x, y) of
t(x, y) showing all 64 regions in pseudocolor
Generation of the template image; (a) image q(x, y) determined by
is related by
(7)
(6)
(8)
(9)
y
Sy (y)
=
Lq(x,y).
t(x, y) = b(x, y) @ g(x, y).
C(u (x, y)) = lit ((x, y)  u(x, y))  p(x, y) 112,
(10)
highlighting the corresponding patterns in h(x, y) as shown in
Fig. 4(a). We then calculate projections of q(x, y) along the
vertical and horizontal axes as:
where u(x, y) = [Ul(X, y), U2(X, y)] is a 2D vector function,
and uland U2 are the coordinate components of u respectively.
The template obtained for the flood histogram under consid
eration is shown in Fig. 4(b). Since the peak locations are
known, t(x, y) can be segmented by horizontal and vertical
lines drawn midway between the peak locations. The resulting
segmentation w(x, y) with region labels in pseudocolor is
shown in Fig. 4(c).
3) Registration ofthe template to the flood histogram: For
registering image t(x, y) to p(x, y), we use an intensitybased
warping scheme with polynomial bases. The objective of the
registration scheme is to find a deformation field u(x, y) such
that the root mean square (RMS) intensity difference between
the target image p(x, y) and the deformed template image
t((x, y)  u(x, y)) is minimized. The RMS cost functional
is given by
From the one dimensional Sx (x), we compute the location
of exactly Dx peaks. This is done by computing the zero
crossing locations of \7Sx (x) and from those, determining a
subset where \72sx (x) is negative. In the same way, Dypeak
locations are determined from Sy (y). Let Ax denote the set
of the Dx peak locations in the horizontal direction and Ay
denote the set of Dypeak locations in the vertical direction.
We then create a binary image b(x, y) such that
{
I,
if x E Ax and y E Ay;
b(x, y) =
0,otherwise.
This binary image is then smoothed by convolving it with a
2D spatial Gaussian filter g(x, y) whose standard deviation is
set to be one third of the shortest distance between adjacent
peaks in the horizontal and vertical directions. The resulting
image is what we call the template t(x. y):
(3)
(4)
h(x,y)
p(x,y) = Ik(x,y)I'
k(x,y)
Ik(x, y)1 is a smoothed version of h(x, y) and highlights the
areas of high and low intensities in the flood histogram as
is seen in Fig. 3(b). The number 7 for Fourier coefficients
was chosen empirically and need not be modified for different
devices. We then compute the intensity corrected image p(x, y)
as
l'vll Nl
L
H(lx,ly)
Lh(x, y)ej21f1'tX
e _ j 2 1 f ~ Y Y ,
(1)
x=o y=O
Ml Nl
Lh(x, y)
=
LH(lx, ly)ej21fltX e j 2 r r ~ y y ,
(2)
fx=O fy=o
where H(Ix, Iy) denotes the DFf of h(x, y). In the discussion
that follows, we use Dx and Dy to denote the number of
crystals in the detector array in the horizontal and vertical
directions respectively.
1) Intensity compensation: There may be large variations
in the efficiencies of the crystals in the detector array [18].
As a result, crystals with high efficiencies would produce
brighter spots in the flood histogram compared to those with
low efficiencies. Intensity compensation is desirable to reduce
the computational burden on the segmentation algorithm. To
achieve this, we first compute H(lx, Iy). We then compute a
low pass filtered version k(x, y) of h(x, y) using the following
equation:
where the division is elementwise. The intensity corrected
image p(x, y) for the flood histogram in Fig. 3(a) is shown in
Fig. 3(c).
2) Generation of the template: Let P(lx,ly) represent
the DFf of p(x, y). The horizontal components in P(lx,ly)
correspond to vertical patterns in p(x, y), while the vertical
components in P(lx,ly) correspond to horizontal pattens in
p(x, y). We form Q(lx, Iy) using the following:
{
P(lx,ly),
if Ix =°or Iy = 0;
otherwise.
