^{24}], as skeletal structures, or hierarchies. Examples encountered in image analysis and computational biology are airway trees [

^{14}], [

^{25}], [

^{37}], vascular systems [

^{7}], shock graphs [

^{3}], [

^{31}], [

^{36}], scale space hierarchies [

^{8}], and phylogenetic trees [

^{5}], [

^{17}], [

^{28}].

^{33}], [

^{40}] could improve tools for computer-aided diagnosis and prognosis.

^{14}], [

^{25}], object recognition [

^{8}], and machine learning [

^{15}], [

^{18}], [

^{29}] based on intertree distances. However, most existing tree-distance frameworks are algorithmic rather than geometric. Very few attempts [

^{2}], [

^{27}], [

^{39}] have been made to build analogues of the theory for landmark point shape spaces using manifold statistics and Riemannian submersions [

^{20}], [

^{21}], [

^{32}]. There exists no principled general approach to studying the space of tree-structured data, and as a consequence, the standard statistical properties are not well defined. As we shall see, difficulties appear even in the basic problem of finding the average of two tree-shapes. This paper starts to fill the gap by introducing a shape-theoretical framework for geometric trees, which is suitable for statistical analysis.

*mean*(or prototype) for a dataset , which can be defined as the minimizer of the sum of squared distances to the data points:

(1)

*Fréchet mean*, assumes a space of tree-shapes endowed with a distance , and is closely connected to geodesics, or shortest paths, between tree-shapes. For example, the midpoint of a geodesic from to is a mean for the two-point dataset . Hence, if there are multiple geodesics connecting to , with different midpoints, then there will also be multiple means.

*As a consequence, without (local, generic) uniqueness of geodesics, most statistical properties are fundamentally ill posed!*

^{16}], we show that QED generically gives locally unique geodesics and means, whereas finding geodesics and means for TED is ill-posed even locally. We explain why using TED for computing average trees must always be accompanied by a carefully engineered choice of edit paths to give unique results, choices which can yield average trees that are substantially different from the trees in the dataset [

^{36}]. The QED approach, on the contrary, allows us to investigate statistical methods for tree-like structures that have previously not been possible, like different well-defined concepts of average tree. This is our motivation for studying the QED metric.

^{31}], [

^{36}]. TED and, more generally, graph edit distance (GED) are also popular in the pattern recognition community and are still used for distance-based pattern recognition approaches to trees and graphs [

^{15}], [

^{29}]. The TED and GED metrics will nearly always have infinitely many shortest edit paths, or geodesics, between two given trees because edit operations can be performed in different orders and increments. As a result, even the problem of finding the average of two trees is not well posed. With no kind of uniqueness of geodesics, it becomes hard to meaningfully define and compute average shapes or modes of variation. This problem can be solved to some extent by choosing a preferred edit path [

^{15}], [

^{29}], [

^{36}], but there will always be a risk that the choice has negative consequences in a given setting. Trinh and Kimia [

^{36}] face this problem when they use TED for computing average medial axes using the simplest possible edit paths, leading to average shapes that can be substantially different from most of the dataset shapes.

^{39}] study metric spaces of trees and define a notion of average tree called the median-mean as well as a version of PCA which finds modes of variation in terms of

*tree lines*, encoding the maximum amount of structural and attributal variation. Aydin et al. [

^{2}] extend this work by finding efficient algorithms for PCA. This is applied to analysis of brain blood vessels. The metric defined by Wang and Marron does not give a natural geodesic structure on the space of trees as it places a large emphasis on the tree-topological structure of the trees. The metric has discontinuities in the sense that a sequence of trees with a shrinking branch will not converge to a tree that does not have that branch. Such a metric is not suitable for studying trees with continuous topological variations and noise, such as anatomical tree structures extracted from medical images, because the emphasis on topology makes the metric sensitive to structural noise.

^{4}] analyze the geometry and topology of shapes from their skeletal graphs using Morse theory. Jain and Obermayer [

^{18}] study metrics on attributed graphs, represented as incidence matrices. The space of graphs is defined as a quotient of the euclidean space of incidence matrices by the group of vertex permutations. The graph space inherits the euclidean metric, giving it the structure of an orbifold. This construction is similar to the tree-space presented in this paper in the sense that both spaces are constructed as quotients of a euclidean space. The graph-space framework does not, however, give continuous transitions in internal graph-topological structure, which leads to large differences between the geometries of the tree and graph spaces.

