In this supplemental material, we provide more details on network architecture and more visualization results, including shape reconstruction/comparison, shape Generation, and shape Interpolations. Furthermore, some results on scene reconstruction and comparison with Local Implicit Grid [3] are presented to demonstrate our superiority on large data representation thanks to the hierarchical tree structure of our proposed OctField representation. All sections are listed as follows: Section 1 provides the details of network architecture and training. Section 2, Section 3 and Section 4 provide more visualization results on a number of 3D modeling tasks, including shape reconstruction, generation and interpolation. Section 5 conducts four ablation studies, including with or without overlapping of adjacent octants, the training strategy, the distinction of latent codes and the subdivision parameter τ .

Recent advances in localized implicit functions have enabled neural implicit representation to be scalable to large scenes. However, the regular subdivision of 3D space employed by these approaches fails to take into account the sparsity of the surface occupancy and the varying granularities of geometric details. As a result, its memory footprint grows cubically with the input volume, leading to a prohibitive computational cost even at a moderately dense decomposition. In this work, we present a learnable hierarchical implicit representation for 3D surfaces, coded OctField, that allows high-precision encoding of intricate surfaces with low memory and computational budget. The key to our approach is an adaptive decomposition of 3D scenes that only distributes local implicit functions around the surface of interest. We achieve this goal by introducing a hierarchical octree structure to adaptively subdivide the 3D space according to the surface occupancy and the richness of part geometry. As octree is discrete and non-differentiable, we further propose a novel hierarchical network that models the subdivision of octree cells as a probabilistic process and recursively encodes and decodes both octree structure and surface geometry in a differentiable manner. We demonstrate the value of OctField for a range of shape modeling and reconstruction tasks, showing superiority over alternative approaches.

Abstract-- Estimation of signed distance functions (SDFs) from point cloud data has been shown to benefit many robot autonomy capabilities, including localization, mapping, motion planning, and control. Methods that support online and large-scale SDF reconstruction tend to rely on discrete volumetric data structures, which affect the continuity and differentiability of the SDF estimates. Recently, using implicit features, neural network methods have demonstrated high-fidelity and differentiable SDF reconstruction but they tend to be less efficient, can experience catastrophic forgetting and memory limitations in large environments, and are often restricted to truncated SDFs. This work proposes -SDF, a hybrid method that combines an explicit prior obtained from gradient-augmented octree interpolation with an implicit neural residual. Our method achieves non-truncated (Euclidean) SDF reconstruction with computational and memory efficiency comparable to volumetric methods and differentiability and accuracy comparable to neural network methods. Extensive experiments demonstrate that -SDF outperforms the state of the art in terms of accuracy and efficiency, providing a scalable solution for downstream tasks in robotics and computer vision. Accurate and differentiable geometric environment representations are critical for many functions in robot autonomy and computer vision, including simultaneous localization and mapping [1]-[3], rendering and AR/VR [4]-[6], autonomous navigation [7], [8] and manipulation [9]-[11].

We present a novel approach to graph drawing based on reinforcement learning for minimizing the global and the local crossing number, that is, the total number of edge crossings and the maximum number of crossings on any edge, respectively. In our framework, an agent learns how to move a vertex based on a given observation vector in order to optimize its position. The agent receives feedback in the form of local reward signals tied to crossing reduction. To generate an initial layout, we use a stress-based graph-drawing algorithm. We compare our method against force- and stress-based (baseline) algorithms as well as three established algorithms for global crossing minimization on a suite of benchmark graphs. The experiments show mixed results: our current algorithm is mainly competitive for the local crossing number. We see a potential for further development of the approach in the future.

In this supplemental material, we provide more details on network architecture and more visualization results, including shape reconstruction/comparison, shape Generation, and shape Interpolations. Furthermore, some results on scene reconstruction and comparison with Local Implicit Grid [3] are presented to demonstrate our superiority on large data representation thanks to the hierarchical tree structure of our proposed OctField representation. All sections are listed as follows: Section 1 provides the details of network architecture and training. Section 2, Section 3 and Section 4 provide more visualization results on a number of 3D modeling tasks, including shape reconstruction, generation and interpolation. Section 5 conducts four ablation studies, including with or without overlapping of adjacent octants, the training strategy, the distinction of latent codes and the subdivision parameter τ .

Recent advances in localized implicit functions have enabled neural implicit representation to be scalable to large scenes. However, the regular subdivision of 3D space employed by these approaches fails to take into account the sparsity of the surface occupancy and the varying granularities of geometric details. As a result, its memory footprint grows cubically with the input volume, leading to a prohibitive computational cost even at a moderately dense decomposition. In this work, we present a learnable hierarchical implicit representation for 3D surfaces, coded OctField, that allows high-precision encoding of intricate surfaces with low memory and computational budget. The key to our approach is an adaptive decomposition of 3D scenes that only distributes local implicit functions around the surface of interest. We achieve this goal by introducing a hierarchical octree structure to adaptively subdivide the 3D space according to the surface occupancy and the richness of part geometry. As octree is discrete and non-differentiable, we further propose a novel hierarchical network that models the subdivision of octree cells as a probabilistic process and recursively encodes and decodes both octree structure and surface geometry in a differentiable manner. We demonstrate the value of OctField for a range of shape modeling and reconstruction tasks, showing superiority over alternative approaches.

Establishing the correspondences between newly acquired points and historically accumulated data (i.e., map) through nearest neighbors search is crucial in numerous robotic applications.However, static tree data structures are inadequate to handle large and dynamically growing maps in real-time.To address this issue, we present the i-Octree, a dynamic octree data structure that supports both fast nearest neighbor search and real-time dynamic updates, such as point insertion, deletion, and on-tree down-sampling. The i-Octree is built upon a leaf-based octree and has two key features: a local spatially continuous storing strategy that allows for fast access to points while minimizing memory usage, and local on-tree updates that significantly reduce computation time compared to existing static or dynamic tree structures.The experiments show that i-Octree surpasses state-of-the-art methods by reducing run-time by over 50% on real-world open datasets.

The work presented here involves the design of a Multi Layer Perceptron (MLP) based classifier for recognition of handwritten Bangla alphabet using a 76 element feature set Bangla is the second most popular script and language in the Indian subcontinent and the fifth most popular language in the world. The feature set developed for representing handwritten characters of Bangla alphabet includes 24 shadow features, 16 centroid features and 36 longest-run features. Recognition performances of the MLP designed to work with this feature set are experimentally observed as 86.46% and 75.05% on the samples of the training and the test sets respectively. The work has useful application in the development of a complete OCR system for handwritten Bangla text.