IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 16, NO. 9, SEPTEMBER 2007
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Geometric Direct Search Algorithms
for Image Registration
Seok Lee, Minseok Choi, Hyungmin Kim, and Frank Chongwoo Park
Abstract—A widely used approach to image registration involves
finding the general linear transformation that maximizes the mutual information between two images, with the transformation
(3)] or volume-preserving
being rigid-body [i.e., belonging to
[i.e., belonging to
(3)]. In this paper, we present coordinate-invariant, geometric versions of the Nelder–Mead optimization
(3),
(3), and their various subalgorithm on the groups
groups, that are applicable to a wide class of image registration
problems. Because the algorithms respect the geometric structure
of the underlying groups, they are numerically more stable, and
exhibit better convergence properties than existing local coordinate-based algorithms. Experimental results demonstrate the
improved convergence properties of our geometric algorithms.
Index Terms—Image registration, mutual information,
Nelder–Mead, optimization, transformation group.
I. INTRODUCTION
O
NE of the fundamental steps in image fusion is the alignment of two or more sets of volumetric image data, commonly referred to as the image registration problem. In one
popular formulation of this problem, the 3-D linear transformation is sought that optimizes some fitting criterion between two
given image volumes; quite often the transformation is assumed
to be either of the rigid body type, i.e., belonging to the special
, or of a more general volume preserving
Euclidean group
type, i.e., belonging to the special linear group
. Two-dimensional versions of the above also arise when the data sets are
restricted to be planar; in this case, the relevant transformation
and
. A popular choice of fitting critegroups are
rion is the mutual information between the two image volumes
[5], [19], [23]. In this paper, we shall not address more general formulations that seek nonlinear diffeomorphisms between
two sets of volume data (e.g., [6], [16], and [22]), although such
problems can, in some instances, be locally decomposed into a
series of subproblems of the above type.
Two features of the ensuing optimization make this problem
challenging: 1) evaluating the mutual information objective
function is computationally very expensive, and 2) the search
,
,
space is nonlinear, in the sense that the groups
and their various subgroups do not admit the structure of a
vector space. Because of the expense in evaluating the objective
Manuscript received December 15, 2006; revised April 10, 2007. This work
was supported in part by the Center for Intelligent Robotics, Cybermed, Inc.,
IAMD-SNU, and in part by the BK21 Program in Mechanical Engineering at
Seoul National University. The associate editor coordinating the review of this
manuscript and approving it for publication was Dr. Til Aach.
The authors are with Seoul National University, Seoul, Korea (e-mail:
[email protected]; [email protected]; [email protected];
[email protected])
Digital Object Identifier 10.1109/TIP.2007.901809
function, classical descent methods that rely on finite difference
approximations of the gradient are highly inefficient: at each
iteration they would require several evaluations of the objective function. Methods based on gradient-free, direct search
methods, such as the Nelder–Mead algorithm [18], are a practical alternative. While it is true that the convergence properties
of these algorithms are difficult to characterize analytically
except in the simplest cases [12], in practice, they have proven
to be very effective, particularly when the objective function
is reasonably well-behaved, in the sense of being sufficiently
smooth, without a large number of local extrema, and an initial
guess reasonably close to local optima can be provided [20].
In [20], a Nelder–Mead algorithm formulated in terms of
local coordinates is presented for addressing the rigid-body
version of the image registration problem. To find the
rotation matrix component of the optimal rigid body transformation, the original vector space Nelder–Mead algorithm is
formulated in terms of the chosen local coordinates (the Euler
angles in this case) that parametrize the rotation group. An
obvious drawback with this approach is that local coordinates
contain singularities; whenever the iteration ventures near such
a singularity, it becomes necessary to transfer to another set of
local coordinates, thereby complicating the algorithm structure.
Optimistically, one might regard this merely as a tolerable inconvenience. More critically, however, a local coordinate-based
algorithm that fails to properly take into account the geometric
structure of the search space—some regions may be highly distorted relative to other regions, for example, requiring smaller
steps and more numerical iterations—will inevitably lead to
very inconsistent, in some cases even catastrophic performance.
Several papers, e.g., [7] and [9], show how the geometry of the
underlying space needs to be taken into account in the general
setting of optimization and numerical analysis on Riemannian
manifolds and Lie groups. Similarly, in previous work by the
authors [8], classical steepest descent and Newton methods are
generalized in a local coordinate-invariant way to quadratic
, and the advantages explicitly
objective functions on
demonstrated.
