# Data-Driven Field Representations and Measuring Processes

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### Neural Networks as Function Approximator

## 3. Computational Model of Learnable Field

#### 3.1. Neural Network-Based Approximator

#### 3.2. Eulerian Representation via Parameters on Spacial Grids

## 4. Computational Models of the Measurement Process

#### 4.1. Integration Adjustments

#### 4.2. Discretization of the Integration

#### 4.3. Computational Process of Summing over Sampled Points

## 5. Conclusions

- Global model with shared parameters: The most common choice in this way is neural networks, which update the entire field. It tends to smooth out local changes or details in the data and may face challenges in capturing and representing small variations. Also, its time consumption has been shown to be heavy in the studies mentioned above.
- Local model with parametric grid: Regardless of the smallest unit of a grid, such as an indexed cube or an element in tensor, in the grid-based method, the field space is arranged, offering efficient spatial querying, allowing for the rapid retrieval of objects or data points within a specific region of interest. When a change is observed, only its surrounding area needs to be updated. Furthermore, this approach allows for skipping or pruning unimportant areas and the adaptive concentration of parameters in areas containing more significant and informative data. However, this approach may not be suitable for certain physical phenomena, such as light phenomena, which often require additional techniques like spherical harmonics for accurate representation [33]. Additionally, the resolution limitations of the grid can restrict its ability to capture fine-grained details, and the memory requirements can grow rapidly as the complexity of the problem increases.

- In the area of integration adjustments, our synthesis of the literature revealed the diverse reparameterization strategies employed to accommodate varying scene scopes. This encompasses scenarios ranging from a limited depth of the scene to an infinite depth and extends to the intricate challenges posed by a 360-degree unbounded scene.
- When it comes to the discretization of integration, our exploration encompasses a discussion on how existing methods strategically sample from space. This sampling is intricately tied to the specific design choices made in integration adjustments, forming a crucial aspect of the computational framework.
- In describing the computational processes, we delved into the widely adopted method and the nuanced numerical integration approach embraced by AutoInt.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Different computational architectures to implement the function family in Equation (2). (

**a**) Different field functions with different physical properties can be learned by a shared neural network, such as the combination of ${f}_{\sigma}$ interested in density and ${g}_{c}$ interested in appearance in Equation (6). However, the approximators are not limited to geometry and appearance. One of the “extra conditions” can be time, and the field function can approximate time-related changes [20]. It should be mentioned that the order of different fields in this figure is just an example. In practice, the order must be carefully considered per the requirements or demands. (

**b**) Each field can be approximated by individual neural networks, which can be updated and adjusted separately. However, not all field functions can be simply fused. Both physical relations and computational skills need to be considered in real implementations.

**Figure 2.**Grid-based approximator of a field. If the smallest cell in a grid is a cube. The learnable parameters can be assigned to the eight vertices, and the local properties are represented within the cube. The feature of any position p in the cube is computed using the selected interpolation strategy. The grid arrangement provides spatial indexing, offering efficient spatial querying, rapidly retrieving objects or points within a specific region of interest.

**Figure 3.**Octree representation of a field. Consider the grid-based field in Equation (7). The root level represents the bounding box for the entire field, while the finest leaf nodes correspond to the smallest units determined by settings, such as the vertices of a cube area. A white node indicates that all its child nodes are empty, allowing it to be skipped during the computing process. Conversely, blue nodes represent important spaces where all the vertices possess valuable features and should not be skipped. Partial nodes indicate that only certain nodes are empty and cannot be skipped. The full and partial nodes can be subdivided into smaller cells until a termination condition is met. Regions with more significant details or rapid changes are subdivided with more nodes, while less important regions are represented with fewer nodes. As all the nodes have corresponding learnable parameters, this adaptive nature enables a higher density of parameter distribution in important regions.

**Figure 4.**Tensor factorization: G represents a field function. A tensor field T is the approximation of G. According to the X, Y, and Z axes of a field, three modes vectors (tensors) ${v}^{X},{v}^{Y},{v}^{Z}$ or matrices ${M}^{X,Y},{M}^{Y,Z},{M}^{X,Z}$ are used for the decomposition. (

**a**) The CP decomposition represents canonical polyadic decomposition [37], which is vector-only decomposition, and (

**b**) VM decomposition refers to vector–matrix decomposition [36]. Tensor decomposition can be expressed as the sum of R outer products of vectors or vector–matrix. A smaller value of R indicates a more compact field representation, but accuracy is sacrificed.

**Figure 5.**The correlation between fields and 2D observations. The diagram provides a visual representation of how data-driven field models interact with and manifest in observable 2D data.

**Figure 6.**Inverted sphere parameterization, illustrating the scene parameterization, showcasing a mapping of coordinates from Euclidean space to a new spatial representation. Specifically, the coordinates in $r<1$ are not changed, while the coordinates in $r>1$ are mapped to $({x}^{\prime},{y}^{\prime},{z}^{\prime},1/r)$.

**Figure 7.**Integration process along a 3D cone. Compared to integrating along a line, which only utilizes the information in the line, integrating along a cone, which utilizes the information within a 3D conical frustum, captures more information.

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Hong, W.; Zhu, S.; Li, J.
Data-Driven Field Representations and Measuring Processes. *Foundations* **2024**, *4*, 61-79.
https://doi.org/10.3390/foundations4010006

**AMA Style**

Hong W, Zhu S, Li J.
Data-Driven Field Representations and Measuring Processes. *Foundations*. 2024; 4(1):61-79.
https://doi.org/10.3390/foundations4010006

**Chicago/Turabian Style**

Hong, Wanrong, Sili Zhu, and Jun Li.
2024. "Data-Driven Field Representations and Measuring Processes" *Foundations* 4, no. 1: 61-79.
https://doi.org/10.3390/foundations4010006