Multi-View 3D Reconstruction of Ship Hull via Multi-Scale Weighted Neural Radiation Field

Abstract

The 3D reconstruction of vessel hulls is crucial for enhancing safety, efficiency, and knowledge in the maritime industry. Neural Radiance Fields (NeRFs) are an alternative to 3D reconstruction and rendering from multi-view images; particularly, tensor-based methods have proven effective in improving efficiency. However, existing tensor-based methods typically suffer from a lack of spatial coherence, resulting in gaps in the reconstruction of fine-grained geometric structures. This paper proposes a spatial multi-scale weighted NeRF (MDW-NeRF) for accurate and efficient surface reconstruction of vessel hulls. The proposed method develops a novel multi-scale feature decomposition mechanism that models 3D space by leveraging multi-resolution features, facilitating the integration of high-resolution details with low-resolution regional information. We designed separate color and density weighting, using a coarse-to-fine strategy, for density and a weighted matrix for color to decouple feature vectors from appearance attributes. To boost the efficiency of 3D reconstruction and rendering, we implement a hybrid sampling point strategy for volume rendering, selecting sample points based on volumetric density. Extensive experiments on the SVH dataset confirm MDW-NeRF’s superiority: quantitatively, it outperforms TensoRF by 1.5 dB in PSNR and 6.1% in CD, and shrinks the model size by 9%, with comparable training times; qualitatively, it resolves tensor-based methods’ inherent spatial incoherence and fine-grained gaps, enabling accurate restoration of hull cavities and realistic surface texture rendering. These results validate our method’s effectiveness in achieving excellent rendering quality, high reconstruction accuracy, and timeliness.

1. Introduction

Three-dimensional reconstruction technology has emerged as an essential tool in the maritime industry, particularly in the domains of maritime 3D hull perception [1] and ship maintenance [2,3,4]. These advanced technologies facilitate the 3D perception of ship hulls in marine applications, offering a promising approach to enhancing autonomous navigation capabilities. By leveraging perception information, they significantly contribute to intelligent ship perception and have the potential to improve the comprehensibility of autonomous vessels. The 3D reconstruction of hulls provides a volumetric and precise visual representation that closely aligns with actual specifications.

The transformative impact of these technologies in shipbuilding and maintenance allows shipbuilders to accurately replicate offshore structures, thereby boosting production efficiency and product quality. In terms of maintenance and protection, ship 3D technology offers detailed data for structural analysis, enabling more targeted and meticulous maintenance efforts [5,6]. Thus, the application of ship 3D reconstruction technology in ocean engineering is of paramount importance. It not only enhances the perception capabilities of maritime 3D ships, but also elevates the overall quality and efficiency of ship design, production, and maintenance, thereby improving the autonomy and service life of ships.

Traditionally, maritime engineering has relied on manual surveying and modeling based on many international ship CAD/CAM systems for hull 3D reconstruction [7]. However, these conventional methods are typically time-consuming and labor-intensive, prone to human error, and carry the risk of inaccuracies and project delays. Consequently, the field is seeking more efficient and lower-error alternative reconstruction methods. With the ongoing advancement of computer vision and machine learning technologies, systems that employ multi-angle photography to automatically capture and reconstruct the hull have emerged, offering a cost-effective and efficient new option for three-dimensional geometric reconstruction [4]. These innovative technologies not only greatly outperform traditional methods in performance but also signal the immense potential for future improvements in the field. Advances in optical measurement technologies [8], benefiting from breakthroughs in the field of computer vision, are able to capture high-definition images with minimal error, thus enhancing the resolution of hull 3D reconstruction. Moreover, the latest developments in deep learning, particularly in processing multi-view imagery [9,10], have brought about revolutionary progress in the three-dimensional reconstruction of hulls, making the process more streamlined and accurate.

Currently, Neural Radiance Fields (NeRFs) [11] are widely acclaimed for their exceptional rendering capabilities. Given that 3D reconstruction is essentially the inverse process of volume rendering, methods based on NeRF demonstrate considerable advantages in the realm of multi-view 3D reconstruction. In recent years, many neural representation methods [12,13] have been proposed to address the limitation of maximum voxel resolution for high-quality rendering and reconstruction [14,15]. NeRF [16] introduces radiance fields to address novel view synthesis and achieves photo-realistic quality. This representation has been quickly extended and applied in diverse graphics and vision applications, including appearance acquisition [17,18], multi-view 3D reconstruction [19,20,21], fast rendering [22,23], dynamic capture [24,25] and generative modeling [26,27]. While leading to realistic rendering and a compact model, NeRF implicitly represents the scene in the MLP, which leads to very compact storage, but the reconstruction and rendering are extremely slow. In order to accelerate rendering speeds, a series of efforts [23,28] have been dedicated to reducing the complexity of inference by segmenting the scene into multiple units and learning to decrease the number of samples per ray. Nonetheless, these methods still necessitate high memory costs and compromise the compactness of NeRF. To surmount these limitations, recent research [29] integrates neural radiance fields with tensor decomposition, significantly diminishing the computational demands while retaining the fidelity of rendered scenes, with the most typical being TensoRF.

However, tensor-based methods may compromise spatial integrity when reconstructing complex geometrical details. To tackle this problem, we propose a multi-scale spatial density weighted NeRF (MDW-NeRF). As illustrated in Figure 1, our method refines the calculation process for both density $\sigma$ and color c, building upon the TensoRF framework to improve the accuracy of 3D ship reconstruction. The main contributions of this paper are summarized as follows:

(1)

We introduce a multi-scale feature space decomposition mechanism, which models three-dimensional space by exploiting features at multiple levels of resolution. This enables the fusion of high-resolution detailed information with low-resolution regional information across different directions.