This paper makes two contributions: 1) a geometric—in the
sense of being invariant with respect to choice of local coordinates, and respecting the chosen metric structure of the underlying space—generalization of the Nelder–Mead algorithm to
and
(and by extension
the transformation groups
their subgroups), and 2) applications of these geometric algorithms to several classes of image registration problems. Our
image registration applications treat in detail the various numerical issues encountered in a practical implementation, and we
, that improve the
suggest modifications, particularly for
computational performance of the algorithm.
1057-7149/$25.00 © 2007 IEEE
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Experimental results indicate that, conservatively, performance improvements of around 15% in computational
efficiency can be achieved relative to existing local coordinate-based methods. More significantly, our geometric
algorithm is shown to be more numerically robust (with respect
to, e.g., input image data and initial conditions) than existing
methods; for several cases in which local coordinate-based
methods failed to converge, our geometric algorithm was able
to converge to a physically meaningful solution.
This paper is organized as follows. After establishing some
geometric preliminaries in Section II, in Section III, we present
the geometric version of the Nelder–Mead algorithm on the Lie
and
. In Section IV, we detail the appligroups
cation of this algorithm to the image registration problem. Section V concludes with a summary and suggestions for further
extensions.
II. GEOMETRIC PRELIMINARIES
In this section, we discuss the geometric structure of the Lie
and
, paying particular attention to the
groups
metric space structure, and computational formulas for the disare well
tance metric and mass centers. The results for
known and have been reported in the literature, e.g., [10]; here,
we offer only a brief summary of the main results. The results
, which, to our knowledge, have not been reported in
for
the literature, are derived from first principles. An overview of
spatial transformations in medical imaging is also given in [1].
A. Rotation and Rigid-Body Transformations
To generalize the Nelder–Mead optimization algorithm to the
group of proper rotations
and the rigid-body transfor, appropriate notions of the “mean” of a set elemations
ments in the group, and the “straight line” connecting two elements of the group, are first needed. We begin with precise formulations of these notions. Recall that the special orthogonal
consists of the 3 3 real matrices that satisfy
group
and
. The special Euclidean group
consists of the 4 4 real matrices of the form
with
consists of the 4
4 real matrices of the form
(3)
.
where
One way in which a Lie group and its Lie algebra are related
is by the (matrix) exponential and logarithm mappings. Given
, where
is a unit vector and
, the
is defined by the formula
map
(4)
The formula for the logarithm
is given by
(5)
where is any scalar satisfying
; restricting
enforces a unique value for the logarithm (with the
, two solutions—antipodal points on the
exception that if
sphere of radius in —exist). Every
admits a
. Although
representation as the exponential of some
we will not have use for them here, corresponding formulas for
and
are available
in, e.g., [17].
With the above preliminaries, the natural distance metric on
is constructed as follows (see [10] for details). Given two
, the distance between these two rotarotations
tions, denoted
is evaluated according to the formula
(6)
where
denotes the matrix Frobenius norm, i.e.,
. The above distance metric is natural in the sense
of being invariant (up to scale factor) with respect to choice of
for
reference frames; that is,
.
any
Given this frame-invariant notion of length on
, it can
be established that the shortest curve connecting two rotations
is of the form
(1)
and
, and the lower-left zero element
where
and
have
indicates the 3-D zero row vector. Both
the structure of a differentiable manifold and an algebraic group
(under matrix multiplication), and are examples of Lie groups.
Each Lie group has associated with it a Lie algebra, typically
consists of the
denoted in small letters. The Lie algebra
real 3 3 skew-symmetric matrices of the form
(7)
so that
and
, and
. This curve
and .
corresponds to the minimal geodesic between
the minimal geodesics are one-parameter subOn
groups of the form
, with
and a scalar parameter.
; given
of
This is not quite the case for
the form
(8)
(2)
Given
, we shall denote its skew-symmetric matrix representation by
. The Lie algebra
associated
the minimal geodesic connecting
is given by
(at
) and
(at
)
(9)
LEE et al.: GEOMETRIC DIRECT SEARCH ALGORITHMS FOR IMAGE REGISTRATION
where
, i.e., the rotation component is simply
between
and . Similarly,
the minimal geodesic on
the translation component is the straight line connecting and
. The associated distance metric on
is
(10)
It should be noted that this metric is only invariant with respect
to choice of fixed (global) reference frame (i.e., left-invariant).
