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This paper addresses the medical imaging problem of joint detection in the upper limbs, viz. elbow, shoulder, wrist and finger joints. Localization of joints from X-Ray and Computerized Tomography (CT) scans is an essential step for the assessment of various bone-related medical conditions like Osteoarthritis, Rheumatoid Arthritis, and can even be used for automated bone fracture detection. Automated joint localization also detects the corresponding bones and can serve as input to deep learning-based models used for the computerized diagnosis of the aforementioned medical disorders. This in-creases the accuracy of prediction and aids the radiologists with analyzing the scans, which is quite a complex and exhausting task. This paper provides a detailed comparative study between diverse Deep Learning (DL) models - YOLOv3, YOLOv7, EfficientDet and CenterNet in multiple bone joint detections in the upper limbs of the human body. The research analyses the performance of different DL models, mathematically, graphically and visually. These models are trained and tested on a portion of the openly available MURA (musculoskeletal radiographs) dataset. The study found that the best Mean Average Precision (mAP at 0.5:0.95) values of YOLOv3, YOLOv7, EfficientDet and CenterNet are 35.3, 48.3, 46.5 and 45.9 respectively. Besides, it has been found YOLOv7 performed the best for accurately predicting the bounding boxes while YOLOv3 performed the worst in the Visual Analysis test. Code available at https://github.com/Sohambasu07/BoneJointsLocalization

Diabetic Retinopathy (DR) is a complication of long-standing, unchecked diabetes and one of the leading causes of blindness in the world. This paper focuses on improved and robust methods to extract some of the features of DR, viz. Blood Vessels and Exudates. Blood vessels are segmented using multiple morphological and thresholding operations. For the segmentation of exudates, k-means clustering and contour detection on the original images are used. Extensive noise reduction is performed to remove false positives from the vessel segmentation algorithm's results. The localization of Optic Disc using k-means clustering and template matching is also performed. Lastly, this paper presents a Deep Convolutional Neural Network (DCNN) model with 14 Convolutional Layers and 2 Fully Connected Layers, for the automatic, binary diagnosis of DR. The vessel segmentation, optic disc localization and DCNN achieve accuracies of 95.93%, 98.77% and 75.73% respectively. The source code and pre-trained model are available https://github.com/Sohambasu07/DR_2021, Comment: RAAI 2020, 11 pages, 12 figures, 2 tables

Achieving large bandwidth at strong mode conversion strengths is a central problem in few-mode optical waveguides. The most uncomplicated modification of a standard mode-converting grating is to incorporate a single extra gap. Such simple design has not been applied yet for broadening conversion spectrum between core modes of any few-mode waveguide e.g. few-mode fiber, possibly due to lacking knowledge of critical parameter control: (a) phase shift value at resonance wavelength, and (b) deterministic method for placement of the phase shift(s) accommodating the change in waveguide properties during grating writing. From semi-analytical considerations, we show in full generality that our design of a single-phase-shifted grating increases the bandwidth at 99 % conversion strength by more than 2.9 times, together with elucidating crucial parameter tolerances. The phase-shift placement is determinable on-the-go from the conversion spectra, thereby obviating any estimation of modified waveguide properties during grating-writing. This offers the longed on-demand in-fiber solution for dispersion control in fiber laser cavities and communication systems.

The resonance wavelength, where a fiber mode converter grating written using periodic external perturbations achieves phase matching, is both a critical design parameter and a device parameter. However, a method to precisely predict the resonance wavelength for any new fiber and grating writing apparatus has been missing so far. The missing link was the lack of direct experimental methods to estimate the modified intermodal phase after writing with external perturbations. The presented method can make this estimation from a single experiment, over a broad wavelength range, based on a novel mathematical connection between two-mode interference and mode conversion. Using the novel methods, experimentally measured resonance wavelengths for different pitch and irradiation conditions have been predicted within relative errors of 0.004.

Given any polynomial with real coefficients, the existence of a real quadratic polynomial factor is proven using only basic real analysis. The aim is to provide an approachable proof to anybody who is familiar with the least upper bound property for real numbers, continuity and growth property of polynomials, and unfamiliar with complex numbers, field extension or advanced topology.

Carl Friedrich Gauss is often given credit for providing the first correct proof of the fundamental theorem of algebra in his 1799 doctoral dissertation. However, Gauss's proof contained a significant gap. In this paper, we give an elementary way of filling the gap in Gauss's proof., Comment: 9 pages, 1 figure. To appear in American Mathematical Monthly