The present paper introduces a novel object of study - a language fractal structure. We hypothesize that a set of embeddings of all $n$-grams of a natural language constitutes a representative sample of this fractal set. (We use the term Hailonakea to refer to the sum total of all language fractal structures, over all $n$). The paper estimates intrinsic (genuine) dimensions of language fractal structures for the Russian and English languages. To this end, we employ methods based on (1) topological data analysis and (2) a minimum spanning tree of a data graph for a cloud of points considered (Steele theorem). For both languages, for all $n$, the intrinsic dimensions appear to be non-integer values (typical for fractal sets), close to 9 for both of the Russian and English language.

In recent years, multiple Light Detection and Ranging (LiDAR) systems have grown in popularity due to their enhanced accuracy and stability from the increased field of view (FOV). However, integrating multiple LiDARs can be challenging, attributable to temporal and spatial discrepancies. Common practice is to transform points among sensors while requiring strict time synchronization or approximating transformation among sensor frames. Unlike existing methods, we elaborate the inter-sensor transformation using continuous-time (CT) inertial measurement unit (IMU) modeling and derive associated ambiguity as a point-wise uncertainty. This uncertainty, modeled by combining the state covariance with the acquisition time and point range, allows us to alleviate the strict time synchronization and to overcome FOV difference. The proposed method has been validated on both public and our datasets and is compatible with various LiDAR manufacturers and scanning patterns. We open-source the code for public access at https://github.com/minwoo0611/MA-LIO.

For a wide range of clinical applications, such as adaptive treatment planning or intraoperative image update, feature-based deformable registration (FDR) approaches are widely employed because of their simplicity and low computational complexity. FDR algorithms estimate a dense displacement field by interpolating a sparse field, which is given by the established correspondence between selected features. In this paper, we consider the deformation field as a Gaussian Process (GP), whereas the selected features are regarded as prior information on the valid deformations. Using GP, we are able to estimate the both dense displacement field and a corresponding uncertainty map at once. Furthermore, we evaluated the performance of different hyperparameter settings for squared exponential kernels with synthetic, phantom and clinical data respectively. The quantitative comparison shows, GP-based interpolation has performance on par with state-of-the-art B-spline interpolation. The greatest clinical benefit of GP-based interpolation is that it gives a reliable estimate of the mathematical uncertainty of the calculated dense displacement map.