]
Moreover, this metric [and for that matter any metric on
depends on the choice of length scale chosen for physical space.
We refer the reader to [10] for proofs, and a discussion of the
physical implications, as well as heuristics for choosing length
scales meaningful for the application at hand. In the event that
one desires to weight the orientation and position equally in
some sense, one possible heuristic is to choose the length scale
such that the volume of the position and orientation workspaces
with respect to
are equal; in this regard, the volume of
its natural volume form is evaluated to be
.
Based on the above, a precise notion of means on the rotation
and Euclidean groups can now be established. First, given a set
in
, one way of defining the
of rotations
mean rotation is the element
that minimizes the
criterion
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B. Group of Volume-Preserving Transformations
adThe group of volume-preserving transformations
mits the structure of a matrix Lie group, consisting of the real
that satisfy
. The associ3 3 matrices
consists of the real 3 3 matrices of trace
ated Lie group
zero. We first discuss a natural choice of local coordinates for
, followed by its metric space structure.
It is known that any element of the general linear group
of nonsingular matrices can be decomposed via the Iwa, where
,
sawa decomposition [3] as
is diagonal with positive entries, and is upper-triangular with
unit diagonal entries. Restricting
to have positive determito be an element of
. The above
nant in turn forces
decomposition reflects the well-known physical property that a
general linear transformation can be decomposed into a combination of rotation, scaling, and shearing. The scaling transfor,
mation can be further decomposed as
so that scaling can be physically decoupled into a scale factor
and stretching component.
, an
Restricting the above Iwasawa decomposition to
can be decomposed as
, where
element
,
is diagonal with unit determinant, and
is
upper-triangular with unit diagonal entries. For example, any
volume-preserving transformation in two dimensions can be decomposed as
(11)
According to Moakher [15], the optimal that globally minimizes , denoted the extrinsic mean, is the orthogonal projeconto
tion of
(12)
Using a result of [11], it can be established that as long as
is
nonsingular (which is almost always true except for pathological
cases), the mean can be expressed as
Three parameters corresponding to rotation ( ), stretching ( ),
and shearness ( ) can be used to parametrize any element of
. In three dimensions, any element
can be
, where
can be paparametrized as
rametrized via the usual exponential representation
, where
, and the stretching transformation and shearing transformation are, respectively, of the form
(14)
(13)
where
where and are obtained from the singular value decompo, and
if
, and
othersition
wise. Alternatively, one could replace the objective function of
;
(11) by the intrinsic distance metric, i.e.,
in this case the minimizer is often referred to as the intrinsic,
or geometric, mean. Because a general closed-form formula for
is not available, and the associated
the intrinsic mean on
numerical procedure for the intrinsic mean is more involved, in
our application we will make use of the extrinsic mean above.
for a set
The corresponding formula for the mean
on
is now straightforward: the
of elements
corresponding rotation
is given by (13), while the
is given by its arithmetic mean. The same
translation
, and
remarks above also apply for the intrinsic mean on
we, therefore, use the extrinsic mean instead.
and
are generated as
(15)
(16)
Developing a metric space structure on
is complicated
by the fact that, unlike
, simple analytic formulas for
minimal geodesics between two arbitrary points are no longer
is no
available. The exponential map
longer surjective, and is bijective only in a neighborhood of
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the identity. Moreover, in the appendix we show that, given
and
on
, the minendpoints
imal geodesic (with respect to the standard left-invariant metric)
connecting
and
must satisfy
(17)
(18)
. Finding the minimal geodesic thus
where
involves the solution of a two-point boundary value problem on
and
, for which multiple solutions typically exist.
The geodesics on
are also given by the same set of (18),
and
.
but this time with
are also geodesics
It readily follows that geodesics on
. The computational cost of determining the minimal
on
geodesic obviously needs to be considered in the overall performance analysis of any geometric direct search algorithm.
