arXiv subjects

Oona Rainio

Publications and source records attributed to Oona Rainio.

At least 19 recordsLinked to original sources

arXiv2025

Fast capacity computation for maze-like configurations

We study the conformal capacity ${\rm cap}(Ω,K)$ where $Ω$ is a bounded domain of $\mathbb{R}^2$ and $K$ is a compact connected set in $Ω$. Because the exact numerical value of the capacity is known only in a handful of special cases, it is important to find estimates for the capacity in terms of domain functionals, simpler than the capacity itself. Here, we study condensers of maze-like structure and compute their capacity by means of a high-order $hp$- finite element method. We compare these numerical results to the estimates given by the quasihyperbolic length and perimeter of the compact set. In particular, we consider the behaviour of these value pairs, numerical results and estimates, when the structure parameters vary and the walls of the maze approach the compact set. Over the configurations covered in the numerical experiments, the quasihyperbolic estimates are shown to have the desired asymptotic properties and superior computational efficiencies once the case-specific analysis is completed.

math.NA↗

arXiv2025

Visual angle metric in the upper half plane

We prove an identity which connects the visual angle metric $v_{\mathbb{H}^2}$ and the hyperbolic metric $ρ_{\mathbb{H}^2}$ of the upper half plane $\mathbb{H}^2$. The proof is based on geometric arguments and uses computer algebra methods for formula manipulation. We also prove a sharp Hölder continuity result for quasiregular mappings with respect to the visual angle metric.

math.MG↗

arXiv2025

Quantitative evaluation of unsupervised clustering algorithms for dynamic total-body PET image analysis

Background. Recently, dynamic total-body positron emission tomography (PET) imaging has become possible due to new scanner devices. While clustering algorithms have been proposed for PET analysis already earlier, there is still little research systematically evaluating these algorithms for processing of dynamic total-body PET images. Materials and methods. Here, we compare the performance of 15 unsupervised clustering methods, including K-means either by itself or after principal component analysis (PCA) or independent component analysis (ICA), Gaussian mixture model (GMM), fuzzy c-means (FCM), agglomerative clustering, spectral clustering, and several newer clustering algorithms, for classifying time activity curves (TACs) in dynamic PET images. We use dynamic total-body $^{15}$O-water PET images collected from 30 patients with suspected or confirmed coronary artery disease. To evaluate the clustering algorithms in a quantitative way, we use them to classify 5000 TACs from each image based on whether the curve is taken from brain, right heart ventricle, right kidney, lower right lung lobe, or urinary bladder. Results. According to our results, the best methods are GMM, FCM, and ICA combined with mini batch K-means, which classified the TACs with a median accuracies of 89\%, 83\%, and 81\%, respectively, in a processing time of half a second or less on average for each image. Conclusion. GMM, FCM, and ICA with mini batch K-means show promise for dynamic total-body PET analysis.

stat.AP↗

arXiv2024

Collinearity of points on Poincaré unit disk and Riemann sphere

We study certain points significant for the hyperbolic geometry of the unit disk. We give explicit formulas for the intersection points of the Euclidean lines and the stereographic projections of great circles of the Riemann sphere passing through these points. We prove several results related to collinearity of these intersection points, offer new ways to find the hyperbolic midpoint, and represent a formula for the chordal midpoint. The proofs utilize Gröbner bases from computer algebra for the solution of polynomial equations.

math.MG↗

arXiv2024

Inequalities for a new hyperbolic type metric

We study a new hyperbolic type metric recently introduced by Song and Wang. We present formulas for it in the upper half-space and the unit ball domains and find its sharp inequalities with the hyperbolic metric and the triangular ratio metric. We also improve existing ball inclusion results and give bounds for the distortion of this new metric under conformal and quasiregular mappings.

math.MG↗

arXiv2024

AI for the prediction of early stages of Alzheimer's disease from neuroimaging biomarkers -- A narrative review of a growing field

