1 Introduction
Matching a shape template to an image is a well studied problem in computer vision and image analysis. It gives rise to a wide range of applications, including image segmentation and object detection. An early approach for the detection of lines and parametrised curves in images is the votingbased Hough transform
[14], which was later generalised to the detection of arbitrary shapes [1].Whilst the Hough transform considers rigid shapes, the utilisation of shape information in image segmentation tasks has also been addressed in the nonrigid case, including methods based on active shape models [10], level sets [12], convex shape spaces [13], multiphase graph cuts [58], or statistical shape models [19, 63]. For the reconstruction of the shape of an object from a single 2D image, shapefromtemplate approaches aim to match a given 3D template to the image via a 3Dto2D projection [50, 38, 46]. Several authors have considered combinatorial formulations of the nonrigid shapetoimage matching problem for certain classes of shapes. For the case of matching contours [11, 53], or 2D chordal graph polygons [16], the resulting optimisation problems can be solved globally. However, a generalisation of these methods to 3D shapes is nontrivial and we are not aware of previous work addressing this issue. The purpose of this work is to fill this gap by presenting a combinatorial formulation for the nonrigid matching of a 3D shape template to a 3D image. For that, we model the shape as a triangular mesh and allow each triangle of this mesh to be independently transformed via a rigid transformation . Using a discretisation of the unbounded 6dimensional Lie group , we formulate the matching task as a manifoldvalued multilabelling problem that can be cast as minimising the energy
(1) 
Here, the data term takes the image information into account, while the smoothness term measures the dissimilarity between the observed shape and the modelled shape prior. By penalising the distance and the relative rotation of neighbouring triangles our matching compromises between image and shape information. In general, minimising functions of the form in (1) is NPhard [26].
1.1 Related Work
To the best of our knowledge the present paper is the first one that considers a combinatorial formulation of the nonrigid 3D shape to 3D image matching problem. In the following we will summarise methodologies that are most relevant to our work.
Continuous Optimisation: In many scenarios it is natural to assume that image or shape deformations are spatially continuous and smooth. Frequently, such problems are formulated in terms of optimisation problems over the space of diffeomorphisms [15, 40, 2, 41]
. Commonly, gradient descentlike methods are used to obtain (local) optima of the (typically nonconvex) problems. However, a major shortcoming of these methods is that a good initial estimate is crucial and in general there are no bounds on the optimality of the solution. To deal with the nonconvexity of a 2D shapetoimage matching problem that is formulated in terms of optimal transport, the authors in
[52] propose to use a branch and bound scheme.Shortest Paths and Dynamic Programming: In contrast to the continuous local optimisation methods, many vision problems can be formulated in a discrete manner such that they are amenable to solutions based on graph algorithms and dynamic programming (DP) [17]. Since curves are intrinsically onedimensional, various curve matching formulations can also be reduced to finding a shortestpath in a particular graph. Moreover, based on a recursive formulation using easiertosolve subproblems, matching problems with templates that have a tree structure can frequently be tackled by DP. For a deformable matching of an open contour to a 2D image, a global solution based on DP has been proposed in [11]. Also based on DP, in [16] the authors present a method for solving the problem of deformably matching a 2D polygon to a 2D image for chordal graph polygons. In [53], the authors propose a globally optimal approach for matching a closed contour to a 2D image based on cycles in a product graph of the contour and the image. A related formulation that is also based on a product graph has recently been introduced in [30] for deformable contour to 3D shape matching.
Graphcuts: It is well known (see e.g. [4]) that any cut of a graph can be interpreted as finding a closed manifold of codimension 1 in the ambient space (e.g., closed curve in 2D, closed surface in 3D, etc.). One such example is the reconstruction of a 3D shape from a set of sparse 3D points, where the latter is represented on a discrete 3D grid [35].
Labelling Problems: Labelling problems are ubiquitous in computer vision and appear both in continuous and discrete settings [62]. The popular Markov Random Field (MRF) framework offers a Bayesian treatment thereof [37]
. Also, linear programming relaxations of MRFs have been studied
[59]. The continuous approaches to multilabelling include various convex relaxations [47, 32, 55, 18], multilabelling problems with total variation regularisation of functions with values on manifolds [33], as well as sublabelaccurate convex relaxations [42, 31]. Among the discrete multilabelling methods are the previouslymentioned graphcuts, which can be used to find global solutions for certain binary labelling problems, including problems with submodular pairwise costs [25]. For a subclass of multilabelling problems a global solution can also be found [25]. This subclass includes pairwise costs that are convex in terms of totally ordered labels [22]. In addition, efficient algorithms for finding local optima of general multilabelling problems have been proposed [6, 27], which even have theoretical optimality guarantees. A more detailed description of the energy functions that can be optimised using graphcuts is given in [26, 24, 25].1.2 Main Contributions
The main contribution of this paper is to present for the first time a combinatorial formulation of the nonrigid 3D shape to 3D image matching problem. Whilst our problem is a natural extension to the aforementioned “dimension one” matching approaches [11, 16, 53, 30], a generalisation to (intrinsic) dimension two problems is more intricate. Our main contributions are:

By using a surface mesh transformation model that makes use of pertriangle rigid transformations, we formulate the 3D shape to 3D image matching problem in terms of a manifoldvalued multilabelling problem.

We introduce a pairwise term that defines a metric on the label space , which itself is a highdimensional Lie group. With that, our energy function is amenable to be minimised by the expansion algorithm [6], which has been shown to work well in practice, is efficient even for very large label spaces, and has theoretical optimality guarantees.

In contrast to continuous optimisation methods that use gradient descentlike algorithms, our combinatorial method does not require a good initialisation.

In order to deal with the computationally challenging discretisation of , we propose to use a coarsetofine discretisation of the Lie group.
2 NonRigid 3D ShapetoImage Matching
In this section we first specify our objective, followed by a description of the data term and smoothness term. After introducing the combinatorial problem, we describe the discretisation of the label space and we discuss the algorithmic solution of the problem.
2.1 Objective Function
In the following, we assume that a 3D shape is given as a triangular mesh. This means we have triangles such that
(2) 
We use the set to define the neighbourhood between pairs of (different) triangles. We assume that for all the neighbouring triangles and are nondisjoint and that the intersection results either in a common edge or a common vertex. Also, w.l.o.g. we assume that for each it holds that , i.e. .
Our objective is it now to match the 3D shape onto a volumetric image , where denotes the compact image domain and describes the amount of image channels. While we are interested in a nonrigid shapetoimage matching, we like to favour matchings that are asrigidaspossible, similar to the approach in [54] that applies (locally regularised) rigid transformations to each vertex. However, in our case this is done by applying to each triangle a rigid transformation
(3) 
where represents the rotational part and represents the translational part of . The task of finding the best matching can be formulated as minimising the energy
(4) 
In Section 2.2 we define the data term that evaluates how well the transformed triangle fits to the image data. In Section 2.3 we define the smoothness term that measures the geometric dissimilarity between the shape model and the transformed shape
(5) 
Using the proposed piecewise rigid transformation model we may end up with a model having (small) gaps or intersections between neighbouring triangles. We will later address this issue and present a simple yet effective way of dealing with this irregularity.
2.2 Data Term
The data term measures how well the transformed triangle fits to the image data . For that, we introduce the score image that is derived from the image
(e.g. a gradient magnitude image, or more advanced predictors based on neural networks). For a triangle
, we define(6) 
With that, the value indicates how well the triangle fits to the image data, where a high value of the score image indicates a good fit. The data term is then given by
(7) 
In the discrete setting, the data term is computed by a weighted sum of function values over the triangle. The weights take the rasterisation of the deformed triangle in the image into account.
2.3 Smoothness Term
The pairwise term penalises the disagreement between neighbouring triangles and after they have been transformed by and , respectively.
For defining the pairwise term we first introduce suitable distances. The expression
(8) 
is the geodesic distance between the rotations and on [21], with matrix logarithm . For and being the quaternion representations of and , one can efficiently compute the distance as , where is the quaternion innerproduct [21].
For defining a distance between neighbouring triangles, we make use of the concept of group actions. To be more specific, we define
(9) 
where the group acts on the nonempty compact set . In our case we use such that can be seen as distance between the deformed triangles and . In this case the maximum in is achieved at the common vertices of and , which is attractive from a computational point of view.
Using the introduced distances, we define our pairwise term as a weighted sum thereof, i.e.
(10)  
The purpose of the bending term, weighted by , is to ensure that the rotations of neighbouring triangles are similar. The stretching term, weighted by , ensures that neighbouring triangles stay close together.
2.4 Combinatorial Formulation
A matching of shape to the image is given by a solution of the optimisation problem
(11) 
Due to the nonconvexity of the feasible set , it follows that Problem (11) is nonconvex. This nonconvexity makes it difficult to solve the problem directly over the unbounded continuous space . Our approach is to optimise instead over a discretisation of the search space. With that, we obtain a multilabelling problem, for which efficient and effective algorithms are available.
For the discretisation of we make use of the fact that it is a product space of and . Thus, we define to be the (finite) manifoldvalued label space that contains elements of .
Translations: The Lie group is noncompact due to the translational part being encoded by . However, since the image domain is compact, the image size provides natural bounds for a discretisation of the translations. Let be the number of voxels of the image and be the number of labels for the , , translations. For convenience, and w.l.o.g., we assume that our template is defined relative to the centre of the image domain, i.e. the template’s centreofgravity coincides with the centre of the image. Moreover, w.l.o.g., we assume that we are looking for a matching such that a substantial part of the (transformed) template lies inside the image^{1}^{1}1If this is not the case, one can increase the image size accordingly.. Let us define to be the set containing evenlyspaced elements with centre , where
is an odd positive integer. The
diameter defines the difference between the largest and the smallest elements. A discretisation of the translations is given by the set with .Rotations: Various works that are related to the discretisation of have previously been presented. These include sampling strategies for rigidbody path planning [29], an approximation of the neighbourhood in
based on vector distances
[36], or an analysis of various metrics for 3D rotations [21]. Our discretisation of is based on the Hopf fibration, which describes in terms of the circle and the 2sphere . The intuition of this approach is to transfer a discretisation of and to the space of rotations. We refer the interested reader to [61] for a detailed description. Let denote the soobtained set of a uniform sampling of containing elements.By optimising over the label space , we now obtain the combinatorial optimisation problem as
(12) 
3 Algorithm
In order to solve Problem (12), we use expansion [6, 26, 5], which greedily updates only one label at a time. We note that there are also potential alternatives to expansion (e.g. for nonmetric pairwise terms [27, 55], or fusion moves [34]). Whilst expansion has the requirement that the pairwise term is a metric, it is appealing both from a practical and a theoretical point of view. To be more specific, it is efficient, robust with respect to initialisation, the obtained local optimum is guaranteed to lie within a factor of the global optimum, and an efficient implementation that supports the online computation of the smoothness term is available [6, 26, 5], which is crucial for the size of problems that we are solving. We now show that our pairwise term is a metric and thus expansion is applicable.
Lemma 1
Let be a nonempty compact set and . The stretching term
is a pseudometric, i.e. it satisfies