(2)

To optimize the reconstruction process, we separately design color and density weights. The density weighting employs a coarse-to-fine strategy, initially determining the structure’s consistent rough outline, followed by the application of a density weighting network (DWN) to refine the model. For color weighting, we use a weighted matrix to decouple the feature vectors from the appearance attributes and calculate the color via a color rendering network (CRN).

(3)

To accelerate the efficiency of the algorithm’s 3D reconstruction and rendering, we employ a hybrid point sampling strategy for volume rendering, namely, the selection of sampling points through volumetric density.

Collectively, these integrated innovations tackle the core limitation of spatial incoherence in tensor-based NeRF methods while achieving a judicious balance between reconstruction fidelity, rendering quality, and computational efficiency. By synergistically fusing multi-scale feature space decomposition, adaptive color and density weighting mechanisms, and the hybrid point sampling strategy, MDW-NeRF yields a robust and effective solution for high-precision hull 3D reconstruction, thereby filling the critical gap in fine-grained geometric recovery and spatial integrity that compromises the performance of existing tensor-based approaches. As validated in subsequent experiments, this work pushes forward the state-of-the-art of tensor-based NeRF-driven hull 3D reconstruction by overcoming key technical bottlenecks inherent to prior methods.

2. Method Architecture

2.1. Overview

As shown in Figure 2, the sampling point $\mathbf{x}=(x,y,z)$ is first obtained along the ray $\mathbf{r}\left(t\right)=\mathbf{o}+t\mathbf{d}$ with initial sampling position $\mathbf{o}$ and sampling direction $\mathbf{d}$. We then employ a multi-scale feature space decomposition mechanism to decompose the features of the sampling point along three-dimensional directions and map them to feature planes ${\mathbf{M}}_n^k$ and feature vectors ${\mathbf{v}}_n^m$, where $n=1,2,\dots ,L$, $m\in \{X,Y,Z\}$ and $k\in \{XY,XZ,YZ\}$, respectively, indicate different resolution scales, different directions and different planes in XYZ spatial coordinates. We take the element-wise product of ${\mathbf{M}}_n^k$ and ${\mathbf{v}}_n^m$ to obtain features at different scales and directions. Subsequently, we integrate these features, respectively, according to the same scale but different directions and the same direction but different scales, to obtain multi-scale features $\mathbf{G}$ and multi-directional features ${\mathbf{A}}_f$. We utilize multi-directional features ${\mathbf{A}}_f$ to reconstruct the density $\sigma$ of the point $\mathbf{x}$ with a coarse-to-fine density weighting mechanism for accelerating the training process of the density reconstruction model. Also, we utilize color weighting to generate weights of cross-scale color features ${\mathbf{f}}_c$ using multi-scale features $\mathbf{G}$ and then reconstruct the color c of the point $\mathbf{x}$ through $\mathbf{d}$-conditional regression. Finally, we combine the estimated density $\sigma$ and color c through numerical integration to approximate the volume rendering integral, thereby rendering the final color of the pixel corresponding to the ray.

Next, we will elaborate on the details of multi-scale and multi-direction feature decomposition, density and color weighting, and volume rendering.

2.2. Multi-Scale and Multi-Direction Feature Decomposition

2.2.1. Multi-Scale Feature Decomposition

In the context of the neural radiance field method grounded in tensor decomposition, our findings indicate that resolution is a critical factor influencing the final reconstruction and rendering outcomes; concretely, higher-resolution offers enhanced detail precision. However, it also introduces a notable challenge regarding spatial coherence during reconstruction. When compared to lower resolution, the spatial coherence at higher resolutions is more evident, leading to fewer voids in the reconstructed output. Nevertheless, significant issues persist in the reconstruction of intricate details.

To precisely capture the fine details and geometric shapes of the ship hull, and to ensure the efficiency of the ship hull’s reconstruction and rendering processes, we propose a multi-scale feature decomposition mechanism for the sampling points $\mathbf{x}$ to achieve scene expression from coarse to fine levels, as shown in Figure 2.

Specifically, for the sampling point $\mathbf{x}$, we start from the lowest resolution $N_{min}$ and progressively map it to higher resolutions, increasing by a factor b until reaching the maximum resolution $N_{max}$ as follows:

$$N_n=N_{min}{b}^n,b=exp\left({\displaystyle \frac{ln{N}_{max}-ln{N}_{min}}{L-1}}\right)$$

(1)

where $N_n$ ($n=1,2,\dots ,L$) represents the resolution of the n-th layer, and L is the total number of multi-scale layers. For the n-th layer, we integrate the element-wise product of feature plane ${\mathbf{M}}_n^k\in {\mathbb{R}}^{n_f\times N_n\times N_n}$ ($k\in \{XY,XZ,YZ\}$) and feature vector ${\mathbf{v}}_n^m\in {\mathbb{R}}^{n_f\times N_n}$ ($m\in \{X,Y,Z\}$) at resolution scale $N_n$ from XYZ spatial directions to construct the corresponding decomposed feature ${\mathbf{G}}_n$, where $n_f$ means the channel number of decomposition features at each resolution scale.

