We describe a simple and robust algorithm for estimating 3D shape given a number of silhouette points obtained
from two or more viewpoints and a parametric model of the shape. Our algorithm minimizes (in the least
squares sense) the distances from the lines obtained by unprojecting the silhouette points to 3D to their closest
silhouette points on the 3D shape. The solution is found using an iterative approach. In each iteration, we
locally approximate the least squares problem with a degree-4 polynomial function. The approximate problem
is solved using a nonlinear conjugate gradient solver that takes advantage of its structure to perform exact and
global line searches. We tested our algorithm by applying it to reconstruct patient-specific femur shapes from
simulated biplanar X-ray images.
We describe a method for segmenting airway trees from greyscale 3D images such as CT (Computed Tomography)
scans. Our approach is based on topological analysis of sets obtained by thresholding from <i>thick slices</i>, i.e. sub-images
consisting of a small number of consecutive slices. From each thick slice under consideration, we select
all sets <i>S</i> obtained from that thick slice by thresholding that have simple enough topological structure. As the
selection criterion, we use a simple algebraic condition involving the numbers of connected components in the
intersection of the set <i>S</i> with every slice in the thick slice. The condition basically asserts that the intersections
of <i>S</i> with each of the slices is small and attempts to limit the number of the branching points of <i>S</i> within the
The output 3D model of the airway tree is obtained as the largest connected component of the union of all
selected sets, extracted from several overlapping thick slices. Experiments with a number of chest CT scans show
that the method leads to promising results.
We propose a simple method for reconstructing thin, low-contrast blood vessels from three-dimensional greyscale images. Our algorithm first extracts persistent maxima of the intensity on all axis-aligned two-dimensional slices through the input volume. Those maxima tend to concentrate along one-dimensional intensity ridges,
in particular along blood vessels. Persistence (which can be viewed as a measure of robustness of a local maximum with respect to perturbations of the data) allows to filter out the `unimportant' maxima due to noise or inaccuracy in the input volume. We then build a minimum forest based on the persistent maxima that uses edges
of length smaller than a certain threshold. Because of the distribution of the robust maxima, the structure of this forest already reflects the structure of the blood vessels. We apply three simple geometric filters to the forest in order to improve its quality. The first filter removes short branches from the forest's trees. The second filter adds edges, longer than the edge length threshold used earlier, that join what appears (based on geometric criteria) to be pieces of the same blood vessel to the forest. Such disconnected pieces often result from non-uniformity of contrast along a blood vessel. Finally, we let the user select the tree of interest by clicking near its root (point from which blood would flow out into the tree). We compute the blood flow direction assuming that the tree is of the correct structure and cut it in places where the vessel's geometry would force the blood flow direction to change abruptly. Experiments on clinical CT scans show that our technique can be a useful tool for segmentation of thin and low contrast blood vessels. In particular, we successfully applied it to extract coronary arteries from heart CT scans. Volumetric 3D models of blood vessels can be obtained from the graph described above by adaptive thresholding.