, ,

symmetry: , and

the triangle inequality:
Proof: (i) and (ii) follow directly from the definition. The triangle inequality holds, since
Proposition 1
For and , the pairwise term defined in (10) is a metric.
Proof: Due to the assumption in Section 2.1, for it follows that is compact. Thus, is a pseudometric (Lemma 1). Whilst is known to be a metric on , it is only a pseudometric on . Since is a positive linear combination of two pseudometrics, is also a pseudometric. To show that is a metric, we show that implies . For , it holds that , which implies . Moreover, with and , it holds that , which implies . Hence .
3.1 CoarsetoFine Processing
In practice, for a reasonably large number of labels , a direct solution of Problem (12) is intractable. In order to cope with this issue we propose to use a coarsetofine strategy that (approximately) solves Problem (12) at different levels of the label space. Let denote the coarsest (initial) level and the finest (final) level. Once a solution has been obtained at level , for running the algorithm at level the labelling is initialised with and the label space is updated accordingly. For computational efficiency, in the coarsetofine approach each triangle has its own feasible label space. Let denote this feasible label space for the th triangle at level . Initially, on the base level , the label spaces are the same for each triangle, i.e. . The general idea for obtaining is to consider a (uniform) discretisation of the neighbourhood of the transformation at level , where the radius of the neighbourhood decreases across the levels. Let us introduce a neighbourhood on :
Definition 1
(ball on )
The ball on with radius and centre is defined as
.
Next, we describe the coarsetofine structure of the label space based on its product space nature. Since each label can be written as
(13) 
we can consider the translations and rotations independently. Note that is not necessarily a group anymore. By enforcing that , the solution at level is also contained in the new label space. Thus, the energy cannot increase from level to .
Translations: We define (cf. Section 2.4). For obtaining the set of translations at level for triangle , the new translation grid at level is centred at . Moreover, the diameter from level is reduced by a factor of two, leading to
(14)  
where the vectorset addition is elementwise. Initially, and .
Rotations: Let denote a discretisation of (the entire) at resolution containing elements, where increases with increasing . should not be confused with , which is the (rotation) label space of the th triangle at level that is to be defined below. Following the construction in [61], we obtain resolutions of the discretisation for . After including the identity in , the number of elements ranges from to . Let us define a set that contains a fixed number of elements from that are “closest” to the identity, i.e.
(15) 
where for each resolution the smallest that fulfils is used as radius.
Now, we define , and
(16) 
which is the set of compositions of with all rotations in . We use for all levels . For the predefined griddings at resolutions there always existed an such that the above inequality is tight, i.e. .
By including the identity in , we make sure that . Since , it follows that . Thus, we have , ensuring that the energy cannot increase when moving from level to .
3.2 Practical Considerations
In this section we describe some aspects for the application of the proposed method in practice.
3.2.1 Mesh Connectivity
After applying an individual rigidbody transformation to each triangle of the template mesh, in general the resulting mesh may have gaps or intersections between neighbouring triangles. However, due to the introduced regulariser, these gaps or intersections can be expected to be rather small. For recovering the original mesh topology, we replace each subset of vertices having the same position in the mesh template by their centreofgravity after the transformation.
3.2.2 Memory Requirements
For running our algorithm we precompute the data term, requiring memory of (the online computation has constant memory requirements but leads to a significantly increased runtime). Since precomputing the full pairwise term requires memory of , we only precompute the bending term , requiring memory of . The stretching term is computed online.
4 Results
For the evaluation of our method we focus on demonstrating the general applicability of our approach. Both pointset registration [3, 48, 8, 36, 44, 43, 23, 20, 49, 39] and the related correspondence problem for 3D shapes [56, 60, 28, 7] can be tackled using our method by recasting them as shapetoimage matching problem. Hence, in our evaluation below, in addition to 3D image segmentation, we also consider the case of deformable mesh registration.
4.1 Deformable Mesh Registration
In the first set of experiments we demonstrate that our method can be used to perform deformable mesh registration. To emphasise that our method is insensitive to initialisation, we compare it exemplarily with the Coherent Point Drift (CPD) algorithm [44, 43], a widelyused pointcloud registration method based on Expectation Maximisation.
Template and Target: For the evaluation we use a lowresolution mesh of the Stanford bunny as template (), as shown in Fig. 1 (top left). A total of deformed versions of the bunny mesh, each with a random pose, are used as registration targets. For that, we synthetically create deformed versions of a highresolution bunny mesh (Fig. 1, top right) based on random 3D displacement vectors defined at
control points on a cubic grid. These displacement vectors are then transferred to the mesh using a splinebased interpolation in order to achieve a smooth and nonlinear deformation. One such deformed mesh is shown in Fig.
1 (bottom left). Eventually, a random pose transform is applied to the deformed shape, as shown in Fig. 1 (bottom right).