Q(lx, Iy) =
(5)
0,
From Q(Ix, Iy), we compute its 2D inverse Fourier transform
q(x, y). q(x, y) is made up of horizontal and vertical lines
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Page 5
By writing the deformation fields in terms of polynomials, we
get
To test the invertibility of the deformation field, we make sure
that J (q» has a positive determinant at every step [21]. If this
is not the case, a lower order polynomial basis is chosen to
approximate the transform. The iterations are stopped when
the determinant becomes nonpositive. The diffeomorphic
mapping cI> thus calculated is applied to w(x, y) to obtain
the segmentation of the flood histogram. The algorithm is
implemented in MATLAB® (The MathWorks, Inc., Natick,
MA, USA).
where n is the rank of the polynomial bases and a ~
are coefficients of the polynomial bases.
In order to minimize C(u(x,y)) in (10) as a function of
the polynomial coefficients, we use the conjugate gradient
algorithm [19]. The linesearch used in the conjugate gradient
algorithm searches in the descent direction following the
Armijo rule for stepsize reduction [20]. The gradient of the
cost functional C with respect to the coefficients a ~
computed as follows:
a ~ ~ JJ(t((x, y)  u(x, y))  p(x, y))2dxdy,
JJ 2(t((x, y)  u(x, y))  p(x, y))
8t(x  u(x, y)) kr
.!1( ()) X
u x  U1 x, Y
and b ~
and b ~ is
III. RESULTS
A. Spatial distortion correction and segmentation
We show our spatial distortion and segmentation results
in Fig. 5(a) and (c) where the segmented region boundaries
E. Iterative determination of a and ()
The a parameter for each detector is determined when the
magnetic field is switched off. For iteratively calculating a,
we first start with a = 0 in the adaptive formulae (table I,
row 3). Therefore, our formulae become equivalent to Zhang's
formulae. After computing the resulting flood histogram from
Xc and Yc, we compute the intensity corrected image p(x, y)
as described in Section IIC1. We then compute sx(x) and
Sy(y) from equations (6) and (7), and try to identify Dx
and Dy peaks respectively in them using the DFf based
procedure outlined in Section IIC2. If we are not successful,
we increment a by 0.05 each time and repeat the peak
identification procedure. We terminate the procedure when all
D x peaks in the horizontal direction and D y peaks in the
vertical direction are identified. The a value for each device
is stored for future use.
The rotation () in table I, row 4 is determined for each
detector when the magnetic field is switched on. The procedure
is similar to that for choosing a, except that we start with an
initial guess for e. Equations from table I, row 4 are used
with the earlier computed a. We then vary () over ±10°, 10
each time, and attempt to identify Dxand Dypeaks. We stop
when all peaks are appropriately identified. If an a or () that
yields satisfactory results is not found, we switch to the manual
correction tool.
F. Studies comparing manual segmentation to the proposed
automatic method
For comparative studies, flood histograms were segmented
using a manual method and automatic method. In the manual
method, the flood histogram was obtained using Anger's
equations. A user blind to the findings ofthe automatic method
clicked on the locations of Dx x Dy peaks. The clicked
locations then were used to create a binary file, which was
subjected to segmentation using the watershed method. For
the automatic method, a and () were determined iteratively
and the distortion corrected flood histogram was generated.
Segmentation was then carried out automatically using the
procedure described in Section IIC. All five data sets were
segmented using both manual and automatic methods.
D. Tool for manual correction.
For exceptional cases where peaks are incorrectly iden
tified, we have developed a graphical tool where the user
can manually click on those crystals in p(x, y) that are not
delineated accurately. As a result, p(x, y) is directly modified
by artificially drawing spots and the segmentation procedure
is repeated. The most number of individual clicks that are
required for this procedure (assuming all crystals are misclas
sified) is Dx+ Dy  1 (Dx clicks horizontally and Dy  1
clicks vertically), which is still substantial saving compared to
Dx x Dyclicks required for manual segmentation.
