Capturing Anatomical Shape Variability Using
B-spline Registration
Thomas H. Wenckebach, Hans Lamecker, and Hans-Christian Hege
Zuse Institute Berlin (ZIB),
Takustr. 7, 14195 Berlin, Germany,
{wenckebach,lamecker,hege}@zib.de, www.zib.de/visual
Abstract. Registration based on B-spline transformations has attracted
much attention in medical image processing recently. Non-rigid registration
provides the basis for many important techniques, such as statistical
shape modeling. Validating the results, however, remains difficult - especially
in intersubject registration. This work explores the ability of Bspline
registration methods to capture intersubject shape deformations.
We study the effect of different established and new shape representations,
similarity measures and optimization strategies on the matching
quality. To this end we conduct experiments on synthetic shapes representing
deformations which typically may arise in intersubject registration,
as well as on real patient data of the liver and pelvic bone. The
experiments clearly reveal the influence of each component on the registration
performance. The results may serve as a guideline for assessing
intensity based registration.
1 Introduction
Motivation. Detailed analysis of anatomical shape variability frequently depends
on identification of corresponding points on different shapes. Morphological
studies, like neuroanatomical studies of the brain, generation of anatomical
atlases, and many other applications demand such information. In recent times
statistical shape modeling has been proven a successful method in medical image
processing. The performance of statistical shape models crucially depends
on the way anatomical regions of different shapes are mapped to each other.
Anatomical correspondence across different subjects is not well understood, and
hence much harder to validate than in intrasubject matching.
Volumetric registration of medical data using B-spline transformations has been
widely applied in medical image processing [1–5]. In many cases registration is
performed directly on (tomographic) image data which implicitly contains the
shape of the object. In this work we will explore the capability of B-spline based
registration methods to capture shape variability. Therefore we focus on registration
of surfaces, where the deformation model itself can be studied more
accurately without interference originating from image-related mismatches. Particularly
in intersubject registration large deformations may occur. Anatomically
corresponding structures may differ geometrically or may be separated widely,
see for instance Fig. 1.
Fig. 1. Comparison of different liver registrations with equal surface distance: Left:
Template. Mid: Triangulation registration with boundary constraints (similar result
for distance field registration), Right: Triangulation registration without boundary constraints
(similar result for label field) leads to incorrect anatomical matching
Contributions. The aim of this work is to study the influence of different shape
representations, similarity measures as well as optimization strategies on the performance
of the B-spline based registration framework. To this end we define a
set of pairs of synthetic shapes that represent deformations which typically may
arise in intersubject registration. Moreover, we consider two anatomical shapes
of different variability: liver consists of soft tissue, and its shape is subject to
respiratory state, patient pose, and configuration of neighbouring organs, while
pelvic bone is basically a rigid structure. Here, additional anatomical expert
knowledge is available for validation. We evaluate the performance measured in
terms of surface distance after registration, regularity of the deformation map,
robustness and landmark placement. These experiments clearly reveal strengths
and weaknesses of the different components under investigation.
Previous Work. Fleute et al. [6] first used intersubject non-rigid registration
for building a statistical shape model of the knee. They employed the algorithm
by Szeliski and Lavallée [1] using asymmetric surface distance as similarity measure.
Frangi et al. applied Rückert’s registration [2] based on label fields for CT
bone, MRI brain data [7], and cardiac images [8]. They compare their method
to the work of Brett et al. [9], who use a symmetric variant of the rigid iterative
closest-points algorithm (ICP) [10] for brain data. Non-rigid extensions to ICP
have been reported recently [11, 12]. Rohlfing et al. [3] employed the algorithm
by Rückert for construction of an anatomical atlas of the honey bee.
The capability of the deformation model has not been analyzed thoroughly up
to now. It was assumed that correspondences based on B-spline registration are
fold-over free [7] as opposed to those obtained by the ICP approach of Brett
et al. We show this to be generally not the case. Usually, validation is performed
indirectly by assessing the performance of the derived shape models or
in terms of the implemented similarity measures. Rohlfing et al. evaluate the
quality (sharpness, entropy) of the averaged intensity image obtained by their
registration. Frangi et al. [8] consider landmark correspondence.
Instead of surfaces, lower dimensional structures such as landmarks [13] or feature
curves [14] are in use. Unfortunately, for many organs like the liver such
descriptions are difficult to derive due to a lack of characteristic shape features.
Incorporation of geometric features into ICP can be found in [15, 16]. Wang et
al. [17] base their semi-automatic matching on curvature classifiers.