Objectives: The objectives of this narrative review are to summarize the current state of AI applications in neuroimaging for early Alzheimer's disease (AD) prediction and to highlight the potential of AI techniques in improving early AD diagnosis, prognosis, and management. Methods: We conducted a narrative review of studies using AI techniques applied to neuroimaging data for early AD prediction. We examined single-modality studies using structural MRI and PET imaging, as well as multi-modality studies integrating multiple neuroimaging techniques and biomarkers. Furthermore, they reviewed longitudinal studies that model AD progression and identify individuals at risk of rapid decline. Results: Single-modality studies using structural MRI and PET imaging have demonstrated high accuracy in classifying AD and predicting progression from mild cognitive impairment (MCI) to AD. Multi-modality studies, integrating multiple neuroimaging techniques and biomarkers, have shown improved performance and robustness compared to single-modality approaches. Longitudinal studies have highlighted the value of AI in modeling AD progression and identifying individuals at risk of rapid decline. However, challenges remain in data standardization, model interpretability, generalizability, clinical integration, and ethical considerations. Conclusion: AI techniques applied to neuroimaging data have the potential to improve early AD diagnosis, prognosis, and management. Addressing challenges related to data standardization, model interpretability, generalizability, clinical integration, and ethical considerations is crucial for realizing the full potential of AI in AD research and clinical practice. Collaborative efforts among researchers, clinicians, and regulatory agencies are needed to develop reliable, robust, and ethical AI tools that can benefit AD patients and society.

cs.LG↗

arXiv2024

Landen transformations applied to approximation

We study computational methods for the approximation of special functions recurrent in geometric function theory and quasiconformal mapping theory. The functions studied can be expressed as quotients of complete elliptic integrals and as inverses of such quotients. In particular, we consider the distortion function $φ_K(r)$ which gives a majorant for $|f(x)|$ when $f: \mathbb{B}^2 \to \mathbb{B}^2, f(0)=0,$ is a quasiconformal mapping of the unit disk $\mathbb{B}^2.$ It turns out that the approximation method is very simple: five steps of Landen iteration is enough to achieve machine precision.

math.CV↗

arXiv2024

Uniformly perfect sets, Hausdorff dimension, and conformal capacity

Using the definition of uniformly perfect sets in terms of convergent sequences, we apply lower bounds for the Hausdorff content of a uniformly perfect subset $E$ of $\mathbb{R}^n$ to prove new explicit lower bounds for the Hausdorff dimension of $E.$ These results also yield lower bounds for capacity test functions, which we introduce, and enable us to characterize domains of $\mathbb{R}^n$ with uniformly perfect boundaries. Moreover, we show that an alternative method to define capacity test functions can be based on the Whitney decomposition of the domain considered.

math.CV↗

arXiv2024

Inequalities for geometric mean distance metric

We study a hyperbolic type metric $h_{G,c}$ introduced by Dovgoshey, Hariri, and Vuorinen. We find the best constant $c>0$, for which this function $h_{G,c}$ is a metric in specific choices of $G$. We give several sharp inequalities between $h_{G,c}$ and other hyperbolic type metrics, and also offer a few results related to ball inclusion.

math.MG↗

arXiv2024

Lipschitz constants and quadruple symmetrization by Möbius transformations

Due to the invariance properties of cross-ratio, Möbius transformations are often used to map a set of points or other geometric object into a symmetric position to simplify a problem studied. However, when the points are mapped under a Möbius transformation, the distortion of the Euclidean geometry is rarely considered. Here, we identify several cases where the distortion caused by this symmetrization can be measured in terms of the Lipschitz constant of the Möbius transformation in the Euclidean or the chordal metric.

math.MG↗

arXiv2024

Conformally invariant metrics and lack of Hölder continuity

The modulus metric between two points in a subdomain of $\mathbb{R}^n, n\ge 2,$ is defined in terms of moduli of curve families joining the boundary of the domain with a continuum connecting the two points. This metric is one of the conformally invariant hyperbolic type metrics, which have become a standard tool in geometric function theory. We prove that the modulus metric is not Hölder continuous with respect to the hyperbolic metric.