(15)
n
L L a ~ x r  q y q ,
r=Oq=O
n
LL
r=Oq=O
r
U1(X,y)
(11)
U2(X, y)
b ~ x r  q yq,
(12)
r
y dxdy.
(13)
a ~ ~ JJ (t((x, y)  u(x, y))  p(x, y)fdxdy,
JJ 2(t((x,y)  u(x,y))  p(x,y))
8t(xu(x,y)) kr
r
.8(())x y dxdy.
x  U2 x,y
(14)
Similarly,
BC
8 b ~
8C
8 a ~
Here 8t((x,y)u(x,y)) and 8t((x,y)u(x,y)) are 'It(x y) inter
8(xuI(x,y))8(X U 2(X,y))
polated at (x, y)  U1 (x, y) and (x, y)  U2(X, y) respectively.
The derivative operators in the above equations are discretized
by using the central difference approximation.
Since the order of the polynomials used is very low, the
resulting displacement is sufficiently smooth and regularizers
such as linear elastic energy are not required in practice. Let
cI> = (4)1, <P2) denote the mapping resulting from this trans
formation, i.e. ~
: (x, y) ~
(x, y)  u(x, y). The Jacobian of
this mapping is given by
,
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Page 6
Fig. 5.
boundaries; (a) flood histogram when the PET detector was outside the
magnetic field (0 = 0.7, () = 0°), (b) flood histogram generated from the
same data set as in (a) but with Anger logic and segmented using the manual
clicks on the peaks followed by the watershed methodbased segmentation,
(c) flood histograms when the PET detector was inside the 7T magnetic field
(0 = 0.7, 0 = 21°). Segmented region boundaries are denoted by white
lines. Slight missegmentations may happen for either case. Higher order
polynomials may be used in our case to to approximate the deformation fields.
We describe the tradeoffs of this prospect in Section IV.
Distortion corrected flood histograms with overlaid segmentation
90
80........10.''''""......,
1020 30
Crystal number
60
50
40
Fig. 6.
segmentation in Fig. 5(b) and with the proposed automatic method (Fig. 5(a».
The error bars indicate the standard deviation over the five sets of data.
Mean photopeak positions for all 64 crystals obtained using the
(c)
(b)
(a)
(16)
are overlaid on the distortioncorrected flood histograms. Both
underlying flood histograms are for the same detector when
placed outside and in the magnetic field respectively. The
iteratively determined ex value for this detector was 0.7.
Since the flood histogram in Fig. 5(a) was obtained when
the magnetic field was off, () = 0
when the detector was in the 7T magnetic field (Fig. 5(c»,
we iteratively determined () = 21
21 0 rotation remained unchanged for other PET detectors in
the scanner that had the same orientation. For detectors that
showed an anticlockwise rotation, we found () = 21
an Intel Xeon, 2.33 GHz computer, the iterative computation
of ex and () took about 1.1 sec each. Template generation took
0.2 sec. The segmentation procedure for results shown in Fig.
5 took on an average of 28 sec each. All 64 crystals were
automatically identified. This procedure was repeated for all
five data sets. The results for each set were very similar to
those shown in Fig. 5 and hence are not shown. In Fig. 5(b),
we show a flood histogram corresponding to the same data
set as Fig. 5(a) except that it was generated using Anger logic
and segmented using manual clicking on a computer screen
64 times, followed by watershed methodbased segmentation.