On the other hand, it is known (e.g., [21]) that near the identity, the minimal geodesics are in fact given by left and right
,
,
translations of the one-parameter subgroups
. To take advantage of this result, we assume that image registration has already been performed over the rigid-body transformations using, e.g., our gradient-free optimization routine on
, and that the image data have been appropriately transformed via this rigid-body transformation. To further refine this
solution we now perform a second optimization over
with
or
; in this case, we can expect that the obtained solution will be reasonably close to the identity.
is within the injectivity radius of
(that
As long as
centered at over which
is, the radius of the open ball in
is surjective), and
can be connected via a one-parameter subgroup. Since an analytic charis not available,
acterizations of the injective radius on
we resort to the following numerical experiment to get a sense
of the size of the injectivity radius. We first parametrize an elby the three parameters
described
ement
earlier, along a line from ( 1.5, 1.5, 1.5) to (1.5, 1.5, 1.5);
along this line is then calculated.
the error
The direction of the line is now changed randomly 2000 times
by sampling from a uniform distribution over the unit sphere.
The error along each line is calculated in the same fashion; the
mean errors over the 2000 randomly generated lines are shown
in Fig. 1. Due to finite precision an exact match (i.e., zero error)
is obtained only at the identity as expected, but the errors are ex. These results suggest that
tremely small, on the order of
that deviate by as much as ( 1.5,
even for elements of
Fig. 1. Numerical plot of injectivity radius of SL(2).
1.5, 1.5) from the origin in
coordinates, the exponential and logarithm maps are nearly bijective.
For completeness, we list the known analytic formulas for the
(see, e.g., [2]): given
of the form
exponential on
we have (19), shown at the bottom of the page.
onto its
Now we present the projection formula of
. Because numerically computed values of
subgroup
invariably drift off
to
, a projection back
is often required. The projecto the nearest element in
tion problem can be formulated as the following minimization:
, we seek the
that minimizes
given
(20)
The optimal solution admits the following simple characterization.
admit the singular value deProposition 1: Let
, where
. The optimal
that
composition
, with and as
minimizes (20) is given by
obtained above, and
a real positive diagonal matrix of unit
.
determinant that minimizes
Proof: See Appendix II.
Using this proposition the projection is simplified to finding
the unit determinant diagonal matrix with positive entries that
. On
, given
, we
minimizes
,
that minimizes
seek the
. From the necessary conditions for optimality
must satisfy
(21)
if
if
if
(19)
LEE et al.: GEOMETRIC DIRECT SEARCH ALGORITHMS FOR IMAGE REGISTRATION
On
, given
,
for optimality,
and
, we seek the
that minimizes
. From the necessary conditions
must satisfy
(22)
(23)
Although not optimal in the above sense, one convenient
to
is as follows. Supposing
means of projecting
is decomposed as
, the
as before that
projection can be obtained by normalizing the elements of to
, the corresponding projection
have unit determinant. For
is
(24)
while in the case of
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denote a nondegenerate simplex consisting of seven
Let
on
. Define the worst point
elements
to be the point in the simplex at which the objective function
attains its largest value. Similarly, the best point is defined to be
the point in the simplex at which the objective function attains
its smallest value. Further define the four scalar parameters
representing the amount of reflection, expansion, contraction,
and shrinkage by , , , and , respectively; some widely
,
,
, and
.
used values are
, at each -th iteration one of four operaGiven a simplex
tions (reflection, expansion, contraction, or shrinkage) is performed. When a reflection, expansion, or contraction is performed, a new vertex is obtained to replace the worst point of
. When a shrink is performed, three new vertexes are obtained, which together with the current best point, constitute the
updated simplex for the next iteration. The detailed algorithm is
as follows.
1) (Ordering) Relabel the simplex vertexes
in increasing order of the objective function value, i.e., such
.
that
the projection is given by
(25)
In the case of
where
deviates strongly from 1,
it can happen that the result is far from the optimal solution.
Under the assumption that rigid registration has already been
performed and the given is reasonably close to identity, it is
. With this formula similar perfair to assume that
formance is obtained, but with better computational efficiency
than solving the optimality condition.
We close this section with a formula for the sample mean
. Given a set of
elements
, the
on
(extrinsic) mean
that minimizes the criterion
(26)
coincides with
(27)
can, in turn, be obtained via the projection of
onto
, whose solution we indicate earlier.
The result can be verified via straightforward calculation,
whose details we omit.
III. GEOMETRIC NELDER–MEAD ALGORITHM
We now describe the Nelder–Mead direct search optimization
, where is an
algorithm for minimizing some function
element of
,
, or any of their various subgroups. For
a detailed description of the classical Nelder–Mead algorithm
on vector spaces, see the original [18] and the more recent [12]
and [24].
