Online Reconstruction of 3D Objects from Arbitrary Cross-Sections

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Online Reconstruction of 3D Objects from Arbitrary Cross-Sections

AMIT BERMANO, AMIR VAXMAN, and CRAIG GOTSMAN

Technion – Israel Institute of Technology

We describe a simple algorithm to reconstruct the surface of smooth three-

dimensional multilabeled objects from sampled planar cross-sections of

arbitrary orientation. The algorithm has the unique ability to handle cross-

sections in which regions are classified as being inside the object, outside

the object, or unknown. This is achieved by constructing a scalar function

on R3, whose zero set is the desired surface. The function is constructed in-

dependently inside every cell of the arrangement of the cross-section planes

using transfinite interpolation techniques based on barycentric coordinates.

These guarantee that the function is smooth, and its zero set interpolates

the cross-sections. The algorithm is highly parallelizable and may be imple-

mented as an incremental update as each new cross-section is introduced.

This leads to an efficient online version, performed on a GPU, which is

suitable for interactive medical applications.

Categories and Subject Descriptors: I.3.5 [Computer Graphics]: Com-

putational Geometry and Object Modeling; I.3.7 [Computer Graphics]:

Three-Dimensional Graphics and Realism

General Terms: Algorithms, Design, Measurement



ACM Reference Format:

Bermano, A., Vaxman, A., and Gotsman, C. 2011. Online reconstruction of

3D objects from arbitrary cross-sections ACM Trans. Graph. 30, 5, Article

113 (October 2011), 14 pages.

DOI = 10.1145/2019627.2019632

http://doi.acm.org/10.1145/2019627.2019632

1. INTRODUCTION

The reconstruction of a surface from a set of planar cross-sections

S (also called slices), such that the surface interpolates, or ap-

proximates, S in the planes, has been thoroughly studied in the

past decades. This problem arises mainly in the fields of medical



imaging (MRI, CT, ultrasound, etc.) and geographical information

systems (for terrain reconstruction). The input is assumed to have

been segmented in a preprocessing step, to create a set of closed

two-dimensional contours, separating the “inside” and “outside” of

the object on each slice. A reconstruction algorithm typically cre-

ates a surface, enclosing a solid volume, which interpolates these

contours and is consistentwith the given inside/outside information.

See Figure 1 for an illustration of the scenario.

The prior art related to this problem typically assumes that entire

cross-sections are available and that the contour information in them

is complete (i.e., a set of closed contours) and segmented correctly.

However, in most practical cases, the input images are noisy, so

there might be regions of uncertainty (see Figure 1) which cannot

be reliably segmented. Thus, instead of a complete inside/outside

classification per cross-section, there will also be regions which are

unclassified. We deal with this general scenario.

The algorithm we describe constructs a signed “distance func-

tion,” whose zero set is the surface. It consists of two steps: com-

pleting the regions of unknown information on the cross-section

planes, and then using these to define a function on R3

, from which

it extracts a zero set. Our construction is based on the definition

of near-binary functions on the individual planes and their inter-

polation onto R3

using transfinite interpolation methods based on

barycentric coordinates.

Our algorithm solves the most general version of the problem,

with every setting of cross-sections and contours, as well as the

multilabeled contour problem (i.e., the contours within each plane

may belong to separate components of the object, and are labeled

accordingly). The resulting surface, obtained as the zero set of a

smooth function, is naturally smooth. To the best of our knowledge,

the only method that attempts to solve this problem at this level of

generality is that of Barequet and Vaxman [2009], which, as we will

see later, takes an entirely different, more complicated approach.

Our algorithm handles all cases of noise and completion of cross-

sections in a simple and uniform manner, without extra complexity

due to irregular topologies. The resulting surfaces are smooth and

well-behaved, requiring minimal postprocessing.

In addition, we deal with the online variant of the problem, in

which the slices are not all available at reconstruction time, rather

given individually over time, and the reconstruction is to be updated

after receiving the new slice. This online reconstruction problem is

typical of freehand ultrasound scanning applications, where the

physician scans a patient and sees the anatomy build up as s/he

scans more and more. Our algorithm easily adapts to this scenario,

as we explain in Section 7.



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