^{17}] visualize sets of phylogenetic trees using multidimensional scaling. Billera et al. [

^{5}] invented a geodesic phylogenetic tree-space for analysis of phylogenetic trees, and Owen and Provan [

^{28}] have developed fast algorithms for computing geodesics in phylogenetic tree-space. Nye [

^{27}] developed the first notion of PCA in phylogenetic tree-space, but makes strict assumptions on principle components being “simple lines” for the sake of computability. Phylogenetic trees are not geometric, and have fixed, labeled leaf sets, making the space of phylogenetic trees much simpler than the space of tree-shapes.

^{12}], [

^{13}] studied geodesics between small tree-shapes in the same type of singular shape space as studied here, but most proofs have been left out. In [

^{11}], we study different algorithms for computing average trees based on the QED metric. This paper extends and continues [

^{12}], giving proofs, in-depth explanations, and more extensive examples illustrating the potential of the QED metric.

*a geodesic passing through the trivial one-vertex tree should indicate that the trees being compared are maximally different.*We want to compare trees where the desired edge matching is inconsistent with tree topology, as in Fig. 1a, which requires geodesic deformations in which the tree topology changes, e.g., as in Fig. 1b.

*same*tree to encode tree topology. By choosing a sufficiently large , we can represent all the trees in our dataset by filling out with empty (collapsed) edges. We call

*the maximal tree*.

*binary*maximal trees . Tree-shapes which are not binary are represented by the binary tree in a natural way by allowing constant, or collapsed, edges, represented by the zero scalar or vector attribute. In this way,

*an arbitrary attributed tree can be represented as an attributed*

**binary**

*tree*, see Fig. 4a. This is geometrically very natural. Binary trees are geometrically

*stable*in the sense that small perturbations of a binary tree-shape do not change the tree-topological structure of the shape. Conversely, a trifurcation or higher-order vertex can always be turned into a series of bifurcations sitting close together by an arbitrarily small perturbation. Thus, in our representation, trifurcations are represented as two bifurcations sitting infinitely close together, and so on.

*ordered*whenever each set of sibling edges in the tree is endowed with such a total order. Conversely, an ordered combinatorial tree always has a unique, implicit embedding in the plane, where siblings are ordered from left to right. For this reason, we use the terms “planar tree” and “ordered tree” interchangingly. We first study metrics on the set of ordered binary trees, which are heavily studied in pattern recognition and vision [

^{3}], [

^{31}], [

^{36}]. Next, we induce distances between unordered trees by considering all possible orders, allowing us to model trees in . Considering all orders leads to computational challenges, which are discussed in Section 5.

^{20}].

*the same*when their collapsed ordered, attributed versions are identical, as in Fig. 4a. More precisely, given a preshape tree , denote by

(2)

(3)

(4)

^{6}, Chapter 1.5]. The geometric interpretation of the tree-space quotient is that we “glue” the preshape space along the identified subspaces; that is, when , we “glue” the two points and together. See Fig. 4c for an illustration.

(5)

(6)

^{6}] induced on the quotient space , defined by

(7)

^{26}]. This is why is called a

*pseudo*metric, and it remains to prove that it actually is a metric, that is, that implies .

(8)

(9)

**Theorem 1.** *The distance function is a metric on which is a contractible, complete, proper geodesic space.*

**2.3.1 Geometric Interpretation of the Metrics**It follows from the definition that the metric coincides with the classical TED metric for ordered trees. In this way, the abstract, geometric construction of tree-space gives a new way of viewing the intuitive TED metric.

The metric is a descent of the euclidean metric on , and geodesics in this metric are concatenations of straight lines in flat regions. A qualitative comparison of the TED and QED metrics is made in Section 5.

^{6}], [

^{9}]. It is crucial for the proof that we are only identifying subspaces of the euclidean space which are spanned by euclidean axes, and these are finite in number. This induces a well-behaved projection , which carries many properties from to .