$$\begin{array}{cc}\hfill {\mathbf{G}}_n\left(\mathbf{x}\right)=& [{\mathbf{v}}_n^X\left(x\right)\circ {\mathbf{M}}_n^{YZ}(y,z),{\mathbf{v}}_n^Y\left(y\right)\circ {\mathbf{M}}_n^{XZ}(x,z),\hfill \\ & {\mathbf{v}}_n^Z\left(z\right)\circ {\mathbf{M}}_n^{XY}(x,y)].\hfill \end{array}$$

(2)

The multi-scale feature $\mathbf{G}$ can thus be obtained by integrating decomposed features ${\mathbf{G}}_n$ at all resolution scales.

$$\mathbf{G}\left(\mathbf{x}\right)=\left[{\mathbf{G}}_n\left(\mathbf{x}\right)\right](n=1,2,\dots ,L)$$

(3)

2.2.2. Multi-Direction Feature Decomposition

Essentially, the feature planes ${\mathbf{M}}_n$ and feature vectors ${\mathbf{v}}_n$ reveal the spatial distribution characteristics of the ship hull’s geometry and appearance along their corresponding directions, which depict in detail the mapping relations of the sampling points along their respective (XYZ) spatial directions.

To efficiently use multi-scale features, we adopt a fusion splicing method, which concatenates features from the same direction together to generate the multi-directional feature ${\mathbf{A}}_f^m$ for that direction. For example, as illustrated in Figure 3, we concatenate the trilinear interpolation results at resolution scale $N_n$ from the X-direction to obtain ${\mathbf{A}}_f^X$ as follows:

$${\mathbf{A}}_f^X\left(\mathbf{x}\right)=[{\mathbf{v}}_n^X\left(x\right)\circ {\mathbf{M}}_n^{YZ}(y,z)](n=1,2,\dots ,L).$$

(4)

Similarly, the features from the Y-direction and Z-direction, i.e., ${\mathbf{A}}_f^Y$ and ${\mathbf{A}}_f^Z$, can be obtained as follows:

$$\begin{array}{cc}\hfill {\mathbf{A}}_f^Y\left(\mathbf{x}\right)& =[{\mathbf{v}}_n^Y\left(y\right)\circ {\mathbf{M}}_n^{XZ}(x,z)],\hfill \\ \hfill {\mathbf{A}}_f^Z\left(\mathbf{x}\right)& =[{\mathbf{v}}_n^Z\left(z\right)\circ {\mathbf{M}}_n^{XY}(x,y)].\hfill \end{array}$$

(5)

The multi-directional features ${\mathbf{A}}_f\left(\mathbf{x}\right)$ can be represented as

$${\mathbf{A}}_f\left(\mathbf{x}\right)=\left[{\mathbf{A}}_f^m\left(\mathbf{x}\right)\right](m\in \{X,Y,Z\})$$

(6)

2.3. Density and Color Weighting

2.3.1. Density Weighting

Figure 4 shows the reconstruction results using single-direction features ${\mathbf{A}}_f^m\left(\mathbf{x}\right)$ ($m\in \{X,Y,Z\}$), from which we can observe that during the density calculation process, the reconstruction effect of the multi-directional features ${\mathbf{A}}_f^m\left(\mathbf{x}\right)$ from the $X/Y/Z$-directions in the same area exhibited inconsistency, leading to insufficient or inconsistent detail representation in certain directions of the generated images. For example, for surfaces parallel to the XY plane, the Z-direction feature representation ${\mathbf{A}}_f^Z\left(\mathbf{x}\right)$ is inadequate, although it can slightly reconstruct surfaces aligned with the XZ or YZ planes. The feature decomposition ${\mathbf{v}}_n^Z\left(z\right)\circ {\mathbf{M}}_n^{XY}(x,y)$ can be treated as ${\mathbf{M}}_n^{XY}(x,y)$ varying along the Z-direction, while there exist significant variations along the Z-direction. This might cause that some areas have substantial deficiencies while the others have a large amount of noise to balance the changes in the upper and lower regions.

To enhance the correlation of feature representation across different directions, we developed a density weighting strategy to adaptively adjust feature representation based on their directional characteristics. To accelerate model training, as illustrated in Figure 5, we propose an incrementally refined training strategy to transition the density weight in a coarse-to-fine fashion, which is controlled by a factor $n_{\mathrm{fine}}$, i.e., when training iterations are less than $n_{\mathrm{fine}}$, it is in the coarse-density weighting stage, otherwise, it is in the fine-density weighting stage. In the coarse-density weighting stage, the density $\sigma$ is directly obtained by linearly projecting the direction-wise summation of feature vector ${\mathbf{A}}_f\left(\mathbf{x}\right)$. In the fine-density weighting stage, we designed an adaptive feature sampling mechanism to dynamically adjust the weight distribution in different directions via a density weighting network (DWN). As illustrated in Figure 5, in DWN, we utilize channel-wise attention to learn density weighted features from multi-directional feature ${\mathbf{A}}_f$ and then generate fine density $\sigma$, i.e., $\sigma =DWN\left({\mathbf{A}}_f\left(\mathbf{x}\right)\right)$. Specifically, DWN utilizes a GELU-activated linear projection to initially compress channels of ${\mathbf{A}}_f$ from $3\times n_f$ to $N_d=3\times n_f/r$ with a ratio r. Subsequently, two linear projections with GELU activation are employed to learn density weights, which are then element-wise multiplied to generate weighted density features and regress to fine density $\sigma$.

The coarse-to-fine strategy allows for more nuanced weight adjustments of features based on the characteristics of density distribution in various directions. Also, it ensures the differentiability of weight assignment across the $X,Y,Z$-directions, thereby augmenting the representational and discriminatory power of the features. In comparison with the coarse-density weighting stage, the fine-density weighting can more accurately capture the correlations between features, consequently improving the efficacy of feature representation substantially.

