Partial-Matching and Hausdorff RMS Distance Under Translation: Combinatorics and Algorithms

We consider the RMS distance (sum of squared distances between pairs of points) under translation between two point sets in the plane, in two different setups. In the partial-matching setup, each point in the smaller set is matched to a distinct point in the bigger set. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision of the plane and derive improved bounds on its complexity. These results lead to the best known algorithm for finding a translation for which the partial-matching RMS distance between the point sets is minimized. In addition, we show how to compute a local minimum of the partial-matching RMS distance under translation, in polynomial time. In the Hausdorff setup, each point is paired to its nearest neighbor in the other set. We develop algorithms for finding a local minimum of the Hausdorff RMS distance in nearly linear time on the line, and in nearly quadratic time in the plane. These improve substantially the worst-case behavior of the popular ICP heuristics for solving this problem.Comment: 31 pages, 6 figure

The 2-Center Problem in Three Dimensions

Let P be a set of n points in R³. The 2-center problem for P is to find two congruent balls of the minimum radius whose union covers P. We present two randomized algorithms for computing a 2-center of P. The first algorithm runs in O(n 3 log 8 n) expected time, and the second algorithm runs in O(n 2 log 8 n/(1 − r ∗ /r0) 3) expected time, where r ∗ is the radius of the 2-center of P and r0 is the radius of the smallest enclosing ball of P. The second algorithm is faster than the first one as long as r ∗ is not very close to r0, which is equivalent to the condition of the centers of the two balls in the 2-center of P not being very close to each other

The 2-Center Problem in Three Dimensions

Let P be a set of n points in R 3. The 2-center problem for P is to find two congruent balls of minimum radius whose union covers P. We present two randomized algorithms for computing a 2-center of P. The first algorithm runs in O(n 3 log 5 n) expected time, and the second algorithm runs in O((n 2 log 5 n)/(1 − r ∗ /r0) 3) expected time, where r ∗ is the radius of the 2-center balls of P and r0 is the radius of the smallest enclosing ball of P. The second algorithm is faster than the first one as long as r ∗ is not too close to r0, which is equivalent to the condition that the centers of the two covering balls be not too close to each other

Computing the Discrete Fréchet Distance in Subquadratic Time

The Fréchet distance measures similarity between two curves f and g that takes into account the ordering of the points along the two curves: Informally, it is the minimum length of a leash required to connect a dog, walking along f, and its owner, walking along g, as they walk without backtracking along their respective curves from one endpoint to the other. The discrete Fréchet distance replaces the dog and its owner by a pair of frogs that can only reside on m and n specific stones, respectively. The stones are in fact sequences of points, typically sampled from the respective curves f and g. These frogs hop from one stone to the next without backtracking, and the discrete Fréchet distance is the minimum length of a “leash ” that connects the frogs and allows them to execute such a sequence of hops from the starting points to the terminal points of their sequences. The discrete Fréchet distance can be computed in O(mn) time by a straightforward dynamic programming algorithm. We present the first subquadratic algorithm for computing the discrete Fréchet distance between mn log log n two sequences of points in the plane. Assuming m ≤ n, the algorithm runs in O ( ) time, log

Hemodynamic response imaging: a potential tool for the assessment of angiogenesis in brain tumors.

Blood oxygenation level dependence (BOLD) imaging under either hypercapnia or hyperoxia has been used to study neuronal activation and for assessment of various brain pathologies. We evaluated the benefit of a combined protocol of BOLD imaging during both hyperoxic and hypercapnic challenges (termed hemodynamic response imaging (HRI)). Nineteen healthy controls and seven patients with primary brain tumors were included: six with glioblastoma (two newly diagnosed and four with recurrent tumors) and one with atypical-meningioma. Maps of percent signal intensity changes (ΔS) during hyperoxia (carbogen; 95%O2+5%CO2) and hypercapnia (95%air+5%CO2) challenges and vascular reactivity mismatch maps (VRM; voxels that responded to carbogen with reduced/absent response to CO2) were calculated. VRM values were measured in white matter (WM) and gray matter (GM) areas of healthy subjects and used as threshold values in patients. Significantly higher response to carbogen was detected in healthy subjects, compared to hypercapnia, with a GM/WM ratio of 3.8 during both challenges. In patients with newly diagnosed/treatment-naive tumors (n = 3), increased response to carbogen was detected with substantially increased VRM response (compared to threshold values) within and around the tumors. In patients with recurrent tumors, reduced/absent response during both challenges was demonstrated. An additional finding in 2 of 4 patients with recurrent glioblastoma was a negative response during carbogen, distant from tumor location, which may indicate steal effect. In conclusion, the HRI method enables the assessment of blood vessel functionality and reactivity. Reference values from healthy subjects are presented and preliminary results demonstrate the potential of this method to complement perfusion imaging for the detection and follow up of angiogenesis in patients with brain tumors