**2.4.1 Details and Terminology**We say that ordered tree-shapes whose (collapsed) ordered structure is the same belong to the same

*combinatorial tree-shape type*. For each combinatorial type of ordered tree-shape ( ) that can be represented by collapsing edges in the maximal tree , there is a family of subsets of , the corresponding to the th representation of any tree-shape of type . That is, any will induce the particular type when the edges in ( ) are endowed with nonzero attributes, leaving all other edges with zero attributes. These subsets are characterized by the properties:

That is, the subset for any lists the set of edges in which have nonzero attributes for the th representation of any shape of type . Corresponding to each is the linear subspace of given by

(10)

and by condition 2 we can define isometries by ignoring the zero entries in and keeping the depth-first coordinate order. We generate the equivalence on by asking that whenever for some . We now define the space of ordered tree-shapes as the quotient , and define the quotient map .

**2.4.2 The Pseudometric Is a Metric**

**Proposition 1.** *Let denote or . The pseudometric is a metric on .*

**Proof.** It is easy to show that for any , so it suffices to show that implies . Hence, from now on, write for , and assume that for two tree-shapes and .

(11)

that is, is smaller than the size of any of the nonzero edges in and .

We may assume that because otherwise we may assume by symmetry that and .

Denote by the image of under the quotient projection for any .

We may assume that and belong to the same identified subspace, that is, there exist such that

(12)

since otherwise

(13)

Because is a finite set and is a closed set, (13) implies . In this case, the path will have to go through some that does not contain points equivalent to and because in order to reach , we need to remove edge attributes that are nonzero in , and for all . Thus, (12) holds, and in fact, it holds for all the intermediate path points from (7).

But if the path points stay in , then the path consists of changing the nonzero edge attributes of the trees in question. This will only give a sum if the trees are identical and the path is constant.

**2.4.3 Topology of the Space of Tree-Shapes**Here, we prove the rest of Theorem 1, namely, that the tree-shape space is a complete, proper geodesic space and is contractible. First, we note that although is not a vector space, there is a well-defined notion of size for elements of , induced by the norm on .

**Lemma 1.** *Note that if , we must have ; hence we can define .*

**Proof.** The equivalence is generated recursively from the conditions whenever either , indicating , or , indicating since the are isometries. Hence, the lemma holds by recursion.

We will prove that is a proper geodesic space using the Hopf-Rinow theorem for metric spaces [ ^{6}], which states that every complete locally compact length space is a proper geodesic space. A length space is a metric space in which the distance between two points can always be realized as the infimum of lengths of paths joining the two points. Note that this is a weaker property than being a geodesic space, as the geodesic joining two points does not have to exist; it is enough to have paths that are arbitrarily close to being a geodesic. It follows from [ ^{6}, Chapter I, L. 5.20] that is a length space for *any* metric on where is a metric.

The tree-shape space is locally compact: Since the projection is finite-to-one, any open subset of has as preimage a finite union of open subsets of . Thus, and is compact whenever is bounded.

We also need to prove that is complete:

**Proposition 2.** *Let denote either of the metrics and . The shape space is complete.*

The proof needs a lemma from general topology.

**Lemma 2.** *[ ^{9}, Chapter XIV, Theorem 2.3] Let be a metric space and assume that the metric has the following property: There existssuch that for allthe closed ballis compact. Then, is complete.*

Using the projection , we can prove:

**Lemma 3.** *Bounded closed subsets of are compact.*

**Proof.** Since Lemma 1 defines a notion of size in , any closed, bounded subspace in is contained in a closed ball in for some , where is the image . Since , it follows that , which is a closed and bounded ball in . Since closed, bounded subsets of are compact, is compact. By continuity of , is compact. Then, is compact too.

It is now very easy to prove Proposition 2:

**Proof of Proposition 2.** By Lemma 3, all closed and bounded subsets of are compact, but then by Lemma 2 the metric must be complete.

Using the Hopf-Rinow theorem [ ^{6}, Chapter I, Proposition 3.7], we thus prove that is a complete, proper geodesic space. We still miss contractibility.

**Lemma 4.** *Let be a normed vector space and let be an equivalence on such that implies for all . Then, is contractible.*

**Proof.** Define a map by setting . Now, is well defined because of the condition on and , so is a homotopy from to the constant zero map.