A fundamentally different approach to non-rigid matching is based on mappings
of two-dimensional manifolds [18–20], as opposed to volumetric mappings.
2 Algorithmic Overview
The task of volumetric registration is to find a spatial mapping T : R 3 → R 3
between a template shape X and a target shape Y , such that X and Y resemble
each other as much as possible. The similarity of X and Y is defined by some
cost function E(T,X,Y ). We will study the following shape representations:
Label fields (LF) A : ΩA → LA with LA ⊂ Z implicitly contain the boundary
along voxels belonging to different segments LA (= {0,1} for binary images).
Label fields with smooth boundaries are generated by scan-conversion of triangulated
surfaces.
Signed distance fields (SDF) A : ΩA → DA with DA ⊂ R encoding for each
voxel the spatial distance to the closest point of a surface. It‘s level sets implicitly
represent a family of shapes. A is computed via euclidean distance mapping [21].
Triangulated surfaces (TS) SA ⊂ R 3 are the only parametric shape representations
considered in this work. Triangulated surfaces are typically generated
by segmentations of tomographic data using the marching cubes algorithm.
In the framework of parametric registration, the transformation T is composed
of an affine transformation Taffine as well as a B-spline deformation TB−spline.
The latter is defined on a 3D discrete uniform control point grid (CPG) with cubic
B-spline interpolation between adjacent control points, see [2, 4] for details.
The B-spline deformation model appears suitable for intersubject registration,
because it provides smooth deformations when a physical model is not known.
The optimal transformation T for a given cost function E is determined by a
nonlinear multilevel optimization scheme. The CPG is refined iteratively, providing
a parameter pyramid, while at the same time there is a data pyramid
consisting of several sampled versions of the shapes. The main intention of this
approach is to prevent optimization from being trapped in a local minimum
of the similarity criterion. By means of B-spline CPG refinement global deformations
are corrected at the beginning, while local deformations are iteratively
resolved later on. The minimum T ∗ of the cost function E is found by employing
a gradient descent-like search strategy. Instead of numerically approximating the
gradient, a search-direction is computed by scanning the whole parameter space
within some capture range depending on the level within the data pyramid [22].
The general cost function of the registration consists of a term measuring shape
similarity D, a regularization term R smoothing the transformation, plus additional
boundary constraints:
E(T, X, Y ) = D(T(X), Y ) + λR(T) + boundary constraints , (1)
where the shape similarity D consists of a weighted sum of different measures:
Similarity Measure 1 (Label Consistency). Given label fields A, B, with labels
L = {1,...,L} and image domain Ω, let pAB(l,m) denote the probability of
Fig. 2. Problem of asymmetric distance measures. Left: Initial template and target.
Mid: Result with one-sided surface distance. Right: Schematic view
cooccurence of labels l,m ∈ L in the overlap domain ΩA,B. Label consistency [8]
is measured by
L�l =1
1
|ΩA,B|�
DL(A, B) =
pAB(l, l) . (2)
Similarity Measure 2 (Grey-Value Difference). For distance fields A and
B, grey-value difference is defined as
DD(A, B) =
x∈ Ω A,B
[A(x) − B(x)] 2 . (3)
For triangulations SA and SB let ds(p, SB) = minq∈B ||p − q||2 denote the
distance of a point p on SA to the surface SB. Based on ds, we define the closest
point c on SB to p on SA by c(p, SB) = arg minq∈SB ||p − q||2. Obviously, this
correspondence is asymmetric (cf. Fig. 2, right). We propose to use a symmetric
surface distance as the fundamental similarity measure for shapes:
Similarity Measure 3 (Surface Distance) is defined by
Ds(SA, SB) =
|SB|�×�
1
|SA| +
p∈SA ds(p, SB) 2 +×�
q∈S B
ds(q, SA) 2�. (4)
Note that measures employed on triangulations should be symmetric in order
to match convex or concave regions as illustrated in Fig. 2, left. This implies a
considerable algorithmic complexity in contrast to the conventional asymmetric
scheme: the latter can be implemented efficiently using a distance map of the
target surface, while the former requires at any partial derivative calculation a
search for closest points.
