topology-constrained surface reconstruction taoju/research/ this work we detail the ﬁrst algorithm
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Topology-Constrained Surface Reconstruction From Cross-sections
Ming Zou Michelle Holloway Nathan Carr Tao Ju
Washington U. in St. Louis Washington U. in St. Louis Adobe Washington U. in St. Louis
Figure 1: Non-parallel cross-section curves delineating a developing chicken heart from a CT volume (a) and the reconstructed surfacesusing the method of Lu et al.  (b), method of Bermano et al.  (c), and our method with genus-1 constraint without utilizing theCT volume (d). The first two surfaces contain numerous topological tunnels, while ours correctly captures the shell-like shape of the object.
In this work we detail the first algorithm that provides topologi-cal control during surface reconstruction from an input set of pla-nar cross-sections. Our work has broad application in a numberof fields including surface modeling and biomedical image anal-ysis, where surfaces of known topology must be recovered. Givencurves on arbitrarily oriented cross-sections, our method produces amanifold interpolating surface that exactly matches a user-specifiedgenus. The key insight behind our approach is to formulate thetopological search as a divide-and-conquer optimization processwhich scores local sets of topologies and combines them to sat-isfy the global topology constraint. We further extend our methodto allow image data to guide the topological search, achieving evenbetter results than relying on the curves alone. By simultaneouslysatisfying both geometric and topological constraints, we are ableto produce accurate reconstructions with fewer input cross-sections,hence reducing the manual time needed to extract the desired shape.
CR Categories: I.3.5 [Computer Graphics]: Computational Ge-ometry and Object ModelingGeometric algorithms, languages,and systems
Keywords: Surface reconstruction, cross-section interpolation,contour stitching, topology constraint, dynamic programming
The need to create surfaces depicting a 3D object from its cross-sections arises in many applications. For instance, in order to re-construct the surface of an organ captured by CT or MRI, radiolo-gists typically start by manually delineating the organ boundary onselected 2D slices of the 3D image volume. The delineated planarcurves then need to be connected to form a closed surface. Figure1 (a) shows an example stack of the delineated curves of a layer ofa developing chicken heart from a CT volume (images on the rightshow two slices through the CT volume).
In many application domains, there exists strong prior informationabout the object to be depicted. One piece of such information istopology, which describes the objects connected components andtheir genus. For example, a biological structure usually has a knowntopology. Most structures have the topology of a sphere, whichis a single component with zero genus. Some structures have amore complicated topology. A layer of a developing chicken heartis a cylindrical shell with genus 1. Bones in our body, such asvertebrae and the hip bone, can have a higher genus. Accuratelycapturing such topology not only helps with recovering the shapeof the object but is also important for downstream applications ofthe reconstructed surface, such as shape matching, fluid dynamicssimulation, and mechanical analysis.
Much research has been done for creating surfaces from cross-sections (see a more detailed review in Section 2). While thestate-of-the-art algorithms are capable of creating watertight sur-faces from even the most complex inputs, no method to date canguarantee that their output has a predefined topology. In fact, it isquite common for these methods to create topological errors, par-ticularly when the cross-sections are not dense (see Figure 1 (b,c)).While it is possible for these methods to achieve the correct topol-ogy by providing them with more cross-sections, this will requireadditional manual labor and the data may not be available at all.
In this paper, we present a novel algorithm for reconstruction fromcross-sections that allows precise control over surface topology
Figure 2: Reconstructed surfaces of our method with genus-1 con-straint, without (a) and with (b) the use of the original CT volume,from the same stack in Figure 1 after removing two cross-sections.
(genus). We modify the classical divide-and-conquer strategy forsolving such problems and combine it with global combinatorialoptimization. Like previous methods, we consider the arrangementof the cross-sections, which divides the space into polyhedral vol-umes (cells). Instead of creating a single surface (called a tile in thiswork) within each cell as previous methods do, we explore a familyof distinct topologies of tiles and compute a score that assesses thelikelihood of each topology. In the optimization step, one topologywithin each cell is chosen so that the overall surface has the de-sired genus while the cumulative score from all cells is maximized.Our method can handle both parallel or non-parallel cross-sections,and each cross-section may contain multiple curves with arbitrarynesting relations. The result of our method for the same stack ofchicken heart cross-sections is shown in Figure 1 (d).