0was used. For the case
0
• We also found that the
0
• On
B. Quantitative comparison between manual segmentation
and the automatic method
To quantitatively compare results obtained using the distor
tion correction scheme and the semiautomatic segmentation
method (Fig. 5(a» with manual segmentation of histograms
obtained using Anger logic (which in this case, may be
considered a gold standard, Fig 5(b», we analyzed three
parameters for all 64 crystals in the detector arrays, namely, (i)
photopeak positions, (ii) energy resolution, and (iii) uniformity
of counts. This was done for detectors both with magnetic
field switched off and on. In the following three subsections,
we show our results. Since the results with magnetic field
switched on were similar to those when magnetic field was
off, we focus on the case when magnetic field was off. The
crystals in Fig. 5(a) and (b) are numbered such that the crystal
in the top left corner is assigned #1. The crystal number is then
incremented by 1 horizontally moving lefttoright. When the
end of the line is reached, the counting continues with the
leftmost crystal in the next horizontal line. Thus, crystal 1, 8,
57 and 64 are corner crystals, while crystals 18, 9, 16, 17,
24, 25, 32, 33, 40, 41, 48, 49, 56, 5764 are edge crystals.
1) Photopeak position: For each crystal in the array, the
energy histogram was obtained and the photopeak location
(Ey) was determined. In Fig. 6, we show measured photopeak
positions for all 64 crystals obtained from segmentations
shown in Fig. 5(a) and (b). The differences between the two
schemes are statistically insignificant for the five data sets.
2) Energy
maximum (FWHM) (denoted as ~
around the photopeak for each crystal was determined. The
energy resolution was then computed using
resolution(ER):
The
of the energy spectrum
fullwidthathalf
E)
~E
ER(%) = 
x 100,
Ey
Fig. 7 shows a plot of the measured energy resolutions for
all 64 crystals using the two methods. The average energy
resolution using manual segmentation for the flood with Anger
logic was measured at 21.11 ± 1.75%. In the case where
the proposed distortion correction and automatic segmentation
schemes were used, the average energy resolution was 20.81±
1.53%. The overall differences are statistically insignificant.
3) Uniformity of counts: Fig. 8 shows a plot of the total
counts measured in each crystal for the five data sets. Overall,
we observe that the corner crystals tend to have more counts
when the proposed positioning method is used. However, the
same cannot conclusively be said about edge crystals. We thus
conclude that overall, comparable uniformity is obtained using
the proposed method and the manual segmentation scheme.
Zhang et al. [8] found that their event positioning method
led to better uniformity of counts based on only those counts
that are within the FWHM of the photopeak position. In our
analysis, we instead consider all measured counts per crystal,
and thus, calculate overall crystal efficiencies in the 350650
keV energy window.
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Page 7
10
15
Fig. 8.
methods. The error bars indicate standard deviation over the five data sets.
Total measured counts in each crystal after segmentation for the two
istration schemes available in public domain (e.g. Automated
Image Registration [22], [23]) can be employed. To nullify the
effect of ambiguities (since invertibility will not necessarily
be enforced), one may be able to apply a median filter to
the resulting image followed by nearest neighbor interpolation
and potentially remove errors due to ambiguities. However,
this scheme needs thorough evaluation. Additionally, if the
detectors do not undergo major design changes and have
consistent performance, one can store the template image
permanently and reuse it to compute the segmentation when
need be. We also note that even theoretically, spatial distor
tions in flood histograms for PSAPDs cannot completely be
corrected [9]. If they could be corrected, one would simply
use the segmentation of the template as the final segmentation
of the flood histogram and the warping procedure would not
be required. However, since residual distortion remains, the
warping procedure is necessary.
We conducted quantitative studies comparing the conven
tional method (flood histograms generated using Anger's
equation followed by watershedbased segmentation) to the
distortion correction scheme and semiautomatic segmentation
method described in this paper. The two schemes were first
compared based on crystalwise photopeak position and energy
resolution. Insignificant differences were found in photopeak
positions. The very minor improvement in energy resolution
that was observed for the proposed method may be attributed
to improved photon statistics in comer crystals. Over all
five data sets, however, this improvement was smaller than
individual variability.
We also compared the total number of counts registered in
individual crystals for the two schemes. Uniformity of counts
in crystals not only depends on how well the distortion is
corrected but also on how accurate the segmentation is. For
example, some crystals in the resulting segmentation in Fig.