Local geometric characterizations of surfaces are often included in the cost function
to improve the matching. Particularly in intersubject registration, situations
are common where anatomical structures are pronounced to a highly different
degree, yet spatially aligned closely (cf. Fig. 3). Such problems may be avoided
by incorporating the normal vector fields nA and nB of the surfaces:
Similarity Measure 4 (Normal Deviation) is defined by
Dn(SA, SB) =
|SB|��
1
dn(p, SB)
|SA| +
p∈SA 2 +�dn(q,
SA)
q∈SB 2�, (5)
Fig. 3. Ambiguity induced by a spatial attractor. The anatomically corresponding
region in the middle (l. quadr.) is hardly pronounced on the target. Left: Overview. Mid:
Close-up: displacements for correct solution. Right: Displacements for bad solution.
with dn(p, SB) = 1 − nA(p) · nB(c(p, SB)).
Other commonly used local geometric characterizations are based on the prin-
cipal curvatures κ1 and κ2 of a surface. Koenderink and van Doorn [23] introduced
two suitable classifiers: the so-called shape index S = 2
κ1+κ2
π arctan κ2−κ1 with
κ2 �= κ1 separates a surface into convex, hyperbolic and concave areas, and
transitions between these; the range is continuous within [−1,1]. Note that S
is invariant under global scaling of the surface. The curvedness C =
� κ 2 1 +κ 2 2
2 is
a suitable classifier when scale is of interest. It also has some advantages over
the mean curvature: the mean curvature vanishes at points where κ1 = −κ2 and
its magnitude is not intuitive, since it does not grow proportionally with the
radius of a sphere. Both defects are cured by the curvedness C. Misregistrations
as shown in Fig. 4 can be avoided by using such information.
Similarity Measure 5 (Curvature Similarity). Let curv denote either the
shape index S or the curvedness C. Curvature similarity is defined as
Dcurv(SA, SB) =
|SB|��
1
dc(p, SB)
|SA| +
p∈SA 2 +� q∈SB with dc(p, SB) = curv(p) − curv(c(p, SB)).
dc(q, SA) 2�, (6)
Fig. 4. Plain LF registration. Left: Target. Mid: Template. Right: Misregistration
(shape index colouring of original template is transferred to deformed template)
Regularization (Grid Energy). Large deformations in intersubject registration,
as well as over-refinement of the CPG may lead to irregular B-spline deformations.
Therefore, in some applications we use a regularization term R in
the cost function, which models the bending energy of a thin metal plate (biharmonic
model, see [1, 2]).
Boundary Constraint (Landmarks). We encountered situations in intersubject
registration where all of the above similarity measures with or without
regularization fail to achieve a reasonable registration. In this case, boundary
constraints expressed by the sum of squared differences of manually specified
corresponding landmarks on the template and the target shape may guide the
optimization towards a better solution. In practice, reliable placement of landmarks
on organs like the liver is possible in rare cases, only.
3 Results
Implementation. All components of the algorithm are implemented in one
software framework in optimized C++ code. Increased performance is achieved
by exploiting separability of B-spline interpolation and incremental evaluation
of similarity measures. Adaptive CPG refinement is accomplished by switching
off control points away from the template surface (for TS and LF registration
only).
As a benchmark for evaluating the matching capability of the B-spline registration
framework we identify three classes of deformations typically arising in
intersubject registration:
Large deformations occur when corresponding structures lie spatially far
apart, even after affine registration. As an example, consider Fig. 4: the concave
structure (“valley”) should be coloured in blue after successful matching.
Instead, it is mapped to the convex region to the right. The “cigar”/“banana”
pair in Fig. 5 exemplifies this problem.
Spatial attractors may cause ambiguities in the registration. In Fig. 3 the lobus
quadratus of the template shape might be deformed towards the left (lobus dexter),
or down towards the anatomically correct region, which is hard to detect
on the target. This situation is represented by “two hills” shapes in Fig. 5.
Absent features on one shape, which exist on the other, inevitably introduce
some degree of arbitrariness. In the liver, neighbouring organs or vessels often
cause deformations to a very different degree (cf. Fig. 9). Such cases are accentuated
in an extreme fashion by the “muffin” shapes in Fig. 5.
We consider the following criteria in our evaluation of registration performance:
a necessary requirement for a correct matching is a value for surface distance Ds
close to zero. Moreover, the transformation T should be regular, i.e. the determinant
of the Jacobian |JT | must be positive for each CPG cell. Additionally, for
synthetic shapes we examine the euclidean distance of manually defined target
points (target point deviation dt in percentage of the shape diameter).