If 3D image data is available (e.g., original MRI or CT images fromwhich the cross-section curves are delineated), our algorithm caneasily utilize it to achieve ever better reconstruction. This is doneby biasing the set of tile topologies as well as their scores to respectthe strong boundaries in the image. As an example, we tested ouralgorithm on the same input as in Figure 1 (a) after removing twocross-sections (2nd and 4th from top). Without using the originalCT volume, our method creates a surface with the correct genusbut fails to capture the shell-like shape of the object (Figure 2 (a)).Utilizing the CT volume allows our algorithm to produce a surfacewith both the correct topology and shape (Figure 2 (b)). Note thatthe quality of a real-world biomedical image (as seen in Figure 1(a)) is often too low for any automatic reconstruction method toproduce a reasonable output.
Contributions: We present the first surface reconstruction algo-rithm from cross-sections that allows global topology (genus) con-trol. Our method enriches the typical divide-and-conquer strategywith an optimization component, and addresses two novel compu-tational problems:
1. How to explore the potentially enormous space of tile topolo-gies within a cell?
2. How to optimally select tile topologies in individual cells un-der a global topology constraint?
For the first problem, we reduce the exploration space by consid-ering the topology of level sets in a scalar function. We propose adefinition of the function based on random walks, which can easilyincorporate additional image data, and present algorithms for enu-merating and scoring distinct level set topologies. For the second
problem, we solve it with a bottom-up dynamic-programming al-gorithm that is guaranteed to find the optimal solution meeting thetopology constraint (if it exists).
2 Related works
Surface from cross-sections: Most methods for this problemtake a divide-and-conquer approach based on the arrangement ofthe cross-sectional planes, and they differ by how the tiles withineach cell is computed. If a cell is bounded by two parallel cross-sections and each has one curve, the problem can be formulatedas searching for the best correspondence between the vertices onthe two curves that optimizes some quality measure [Keppel 1975;Fuchs et al. 1977; Meyers et al. 1992]. However, this formu-lation no longer applies to cells with multiple, possibly nestedcurves. There are three common strategies proposed for arbitrarycurves. One can consider the Delaunay triangulation of the cell con-strained by the cross-section curves [Boissonnat 1988; Cheng andDey 1999]. This strategy has been successfully extended to cellsbounded by non-parallel cross-sections [Boissonnat and Memari2007]. The second strategy partitions the cell interior by projectingthe curves from the cell boundary onto some set of middle sheetsof the cell (e.g., medial axis or straight skeleton) [Bajaj et al. 1996;Barequet and Sharir 1996; Oliva et al. 1996; Barequet et al. 2004;Barequet and Vaxman 2007]. Besides non-parallel cross-sections,this strategy can reconstruct non-manifold surface networks thatpartition the space into regions with multiple labels [Ju et al. 2005;Liu et al. 2008; Barequet and Vaxman 2009]. The last strategy, towhich our method belongs, extracts the tile as the level set surfaceof a scalar function inside the cell that interpolates the labellingon the cell boundary. The function can be defined using linear in-terpolation [Herman et al. 1992] (limited to cells bounded by twocross-sections), radial basis functions [Turk and OBrien 1999], ormean-value interpolation [Bermano et al. 2011]. Note that somemethods using the projection strategy (e.g., [Liu et al. 2008]) mayalso be considered as a function-based method where the functionis defined by nearest-neighbor interpolation.
All of the above methods offer only one choice of tile per cell. Thetopological correctness of this tile is highly dependent on the den-sity of the cross-sections; a denser sampling usually leads to morelikely topology. An excessive number of cross-sections may berequired, particularly for thin objects and cross-sections at a non-optimal angle (see a more formal study of sampling conditions byAmini et al. ). Figure 3 demonstrates two typical failurecases for the projection strategy: a thin curve that migrates slightlyfrom one cross-section to the next disconnects the surfaces (top),whereas a change in the shape of the curve introduc