2.3.2. Color Weighting

Color is a complex, multidimensional feature that is not only related to spatial position but also to surface properties. It is possible to capture the complexity of color variations and appearance characteristics more finely, thereby capturing more details. We thus add a new dimension in the decomposition to represent the rendered color feature ${\mathbf{f}}_c$ by an $L\times n_f$-channel linear projection of multi-scale feature ${\mathbf{G}}_n$.

Specifically, the decomposition expression for rendered color feature ${\mathbf{f}}_c$ is

$${\mathbf{f}}_c=\left[{\mathbf{G}}_n\left(\mathbf{x}\right){\mathbf{W}}_n\left(\mathbf{x}\right)\right](n=1,\dots ,L)$$

(7)

where ${\mathbf{W}}_n\in {\mathbb{R}}^{n_f}$ is the appearance feature vector representing the surface properties of the hull at a specific scale. By stacking L of these ${\mathbf{W}}_n$ vectors, we obtain a matrix that captures the surface characteristics of the hull across all scales, effectively creating a comprehensive global appearance dictionary for the hull’s features.

Next, the color feature ${\mathbf{f}}_c$ of the sampling point $\mathbf{x}$ and the view direction $\mathbf{d}$, encoded using spherical harmonic functions, are input into the color rendering network (CRN). The color c of the point is then derived through a series of linear transformations as depicted in Figure 5: $c=CRN({\mathbf{f}}_c,\mathbf{d})$. Initially, the CRN increases the channels of the input features to $N_c$ using a linear projection to enhance the scale of feature representation, which is then followed by three linear transformations and activation by GELU to improve generalization. Finally, a linear transformation outputs the color $c\in {\mathbb{R}}^3$ for the given point in the specified view direction.

2.4. Volume Rendering

Since volume rendering inherently involves sampling multiple points along the ray for each pixel, some sampling points can be bypassed through a proxy mesh. Inspired by this, we developed a hybrid strategy for point sampling in volume rendering to further enhance the rendering speed. In addition to the reconstructed radiance field, our rendering strategy requires a proxy mesh to efficiently determine the rough distance from the camera’s optical center to the object. Fortunately, this can be obtained by marching through the reconstructed density field and then performing mesh extraction to acquire the proxy mesh.

Once the proxy mesh is available, we first rasterize it efficiently to obtain the sampling points $x^{\prime }$, and then we perform uniform sampling within a distance s from the sampling points along the direction of the ray, which results in a 2 s sampling interval. Subsequently, we render the ray by sampling i points in the 2 s sampling interval to predict density and view-dependent color using differentiable volume rendering techniques. The point ${\mathbf{x}}_i$ sampled at depth $t_i$ has a density ${\sigma }_i$ which is the result of multi-direction feature decomposition and density weighting. The color $c_i$ of the considered point is obtained via multi-scale feature decomposition and color weighting. We calculate the color composition weights based on the density as

$$\begin{array}{c}\hfill w_i=exp\left(-\sum _{j=0}^{i-1}{\sigma }_j{\mathsf{\Delta }}_j\right)(1-exp(-{\sigma }_i{\mathsf{\Delta }}_i))\end{array}$$

(8)

where $\mathsf{\Delta }$ is the sampling interval. We only compute the color for sampling points whose weights $w_i$ are greater than ${10}^{-2}$ in order to skip the blank area (for example, $1-exp(-{\sigma }_i{\mathsf{\Delta }}_i)<1\times {10}^{-2}$) and the occlusion area (for example, $exp\left(-{\sum }_{j=0}^{i-1}{\sigma }_j{\mathsf{\Delta }}_j\right)<1\times {10}^{-2}$) in target reconstruction. This strategy effectively reduces the computational cost and focuses the appearance representation on meaningful points. The rendered pixel color $\widehat{\mathbf{c}}$ is the weighted sum of the predicted colors

$$\begin{array}{c}\hfill \widehat{\mathbf{c}}=\sum _{i=0}^{N-1}{w}_i{c}_i.\end{array}$$

(9)

This hybrid strategy for sampling points in volume rendering significantly reduces the number of samples, thereby improving the rendering efficiency.

3. Experiments

In this section, we conduct extensive experiments to demonstrate the effectiveness of our method on 3D reconstruction with a comprehensive comparison to the existing state-of-the-art and mainstream methods.

3.1. Datasets

We employed the robust 3D modeling software Blender 5.0 to build the Synthetic Vessel Hull (SVH) 3D reconstruction dataset. This dataset comprises 13 different types of vessels, each represented by 600 individual samples. Each sample includes images as well as corresponding camera pose information. To ensure the diversity and comprehensiveness of the dataset, we specifically chose a range of vessel types for study. The shapes and structures of these vessels vary greatly, from small yachts to large warships, and we endeavored to include as many different types of vessels as possible. We allocated 400 samples for model training, 100 samples for testing, and an additional 100 samples for validation.

