Covering many points with a small-area box

Let $P$ be a set of $n$ points in the plane. We show how to find, for a given integer $k>0$, the smallest-area axis-parallel rectangle that covers $k$ points of $P$ in $O(nk^2 \log n+ n\log^2 n)$ time. We also consider the problem of, given a value $\alpha>0$, covering as many points of $P$ as possible with an axis-parallel rectangle of area at most $\alpha$. For this problem we give a probabilistic $(1-\varepsilon)$-approximation that works in near-linear time: In $O((n/\varepsilon^4)\log^3 n \log (1/\varepsilon))$ time we find an axis-parallel rectangle of area at most $\alpha$ that, with high probability, covers at least $(1-\varepsilon)\mathrm{\kappa^*}$ points, where $\mathrm{\kappa^*}$ is the maximum possible number of points that could be covered

A novel MRA-based framework for the detection of changes in cerebrovascular blood pressure.

Background: High blood pressure (HBP) affects 75 million adults and is the primary or contributing cause of mortality in 410,000 adults each year in the United States. Chronic HBP leads to cerebrovascular changes and is a significant contributor for strokes, dementia, and cognitive impairment. Non-invasive measurement of changes in cerebral vasculature and blood pressure (BP) may enable physicians to optimally treat HBP patients. This manuscript describes a method to non-invasively quantify changes in cerebral vasculature and BP using Magnetic Resonance Angiography (MRA) imaging. Methods: MRA images and BP measurements were obtained from patients (n=15, M=8, F=7, Age= 49.2 ± 7.3 years) over a span of 700 days. A novel segmentation algorithm was developed to identify brain vasculature from surrounding tissue. The data was processed to calculate the vascular probability distribution function (PDF); a measure of the vascular diameters in the brain. The initial (day 0) PDF and final (day 700) PDF were used to correlate the changes in cerebral vasculature and BP. Correlation was determined by a mixed effects linear model analysis. Results: The segmentation algorithm had a 99.9% specificity and 99.7% sensitivity in identifying and delineating cerebral vasculature. The PDFs had a statistically significant correlation to BP changes below the circle of Willis (p-value = 0.0007), but not significant (p-value = 0.53) above the circle of Willis, due to smaller blood vessels. Conclusion: Changes in cerebral vasculature and pressure can be non-invasively obtained through MRA image analysis, which may be a useful tool for clinicians to optimize medical management of HBP