math.CV↗

arXiv2023

Image augmentation with conformal mappings for a convolutional neural network

For augmentation of the square-shaped image data of a convolutional neural network (CNN), we introduce a new method, in which the original images are mapped onto a disk with a conformal mapping, rotated around the center of this disk and mapped under such a Möbius transformation that preserves the disk, and then mapped back onto their original square shape. This process does not result the loss of information caused by removing areas from near the edges of the original images unlike the typical transformations used in the data augmentation for a CNN. We offer here the formulas of all the mappings needed together with detailed instructions how to write a code for transforming the images. The new method is also tested with simulated data and, according the results, using this method to augment the training data of 10 images into 40 images decreases the amount of the error in the predictions by a CNN for a test set of 160 images in a statistically significant way (p-value=0.0360).

cs.CV↗

arXiv2023

Hilbert metric in the unit ball

The Hilbert metric between two points $x,y$ in a bounded convex domain $G$ is defined as the logarithm of the cross-ratio of $x,y$ and the intersection points of the Euclidean line passing through the points $x,y$ and the boundary of the domain. Here, we study this metric in the case of the unit ball $\mathbb{B}^n$. We present an identity between the Hilbert metric and the hyperbolic metric, give several inequalities for the Hilbert metric, and results related to the inclusion properties of the balls defined in the Hilbert metric. Furthermore, we study the distortion of the Hilbert metric under conformal mappings.

math.MG↗

arXiv2023

Intrinsic metrics defined with arithmetic and logarithmic mean values

We introduce several new functions that measure the distance between two points $x$ and $y$ in a domain $G\subsetneq\mathbb{R}^n$ by using the arithmetic or the logarithmic mean of the Euclidean distances from the points $x$ and $y$ to the boundary of $G$. We study in which domains these functions are metrics and find sharp inequalities between them and the hyperbolic metric. We also present one result about their distortion under quasiregular mappings.

math.MG↗

arXiv2023

Inequalities for the generalized point pair function

We study a new generalized version of the point pair function defined with a constant $α>0$. We prove that this function is a quasi-metric for all values of $α>0$, and compare it to several hyperbolic-type metrics, such as the $j^*$-metric, the triangular ratio metric, and the hyperbolic metric. Most of the inequalities presented here have the best possible constants in terms of $α$. Furthermore, we research the distortion of the generalized point pair function under conformal and quasiregular mappings.

math.MG↗

arXiv2023

Metrics and quasimetrics induced by point pair function

We study the point pair function in subdomains $G$ of $\mathbb{R}^n$. We prove that, for every domain $G\subsetneq\mathbb{R}^n$, the this function is a quasi-metric with the constant less than or equal to $\sqrt{5}\slash2$. Moreover, we show that it is a metric in the domain $G=\mathbb{R}^n\setminus\{0\}$ with $n\geq1$. We also consider generalized versions of the point pair function, depending on an arbitrary constant $α>0$, and show that in some domains these generalizations are metrics if and only if $α\leq12$.

math.MG↗

arXiv2023

Different coefficients for studying dependence

Through computer simulations, we research several different measures of dependence, including Pearson's and Spearman's correlation coefficients, the maximal correlation, the distance correlation, a function of the mutual information called the information coefficient of correlation, and the maximal information coefficient (MIC). We compare how well these coefficients fulfill the criteria of generality, power, and equitability. Furthermore, we consider how the exact type of dependence, the amount of noise and the number of observations affect their performance.

stat.ME↗

arXiv2023

Möbius metric in sector domains

The Möbius metric $δ_G$ is studied in the cases where its domain $G$ is an open sector of the complex plane. We introduce upper and lower bounds for this metric in terms of the hyperbolic metric and the angle of the sector, and then use these results to find bounds for the distortion of the Möbius metric under quasiregular mappings defined in sector domains. Furthermore, we numerically study the Möbius metric and its connection to the hyperbolic metric in polygon domains.

math.MG↗