Rotational Invariance Research Articles

Computer-Vision-Based Product Quality Inspection and Novel Counting System

In this study, we aimed to enhance the accuracy of product quality inspection and counting in the manufacturing process by integrating image processing and human body detection algorithms. We employed the SIFT algorithm combined with traditional image comparison metrics such as SSIM, PSNR, and MSE to develop a defect detection system that is robust against variations in rotation and scale. Additionally, the YOLOv8 Pose algorithm was used to detect and correct errors in product counting caused by human interference on the load cell in real time. By applying the image differencing technique, we accurately calculated the unit weight of products and determined their total count. In our experiments conducted on products weighing over 1 kg, we achieved a high accuracy of 99.268%. The integration of our algorithms with the load-cell-based counting system demonstrates reliable real-time quality inspection and automated counting in manufacturing environments.

A coarse-to-fine image registration method based on autocorrelation structural difference information

ABSTRACT The automatic registration of synthetic aperture radar (SAR) and optical images is still a challenging problem due to different imaging mechanisms. This letter proposes a coarse-to-fine image registration method that leverages self-similarity structural difference information. In the coarse registration stage, a Scale-Invariant Feature Transformation-based (SIFT-like) method is employed, complemented by an improved Fast Sample Consensus (IFSC) method to eliminate mismatched point pairs by probabilistic and geometric information. This stage ensures robustness against scale and rotational variations. In the fine registration stage, robust feature points are selected by utilizing phase and edge structural information. A descriptor which based on phase consistency and autocorrelation structural difference (ASDPC) is constructed to capture the structural variations between region blocks, and a fine search is carried out within the neighbourhood of the already matched feature points, so as to find more accurate matched feature points and obtain fine registration. The experimental results demonstrate that the proposed method provides robust and accurate registration for optical-to-SAR images.

An efficient unsupervised classification model for galaxy morphology: Voting clustering based on coding from ConvNeXt large model

By combining unsupervised and supervised machine learning methods, we have proposed a framework, called USmorph to carry out automatic classifications of galaxy morphologies. In this work, we update the unsupervised machine learning (UML) step by proposing an algorithm based on ConvNeXt large model coding to improve the efficiency of unlabeled galaxy morphology classifications. The method can be summarized into three key aspects as follows: (1) a convolutional autoencoder is used for image denoising and reconstruction and the rotational invariance of the model is improved by polar coordinate extension; (2) utilizing a pre-trained convolutional neural network (CNN) named ConvNeXt for encoding the image data. The features were further compressed via a principal component analysis (PCA) dimensionality reduction; (3) adopting a bagging-based multi-model voting classification algorithm to enhance robustness. We applied this model to I-band images of a galaxy sample with $I_ mag < 25$ in the COSMOS field. Compared to the original unsupervised method, the number of clustering groups required by the new method is reduced from 100 to 20. Finally, we managed to classify about 53<!PCT!> galaxies, significantly improving the classification efficiency. To verify the validity of the morphological classification, we selected massive galaxies with $M_ M_ sun $ for morphological parameter tests. The corresponding rules between the classification results and the physical properties of galaxies on multiple parameter surfaces are consistent with the existing evolution model. Our method has demonstrated the feasibility of using large model encoding to classify galaxy morphology, which not only improves the efficiency of galaxy morphology classification, but also saves time and manpower. Furthermore, in comparison to the original UML model, the enhanced classification performance is more evident in qualitative analysis and has successfully surpassed a greater number of parameter tests. The enhanced UML method will support the Chinese space station telescope in the future.

Triangular Mesh Surface Subdivision Based on Graph Neural Network

Mesh subdivision is a common mesh-processing algorithm used to improve model accuracy and surface smoothness. Its classical scheme adopts a fixed linear vertex update strategy and is implemented iteratively, which often results in excessive mesh smoothness. In recent years, a nonlinear subdivision method that uses neural network methods, called neural subdivision (NS), has been proposed. However, as a new scheme, its application scope and the effect of its algorithm need to be improved. To solve the above problems, a graph neural network method based on neural subdivision was used to realize mesh subdivision. Unlike fixed half-flap structures, the non-fixed mesh patches used in this paper naturally expressed the interior and boundary of a mesh and learned its spatial and topological features. The tensor voting strategy was used to replace the half-flap spatial transformation method of neural subdivision to ensure the translation, rotation, and scaling invariance of the algorithm. Dynamic graph convolution was introduced to learn the global features of the mesh in the way of stacking, so as to improve the subdivision effect of the network on the extreme input mesh. In addition, vertex neighborhood information was added to the training data to improve the robustness of the subdivision network. The experimental results show that the proposed algorithm achieved a good subdivision of both the general input mesh and extreme input mesh. In addition, it effectively subdivided mesh boundaries. In particular, using the general input mesh, the algorithm in this paper was compared to neural subdivision through quantitative experiments. The proposed method reduced the Hausdorff distance and the mean surface distance by 27.53% and 43.01%, respectively.