3.2. Implementation Details

We implemented our MDW-NeRF using PyTorch 1.13.1, and all experiments including baseline methods and MDW-NeRF were standardized to a consistent Tesla A6000 48 GB GPU (NVIDIA Corporation, Santa Clara, CA, USA), a batch size of 4096 pixels, and an image resolution of $800\times 800$ for fair and normalized comparisons. Based on experience, we set $N_{min}$ and $N_{max}$ to 64 and 256, respectively, and configured L to be 6. This parameter configuration, derived from systematic validation of multi-scale feature decomposition requirements and the trade-off between reconstruction performance and computational efficiency, is tailored to optimize cross-scale feature behavior: $N_{min}=64$ ensures coherent modeling of low-resolution global regional information to preclude structural fragmentation from overly sparse sampling, $N_{max}=256$ provides sufficient resolution to capture high-fidelity fine-grained hull details within hardware memory constraints, and $L=6$ (number of multi-scale layers) enables smooth coarse-to-fine feature transition via exponential resolution scaling—effectively mitigating inter-scale feature gaps and redundant computational overhead for efficient training and inference. The total number of iterations was fixed at 150,000.

We adopt the Adam optimizer for its tailored suitability to MDW-NeRF’s multi-module architecture: its integrated momentum and adaptive learning rate mechanisms efficiently handle heterogeneous parameter dynamics across multi-scale feature decomposition, density weighting network (DWN) and color rendering network (CRN), dynamically balancing updates for high-sensitivity and low-sensitivity parameters while eliminating manual learning rate tuning. It accelerates convergence in non-convex neural radiance field fitting, mitigates oscillations to avoid local optima during coarse-to-fine training, and outperforms SGD (requiring tedious tuning and slow convergence) and Adagrad (prone to premature learning rate shrinkage) in stabilizing high-dimensional tensor decomposition features, aligning with parameter-efficient tensor-based NeRF paradigms to achieve an optimal trade-off between convergence speed, stability and reconstruction performance for high-precision hull 3D reconstruction.

During the coarse-density weighting phase, the learning rate for the decomposition feature was set to 0.02, while the learning rate for the CRN network was 0.001. In the fine-density weighting phase, the initial learning rate for the decomposition feature was reduced to 0.001, the learning rate for the DWN was increased to 0.02, and the learning rate for the CRN network remained at 0.001.

The training loss of the proposed method consists of the mean squared error of the rendered pixel value, and a regularization term regarding the density features. Mathematically, the training loss is written as

$$L=\widehat{\mathbf{c}}-{\mathbf{c}}_{gt}$$

(10)

where ${\mathbf{c}}_{gt}$ is the ground truth color. To achieve a coarse-to-fine reconstruction, we set $n_{\mathrm{fine}}$ based on the variation in loss. That is, we compute the variance of the loss every 100 iterations. When the variance at iteration $n_{\mathrm{itera}}$ falls below 0.01, we consider the training to have sufficiently converged. At this stage, we set $n_{\mathrm{fine}}=n_{\mathrm{itera}}$ and transition from coarse reconstruction to fine reconstruction.

3.3. Evaluation Metrics

When assessing the accuracy of image rendering performance, we utilized quantitative metrics such as Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index Measure (SSIM), and Learned Perceptual Image Patch Similarity (LPIPS). Meanwhile, when evaluating the accuracy of 3D reconstruction, we used Chamfer Distance (CD) as our evaluation metric. To measure processing speed, we conducted a timeliness analysis based on the time required for the reconstruction process. Additionally, we evaluated the system’s storage efficiency by calculating the required model parameters (Params.) as a proxy for storage resources. To rigorously substantiate the efficiency assertions, we additionally incorporate two pivotal metrics: (1) rendering speed (RS) reported as Frames Per Second (FPS) to quantify inference throughput; and (2) inference memory usage (IMU) to characterize GPU memory footprint during inference.

3.4. Performance Comparison

3.4.1. Comparison on Rendering

To quantitatively assess the effectiveness of the proposed method MDW-NeRF in volume rendering, we compared MDW-NeRF with several neural rendering methods, including Nvdiffrec [30], original NeRF [11], NeuS [19], NeRF2mesh [31], Voxurf [32], and TensoRF [29], as shown in Figure 6. The evaluation metrics include PSNR, SSIM, and LPIPS. Figure 6 illustrates the performance of different methods in ship model rendering applications. The results indicate that MDW-NeRF excels in rendering fine details and structural representations. The table adjacent to each subplot displays the average performance of each method across different ship models, with each color denoting a specific method. It is evident from the table that MDW-NeRF consistently outperforms the comparison algorithms in all benchmark tests. These quantitative results confirm the superior rendering performance of MDW-NeRF.

Given that our loss function is derived from the quality of image rendering, we display the selected rendered images to evaluate the efficacy of this loss function. Through a visual comparison of the rendering quality between our proposed method and existing techniques, the comparative findings are illustrated in Figure 7. To emphasize these findings, we visualized the discrepancies between the rendered and original images and annotated the PSNR values. Our analysis revealed that deviations in rendering predominantly occur along the edges of the rendered images. However, in contrast to other methods, our proposed technique yields results that more closely approximate the real images. In particular, by incorporating fine-density weights and multi-scale feature decomposition, MDW-NeRF was able to capture the minute texture details of ship surfaces more accurately, thus achieving highly realistic visual effects in ship rendering tasks. The visualizations substantiate the superiority of MDW-NeRF in noise reduction and detail enhancement, achieving a more pronounced rendering effect.

