# VFLD: Voxelized Fractal Local Descriptor
## 1 Introduction  
The ability to recognize three-dimensional objects is a key feature for a range of different applications. For instance, intelligent and self-driving cars must be able to detect and identify the objects around them in order to enable a specific behavior: they must stop at a red light, engage the emergency brake if a pedestrian suddenly crosses the road or detect traffic signs in order to drive safely. Another notable application for three-dimensional object recognition is social robotics. Robots that are intended to interact with humans should be able to perceive each different person, and also to detect certain objects that may be potentially involved in their tasks. For instance, it might be useful for the robot to recognize household items. Additionally, in computer vision, point cloud registration is an important task in many applications related to robotics, such as simultaneous localization and mapping (SLAM). It consists of finding the transformation that aligns one point cloud to another, considering that there could be potential obfuscation, yielded noise, or only partial representation of the scene [1]. The techniques that explicitly work with the 3D points and generate local descriptors are directly applicable to both tasks, that is, 3D object recognition and registration, and thus we focus on these methods in the present work. Most deep learning methods require large amounts of data that are not easy to obtain, which sometimes makes them practically useless in real scenarios due to the lack of generalization capabilities. On the other hand, many traditional descriptors require the manual adjustment of numerous parameters for each different problem, and their performance tends to degrade when minimal changes occur in the density of the point cloud or when noise is introduced. These limitations necessitate a more robust and adaptable approach, especially for applications like 3D object recognition, where variability and irregularity are common. To address these challenges, we explore the concept of fractal dimension. Fractal dimension is a geometric measure that quantifies the complexity of a shape or pattern by providing a statistical index that describes how detail in the pattern changes with scale. Unlike traditional descriptors, which may rely on rigid structural or geometric assumptions, fractal dimensions can effectively capture the irregularities and self-similar properties found in natural objects and complex shapes [2]. This makes them highly suitable for point cloud data, where the distribution of points can be sparse, noisy, or exhibit intricate structural variations. Fractal dimension has already demonstrated its potential in 2D image processing, where it has been employed to describe textures and patterns with promising results [3; 4; 5]. However, its application in 3D data processing, particularly as a descriptor for 3D object recognition, remains relatively novel. Our recent research [6] has shown that fractal dimensions can serve as effective global 3D descriptors, offering a reliable way to characterize entire objects based on their overall complexity. This global approach yielded encouraging results, particularly in scenarios where objects varied significantly in scale and form. Building on this success, we propose extending the use of fractal dimension from global to local 3D descriptors. By applying fractal dimension analysis at a local level, we aim to capture the fine-grained structural details within specific regions of an object, enhancing the ability to distinguish between objects that may share similar global characteristics but differ in local features. This approach offers several advantages:  
* Scale invariance. Fractal dimensions inherently account for changes in scale, which makes them particularly effective in scenarios where objects may appear at different sizes or resolutions within a point cloud. * Robustness to noise.