A Robust ALOHA and Sequential clustering-based mode estimator for low frequency oscillations in power system using synchrophasors

Accurate estimation of low frequency modes in power system are very much important for improving small signal stability. The parametric model parameters estimator known as Total least square estimation of signal parameters via rotational invariance techniques (TLS-ESPRIT) works effectively even in noisy conditions. However, this model parameter estimator requires prior information about numbers of modes of the signal. There are different Model order (MO) estimation techniques discussed in the recent past, which consider the significant eigenvalues of auto-correlation matrix (ACM) for its estimation. As the eigenvalues of ACM highly affected by bad measurements and Outliers, making these techniques inefficient and harder to automate in real time. So, to overcome aforementioned limitations, this paper proposes a robust mode estimation technique that can precisely detect the signal low frequency modes even in the presence of high variance noise and outliers. So, in this proposed work, an annihilating filter-based low-rank Hankel matrix (ALOHA) technique is implemented to obtain the rank deficient Hankel matrix to nullify the existence of noise and outliers in Phasor measurement unit (PMU) signal, thereafter approximated low rank Hankel matrix is considered for estimation of signals dominant modes. Wherein a sequential clustering technique (Sequential K-Mean++) is implemented for detection of numbers of prominent low frequency modes by segregating eigenvalues of ACM into two opponents i.e the signal and noise subspace. Thereafter the estimated MO is considered for estimation of modes through TLS-ESPRIT. The robustness of the proposed technique is validated by scheduling comparative study with recently developed techniques for synthetic signal, two area data, real PMU data of Western Electricity Coordinating Council (WECC) and oscillatory power data obtained from Western System Coordinating Council (WSCC) 9 bus system and IEEE68 bus system simulated through Real Time Digital Simulator (RTDS).

Fusion or not: Learning visual relocalization with matrix Fisher distribution

Visual relocalization is crucial for applications such as simultaneous localization and mapping (SLAM) and augmented reality. State-of-the-art methods based on multiple view geometry struggles with robust localization in challenging environments due to visual perturbations. Recently, deep learning-based absolute pose regression (APR) has gained significant attention owing to its robust feature extraction capabilities, yet it consistently falls short of matching the localization accuracy achieved by structure-based localization methods. In the paper, a novel visual localization framework is developed, which fuses geometric structure methods with neural network inference via the matrix Fisher distribution on the special orthogonal group SO(3). To be specific, a deep neural network is trained to generate the exponential probability density model on SO(3), i.e., the matrix Fisher distribution, which does not require the implementation of complex functions to constrain the rotational variables and can simultaneously model the uncertainty of the rotation estimates. A Bayesian fusion method is then introduced to fuse the orientation predictions recovered from the Fisher parameters with the rotation estimates computed by a structure-based approach. Moreover, the fusion results are concatenated with the feature encoding of the image to compensate for the correlation with the rotational variables, and the position prediction is output through a fully connected layer. The performance of the developed localization framework is evaluated via public indoor and outdoor datasets, demonstrating that the approach significantly improves localization accuracy while remaining robust in challenging environments.

A Long-duration Superflare on the K Giant HD 251108

Many giant stars are magnetically active, which causes rotational variability, chromospheric emission lines, and X-ray emission. Large outbursts in these emission features can set limits on the magnetic field strength and thus constrain the mechanism of the underlying dynamo. HD 251108 is a Li-rich active K-type giant. We find a rotational period of 21.3 days with color changes and additional long-term photometric variability. Both can be explained with very stable stellar spots. We followed the decay phase of a superflare for 28 days with NICER and from the ground. We track the flare decay in unprecedented detail in several coronal temperature components. With a peak flux around 1034 erg s−1 (0.5–4.0 keV) and an exponential decay time of 2.2 days in the early decay phase, this is one of the strongest flares ever observed, yet it follows trends established from samples of smaller flares, for example, for the relations between Hα and X-ray flux, indicating that the physical process that powers the flare emission is consistent over a large range of flare energies. We estimate a flare loop length about 2–4 times the stellar radius. No evidence is seen for abundance changes during the flare.