Score Image: In order to use our method for mesh registration we create a score image for each target mesh and then fit the template mesh to the score image. For we denote by the distance of position to the boundary of the target mesh. Now, we define the score image as , where we used .
Parameters: We set and , where the normalisation factor takes the problem size into account. The positive number is an upper bound for the largest possible absolute value of the data term for a single triangle (cf. eq. (6)), which we compute by multiplying the largest value of the score image by the area of the largest triangle. The size of the label space is , and the dimensions of the score image range from to , which resulted in an average processing time of minutes per registration on a MacBook Pro (2.5GHz, 16GB).
Results: In the case of CPD, we first solve for a rigid registration and then for a nonrigid registration (performing a nonrigid registration directly performed worse). Since CPD is highly initialisationsensitive it fails in out of the evaluated cases, where one representative failure case is depicted in Fig. 2 (top row). This extreme amount of corrupt registrations emphasises the necessity of a method that is robust with respect to initialisation. In contrast, in all cases our method is able to achieve a good registration, see Fig. 2 (bottom row) for a representative result. In Fig. 3 we present a quantitative evaluation.
Discussion: Whilst we do not claim to present an exhaustive evaluation of mesh registration methods, we demonstrated the insensitivity to initialisation of our method in a proof of concept manner. One advantage of our approach is that we neither have the necessity of compatible mesh topologies, nor of compatible mesh discretisations, since the target is represented in terms of the score image. Since the score image is a discrete representation of the target shape surface, our approach amounts to a surfacebased registration, rather than a pointbased registration as CPD, which is biased towards aligning points to be as close as possible. Moreover, the score image offers further flexibility since additional information can be integrated (e.g. uncertainties, mesh texture, shape features, etc.).
4.2 Segmentation
In the second set of experiments we apply our method to the segmentation of four brain structures (substantia nigra & subthalamic nucleus as single object and the nucleus ruber, both bilaterally) in multimodal T magnetic resonance images. The delineation of the subthalamic nucleus is known to be a challenging task, even for humans [51]. The main difficulties include weak image contrasts and the small size of the brain structures (the structures shown in Fig. 4 are contained in a bounding box of , with the MRI image covering a volume of ).
Template: For capturing the interrelation between the brain structures, we use a multiobject template (), as shown in Fig. 4. The template connects neighbouring brain structures by (degenerate) triangles, referred to as “phantom triangles”, which are used only for the smoothness term and are “free” with respect to the data term.
Parameters: We set and , where is defined as before. The size of the label space is , and the dimension of all score images is , which resulted in an average processing time of minutes per fitting.
Score Image: In order to perform image segmentation with our method, we use a data term that is based on the recently proposed 3D UNet CNN [9]. For all images we train the network in a leaveoneout manner for the prediction of volumetric segmentations. In the centre column of Fig. 6 three examples of sopredicted volumetric segmentations are shown. For this challenging segmentation task the UNet is able to identify the (rough) location of the brain structures, but does in many cases not produce an output that resembles the shape of the brain structures (the first two rows in Fig. 6). Thus, we complement the UNet segmentations with geometric information using our method.
