Score-Based Multibeam Point Cloud Denoising*
^††thanks: *This work was supported by Swedish Maritime Robotics Center (SMaRC).

Given a noisy MBES point cloud $\mathbf{O}=\{\mathbf{o}_{i}\}_{i=1}^{N}$, each $\mathbf{o}_{i}\in\mathbb{R}^{3}$ consists of the XYZ hits of the MBES beam onto the seafloor, the goal is to recover the corresponding clean MBES point cloud $\mathbf{C}=\{\mathbf{c}_{i}\}_{i=1}^{N}$.

To achieve this, we adapt the score-based point cloud denoising from [luoScoreBasedPointCloud2021]. More specifically, we assume that the clean set $\mathbf{C}$ is sampled from an underlying distribution $p$, whilst the noisy set $\mathbf{O}$ is sampled from $p$ convolved with a noise distribution $n$, $p*n$. As such, the mode of $p*n$ represents the noise-free surface. The gradient of the log-probability of $p*n$, also known as the score of $p*n$, can be denoted by $\nabla_{\mathbf{o}}\log[(p*n)(\mathbf{o})$ and can be learned directly from $\mathbf{O}$. Using the score, we can denoise $\mathbf{O}$ by performing gradient ascend, moving the noisy points to the mode of $p*n$. Moreover, the relative magnitude of the score can be used for outlier detection.

For the MBES point cloud, the beams are focused to the surface of an infinite cone, with the main axis typically aligned to the vehicle’s forward direction, and the peak of the cone located at the sensor’s frame origin. In the simplest case, this cone can be reduced to a disk (i.e points lie in a plane). In such case, the X values, or the along-track values, are determined by the sensor and vehicle pose, whilst the Y and Z values are together determined by the range of the bottom detection. In this work, we consider this simplest case. We assume that the X value of the points are fixed, and define a 1D score for the Z dimension. Note that the Y values can be moved according to MBES sensor geometry given Z.

For a noisy point $\mathbf{o}\in\mathbb{R}^{3}$, the ground truth score is defined as $s(\mathbf{o}_{z})=NN_{xy}(\mathbf{o},\mathbf{C})_{z}-\mathbf{o}_{z}$, where $NN_{xy}(\mathbf{o},\mathbf{C})$ denotes the nearest neighbor to point $\mathbf{o}$ in $\mathbf{C}$ and $[]_{z}$ denotes the $z$ component of the point. Intuitively, this score represents the 1D vector from the noisy $z$ component to its noise-free correspondence (see Figure 1).

A score estimation network inspired by [luoScoreBasedPointCloud2021] is then used to learn the local score function $S_{i}(\mathbf{o})$, which corresponds to the gradient field around a point $\mathbf{o}_{i}$. This network takes the form of $\mathcal{S}_{i}(\mathbf{o})=\text{MLP}(\mathbf{o}-\mathbf{o}_{i},\mathbf{h}_{i})$, where MLP denotes multi-layer perceptron, $\mathbf{o}\in\mathbb{R}^{3}$ denotes a point closed to $\mathbf{o}_{i}$, and $\mathbf{h}_{i}$ denotes the feature of $\mathbf{o}_{i}$ extracted by a DGCNN [wangDynamicGraphCnn2019]. The loss function is the L2-norm between $S_{i}(\mathbf{o})$ and $s(\mathbf{o}_{z})$, i.e. $\mathcal{L}=\mathbb{E}_{\mathbf{o}\sim\mathcal{N}(\mathbf{o}_{i})}[||s(\mathbf{o}_{z})-S_{i}(\mathbf{o})||_{2}^{2}]$, with $\mathcal{N}(\mathbf{o}_{i})$ representing the local neighborhood of point $\mathbf{o}_{i}$.

Using the estimated score $S_{i}(\mathbf{o})$, the $z$ component of the noisy point $\mathbf{o}_{i}$, denoted as $z_{i}$, can be iteratively denoised for $t$ steps as follows: let $z_{i}^{(0)}=z_{i}$, the subsequent $z_{i}^{(t)}$ are computed using $z_{i}^{(t)}=z_{i}^{(t-1)}+\alpha_{i}\mathcal{E}_{i}(z_{i}^{(t-1)})$, where $\alpha_{i}$ is the step size at step $t$, and $\mathcal{E}_{i}$ is an ensemble score function using the estimated score of $S_{j}(\mathbf{o})$ for $\mathbf{o}_{j}$ in the local neighborhood of $\mathbf{o}_{i}$. Whilst [luoScoreBasedPointCloud2021] uses mean as ensemble method, we propose to use median to deal with the severe outliers in MBES data.

We will evaluate the proposed method for both denoising and outlier rejection. For denoising, we will report the pointwise Chamfer distance, mean absolute error (MAE) and root-mean-squared error (RMSE) in z, as well as the percentage points with error less than a specified threshold [irisawaHighResolutionBathymetryDeepLearning2024]. For outlier rejection, we will report the common binary classification metrics, including precision, recall, accuracy, etc [stephensUsingThreeDimensional2020, longComprehensiveDeepLearningBased2023].

For outlier rejection, we evaluate Ordinary Kriging [chilesGeostatisticsModelingSpatial2012] used by [irisawaHighResolutionBathymetryDeepLearning2024], PointCleanNet [rakotosaonaPointCleanNetLearningDenoise2019] used by [longComprehensiveDeepLearningBased2023], as well as classical point cloud filtering methods such as statistical outlier removal and radius outlier removal.

A big obstacle in developing and evaluating MBES denoising methods is the lack of ground truth. In this work, we manually clean a MBES dataset collected by a Hugin AUV with Kongsberg’s EM2040 sensor in an area rich of trawling patterns. The dataset covers around 5h of survey time and contains 216k MBES pings, each with 400 beams. After cleaning, we construct a mesh using EIVA NaviModel, and use AUVLib’s draping functionality to construct the ground truth noise-free point cloud. To construct the denoising dataset, we then divide the data into 32-ping patches, resulting in 6777 patches for training and evaluation.