Broadband VHF lightning radiation sources localization by ESPRIT algorithm

Fast and fine radiation source localization algorithm is of vital importance to lightning warning and protection. Current high-accuracy localization techniques, such as the time reversal technique (TR) and the multiple signal classification (MUSIC), are based on traversal search mechanism, which takes a long time. In this paper, the estimation of signal parameters via rotational invariance technique (ESPRIT) is applied to lightning radiation source localization, and the direction of arrival (DOA) can be directly solved through the covariance matrix, so it is very efficient. For the broadband VHF signal, the incoherent signal method (ISM) is combined with ESPRIT. Two classical structures of uniform L-shaped array and uniform circular array are studied and applied to ESPRIT algorithm. As for the uniform circular array, the fourth order cumulant matrix is constructed to find the translation invariant subarrays. To unwrap the phase ambiguity caused by the calculated phase angle of ESPRIT algorithm, the total least difference of slope (TLDS) principle is proposed. The proposed ESPRIT algorithm is more than 1400 times faster than MUSIC while its localization accuracy is still pretty high, which is able to detect the radiation source under -8 dB signal to noise ratio (SNR) according to the result of numerical simulations, and can map continuous and fine lightning development channel in the experiments on the measured lightning data.

Analysis and Effect of Cutting Parameters on The Surface Roughness of Type 4140 Steel in The Dry Milling Process

Currently in the world of metal cutting industry, wet machining is still carried out and machining experts continue to strive to reduce the use of coolant so that benefits can be obtained from economic, environmental and operator safety aspects. One of the criteria for determining machining quality is surface roughness. Surface roughness is influenced by machining parameters such as cutting speed, feed depth and feed rate as independent variables. The aim of the research is to obtain surface roughness values ​​for type 4140 steel workpieces due to spindle rotation speed and feed depth, which uses a carbide endmill as a cutting tool. Research activities were carried out with 9 specimens each for dry machining and wet machining using variations in machine spindle rotation of 1200 rpm, 1400 rpm and 1600 rpm with feed depths of 0.5 mm, 0.75 mm, 1 mm. Data processing uses the statics method with a normal distribution curve to obtain surface roughness data. To determine the roughness value of the machined surface, it is tested using a Surface Test measuring instrument. The surface roughness obtained from machining under optimum conditions for dry machining is 1.699 µm, 2.778 µm and 2.868 µm at a machine spindle rotation of 1600 rpm. while those carried out using wet machining obtained respective results of 1.602 µm, 2.654 µm and 2.815 µm at a machine spindle rotation of 1600 rpm. So optimum cutting/slicing is obtained at a spindle rotation speed of 1600 rpm with Ra = 1.699 µm for dry machining while for wet machining the surface roughness value Ra = 1.602 µm is obtained at a spindle rotation speed of 1600 rpm. The comparison of the results of dry machining (Ra= 1.699 µm) and wet machining (Ra= 1.602 µm) to obtain the most optimum Ra surface roughness value is not significant, so it is a good opportunity that dry machining technology can be applied to the metal cutting industry although Until now, many people still use wet machining processes using cutting fluids to cut steel meta

Power spectra of Dyson’s circular ensembles

The power spectrum is a Fourier series statistic associated with the covariances of the displacement from average positions of the members of an eigen-sequence. When this eigen-sequence has rotational invariance, as for the eigen-angles of Dyson’s circular ensembles, recent work of Riser and Kanzieper has uncovered an exact identity expressing the power spectrum in terms of the generating function for the conditioned gap probability of having k=0,…,N eigenvalues in an interval. These authors moreover showed how for the circular unitary ensemble integrability properties of the generating function, via a particular Painlevé VI system, imply a computational scheme for the corresponding power spectrum, and allow for the determination of its large N limit. In the present work, these results are extended to the case of the circular orthogonal ensemble and circular symplectic ensemble, where the integrability is expressed through four particular Painlevé VI systems for finite N, and two Painlevé III′ systems for the limit N→∞, and also via corresponding Fredholm determinants. The relation between the limiting power spectrum S∞(ω), where ω denotes the Fourier variable, and the limiting generating function for the conditioned gap probabilities is particularly direct, involving just a single integration over the gap endpoint in the latter. Interpreting this generating function as the characteristic function of a counting statistic allows for it to be shown that S∞(ω)∼ω→01πβ|ω|, where β is the Dyson index.

