The exact perimeter of an ellipse involves the complete elliptic integral of the second kind, which lacks a closed-form expression in elementary functions. As a result, analytical approximations have been proposed for applications requiring fast and accurate evaluation of elliptical geometries. In this study, we present a new ultra-accurate and compact closed-form approximation for the ellipse perimeter based on an exponential correction applied to Ramanujan’s second formula. The proposed expression preserves simplicity—using only three exponential functions and six constants—while achieving a maximum relative error of approximately 0.57 ppm observed over the tested grids covering the full eccentricity range. This represents a significant accuracy improvement over classical and modern approximations while maintaining a single-line analytical form with low computational cost. Due to its robustness, quasi-exact behavior at both circular and highly eccentric limits, and its suitability for numerical algorithms and embedded implementations, the proposed approximation is particularly useful in engineering computations involving elliptical boundaries.

This paper presents an automatic classification system for pulmonary diseases in chest X-rays based on geometric normalization. The proposed method consists of three main modules. Module 1: A landmark detector: A ResNet-18 convolutional neural network with coordinate attention mechanism is trained to predict 15 landmarks defining the lung contour, achieving a mean error of 3.61 pixels (median 3.07 pixels) through an ensemble of four models with test-time augmentation. Module 2: Geometric normalizer: a set of landmarks surrounding the lung region is used to geometrically normalize each image. This normalization involves: Generalized Procrustes Analysis used once to obtain a standard lung shape, Delaunay triangulation to build a deformation mesh, and a piecewise affine transformation (warping) to map the original lung region to a standardized region. This process eliminates variations in position, scale, and orientation in the original set. Module 3: Classifier: normalized images are classified into three categories (COVID-19, Viral Pneumonia, and Normal) using a ResNet-18 classifier with transfer learning and a contrast adjustment (using the SAHS (Statistical Asymmetrical Histogram Stretching) method). The classifier was evaluated through five-fold cross-validation on the COVID-19 Radiography Database, demonstrating high stability with 98.60 ± 0.26% accuracy, and 98.00% F1-Macro, confirming the robustness of the approach. Although the classifier trained with original images reached a higher accuracy than using normalized images, Gradient-weighted Class Activation Mapping (Grad-CAM) analysis and the cropping experiment suggest that this advantage is partly driven by acquisition artifacts rather than lung pathology. In contrast, geometrically normalized images outperform their non-aligned artifact-masked/cropped counterparts: 98.60% vs. 96.24% on the COVID-19 Radiography Database and 94.67% vs. 94.17% on a balanced adult–pediatric mixed dataset including pediatric cases from the Kermany dataset, suggesting that anatomical alignment can yield a more reliable and artifact-resistant representation for pulmonary disease recognition.

En este artículo se describe el diseño de un algoritmo capaz de clasificar imágenes térmicas de personas acostadas y no acostadas, con el objetivo de aplicarlo en un sistema de detección de caídas. Se diseñó un algoritmo de normalización automática en rotación, traslación y tamaño, aplicado a una base de datos de imágenes térmicas, con el propósito de obtener un nuevo conjunto de imágenes alineadas y así realizar una reducción de dimensionalidad mediante PCA. Se utilizaron secuencias de 100 fotogramas, y se produjeron tanto secuencias de caídas como de no caídas. Aplicando el clasificador K-NN ponderado para identificar la clase de cada fotograma, se obtuvo un vector de probabilidades de la clase acostado con 100 posiciones. Estos nuevos vectores se utilizaron como ejemplos de entrenamiento para un nuevo clasificador K-NN, el cual contiene ejemplos de vectores de probabilidad de caídas y no caídas. Aplicando la validación cruzada, nuestro sistema es capaz de reconocer caídas con un 91 % de precisión.