Fig. 5. Synthetic test shapes with target points (available for download at
www.zib.de/wenckebach/ipmi05). Left: Problem 1: cigar/banana. Mid: Problem 2:
two hills. Right: Problem 3: muffins
3.1 Experiments: Synthetic Shapes
The optimization starts on a coarse CPG (5×5×5), which is successively refined
to 19 × 19 × 19. The results of all experiments are sensitive to the initial CPG
resolution. As a general rule, we found that matching quality improves when the
resolution of the CPG is adapted to the frequency content of the shape.
The step width for the search direction is iteratively decreased from 10% of
shape diameter on the initial to 0.05% on the final level. A higher initial CPG
resolution was necessary for LF and SDF registration of the muffins. In all cases
the surface distance vanishes after registration except for the muffin shapes using
SDF or LF registration. Regularity is violated whenever using TS registration.
Moreover, all experiments show that registration is not robust with respect to
the weighting of the geometric similarity measures.
Fig. 6. Results banana. Left: TS registration with curvedness (1st level, result). Right:
Combination of SDF and TS registration with curvedness (1st level, result)
Banana. For TS registration, the best result is shown in Fig. 6, left (dt = 3.3%).
Carefully adjusting the weight of curvedness in the cost function improves matching
the cusps. Yet, distortions are spread unevenly over the surface and there
is little regularity. Amplifying the grid energy reduces surface distortion at the
cost of larger target point deviation (dt = 8.3%). SDF registration combined
with curvedness yields better results (dt = 2.8%), cf. Fig. 6 right. Neither shape
index (dt = 47.8%) nor normal deviation (dt = 11.7%) lead to improvements.
Two Hills. Shape representation plays a crucial role in this example: SDF registration
yields nearly perfect target point deviation of dt = 0.7%, whereas LF
works satisfactorily only with a much larger initial search width (dt = 9.3%).
TS registration leads to large mismatches with dt = 13.3%, which is improved
by using normal deviation (dt = 1%) or shape index (dt = 1.3%), cf. Fig. 7.
Curvedness performs better (dt = 0.7%) at the cost of enormous surface distortions;
grid energy alleviates this. Both shape index and curvedness produce
irregular deformations.
Fig. 7. Results two hills. Left: TS registration (1st level, result). Right: TS with normal
deviation (1st level, result)
Muffin. Implicit representations require higher CPG resolution to accomplish a
satisfactory target point deviation (LF: dt = 6.3%, SDF: dt = 7.9%). TS registration
has larger target point deviation of dt = 8.4%, which is improved by
considering normal deviation (dt = 5.8%); curvature measures yield no improvements.
TS registration produces severe surface foldings, while SDF registration
shows massive surface distortions (cf. Fig. 8). SDF registration will often fail in
such cases, since deformations along the surface are ill-defined. Grid energy fails
to alleviate this problem due to unacceptable matching quality.
The performance in terms of CPU time is – in all cases – best using LF or TS
registration with one-sided surface distance. SDF representation is worse by a
factor of 5, while symmetric TS registration increases the runtime by a factor of
20, due to the complexity of evaluating the two-sided surface distance.
Fig. 8. Results muffin. From left to right: LF, SDF (surface distortion), TS registration
(severe surface foldings), TS with normal deviation (improved matching)
3.2 Experiments: Anatomical Shapes
The optimization starts on a CPG of about 100 mm grid spacing, which is successively
refined to 5 mm. For the LF representation an increased search width is
employed. No geometric similarity measures are incorporated into the cost function,
as this would require an extensive parameter study for the relative weights
within the cost function.
Fig. 9. SDF registration for the liver. Anatomically corresponding regions can be identified
by their colour; target point locations indicate matching errors. Left: Template
after affine registration. Mid: Target. Right: Resulting deformed template. The “valleys”
of the template are not matched perfectly, yet the transformation remains regular;
surface distortion is moderate, anatomical matching is satisfactory
Liver. The sample consists of 24 individuals. All shapes are registered to one
target shape. Registration based on distance fields is always performed in combination
with regularization to avoid distortions as present in the muffin example
(cf. Fig. 8). The grid energy is applied adaptively on the last two levels of the
optimization (λ = 0.002/0.01).
Anatomical mismatch is measured by the overlap do of corresponding anatomical
regions (patches) of the liver. The surface distance is computed among corresponding
patches only, and afterwards divided by Ds: the smaller do, the better
the anatomical match. Although these regions cannot in general be specified
uniquely, this measure is less sensitive to errors than individual landmark placement.
