Spatial Distortion Correction and Crystal Identification for MRI-Compatible Position-Sensitive Avalanche Photodiode-Based PET Scanners

Abstract

Position-sensitive 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 MRI-compatible. 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 PSAPD-based PET detectors when placed both inside and outside a MRI scanner. Analytical formulas 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 lower-order spatial approximation of the distortion-corrected PSAPD flood histogram, which we call the ldquotemplaterdquo. The template is then registered to the flood histogram using a diffeomorphic iterative intensity-based 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 l-
ead to a large reduction in manual labor and indeed can routinely be used for calibration and characterization studies in MRI-compatible PET scanners based on PSAPDs.

Full text

Page 1

2008 IEEE Nuclear Science Symposium Conference Record
Spatial distortion correction and crystal
identification for position-sensitive avalanche
photodiode-based 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
M10-298
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 MRI-compatible
I. INTRODUCTION
M
simultaneously-acquired images of morphology, function and
metabolic activity are expected to have a huge positive impact
on both pre-clinical as well as clinical imaging fields [1]-[3].
At DC Davis, a pre-clinical MRI-compatible 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 field-insensitive position-sensitive 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 position-sensitive
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 two-dimensional probabilistic maps
RI-compatible Positron Emission Tomography (PET)
scannersthatproduceanatomicallyco-registered
Abstract-Position-sensitive 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 MRI-compatible. 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
PSAPD-based 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 lower-order
spatial approximation of the distortion-corrected PSAPD flood
histogram, which we call the 'template'. The template is then
registered to the flood histogram using a diffeomorphic iterative
intensity-based 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 MRI-compatible
PET scanners based on PSAPDs.
Index Terms-PETIMRI, PSAPD, spatial distortion correction,
crystal identification
A. 1. Chaudhari, Y. Wu and S. R. Cherry are with the Department of
Biomedical Engineering, University of California-Davis, 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 California-Los 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
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(b)
(c)
978-1-4244-2715-4/08/$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 PSAPD-based 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 barrel-type 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 semi-automatic scheme
involving thresholding the flood histogram to automatically
identifying peaks followed by watershed-based 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 self-organizing maps,
multi-level 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 distortion-corrected flood histogram, we
previously have developed a general purpose semi-automatic
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 polynomial-based 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 MRI-compatible 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 PD2-MFS 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
Pentium-4 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 350-650 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 2-D
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 II-E. 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 II-E. 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 III-A.
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 look-up 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 pseudo-color
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 pseudo-color 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 intensity-based
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'vl-l N-l
L
H(lx,ly)
Lh(x, y)e-j21f1'tX
e _ j 2 1 f ~ Y Y ,
(1)
x=o y=O
M-l N-l
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 element-wise. 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 2-D 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 non-positive. 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 line-search 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)) k-r
.!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 II-C1. 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 II-C2. 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 II-C. 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(x-u(x,y)) k-r
r
.8(())xy 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(x-uI(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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