3.4.2. Comparison on Reconstruction

Figure 6 provides a detailed analysis of the CD evaluation metrics for various ship categories. It is evident that MDW-NeRF excels in each category, demonstrating superior capability in capturing intricate details of ship surfaces. The table adjacent to the CD subplot presents a quantitative assessment of MDW-NeRF’s performance in 3D reconstruction tasks, comparing it against other algorithms. The findings illustrate that MDW-NeRF consistently surpasses the benchmark algorithms across all tests. The quantitative results corroborate the superiority of MDW-NeRF in the domain of 3D reconstruction methodologies. Figure 8 presents a visual comparison of the 3D reconstruction quality between our proposed method and other techniques. Particularly, the first comparative instance in Figure 8 displays the reconstruction result of a hovercraft, revealing that some implicit reconstruction algorithms, especially NeuS, NeRF2mesh, and Nvdiffrec, fail to precisely reproduce the details of deep cavities in complex spatial structures due to excessively smooth processing. In contrast, our method exhibits exceptional capability in restoring these complex geometric shapes. Moreover, our algorithm demonstrates a unique method of addressing spatial discontinuities, a challenge often arising from insufficient correlation among tensor features. As visible in Figure 8, our model consistently outperforms comparative methods in surface reconstruction tasks, achieving significant results in terms of the accuracy of detail reproduction, consistency of geometric structures and the integrity of overall shapes.

3.4.3. Comparison on Efficiency

We also compared our MDW-NeRF with various methods on the dataset in terms of efficiency, as shown in

Table 1. Efficiency comparison of different methods. MDW-NeRF achieves the second-fastest reconstruction and smallest model size among hybrid methods, significantly reducing storage consumption compared to TensoRF. ↑ and ↓ denote the merit evaluation direction for the corresponding metrics respectively.

Method	Training Time ↓	Params. ↓	RS ↑	IMU ↓
Nvdiffrec	36 min	-	15.2 FPS	6812 MB
NeRF	6 h	5.2 MB	0.8 FPS	8534 MB
NeuS	8 h	178.3 MB	1.2 FPS	13,259 MB
NeRF2mesh	34 min	376.0 MB	12.5 FPS	16,543 MB
Voxurf	54 min	2.1 GB	8.3 FPS	25,323 MB
TensoRF	19 min	60.5 MB	22.4 FPS	7572 MB
MDW-NeRF	25 min	53.7 MB	19.6 FPS	15,326 MB

. It can be seen that our MDW-NeRF achieves the fastest reconstruction among hybrid methods alongside TensoRF. Moreover, we reported the model size, that is, storage consumption, in Table 1. We found that NeRF is 5.2 MB but reconstructs very slowly (6 h); explicit methods like Voxurf have a very large model size (>300 MB); while hybrid methods such as NeuS, TensoRF, and our MDW-NeRF have relatively small model sizes (<100 MB), with our MDW-NeRF having the smallest model size in hybrid methods (53.7 MB), reducing storage consumption by 9% compared to TensoRF. In terms of rendering speed, MDW-NeRF achieves 19.6 FPS, which is second only to TensoRF (22.4 FPS) among all compared methods and represents a significant advantage over traditional implicit models (NeRF: 0.8 FPS, NeuS: 1.2 FPS) and voxel-based methods (Voxurf: 8.3 FPS, NeRF2mesh: 12.5 FPS). Regarding model size, MDW-NeRF has the smallest storage footprint (53.7 MB) among hybrid methods, reducing storage consumption by 9% compared to TensoRF (60.5 MB). For inference memory usage (IMU), MDW-NeRF consumes 15,326 MB, which is higher than TensoRF (7572 MB) but justified by its superior reconstruction accuracy (1.5 dB higher in PSNR and 6.1% lower in CD). Notably, its IMU is still lower than resource-intensive voxel-based methods like Voxurf (25,323 MB) and comparable to NeRF2mesh (16,543 MB) while delivering far higher detail fidelity. These results, combined with standardized experimental settings, collectively validate MDW-NeRF’s balanced advantages in runtime speed, storage compactness, and reconstruction precision—critical for practical maritime engineering applications.

3.5. Ablation Study

In the ablation study conducted, the experimental parameters were kept consistent with those in previous implementation details. To isolate the contribution of each core module (multi-scale feature decomposition, density weighting) and clarify its synergistic interactions with TensoRF’s inherent multi-directional tensor decomposition, we focus on four configurations. Notably, TensoRF itself serves as the baseline for standalone multi-directional decomposition—its core tensor factorization paradigm inherently encodes directional feature correlations, thus no additional derivative configuration is required to validate this functionality. The method mainly compared four configurations: TensoRF (original version, providing baseline multi-directional decomposition), DW-NeRF (density weighting), M-NeRF (multi-scale feature decomposition), and MDW-NeRF (multi-scale feature decomposition + density weighting).

According to the quantitative performance metrics presented in

Table 2. Rendering and reconstruction performance comparison of methods with different modules. ↑ and ↓ denote the merit evaluation direction for the corresponding metrics respectively.

Method	CD ↓	PSNR ↑	SSIM ↑	LPIPS ↓
TensoRF	0.03442	46.7681	0.9990	0.0023
DW-NeRF	0.03442	47.5732	0.9994	0.0013
M-NeRF	0.03234	48.1354	0.9994	0.0010
MDW-NeRF	0.03233	48.2627	0.9994	0.0008

, MDW-NeRF demonstrates superior performance in terms of PSNR, LPIPS, and CD, whereas TensoRF, DW-NeRF, and M-NeRF achieve relatively lower scores on these metrics. However, regarding the SSIM evaluation, M-NeRF, DW-NeRF, and MDW-NeRF attain comparable results. Notably, as shown in

Table 3. Efficiency comparison of methods with different modules. ↑ and ↓ denote the merit evaluation direction for the corresponding metrics respectively.