2) (Reflection) Calculate the reflection point
as follows:
(28)
(29)
as
where denotes the mean rotation of
. If
given by (13), and denotes the centroid of
, replace
by
and go to Step 6.
3) (Expansion) If
from
, find the expansion point
(30)
(31)
, replace
by
and go to Step 6.
If
by
and go to Step 6.
Otherwise, replace
4) (Contraction) If
two following contractions.
• (External Contraction) If
the ouside contraction point
, then perform one of the
, evaluate
according to
(32)
(33)
, replace
by
If
Otherwise, proceed to Step 5.
and go to Step 6.
• (Internal Contraction) If
according to
inside contraction point
, evaluate the
(34)
(35)
A. Nelder–Mead on
Denote the rotation and translation components of
by
and
, respectively.
, replace
by
If
Otherwise, proceed to Step 5.
and go to Step 6.
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for
respectively.
5) (Shrinkage) Calculate the shrinkage vertexes about
according to
(36)
(37)
for
respectively.
. Replace
by
6) Go back to Step 1 until
stopping criterion . Here
left-invariant distance metric on
. Replace
by
6) Go back to Step 1 until
stopping criterion .
,
for some preset
To ensure that the one-parameter subgroups calculated above
do not drift off
, the projection formulas derived in the
previous section can be applied.
,
for some preset
and
denotes the
of (10).
B. Nelder–Mead on
We now describe the geometric Nelder–Mead algorithm on
. The
version is similar, the difference being that
the simplex is now composed of four vertexes. As before, given
, at each iteration one of four operations (rea th simplex
flection, expansion, contraction, or shrinkage) is performed as
follows.
IV. APPLICATIONS TO IMAGE VOLUME REGISTRATION
We now present applications of our geometric direct search
algorithm to the mutual information based image registration
problem, based on the formulations given in [14], [4], and
[23]. We assume as given two image volume data sets
and , which we regard as random variables with marginal
,
, and joint distribution
.
distributions
The Shannon entropy for each distribution, which intuitively
reflects the amount of uncertainty in the corresponding random
variable, is defined by
1) (Ordering) Relabel the simplex vertexes
in increasing order of the objective
.
function value, i.e., such that
2) (Reflection) Calculate the reflection point
(43)
(44)
as follows:
(45)
(38)
where
denotes the mean of
, replace
3) (Expansion) If
from
by
. If
and go to Step 6.
, find the expansion point
The mutual information between
is defined by
and
, denoted
,
(46)
(47)
(39)
, replace
by
and go to Step 6.
If
by
and go to Step 6.
Otherwise, replace
4) (Contraction) If
two following contractions.
• (External Contraction) If
the ouside contraction point
, then perform one of the
, evaluate
according to
Given the image intensity values and of a pair of corre,
sponding voxels in the two images, estimates for
, and
can be obtained via normalization of the joint
and marginal histograms of the overlapping parts of both images; see [14] for details.
that maximizes the
We seek the optimal transformation
; the latter demutual information between image and
notes the transformed version of image via
(40)
(48)
, replace
by
If
Otherwise, proceed to Step 5.
and go to Step 6.
• (Internal Contraction) If
according to
inside contraction point
, evaluate the
(41)
, replace
by
If
Otherwise, proceed to Step 5.
and go to Step 6.
5) (Shrinkage) Calculate the shrinkage vertexes about
according to
(42)
In existing approaches, the transformation
is parametrized
is maxvia some suitable local coordinates, and
imized in a vector space setting; for example, in the case of
, the Euler angles are a popular choice of local coordinates. In our experiments below, we compare the results of our
geometric direct search algorithms with those obtained from
local coordinate-based formulations of the Nelder–Mead algorithm. All the simulations and experiments described below are
performed with real medical images. The optimizations are performed using Matlab, running on an IBM Pentium 800 MHz
with 640 MByte of memory. We use a stopping criterion of
.
LEE et al.: GEOMETRIC DIRECT SEARCH ALGORITHMS FOR IMAGE REGISTRATION
Fig. 2. Comparison of convergence speed between
tion.
SL(2) and <
optimiza-
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Fig. 4. Comparison of convergence speed between SE (3) and < optimization.
Fig. 4 shows the results of our volume registration ex. The original MRI volume of size
periments on
128 128 70 is transformed with the following rigid transformation
:
(50)
Fig. 3. Convergence behavior of
larity.