For each of the four brain structures we use an individual score image . Given the binary UNet segmentation for brain structure , we first extract the (predicted) boundary using morphological operations. For we denote by the distance of position to the soextracted boundary. Then, we use a Gaussian kernel to define the score image as
(17) 
The weight and bandwidth are used for incorporating the confidence about the UNet segmentation of brain structure . To this end, for
(18) 
being the onelevel set of that represents the pointcloud of segmented voxels, we use the average of the (percoordinate) median absolute deviation (MAD) as robust dispersion measure, computed as
(19)  
The median is understood in a percoordinate sense, and both the setvector difference and the absolute value are understood elementwise. Now, for denoting the pointcloud of segmented voxels of structure as given by the template, the absolute value of the average MAD difference is given by . With that, we define , scaled by . Thus, if the average MAD for the UNet segmentation and the template are equal, the bandwidth corresponds to , whereas a larger difference in dispersion leads to a larger bandwidth, accounting for more uncertainty in the UNet segmentation. Moreover, we define such that an increased uncertainty in the UNet segmentation of brain structure leads to a decreased weight for its data term.
Results: For the evaluation of the segmentation we use the Dice Similarity Coefficient (DSC) as volumetric overlap measure, which is defined as , for and each being pointclouds of segmented voxels. We compute the DSC for each individual brain structure, and then report the average of the four DSC values. Quantitative results comparing the plain UNet segmentation and our obtained segmentations are presented in Fig. 5. The boxplot on the left reveals that in overall our method achieves higher volumetric overlaps across the cases. Moreover, the plot of sorted DSC differences on the right emphasises that applying our method improves the DSC in most cases (the values above zero), and in only a few cases it is reduced slightly.
In Fig. 6 qualitative results for three segmentation cases that correspond to the best, median and worst cases in Fig. 5 (right) are shown. In the first row of Fig. 6 it can be seen that in some cases our method is even able to achieve a reasonable segmentation based on a poor UNet segmentation. This was possible by putting a stronger emphasis on the shape information relative to the data term, which also biases the method towards the shape information. A related discussion on the biasedness of modeltodata fitting approaches in structurefrommotion can be found in [45].

5 Conclusion
We introduced the first combinatorial method for nonrigidly matching a 3D shape to a 3D image. The key idea is to represent the 3D shape as a triangular mesh and to solve a manifoldvalued multilabelling problem on the set of triangles. We determine an assignment of a rigidbody transformation associated with each triangle by minimising a cost function where the unary terms encode the local matching cost in the image and the pairwise terms penalise the amount of nonrigidity in the deformation. In particular, we propose an efficient discretisation of the unbounded 6dimensional Lie group of rigid motions. Moreover, we solve the large and NPhard optimisation problem with a graph theoretic algorithm that is insensitive to initialisation and has the guarantee that the obtained solution is within a factor of the global optimum [6]. Experimental validation confirms these benefits.
Acknowledgements
The authors would like to thank Özgün Çiçek for helpful feedback regarding the UNet implementation, as well as Benedikt Staffler and Ankush Gupta for general feedback on CNNs. CNNrelated computations presented in this paper were carried out using the HPC facilities of the University of Luxembourg [57]. The authors gratefully acknowledge the financial support by the Fonds National de la Recherche, Luxembourg (6538106, 8864515). This work was supported by the ERC Consolidator Grant 3D Reloaded.
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