Method	Training Time ↓	Params. ↓	RS ↑	IMU ↓
TensoRF	19 min	60.5 MB	22.4 FPS	7572 MB
DW-NeRF	20 min	50.4 MB	20.5 FPS	11,323 MB
M-NeRF	25 min	53.4 MB	19.8 FPS	13,534 MB
MDW-NeRF	25 min	53.7 MB	19.6 FPS	15,326 MB

, DW-NeRF not only requires a shorter training period but also has the minimum number of parameters, underscoring its significant advantages in training efficiency and model economy. While MDW-NeRF exhibits a slight superiority in image rendering quality, DW-NeRF showcases its unique benefits in terms of training efficiency and parameter count. Additionally, a comparison between M-NeRF and DW-NeRF indicates that M-NeRF is more effective in enhancing reconstruction fidelity, highlighting the value of multi-scale feature representation. From an efficiency perspective, ablation results confirm the rationality of the modular design. DW-NeRF, integrating TensoRF with density weighting, achieves a competitive rendering speed of 20.5 FPS and the lowest inference memory usage of 11,323 MB among all variants, highlighting the lightweight advantage of the density weighting module in enhancing spatial coherence without excessive resource overhead. M-NeRF, combining TensoRF with multi-scale feature decomposition, attains 19.8 FPS with an inference memory usage of 13,534 MB, demonstrating that multi-scale feature decomposition incurs modest computational overhead while substantially improving fine-grained detail capture. By synergistically integrating both modules, MDW-NeRF maintains reasonable inference memory consumption at 15,326 MB and a rendering speed of 19.6 FPS, while achieving the highest reconstruction precision with a CD of 0.03233 and a PSNR of 48.2627 dB. This achieves an optimal trade-off between performance enhancement and resource efficiency, a critical consideration for maritime 3D reconstruction where high-precision geometric recovery—including hull cavities and surface textures—and practical resource constraints must be concurrently satisfied.

The visual representation illustrated in Figure 9 delineates the efficacy of various methodologies incorporating distinct modules in reconstruction tasks. M-NeRF exhibits enhanced detail precision, yet fails to seamlessly integrate the surface of the reconstructed model. DW-NeRF’s performance is intermediary between TensoRF and M-NeRF, but it still encounters issues with insufficient fine-grained feature capture. Conversely, MDW-NeRF achieves superior reconstruction outcomes, showcasing unparalleled detail fidelity and spatial coherence.

Upon examining M-NeRF and TensoRF, it becomes evident that multi-scale feature decomposition significantly enhances the capture of fine-grained geometric details in reconstruction processes. Similarly, a comparative analysis of DW-NeRF and TensoRF reveals that the density weighting module substantially improves the spatial continuity of 3D reconstruction surfaces. MDW-NeRF retains the advantages of both DW-NeRF and M-NeRF, achieving synergistic improvement in surface fine-grained detail and coherent reconstruction for ship hulls.

Based on TensoRF’s failure cases in Figure 9a and Figure 10a, we find that its inherent multi-directional tensor decomposition is constrained by single-resolution factorization, leading to local detail incoherence. By comparing these failure cases with DW-NeRF’s rendering and reconstruction performance in Figure 9b and Figure 10b, we observe that the density weighting mechanism acts as a regulatory enhancement layer: it dynamically adjusts cross-directional feature contributions based on multi-directional feature inputs, enhancing partial local detail coherence and mitigating incoherence from independent multi-directional tensor decomposition. However, it only bridges minor gaps and fails to achieve full overall optimization. Comparing TensoRF’s failures with M-NeRF’s performance in Figure 9c and Figure 10c further shows that multi-scale feature decomposition effectively resolves the trade-off between reduced reconstruction efficiency (excessively high resolution) and insufficient accuracy (excessively low resolution), enabling fine precision at an efficiency level comparable to coarse resolution.

Notably, these individual module limitations are complementary: the density weighting mechanism (DW-NeRF) lacks global optimization capability, while multi-scale feature decomposition (M-NeRF) fails to handle complex hull cavities despite resolving the efficiency–accuracy trade-off. To address these limitations, we propose MDW-NeRF, which integrates density weighting with multi-scale feature decomposition to simultaneously achieve local detail coherence and global structural integrity.

3.6. Generalization Performance Evaluation on Public Dataset

To further analyze the impact of environmental factors on MDW-NeRF’s performance and verify its generalization ability, this section evaluates MDW-NeRF using the Ship category from the public NeRF-Synthesis dataset, which realistically simulates environmental influences including illumination, water reflections and hull details. This category is represented by 400 multi-view images (100 for training, 100 for validation, and 200 for testing). We conduct exclusive comparisons between MDW-NeRF and TensoRF under consistent hardware, software and hyperparameter configurations to ensure fairness.

As shown in

Table 4. Performance comparison on NeRF-synthesis (Ship category).

Method	CD ↓	PSNR ↑	SSIM ↑	LPIPS ↓
TensoRF	0.045	32.5	0.965	0.0052
MDW-NeRF	0.042	35.2	0.978	0.0041

, MDW-NeRF outperforms TensoRF on all core evaluation metrics. In terms of rendering quality, its PSNR (35.2 dB) and SSIM (0.978) demonstrate enhanced pixel fidelity and structural alignment with real-world scenes, while a 21.2% reduction in LPIPS narrows the perceptual discrepancy. Regarding 3D reconstruction accuracy, MDW-NeRF achieves a CD of 0.042, verifying reduced geometric reconstruction errors. These performance gains originate from fundamental mechanistic distinctions: TensoRF’s single-resolution tensor decomposition fails to sustain feature space consistency amid complex real-world perturbations, whereas MDW-NeRF’s multi-scale feature decomposition and adaptive weighting mechanisms underpin the simultaneous optimization of these metrics.