SL(2) and <
optimization near a singu-
We first present simulation results for optimization on
.
Fig. 4 shows the optimization result; the right upper image is a
single slice of a size 201 201 brain CT, while the right bottom
image is the deformed image using the following transformation
:
The graph on the left side shows the convergence behavior of
each algorithm. We note that the convergence of the optimization on
is faster than on ; the maximum is obtained
case, compared to 256 iterations
within 305 iterations in the
case.
in the
Fig. 4 shows the results of our volume registration ex. The original MRI volume of size
periments on
128 128 70 is deformed with the following transformation
:
(51)
(49)
The graph on the left side shows the convergence behavior for
each algorithm; note that faster convergence of the algorithm
on
compared to that for . The maximum is obtained
within 78 iterations in the
case, compared to 70 iterations
case. One can also observe more rapid and steady
in the
progress in the objective function maximization for the
case.
In Fig. 4, we use the same set of simulation conditions as
before, except that we now intentionally apply a 2-D rotation
singularity at zero radians, to compare the convergence propand
optimization algorithms near a local
erty of the
coordinate singularity. In the
optimization, zero and
are identified as the same point, whereas this is not the case in
the vector space optimization. As shown in Fig. 4, the perfordeteriorates significantly; the
mance of the optimization in
optimal value is obtained in 130 iterations, which is 86% slower
. In certain singular cases, local
than the optimization on
coordinate-based algorithms can even fail to converge. We note
that this singularity problem is inherent in every local coordinate-based optimization algorithm.
The upper columns on the right show a 3-D view of the original
and deformed volumes. The graph on the left compares the conand
. Here,
vergence behavior of the optimization on
is faster
we also find that the speed of convergence on
than that of .
More than 50 simulations with cases similar to the above were
performed; these 50 trials do not include cases in which the solutions are in the neighborhood of a local coordinate singularity.
Table I shows the average elapsed number of iterations and computation times for the respective algorithms. The optimizations
took approximately 20 min in a Matlab environment,
on
while for selected C++ implementations of corresponding specific cases less than 2 min were required. Based on these results
our expectation is that the actual optimization times for the geoor
will typically be up to ten
metric algorithms on
times faster than listed in Table I, which were obtained without
any consideration of attaining real-time performance.
Based on our experimental comparison results, the geometric
,
, and
are reoptimization algorithms on
spectively 4.5%, 16.4% and 19.6% faster than the corresponding
vector space optimization algorithms with respect to time, and
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TABLE I
CONVERGENCE TIME FOR EACH OPTIMIZATION ALGORITHM
Fig. 5. Comparison of convergence speed between
tion.
SL(3) and <
optimiza-
8.6% 16.9%, and 14.1% more efficient in terms of the number
of iteration.
Fig. 6 shows some representative results of image registra.
tion using our geometric Nelder–Mead algorithm on
An MFC-based program for loading and manipulating DICOM
format medical images was first implemented. Some speed up
techniques in [4] were used and the cubic 6 6 interpolation
method with continuous second derivatives [13] was used to calculate voxel values after transformation. The first and second
columns are planar views of the reference and target volumes,
respectively, while the third column displays the fusion results
after registration. The fourth column shows the 3-D view of the
reference volumes.
V. CONCLUSION
This paper has presented a geometric generalization of the
Nelder–Mead direct search algorithm to the Lie groups
and
, together with applications to image volume registration. The algorithms are geometric in the sense of being local
coordinate-invariant, and respecting the chosen metric structure
of the underlying space. The advantages of the geometric approach are that local coordinate singularities do not require any
special treatment, and also ensure uniform performance of the
algorithm that otherwise might depend on local features of the
coordinate representation (e.g., in the chosen coordinate representation, some regions of the search space may be highly distorted relative to other regions).
Direct search algorithms are most effective for image registration applications in which the evaluations of the objective
function are computationally expensive, and gradient information is unavailable. Simulation and experimental results
evaluating the performance of our algorithms have been obtained using real CT and MRI image volumes. Our experimental
results indicate that, conservatively, performance improvements
of around 15% in computational efficiency can be achieved
Fig. 6. Some 3-D volume registration results using SE (3) optimization.