A statistical-based automatic detection of a low-contrast object in the ACR CT phantom for measuring contrast-to-noise ratio of CT images

Purpose. This study aimed to develop a new method for automated contrast-to-noise ratio (CNR) measurement using the low-contrast object in the ACR computed tomography (CT) phantom.Methods. The proposed method for CNR measurement was based on statistical criteria. A region of interest (ROI) was placed in a specific radial location and was then rotated around 360° in increments of 2°. At each position, the average CT number within the ROI was calculated. After one complete rotation, a profile of the average CT number around the full rotation was obtained. The center coordinate of the low-contrast object was determined from the maximum value of the profile. The CNR was calculated based on the average CT number and noise within the ROI in the low-contrast object and the ROI in the background, i.e., at the center of the phantom. The proposed method was used to evaluate CNR from images scanned with various phantom rotations, images with various noise levels (tube currents), and images from 25 CT scanners. The results were compared to a previous method based on a threshold approach.Results. The proposed method successfully placed the ROI properly in the center of a low-contrast object for variations of phantom rotation and tube current, whereas was not properly located in the center of the low-contrast object using the previous method. In addition, from 325 image samples of the 25 CT scanners, the proposed method successfully (100%) located the ROI within the low-contrast objects of all images used. The success rate of the previous method was only 58%.Conclusion. A new method for measuring CNR in the ACR CT phantom has been proposed and implemented. It is more powerful than a previous method based on a threshold approach.

Robust block tensor PCA with F-norm projection framework

Tensor principal component analysis (TPCA), also known as Tucker decomposition, ensures that the extracted “core tensor” maximizes the variance of the sample projections. Nevertheless, this method is particularly susceptible to noise and outliers. This is due to the utilization of the squared F-norm as the distance metric. In addition, it lacks constraints on the discrepancies between the original tensors and the projected tensors. To address these issues, a novel tensor-based trigonometric projection framework is proposed using F-norm to measure projection distances. Tensor data are first processed utilizing a blocking recombination technique prior to projection, thus enhancing the representation of the data at a local spatio-temporal level. Then, we present a block TPCA with the F-norm metric (BTPCA-F) and develop an iterative greedy algorithm for solving BTPCA-F. Subsequently, regarding the F-norm projection relation as the “Pythagorean Theorem”, we provide three different objective functions, namely, the tangent, cosine and sine models. These three functions directly or indirectly achieve the two objectives of maximizing projection distances and minimizing reconstruction errors. The corresponding tangent, cosine and sine solution algorithms based on BTPCA-F (called tan-BTPCA-F, cos-BTPCA-F and sin-BTPCA-F) are presented to optimize the objective functions, respectively. The convergence and rotation invariance of these algorithms are rigorously proved theoretically and discussed in detail. Lastly, extensive experimental results illustrate that the proposed methods significantly outperform the existing TPCA and the related 2DPCA algorithms.

Robust Multimodal Image Matching Based on Radiation Invariant Phase Correlation

Abstract. Due to the influence of nonlinear radiation distortion and geometric deformation, achieving multimodal image matching remains a challenging task. To address these issues, this paper proposes a method called radiation invariant phase correlation (RIPC) to simultaneously estimate the rotation, scale, and displacement changes of multimodal image pairs. Firstly, based on the local structure characteristics of the image itself, we harness the nonlinear invariance of kernel canonical correlation analysis to devise the multimodal local self-correlation (MLSC) descriptor. This descriptor is resilient to nonlinear radiative differences, as well as local rotation and scale variations. Subsequently, we incorporate the log-polar coordinate transformation to capture the overall rotation and scale changes in the image, enabling independent representation of these factors on the Cartesian coordinate system. Finally, drawing upon the continuity of displacement estimation, as well as rotation and scale estimation, we construct a five-dimensional descriptor tailored for phase correlation. Extensive experiments conducted on five open-source datasets demonstrate that our proposed method surpasses state-of-the-art (SOTA) techniques in matching performance. Furthermore, our RIPC method achieves matching accuracy within 2-pixel threshold, which underscores its effectiveness in multimodal remote sensing image matching.

Robust Symbol Detection Based on Quaternion Neural Networks in Wireless Polarization-Shift-Keying Communications.