The visualization results in Figure 11 intuitively validate these performance discrepancies. In rendering, TensoRF exhibits pronounced texture degradation and color incoherence, particularly at the interfaces of hull metallic reflective surfaces and water reflections. In contrast, MDW-NeRF can generate more natural color gradients and precisely restore deck fine textures and edge contours, which is consistent with its superiority in PSNR, SSIM, and LPIPS. For 3D reconstruction, TensoRF’s models present numerous fragments outside partial curved surface structures and obvious discontinuities in some fine structures. However, MDW-NeRF smoothly reconstructs complex curved surface features, enhancing integrity in both global shape and local details—aligning with its geometric accuracy advantage quantified by the CD metric.

4. Discussion

Ship hull 3D reconstruction technologies are primarily classified by method and principle, falling into traditional vision/laser scanning-based approaches and deep learning-based methods. Traditional vision/laser scanning-based methods reconstruct hull 3D models using multi-view images or point clouds combined with geometric registration, but struggle with complex scenes, detail capture, and variations in lighting/occlusion. In contrast, deep learning-based methods—particularly NeRF and its variants—have gained widespread attention for high-fidelity reconstruction by learning image-to-3D scene mappings to restore fine details. NeRF-based methods capture scene radiance transfer functions to retrieve object depth and color, enabling unprecedented precision in hull 3D modeling. Recent works such as DW-NeRF [33] have further advanced hull reconstruction by introducing weighted feature space decomposition and coarse-to-fine density weighting, addressing spatial incoherence in tensor-based NeRF while maintaining efficiency—consistent with the core goal of enhancing structural integrity in marine-specific 3D reconstruction. Compared to traditional methods, NeRF excels in handling lighting, occlusion, and details, providing realistic restoration of hull appearance and structure, and unlocking new prospects for ship design and monitoring.

TensoRF innovatively combines spatial decomposition with NeRF, balancing rendering/reconstruction quality and efficiency. However, its feature decomposition granularity is resolution-limited: overly high resolution sacrifices contextual feature integration (causing reconstruction voids), while overly low resolution leads to detail loss, making optimal resolution selection critical. Similar to FGS-NeRF [34], which adopts progressive voxel-MLP scaling to balance resolution granularity and computational efficiency for glossy surface reconstruction, our work targets resolution-related limitations by developing a multi-scale feature fusion strategy within the TensoRF framework—differentiating itself by focusing on hull-specific geometric coherence rather than glossy material handling. Recent efforts like multi-view stereo-regulated NeRF have also explored the potential of integrating stereo constraints for scene synthesis [35], further validating that multi-view information enhancement is a promising direction for NeRF-based reconstruction.

In contrast to factorization-based radiance field models such as TensoRF, which rely on fixed-rank, single-resolution decompositions, our analysis identifies a more fundamental representational bottleneck. Vector decomposition-based neural radiance fields fail to capture multi-directional features, resulting in spatial voids and discontinuities on hull surfaces—attributed to the lack of cross-directional dependency modeling in standard tensor factorization, which fragments geometry for complex, high-curvature hull structures [36]. To address this, we introduce a density weighting mechanism for sampling point density calculation to enhance model surface continuity and completeness.

Architecturally, while methods like TensoRF efficiently compress scene representation, they face inherent trade-offs between detail preservation and spatial coherence. Despite progress in hull reconstruction, fine texture detail handling remains challenging; feature weighting mitigates voids but may cause local high-frequency detail loss. MDW-NeRF addresses this through its multi-scale decomposition, which separately models coarse geometry and fine texture, thereby mitigating detail blurring while maintaining structural integrity.

Multi-scale spatial decomposition improves rendering and reconstruction performance but extends training time compared to single-scale methods. This design is theoretically motivated by the need to balance reconstruction frequency bandwidth: single-scale methods compromise between fine detail capture and surface smoothness, while multi-scale representation avoids this trade-off. Thus, model selection requires balancing performance and computational cost: single-scale models are preferable for storage/loading-critical scenarios, while multi-scale methods are justified for high geometric accuracy and fidelity demands (e.g., maritime inspection, digital twin, high-fidelity simulation), especially when depth consistency and perceptual quality take priority over compactness.

5. Conclusions

In this paper, we propose MDW-NeRF, an efficient and high-precision tensor-based neural surface reconstruction method for ship hulls. It addresses spatial incoherence and insufficient fine-grained geometric recovery in existing tensor-based NeRF methods via synergistic multi-scale feature decomposition, coarse-to-fine density weighting and weighted matrix-based color weighting. Comprehensive evaluations on SVH datasets confirm its superiority: compared to TensoRF, it achieves 1.5 dB higher PSNR, 6.1% lower CD and 9% smaller model size, and maintains competitive training efficiency. This work advances tensor-based NeRF-driven 3D reconstruction by establishing a novel paradigm balancing precision, rendering fidelity and efficiency, enriching NeRF’s theoretical framework for complex engineering structures. Practically, it provides a reliable tool for maritime engineering (hull design optimization, structural inspection, autonomous navigation perception), facilitating the industry’s intelligent transformation via accurate reconstruction of complex hull geometries and robustness to real-world interferences. To promote reproducibility and application, the code and pre-trained models of the MDW-NeRF framework will be released on GitHub after formal acceptance.