Combining the results of Section 2.4, we see that the proof of Theorem 1 is complete.

**Theorem 2.** *For induced by either or , the function is a metric and the space is a contractible, complete, proper geodesic space.*

*curvature*, a concept that fundamentally depends on the underlying metric. We shall start out by reviewing the concept of curvature in metric geometry [

^{6}], [

^{16}] before we investigate the curvature of the tree-shape space with the QED and TED metrics. Using curvature, we obtain well-posed local statistical methods for QED. We discuss the implications of locality and explain how locality issues can be dealt with by restricting to smaller tree-spaces that are particularly relevant for a given problem. As part of the exposition, we discuss two conjectures on statistics for global generic datasets.

*geodesic triangles*, which are compared with

*comparison triangles*in model spaces with a fixed curvature . The model spaces are spheres ( ), the plane ( ), and hyperbolic spaces ( ), where the metric space curvature is bounded by the curvature of the model space. In this paper, we shall use comparison with planar triangles, which gives us curvature bounded from above by 0. For an extensive review of metric geometry, we refer to [

^{6}].

*geodesic triangle*in consists of three points and geodesic segments joining them. A planar

*comparison triangle*for the triangle consists of a triangle with vertices in the plane, whose side lengths are identical to those of , see Fig. 5.

**Example 1.**

1. The so-called *open book* obtained by gluing a family of euclidean spaces together along isomorphic affine subspaces is a space. At any smooth point in , the local curvature is 0 because the space is locally isomorphic to the corresponding . At any singular point, the local curvature is ( for every ).

2. The generalized PCA construction by Vidal et al. [ ^{38}] defines a space, giving a potential use of spaces and metric geometry in machine learning.

3. As we are about to see, the space of tree-shapes is locally a space almost everywhere.

*circumcenter*considered in [

^{6}], the

*centroid*considered, among other places, in [

^{5}], and the

*mean*[

^{19}].

**Definition 1.**

1. **Circumcenters**. Consider a metric space and a bounded subset . There exists a unique smallest closed ball in that contains ; the center of this ball is the circumcenter of .

2. **The centroid of a finite set**. Let be a uniquely geodesic metric space (any two points are joined by a unique geodesic). The centroid of a set of elements is defined recursively as a function of the centroids of subsets with elements as follows: Denote the elements of by . If , the centroid of is the midpoint of the geodesic joining and . If , define and for ; these are sets with elements each. If the elements of converge to a point as , then is the centroid of in .

^{5}] and circumcenters [

^{6}] are unique in -spaces. Uniqueness of means follows from a more general theorem by Sturm [

^{34}, Proposition 4.3]; we include an elementary proof here for completeness.

*strictly*convex. Convex coercive functions have minimizers, which are unique for strictly convex functions. Hence, existence and uniqueness of means can be proven by expressing them as minimizers of strictly convex functions.

**Lemma 5.**

*mean*of a finite subset in a metric space is defined as in (1), and is called the

*Fréchet mean*. Local minimizers of (1) are called

*Karcher means*.

**Theorem 3.** *Means exist and are unique in spaces.*

**Proof.** The function given by is convex for any fixed by [ ^{6}, Proposition II.2.2], so the function is *strictly* convex by step 1 in Lemma 5. But then is strictly convex by step 2 in Lemma 5, and a mean is just a minimizer of the function . The function is coercive, so the minimizer exists. Since is strictly convex, the minimizer is unique.

**Definition 2 (Generic Property).** *A generic property in a metric space is a property that holds on an open, dense subset of .*

*nongeneric property*is a property whose “not happening” is generic. This is similarly interpreted as a property that may

*not*hold for randomly selected tree-shapes. A detailed discussion of the relation between genericity and probability is found in Appendix A in the supplemental material, which can be found in the Computer Society Digital Library at http://doi. ieeecomputersociety.org/10.1109/TPAMI.2012.265.

**Example 2.** Being a truly binary tree-shape (i.e., the corresponding collapsed tree-shape is a binary tree) is a generic property. To see this, let be a tree-shape in or that is not truly binary, represented by a maximal binary tree . By adding small noise to the zero attributes on edges of , we can obtain truly binary tree-shapes that are arbitrarily close to . Thus, the set of full truly binary tree-shapes in or is open and dense and contained in the set of binary tree-shapes.

