Technical Report MSC-2011-01

Title: Online Reconstruction of 3D Objects from Arbitrary Cross-Section Data
Authors: Amit Haim Bermano
Supervisors: Chaim Gotsman
Abstract: The reconstruction of a surface from a set of planar cross-sections such that the surface interpolates, or approximates, the input 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).

Many variants of the problem have been considered over time, such as parallel planar samples – as opposed to those of arbitrary orientation, the reconstruction of several objects, consisting of different materials (or "labels") or the handling of partial data.

In this thesis, I will describe a simple algorithm to reconstruct the surface of smooth three-dimensional multi-labeled 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 independently inside every cell of the arrangement of the cross-section planes using transfinite interpolation techniques based on barycentric coordinates. We also present a new closed form solution to mean-value barycentric coordinates on constant valued planar polygons.

These guarantee that the function is smooth, and its zero set interpolates the cross-sections. The algorithm is highly parallelizable and may be implemented as an incremental update as each new cross-section is introduced. This leads to an efficient online version which is suitable for interactive medical applications.

The input to the algorithm 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, as well as regions of different materials. As mentioned before, we allow for regions of uncertainty created during this step, which will be completed to create a smooth reconstruction.

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