(a) Pre- to postoperation image registration result. (b) CT to MRI image of spine
registration result. (c) CT to MRI image of spine registration result—saggital
view. (d) MRI to PET registration result.
relative to existing local coordinate-based methods. More
significantly, our geometric algorithms do not exhibit the poor
convergence behavior near singularities characteristic of local
coordinate-based methods. The results on minimal geodesics
and
, and the projection formula on
,
on
should also be of independent interest.
The direct search algorithms described in this paper are in
principle generalizable to other matrix groups, provided that
precise notions of the sample mean and minimal geodesics can
be characterized. Some natural extensions are to the
, and the affine group
. Efforts are also underway
to use the algorithms in a more general scheme for finding optimal nonlinear diffeomorphisms, by an appropriate decomposition of the global problem into several local versions of the
above.
APPENDIX I
MINIMAL GEODESICS ON
In this appendix, we derive the variational equations for minwith respect to the standard left-inimal geodesics on
variant metric. We formulate this problem as an optimal control
problem, in which the objective is to minimize
(52)
over
subject to
(53)
LEE et al.: GEOMETRIC DIRECT SEARCH ALGORITHMS FOR IMAGE REGISTRATION
with
is perturbed to
and
are given. Assume
, where
with
. Then after some calculation, we have
2223
The original problem can, thus, be recast as a minimization of
. From
and
, it can be seen
that the optimal has the same orthogonal part as that of .
(54)
REFERENCES
In other words, the associated is now perturbed to
. The associated first variation becomes
(55)
Integrating the second term by parts leads to
(56)
Setting this equation equal to zero for all admissible
end up with
, we
(57)
The optimality conditions can, therefore, be expressed as
(58)
(59)
,
, and boundary conditions
given. Note that the above derivation also applies
to
, so that the optimality conditions are identical, but
and
. Clearly, the geodesics
with
are also geodesics on
. It also follows that
on
is constant in time; this implies that the differential
equation for
is Lipschitz, thereby ensuring existence of a
solution.
with
,
APPENDIX II
PROOF OF PROJECTION FORMULA ON
In this appendix, we provide a proof of Proposition 1.
where
Suppose the SVD of
and
with
and
. After
some manipulation
(60)
,
. The maximal value of
,
attained when
,
. Note that the maximal value depends on the
. It follows that (60) is lower bounded by
sign of the
.
if
and
Letting
if
, (60) becomes
where
is
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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 16, NO. 9, SEPTEMBER 2007
Seok Lee received the B.S., M.S., and Ph.D. degrees
in mechanical and aerospace engineering from Seoul
National University, Seoul, Korea, in 2000, 2002, and
2007, respectively.
He currently serves as a R&D staff member in the
Computing and Intelligence Laboratory, Samsung
Advanced Institute of Technology. His research
interests are in robotics, image processing, medical
imaging, and related areas of computer vision.
Minseok Choi received the B.S. degree in mechanical engineering and mathematics from Seoul
National University, Seoul, Korea, in 2002, where
he is currently pursuing the M.S. degree.
From 2002 to 2005, he was a Software Engineer
with Denali IT and Empas. His research interests include motion planning, computer vision, and related
areas of applied mathematics.
Hyungmin Kim received his engineering degree in
mechanical engineering from Seoul National University, Seoul, Korea, in 1998, and the M.S. degree from
the Robotics and Automation Laboratory, Seoul National University. He is currently pursuing the Ph.D.
degree in biomedical engineering at the University of
Bern, Bern, Switzerland.
After his degree courses, he entered himself into
the medical industry, developed medical imaging
softwares and surgical navigation systems at Cybermed, Inc., for six years. He serves as a Research
Associate at MEM Research Center, University of Bern.
Frank Chongwoo Park received the B.S. degree in
electrical engineering and computer science from the
Massachusetts Institute of Technology, Cambridge,
in 1985, and the Ph.D. degree in applied mathematics
from Harvard University, Cambridge, in 1991.
He is a Professor of mechanical and aerospace
engineering AT Seoul National University, Seoul,
Korea. He was an Assistant Professor of mechanical
and aerospace engineering at the University of California, Irvine. His research interests are in robotics
and computer vision, mathematical systems theory,
and related areas of applied mathematics.
Dr. Park currently serves as the 2007 IEEE Distinguished Lecturer for
the Robotics and Automation Society, as a Senior Editor for the IEEE
TRANSACTIONS ON ROBOTICS, and as an Area Editor for the forthcoming
Springer-Verlag Handbook of Robotics.