*will*combine to a new, more restrictive, generic property.

**Theorem 4 (Curvature for Ordered Tree-Shapes).**

1. *Endow with the TED metric . The metric space does not have locally unique geodesics anywhere and the curvature of is everywhere.*

2. *Endow with the QED metric . Having local nonpositive curvature is a generic property in . That is, the set of points that have a neighborhood in which the curvature is nonpositive contains an open, dense set. At the remaining points, the curvature of is , i.e., unbounded from above.*

**Proof.** *Case 1:* Consider any tree-shape , where trees in are spanned by with at least two edges , and induce a second tree-shape such that for all and and . Now, there are at least two geodesics from to : one which first deforms to before deforming to , and one which first deforms to before deforming to . Moreover, the TED distance between and can be arbitrarily small. Since this is possible anywhere in tree-space, geodesics are nowhere locally unique and the curvature of tree-space with TED is unbounded everywhere [ ^{6}, Prop. II 1.4].

*Case 2:* Recall from Section 2.2 how was formed by identifying subspaces , defined in (6). These identified subspaces corresponded to different representations in of the same shapes . The points in can now be divided into three categories:

1. Points that do not belong to the image of an identified subspace because they only have one representative in .

2. Points at which the space is locally homeomorphic to an open book singularity, as in step 1 of Example 1.

3. Points that are preimages of points at which identified subspaces intersect in . An example of such points is the image of the origin in Fig. 4c. Infinitely close to these trees, we will find pairs of trees in between which geodesics are not unique, as in Fig. 4c.

These three classes of points correspond to local curvature 0, , and , respectively. That is, the space is locally at points in category 1; at points from category 2 it is for every , so it has curvature ; and at points from category 3 it is not for any ; hence, the curvature is . It thus suffices to show that the set of points in categories 1-2 contains an open, dense subset. This follows from the fact that the points in category 3 sit in a lower-dimensional subspace of .

**Theorem 5 (Curvature for Unordered Tree-Shapes).** *Nonpositive curvature is a generic property in the space of unordered trees with the QED metric. With the TED metric , however, has everywhere unbounded curvature, and geodesics are nowhere locally unique.*

^{5}], [

^{6}], [

^{19}] will fail for the TED metric. This motivates our study of the QED metric.

**Example 3.**

1. Consider the space of tree-shapes spanned by the rooted binary tree with three leaves. This represents, for instance, the subtree of the central airways that spans the segments of the left upper lobe. In this space, the trees at which the tree-space curvature is are missing two consecutive edges, which rarely happens in airway trees. Moreover, the spatial orientations of the airway segmental branches in different individuals are fairly similar, although the branching order of given branches varies. This means that a typical dataset is quite local. Thus, in this tree-space, the neighborhoods are not locally euclidean, and our airway subtrees will usually be contained in a neighborhood.

2. Consider the tree-shapes and shown in Fig. 6a. These two tree-shapes are joined by two geodesics, and thus and are not contained in the same neighborhood in . However, in a suitably chosen smaller tree-space , as defined in Definition 3 below, they can be.

3. Consider the tree-shape in the space of unordered tree-shapes with attributes in , spanned by the maximal tree as in Fig. 6b. Arbitrarily close to we will find two trees and that are joined by two geodesics, as shown in the figure.

**Definition 3 (Reduced Tree-Shape Space).** *Consider a subset which only contains all representations of trees of certain particular topologies. The th topology is characterized by a subset consisting of the edges in the maximal tree that have nonzero attributes. Associated to is a linear subspace containing representations of all the trees of this particular topology. We include all representations of each tree topology (otherwise some shape space paths may disappear), and obtain a reduced preshape space*

*containing all the trees that have of one of the considered topologies. The equivalence relation on restricts to an equivalence relation on , from which we obtain a reduced tree-shape space . The QED metric on induces a metric on that induces a quotient pseudometric on .*

**Example 4.** Denote by the space of all trees in with leaves; now is the space of ordered tree-shapes with leaves.

**Theorem 6.**

1. *The pseudometric on the reduced space of ordered tree-shapes is a metric and a contractible, complete, proper metric space.*

2. *The same holds for the reduced space of unordered tree-shapes: Assume that is saturated with respect to the reordering group ^{1} . For the corresponding reduced tree-space , the quotient pseudometric is a metric and the space is a contractible, complete, proper geodesic space.*

3. *Local nonpositive curvature is a generic property in and . At points in and where the curvature is not , it is .*

**Proof.** First, we show that the pseudometric is a metric. The pseudometric in (7) defines the distance as the infimum of lengths of paths in connecting and . Any path in is also a path in , so if , then as well, so .

