In this paper, a novel polynomial-time algorithm is described for solving the optimal net surface detection problem on proper ordered multi-column graphs in N-D space (N ≥ 3). The method is applied to searching for optimal object boundaries with arbitrary smoothness constraints in volumetric medical images. By simple transformations, such optimal surface detection problems can be simplified to a problem of computing the minimum s-t cuts in the transformed graphs. An efficient implementation for the 3-D case that can achieve near real-time performance on moderate-sized datasets is presented. We further examine our technique in experiments by segmenting the cylindrical surfaces of human airways from pulmonary volumetric CT images, and compare the results to those produced by previous methods. By allowing full specifications of the costfunction and smoothness constraints without degrading the performance, the new algorithm is more flexible than traditional methods and guarantees global optimality. The multi-dimensional nature of the algorithm maintains continuity in higher dimensions.

Identification of both the luminal and the wall areas of the bronchial tree structure from volumetric X-ray computed tomography (CT) data sets is of critical importance in distinguishing important phenotypes within numerous major lung diseases including chronic obstructive pulmonary diseases (COPD) and asthma. However, accurate assessment of the inner and outer airway wall surfaces of a complete 3-D tree structure is difficult due to their complex nature, particularly around the branch areas. In this paper, we extend a graph search based technique (LOGISMOS) to simultaneously identify multiple inter-related surfaces of branching airway trees. We first perform a presegmentation of the input 3-D image to obtain basic information about the tree topology. The presegmented image is resampled along judiciously determined paths to produce a set of vectors of voxels (called voxel columns). The resampling process utilizes medial axes to ensure that voxel columns of appropriate lengths and directions are used to capture the object surfaces without interference. A geometric graph is constructed whose edges connect voxels in the resampled voxel columns and enforce validity of the smoothness and separation constraints on the sought surfaces. Cost functions with directional information are employed to distinguish inner and outer walls. The assessment of wall thickness measurement on a CT-scanned double-wall physical phantom (patterned after an in vivo imaged human airway tree) achieved highly accurate results on the entire 3-D tree. The observed mean signed error of wall thickness ranged from −0.09 ± 0.24 mm to 0.07 ± 0.23 mm in bifurcating/nonbifurcating areas. The mean unsigned errors were 0.16 ± 0.12 mm to 0.20 ± 0.11 mm. When the airway wall surface was partitioned into meaningful subregions, the airway wall thickness accuracy was the same in most tested bifurcation/nonbifurcation and carina/noncarina regions (p=NS). Once validated on phantoms, our method was applied to human in vivo volumetric CT data to demonstrate relationships of airway wall thickness as a function of luminal dimension and airway tree generation. Wall thickness differences between the bifurcation/nonbifurcation regions were statistically significant (p < 0.05) for tree generations 6, 7, 8, and 9. In carina/noncarina regions, the wall thickness was statistically different in generations 1, 4, 5, 6, 7, and 8.

Abstract. We present efficient algorithms for solving polygonal-path approximation problems in three and higher dimensions. Given an n-vertex polygonal curve P in R d , d ≥ 3, we approximate P by another polygonal curve P of m ≤ n vertices in R d such that the vertex sequence of P is an ordered subsequence of the vertices of P. The goal is either to minimize the size m of P for a given error tolerance ε (called the min-# problem), or to minimize the deviation error ε between P and P for a given size m of P (called the min-ε problem). Our techniques enable us to develop efficient near-quadratic-time algorithms in three dimensions and subcubic-time algorithms in four dimensions for solving the min-# and min-ε problems. We discuss extensions of our solutions to d-dimensional space, where d > 4, and for the L 1 and L ∞ metrics.

We give the firsl.linear-time algorithm for computing single-source shortest paths in a weighted interval or circular-arc graph, when we arc given the model of that graph, i.e" the actual weighted intervals or circular"arcs and the sorted list of the interval endpoints. Our algorithm solves this problem optimally in O(n) time, where n is the nUlIlber of intervals or circular-arcs in a graph. An immediate consequence of our result

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