5(a) and (c) appear partially truncated due to segmentation
boundaries. An obvious way to reduce this error is to use
higher order polynomials to approximate the warping field.
This however, may lead to three potential challenges. Firstly,
the number of bases used for a given degree of the poly
nomial are given by the sum of the corresponding row in
Pascal's triangle. This number grows rapidly, and in tum,
increases computational burden. Secondly, implementation of
diffeomorphic constraints for higher order polynomials can
become nontrivial [24]. And thirdly, while trying to account
for higher order deformations, some constraints on the warping
field are necessary. Thus, regularization schemes need to be
implemented [25]. As a result, both computational complexity
and required time may increase substantially.
Our quantitative results indicate that the performance of the
proposed method is comparable to that of the conventional
method. However, the real major benefit from the proposed
method is in decreasing human effort and time. As an example,
the manual segmentation scheme applied to one PET/MR de
tector required 64 clicks on the computer screen and a total of
about 90 seconds. The same flood histogram was automatically
segmented in 28 sec without human involvement. This saving
in time and effort would potentially be huge when thousands
of crystals in the scanner would require to be identified. In
60
50
[



Manual segmentation
Automatic method
~
20 3040
Crystal number
10
16000
14000
4000
2000
! 12000 .
~u
.c 10000
i
1:
!!c
:18
IV. DISCUSSION
We have extended the use of our adaptive spatial distortion
correction scheme for flood histograms of PSAPDbased PET
detectors for compensating effects of a 7T magnetic field on
the detectors. By choosing an appropriate a parameter to strike
a balance between two existing schemes, flood histograms with
minimal pincushion or barrel distortions are obtained. The
proposed scheme also allows flexibility for choosing a () value
based on the magnetic field to be used for scanning. Both ex
and () parameters are iteratively determined by an automatic
procedure and do not require human involvement. We also
have provided implementation details for our previously pro
posed semiautomatic flood histogram segmentation scheme
[16].
In our semiautomatic segmentation scheme, we generate
a template image and register it to the given flood histogram.
For this purpose, we propose a diffeomorphic warping scheme
that is free from ambiguities due to illconditioned mapping
between the template and the flood histogram. However, this
is more of a matter of convenience than a necessity. After
generating a template as described in this paper, other reg
Fig. 7.
the proposed automatic method for the five sets ofdata. The error bars indicate
standard deviation over the data sets.
Crystalwise energy resolution (%) using manual segmentation and
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Page 8
addition, we also have developed a tool for manual correction
of the flood histogram in exceptional cases. Considerable time
and effort is saved even if this tool is put to use compared to
manual segmentation.
v. CONCLUSIONS
We have developed a distortion correction scheme and
an efficient semiautomatic crystal identification scheme for
PSAPDbased PET detectors and have quantitatively assessed
its performance for use with the UC Davis MRIcompatible
PET scanner. The proposed scheme requires minimum human
involvement while still allowing considerable flexibility and
thus, potentially should accelerate routine detector calibration
and characterization studies. The proposed scheme is generic
and has the potential to be employed for a broader range of
PET scanners based on PSAPDs.
ACKNOWLEDGEMENTS
The authors would like to thank Dr Yongfeng Yang and
Spencer L. Bowen from the Department of Biomedical Engi
neering, University of California  Davis, Davis, CA 95616,
USA for their help in the preparation of this manuscript. This
work was funded in part by the American Cancer Society
award IRG9512507, by the National Institutes of Health
R44CA094385, and by the Susan G. Komen Foundation
award BCTR0707455. This publication was also made pos
sible by Grant Number ULI RR024146 from the National
Center for Research Resources (NCRR), a component of the
National Institutes of Health (NIH), and the NIH Roadmap for
Medical Research. Its contents are solely the responsibility of
the authors and do not necessarily represent the official view
of NCRR or NIH.
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