The proofs of the other claims follow the proofs of Theorems 1, 2, 4, and 5.

**Example 5.** Denote by the space of all trees in with a feasible airway tree topology, as defined in Appendix C, available in the online supplemental material. Now, is the reduced space of airway trees.

**Theorem 7.** *Consider the tree-spaces and . Local uniqueness of means, centroids, and circumcenters is generic in both spaces. That is, the set of points with a neighborhood such that sets contained in have unique means, centroids, and circumcenters contains an open, dense subset. For the TED metric, these statistics are not locally unique anywhere. The same results hold in the reduced tree-spaces and , defined in Definition 3.*

**Proof.** First consider and with the QED metric. By Theorems 4 and 5, and are both locally spaces, and by Theorems 1 and 2 they are both complete metric spaces. We have seen in Theorem 3 that means exist and are unique in spaces, so the statement holds for means. By [ ^{6}, Proposition 2.7], any subset of a complete space has a unique circumcenter. Hence, the statement holds for circumcenters. Similarly, by [ ^{5}, Theorem 4.1], finite subsets of spaces have centroids (unique by definition), so the statement holds for centroids.

For the TED metric, note that for any two-point dataset, all these notions of mean coincide with the midpoint of a geodesic connecting the points. We know that geodesics and midpoints are not unique in the TED metric.

The proofs transfer directly to the spaces and .

^{1}], and we expect this extension from existing models to be nontrivial. In the meantime, as discussed in the previous section, application-specific reduced tree-spaces can be designed, where -neighborhoods are larger.

^{19}] in any convex neighborhood in which geodesics are unique. In tree-space, any convex neighborhood that does not contain points of curvature will be a neighborhood. In tree-space, radius is not a good measure for the size of a neighborhood as a tree may contain both small branches, leading to a small radius, as well as large branches, giving much larger neighborhoods than indicated by the radius.

^{10}], but we can often use geometry and prior knowledge (e.g., anatomy) as a heuristic to reduce the search space. For instance, trees appearing in applications are often not completely unordered, but are semi-labeled.

**Example 6 (Semilabeling of the Central Airway Tree).** The airway tree has about 30 branches that have anatomical names and can be identified by experts. For instance, the root edge is the trachea; the second generation edges are the left and right main bronchi. As these top-generation branches are easily identified, we use their identification to simplify computations of interairway geodesics in Section 6.2.

(14)

*the number of topological transitions in the geodesic*. Note that this approximation is similar to that employed for TED in [

^{35}].

Algorithm 1.Computing approximate QED distances

between ordered, rooted trees with up to

structural transitions through trees of order at most

1: , planar rooted binary trees in .

2: set of ordered isometric pairs of

subspaces of in which the same ordered tree-shapes

are representing, as in eq. (6).

3:forwithdo

4: ,

where and

.

5: .

6:endfor

7:

8:

9:

10: geodesic

11:return

^{31}]. Our experimental results on real and synthetic data illustrate the geometric properties of the QED metric. The experiments on airway trees in Section 6.2 show, in particular, that it is feasible to compute geodesics between real, 3D data trees. In all our experiments, edges are translated to start at and are represented by six landmark points evenly distributed along the edge, the first at the origin.

^{11}], [

^{34}]) and centroid [

^{5}] trees are shown in Fig. 8. The mean and centroid trees clearly represent the main common properties of the dataset trees.

^{23}]. The airway trees were first segmented from low dose screening computed tomography (CT) scans of the chest using a voxel classification-based airway tree segmentation algorithm [

^{22}]. The centerlines were extracted from the segmented airway trees using a modified fast marching algorithm based on [

^{30}]. The method gives a tree structure directly through connectivity of parent and children branches. For simplicity, we only consider the central airways down to the lobar branches.