Quaternion neural networks (QNNs) form a class of neural networks constructed with quaternion numbers. They are suitable for processing 3-D features with fewer trainable free parameters than real-valued neural networks (RVNNs). This article proposes symbol detection in wireless polarization-shift-keying (PolSK) communications by employing QNNs. We demonstrate that quaternion plays a crucial role in the symbol detection of PolSK signals. Existing artificial-intelligence communication studies mainly focus on RVNN-based symbol detection in digital modulations having constellations in complex plane. However, in PolSK, information symbols are represented as the state of polarization, which can be mapped on the Poincare sphere and thus its symbols have a 3-D data structure. Quaternion algebra offers a unified representation to process 3-D data with rotational invariance and, therefore, it keeps the internal relationship among three components of a PolSK symbol. Hence, we can expect that QNNs learn the distribution of received symbols on the Poincare sphere with higher consistency to detect the transmitted symbols more efficiently than RVNNs. We compare PolSK symbol detection accuracy of two types of QNNs, RVNN, existing methods such as least-square and minimum-mean-square-error channel estimations, as well as detection knowing perfect channel state information (CSI). Simulation results including symbol error rate show that the proposed QNNs outperform the existing estimation methods and that they reach better results with two to three times fewer free parameters than the RVNN. We find that QNN processing will bring practical use of PolSK communications.

Biokinetic modeling approach to investigate the impact of rotational speed variations in modified rotating biological contactors for palm oil mill effluent treatment

Palm oil mill effluent (POME) contains high organic content, posing significant environmental challenges that demand effective treatment solutions. Conventional rotating biological contactor (RBC) used for POME treatment are limited by insufficient surface area for microbial growth and inefficient oxygen transfer, limiting their effectiveness. This study enhances RBC systems by incorporating bioballs to increase microbial attachment surfaces and improve treatment performance. Experiments were conducted at various rotational speeds, and biokinetic modeling was applied under unsteady-state conditions. The results showed that the RBC operated at 3 rpm achieved the highest removal efficiencies, with 67.4 % for SCOD, 92.4 % for TSS, and 73 % for NH3. The biokinetic model revealed that rotational speed did not significantly impact oxygen transfer, mainly during the aeration phase. Lower speeds optimized substrate degradation and microbial growth, while higher speeds caused biofilm detachment. The innovative use of bioballs improves POME treatment efficiency at lower speeds, offering a cost-effective solution.

Automated production of batched unclonable micro-patterns anti-counterfeiting labels with strong robustness and rapid recognition speed

Anti-counterfeiting technologies are indeed crucial for information security and protecting product authenticity. Traditional anti-counterfeiting methods have their limitations due to their clonable nature. Exploring new technologies, particularly those based on pixel-level textures, is a promising avenue to address the clonable issue due to high encoding capacity. However, research in this field is still in its early stages. This work introduces a new fluorescent anti-counterfeiting label technology with four key characteristics: efficient laser etching, high-throughput fabrication and segmentation, robustness aided by data augmentation, and an exceptionally high recognition speed. Specifically, the etching achieves a speed of 1,200 labels/3s, the high-throughput procedures yield a rate of 2,400 labels/4 min, and the total count is about 5.2 × 104 labels. The number of labels is further augmented to about 5.2 × 106 by implementing arbitrary rotation and brightness variation to enhance the robustness in the recognition procedure. We divide these labels into 44 categories based on differences in patterns. Utilizing machine learning methods, we have achieved a total recognition (including extraction and search process) time per label averaging 421.96 ms without classification, and 40.13 ms with classification. Specifically, the search process with classification is nearly fiftieth times shorter than the non-classification method, reaching 8.52 ms in average. The overall recognition time is much faster than previous works, and achieves an accuracy of over 98.7%. This work significantly increases the practical significance of pixel-level anti-counterfeiting labels.

A Base-Map-Guided Global Localization Solution for Heterogeneous Robots Using a Co-View Context Descriptor

With the continuous advancement of autonomous driving technology, an increasing number of high-definition (HD) maps have been generated and stored in geospatial databases. These HD maps can provide strong localization support for mobile robots equipped with light detection and ranging (LiDAR) sensors. However, the global localization of heterogeneous robots under complex environments remains challenging. Most of the existing point cloud global localization methods perform poorly due to the different perspective views of heterogeneous robots. Leveraging existing HD maps, this paper proposes a base-map-guided heterogeneous robots localization solution. A novel co-view context descriptor with rotational invariance is developed to represent the characteristics of heterogeneous point clouds in a unified manner. The pre-set base map is divided into virtual scans, each of which generates a candidate co-view context descriptor. These descriptors are assigned to robots before operations. By matching the query co-view context descriptors of a working robot with the assigned candidate descriptors, the coarse localization is achieved. Finally, the refined localization is done through point cloud registration. The proposed solution can be applied to both single-robot and multi-robot global localization scenarios, especially when communication is impaired. The heterogeneous datasets used for the experiments cover both indoor and outdoor scenarios, utilizing various scanning modes. The average rotation and translation errors are within 1° and 0.30 m, indicating the proposed solution can provide reliable localization support despite communication failures, even across heterogeneous robots.

