Automated Topological Correction of Cortical Surfaces

Monica K. HurdalDepartment of Mathematics, Florida State University, Tallahassee, U.S.A.

Introduction• a number of techniques are now available for unfolding and

flattening cortical surfaces [1-4] to assist with the visualization of buried cortical folds and functional activation foci

• all flattening approaches require a triangulated surface mesh that represents the cortical surface and this surface must be a topologically correct 2-manifold (i.e. a topological sphere or disc).

• few algorithms are available for creating topologically correct cortical surfaces; widely used algorithms, such as the marching cubes/tetrahedra algorithms generate surfaces with topological errors

• we present methods for detecting and correcting topological errors in triangulated surfaces.

Methods• a piecewise flat, topologically correct triangulated

surface S consists of flat triangular faces connected along edges

• each edge of S is an interior edge (contained in exactly 2 triangles) or a boundary edge (contained in exactly 1 triangle)

• the surface S is a topological sphere if there are no boundary components; the surface is a topological disc if the boundary edges form a single closed boundary component

• topological problems are detected and repaired using topological invariants and examining the connectivity of the surface

Topological Invariants• if S is a surface then:

– the Euler characteristic of S is defined to be

where v, e, f are the numbers of vertices, edges and faces of S respectively

– the genus of S is defined to be

where m(S) is the number of boundary components of S

• if a surface is topologically correct, then it is a topological sphere if and only if (S) = 2 and it is a topological disc if and only if (S) = 1

• NOTE: a surface with (S) = 2 is NOT necessarily topologically correct

• the number of handles in S = g(S)

)()(22

1)( SmSSg

fevS )(

Topological Surface Problems• a triangulated surface S can be described in terms of:

– its combinatorics, K, where K is a (abstract) simplicial 2-complex representing the connectivity of the vertices vi (i =1,…,V), edges ej (j = 1,…,E) and faces fk (k=1,…,F) where K is given by a list of triangle faces, fk composed of 3 vertices V

– its geometric realization, P, where P is a set of vertex positions 3 (where vi = {xi, yi, zi }) defining the shape of the mesh

• typical topological problems include multiple-connected components, non-manifold edges, holes and handles

Types of Topological Surface Problems• Unused vertices

– vertices which do not belong to any triangle: vi F– checked/corrected by examining the connectivity of the

surface

• Duplicate triangles– triangle faces which are listed twice: fm = fn

– checked/corrected by examining the vertex components of fk

• Non-manifold edges– each edge of S is required to be an interior edge or a

boundary edge– edges which occur 3 or more times are non-manifold edges– problems include walls, ridges, bubbles, holes– checked/corrected using a region growing algorithm

• Multiple-connected components– every triangle belongs to the main part of the surface– checked/corrected using a region growing algorithm

• Triangle orientation– all triangles are required to be oriented in the same direction– by convention, the orientation is assumed to be counter

clockwise so the surface normals point outwards according to the right hand rule

– checked/corrected by examining each interior edge:

– each interior edge must occur in both directions i.e. edge v2-v3 and edge v3-v2

v1 v2

v3 v4

Triangle v1-v2-v3 gives edges v1-v2, v2-v3, v3-v1

Triangle v2-v4-v3 gives edges v2-v4, v4-v3, v3-v2

• Vertex singularity (pinched surface)– the surface touches itself at a single point

– checked/corrected using vertex neighbors which form a vertex flower:

– each vertex should be the center of at most 1 flower

– if vertex vi belongs to flower of vj, then vertex vj must belong to the flower of vi

v1

v2 v3 v4

v5

v6v7v8

vjvi

vi: v1, v2, v3, vj, v7, v8

vj: v3, v4, v5, v6, v7, vi

v

v1 v2

v3vn

closed flowerv: v1, v2, …, vn, v1

vvn

v1 v2

v3

vn-1

open flowerv: v1, v2, …, vn

• Holes– a surface should contain at most one closed boundary component; additional

boundary components are holes– checked/corrected by examining the boundary components and “filling” in

missing regions with additional triangles

• Handles (or Tunnels)– each handle contributes -2 to the Euler characteristic– the number of handles can only be determined after all other topological

problems have been corrected– handles are detected by computing a “running” Euler characteristic during a

region growing algorithm to determine how the Euler characteristic changes over time (see Figure 1)

– a handle can be corrected in 2 possible ways:• cut handle and then “cap” of ends (see Figure 2) or• fill in handle “tunnel”

– unless a priori information about handles are known or assumed, then handle correction should be guided by volumetric data used to create the surface

-1

-3

-5

Figure 1: A runing Euler characteristic is used to detect the location of handles. At each of the indicated regions, the Euler characteristic drops by 2 and never returns to its previous

value, indicating a handle has been formed at that location.

Note that the final Euler characteristic of this

surface is -4, indicating there are 3 handles.

Fixing Handles

Figure 2: A surface handle before correction and the same handle following correction produced by cutting and then “capping” off the ends.

Results and Conclusions• this paper presents algorithms which have been

developed for automatically detecting and correcting topological errors in triangulated surfaces

• these algorithms have been used successfully on cortical surfaces generated from a variety of algorithms

• software is available that can read in and output surfaces in a variety of file formats (including byu, obj, vtk, CARET, CirclePack and FreeSurfer)

• applying these algorithms to cortical data will speed up the processing pipeline for creating surfaces which are topologically correct

SoftwareThe software TopoCV (Topology Checker and Viewer) is

available for download from www.math.fsu.edu/~mhurdal and runs on Linux platforms. This software will also reformat surfaces into the format required by the conformal flattening software CirclePack.

References[1] Drury, H.A. et al. 1996. J. Cog. Neuro. 8:1-28.

[2] Fischl, B. et al. 1999. Neuroimage 9:195-207.

[3] Goebel, R. 2000. Neuroimage 11:S680.

[4] Hurdal, M.K. et al. 1999. Lecture Notes in Computer Science 1679: 279-286.

AcknowledgementsThis work is supported in part by NSF FRG grant DMS-0101329,

NIH Human Brain Project grant EB02013.