^{36}] and QED means (using the weighted midpoints algorithm [

^{11}], [

^{34}]). The dataset is shown in Fig. 9a; the results in Fig. 9b. We clearly see how the choice of the TED geodesic makes the TED mean vulnerable to missing branches in the dataset trees, giving a mean shape whose structure is too simple. The QED mean, on the other hand, represents the data well.

^{31}], [

^{36}] or is unable to simultaneously handle topology and geometry [

^{39}].

^{21}], [

^{26}]. Construction of efficient algorithms is by no means trivial due to the nonsmooth structure of the tree-shape space and the complexity of computing exact geodesics.

# Acknowledgments

*A. Feragen, P. Lo, M. Nielsen, and F. Lauze are with the eScience Center, Department of Computer Science, University of Copenhagen, Universitetsparken 5, 2011 Copenhagan, Denmark.*

*E-mail: {aasa, pechin, madsn, francois}@diku.dk.*

*M. de Bruijne is with the eScience Center, Department of Computer Science, University of Copenhagen, Universitetsparken 5, 2011 Copenhagan, Denmark, and the Biomedical Imaging Group Rotterdam, Department of Radiology and Medical Informatics, Erasmus MC-University Medical Center Rotterdam, Room Ee 2112, PO Box 2040, Rotterdam 3000 CA, The Netherlands. E-mail: marleen@diku.dk.*

*Manuscript received 1 Feb. 2012; revised 26 Sept. 2012; accepted 30 Nov. 2012; published online 19 Dec. 2012.*

*Recommended for acceptance by D. Weinshall.*

*For information on obtaining reprints of this article, please send e-mail to: tpami@computer.org, and reference IEEECS Log Number TPAMI-2012-02-0080.*

*Digital Object Identifier no. 10.1109/TPAMI.2012.265.*

1. is saturated if, for each tree topology appearing in , all reorderings of the same tree topology also appear in . Equivalently, , where .

#### References

**Aasa Feragen**received the PhD degree in mathematics from the University of Helsinki, Finland, in 2010. She is currently Freja Fellow and Assistant Professor at the Department of Computer Science, University of Copenhagen, Denmark. Her research interests include mathematical modeling and computational anatomy for biomedical image analysis, as well as statistics and machine learning for structured data.

**Pechin Lo**received the PhD degree in computer science from the University of Copenhagen, Denmark, in 2010. He is currently a postdoctoral researcher at the University of California, Los Angeles. His research interests include image segmentation, pattern recognition, and machine learning applied to the field of medical image processing.

**Marleen de Bruijne**received the MSc degree in physics and the PhD degree in medical imaging, in 1997 and 2003, both from Utrecht University, The Netherlands. She is an associate professor of medical image analysis at Erasmus MC Rotterdam, The Netherlands, and at the University of Copenhagen, Denmark. She has coauthored more than 100 peer-reviewed papers in international conferences and journals. She was on the program committee of many international conferences in medical imaging and computer vision and is a member of the editorial board of

*Medical Image Analysis*. Her research interests include model-based and quantitative analysis of medical images in various applications.

**Mads Nielsen**received the PhD degree in computer science from the University of Copenhagen, Denmark, in 1995. In 1995-1996, he was a postdoctoal researcher at Utrecht University, The Netherlands, and the University of Copenhagen, where he was an assistant professor during 1997-1999. He became an associate professor in 1999 and a professor in 2002 at the IT University of Copenhagen, where he headed the Image Analysis Group. In 2007, he became a professor and the head of the Image Group, University of Copenhagen. He was a general chair of MICCAI '06 and is a member of the editorial boards of the

*International Journal of Computer Visio*n and

*Journal of Mathematical Imaging and Vision*.

**François Lauze**studied mathematics in France at the University of Nice-Sophia-Antipolis, where he received the PhD degree in algebraic geometry in 1994. He received another PhD degree in 2004 from the IT University of Copenhagen, Denmark, including work on variational methods for motion compensated in-painting and motion recovery. He has worked with variational and geometric methods for image analysis.

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