Research on the Registration of Aerial Images of Cyclobalanopsis Natural Forest Based on Optimized Fast Sample Consensus Point Matching with SIFT Features

The effective management and conservation of forest resources hinge on accurate monitoring. Nonetheless, individual remote-sensing images captured by low-altitude unmanned aerial vehicles (UAVs) fail to encapsulate the entirety of a forest’s characteristics. The application of image-stitching technology to high-resolution drone imagery facilitates a prompt evaluation of forest resources, encompassing quantity, quality, and spatial distribution. This study introduces an improved SIFT algorithm designed to tackle the challenges of low matching rates and prolonged registration times encountered with forest images characterized by dense textures. By implementing the SIFT-OCT (SIFT omitting the initial scale space) approach, the algorithm bypasses the initial scale space, thereby reducing the number of ineffective feature points and augmenting processing efficiency. To bolster the SIFT algorithm’s resilience against rotation and illumination variations, and to furnish supplementary information for registration even when fewer valid feature points are available, a gradient location and orientation histogram (GLOH) descriptor is integrated. For feature matching, the more computationally efficient Manhattan distance is utilized to filter feature points, which further optimizes efficiency. The fast sample consensus (FSC) algorithm is then applied to remove mismatched point pairs, thus refining registration accuracy. This research also investigates the influence of vegetation coverage and image overlap rates on the algorithm’s efficacy, using five sets of Cyclobalanopsis natural forest images. Experimental outcomes reveal that the proposed method significantly reduces registration time by an average of 3.66 times compared to that of SIFT, 1.71 times compared to that of SIFT-OCT, 5.67 times compared to that of PSO-SIFT, and 3.42 times compared to that of KAZE, demonstrating its superior performance.

Efficient Semantic Segmentation for Large-Scale Agricultural Nursery Managements via Point Cloud-Based Neural Network

Remote sensing technology has found extensive application in agriculture, providing critical data for analysis. The advancement of semantic segmentation models significantly enhances the utilization of point cloud data, offering innovative technical support for modern horticulture in nursery environments, particularly in the area of plant cultivation. Semantic segmentation results aid in obtaining tree components, like canopies and trunks, and detailed data on tree growth environments. However, obtaining precise semantic segmentation results from large-scale areas can be challenging due to the vast number of points involved. Therefore, this paper introduces an improved model aimed at achieving superior performance for large-scale points. The model incorporates direction angles between points to improve local feature extraction and ensure rotational invariance. It also uses geometric and relative distance information for better adjustment of different neighboring point features. An external attention module extracts global spatial features, and an upsampling feature adjustment strategy integrates features from the encoder and decoder. A specialized dataset was created from real nursery environments for experiments. Results show that the improved model surpasses several point-based models, achieving a Mean Intersection over Union (mIoU) of 87.18%. This enhances the precision of nursery environment analysis and supports the advancement of autonomous nursery managements.

Probing the effects of broken symmetries in machine learning

Abstract Symmetry is one of the most central concepts in physics, and it is no surprise that it has also been widely adopted as an inductive bias for machine-learning models applied to the physical sciences. This is especially true for models targeting the properties of matter at the atomic scale. Both established and state-of-the-art approaches, with almost no exceptions, are built to be exactly equivariant to translations, permutations, and rotations of the atoms. Incorporating symmetries—rotations in particular—constrains the model design space and implies more complicated architectures that are often also computationally demanding. There are indications that unconstrained models can easily learn symmetries from data, and that doing so can even be beneficial for the accuracy of the model. We demonstrate that an unconstrained architecture can be trained to achieve a high degree of rotational invariance, testing the impacts of the small symmetry breaking in realistic scenarios involving simulations of gas-phase, liquid, and solid water. We focus specifically on physical observables that are likely to be affected—directly or indirectly—by non-invariant behavior under rotations, finding negligible consequences when the model is used in an interpolative, bulk, regime. Even for extrapolative gas-phase predictions, the model remains very stable, even though symmetry artifacts are noticeable. We also discuss strategies that can be used to systematically reduce the magnitude of symmetry breaking when it occurs, and assess their impact on the convergence of observables.