The following regions were defined by medical experts (cf. Fig. 9): (1)
lower left lobe, (2) lower right lobe plus caudate lobe, (3) lower quadratic lobe
and (4) whole upper part of the liver. Cases of large anatomical mismatches
(cf. Fig. 1) could be resolved by using landmark based boundary constraints. A
typical result is shown in Fig. 9. Quantitative results are given in Tab. 1.
Table 1. Results for registration of real patient data (GE grid energy). Maximum
and median of surface distance over the whole set, as well as the mean percentage of
surface distance above a threshold of 2mm is given. Average CPU times refer to a SGI
system with 500 MHz MIPS R14k processor. For liver data, a histogram of anatomical
mismatch (log. scale) is provided
shape method Ds [mm] > 2[%] CPUtime
class max med mean [hh:mm:ss]
LF 0.71 0.49 4.73 00:42:45
liver SDF GE 1.27 0.44 3.84 25:11:33
TS 0.44 0.32 0.71 11:40:49
LF 0.74 0.61 4.75 01:22:04
pelvic SDF 0.83 0.45 1.98 03:10:09
bone TS 0.38 0.34 0.33 31:35:24
Anat. Mismatch
1.4
1.2
1.0
0.8
0.6
0.4
0.2
LF
SDF
TR
0.0
1 4 7 10 13 16 19 22 Livers
Pelvic Bone. The sample comprises 17 male individuals. The target shape is
reconstructed from the male visible human data set. One exemplary result is
shown in Fig. 10. Quantitative results over the whole sample are given in Tab. 1.
The deformation map is fold-over free for all shape representations. Visual inspection
shows virtually no anatomical mismatch in all cases apart from the LF
registrations, where the variability of the bending of the sacrum is not captured
correctly (only 6 correct results).
Fig. 10. SDF registration for pelvic bone. Left: Initial setting, result. Mid, right: Deformed
template (with little surface distortion)
4 Discussion and Conclusion
We studied the ability of a registration framework based on B-spline transformations
to capture intersubject shape variability. To this end we identified three
different classes of typical deformations, which we represented by three different
synthetic shapes. Moreover, two anatomical objects of different degrees of variability
were examined. For our experiments we varied the essential components
of the framework: shape representation, similarity measure and optimization
strategy. The performance was measured in terms of surface distance between
the deformed template and the target shape, regularity of the transformation T,
robustness and correspondence of landmark points or regions.
Optimization Strategies. The resolution of the CPG plays a crucial role.
Although B-spline transformations are indeed capable of capturing large shape
deviations, the control point spacing should be adapted to the frequency content
of the shapes. This was shown by the banana/cigar example, where most of the
deformation takes place around the cusps. Harmonic analysis of the shapes may
be a suitable pre-processing step for building adaptive control grids.
Similarity Measures. Using the symmetric surface distance is very costly, yet
may be needed in cases where deformations are large. Weighting the terms in
the cost function is a difficult task, especially between surface distance and geometric
features. Although it seems fairly clear for the synthetic shapes, which
geometric similarity is feasable to use, one cannot deduce from this the correct
cost function for the anatomical examples. Completely different cost functions
may certainly be more suitable for other applications. We found that the grid
energy as a regularizer often is too restrictive to recover large deformations. To
prevent large surface distortions, yet obtain good matchings, regularizers constraining
deformation in the tangential directions of the shapes rather than in
the normal direction should be employed.
Shape Representations. Of all shape representations used the SDF approach
performed best. Triangulation based registration often lead to irregular transformations,
while label field registration yielded deficient matchings, both in terms
of target point deviation for the synthetic shapes and of anatomical correspondence
for the medical examples. This can be explained by the fact that distance
fields contain more information than triangulated surfaces or label fields: the
latter encode a single 2-dimensional manifold while distance fields contain a
continuous set of manifolds by extending the surface into R 3 , which is beneficial
in the optimization process.
Since the shape of anatomical structures is contained only implicitly in medical
image data (e.g. MRI or CT), it is a-priori uncertain whether B-spline transformations
represent the appropriate deformation model. The results of our experiments
may serve as a guideline for assessing such intensity based registration
tasks.
Future work will be directed towards combining SDF representations with different
regularizers. Moreover, cost functions with different structures than the
one used here should be explored. As a general rule, any a-priori anatomical
knowledge available should be included to support intersubject registration.
Acknowledgements. We thank the developers of Amira [24], a software framework
for advanced visual data analysis, that has been used for this work.
Funding for T.H. Wenckebach and H. Lamecker has been provided by Deutsche
Forschungsgemeinschaft (DFG) through the research center Matheon.
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