PointPCA+: A full-reference Point Cloud Quality Assessment metric with PCA-based features

Contents lists available at ScienceDirect Signal Processing: Image Communication journal homepage: www.elsevier.com/locate/image PointPCA+: A full-reference Point Cloud Quality Assessment metric with PCA-based features Xuemei Zhou a,c,∗, Evangelos Alexiou b, Irene Viola a, Pablo Cesar a,c aCentrum Wiskunde & Informatica, Amsterdam, The Netherlands bTNO Netherlands Organization for Applied Scientific Research, The Hague, The Netherlands cTU Delft, Delft, The Netherlands ARTICLE INFO MSC: 41A05 41A10 65D05 65D17 Keywords: Point cloud Perceptual Quality Assessment PCA Full-reference Feature selection Random forest ABSTRACT This paper introduces an enhanced Point Cloud Quality Assessment (PCQA) metric, termed PointPCA+, as an extension of PointPCA, with a focus on computational simplicity and feature richness. PointPCA+ refines the original PCA-based descriptors by employing Principal Component Analysis (PCA) solely on geometry data; additionally, the texture descriptors are refined through a direct application of the function on YCbCr values, enhancing the efficiency of computation. The metric combines geometry and texture features, capturing local shape and appearance properties, through a learning-based fusion to generate a total quality score. Prior to fusion, a feature selection module is incorporated to identify the most effective features from a proposed super-set. Experimental results demonstrate the high predictive performance of PointPCA+ against subjective ground truth scores obtained from four publicly available datasets. The metric consistently outperforms stateof-the-art solutions, offering valuable insights into the design of similarity measurements and the effectiveness of handcrafted features across various distortion types. The code of the proposed metric is available at https://github.com/cwi-dis/pointpca_suite/. 1. Introduction Point cloud is prevailing among the available 3D imaging formats in recent years. It is essentially a collection of points, where each point has attributes of geometry, color, reflectance, etc. However, through acquisition, compression, transmission, and rendering, the quality of a point cloud can be degraded, which necessitates effective and efficient Point Cloud Quality Assessment (PCQA) metrics. These metrics provide a guide on the design, optimization, and parameter tuning of point cloud processing pipelines. PCQA metrics have been extensively utilized in various applications, including benchmarking visual tasks such as restoration [1,2], compression [3–5], super-resolution [6,7], as well as for quality monitoring in various systems [8–11]. Based on the availability of reference point cloud data, PCQA can be generally divided into three categories, namely Full Reference (FR), Reduced Reference (RR) and No Reference (NR). Compared with RR and NR, FR requires the fully available reference during the execution, which mimics the way the Human Vision System (HVS) perceives quality as a comparison of differences between a degraded and a pristine version [12]. From a methodology standpoint, PCQA is composed of two main components: feature extraction and feature regression [13– 16]. Researchers have utilized both hand-crafted and learning-based ∗Corresponding author. E-mail address: [email protected] (X. Zhou). features, usually combined with a non-linear function or learning-based regressor [17,18]. End-to-end schemes can significantly improve the prediction performance by fitting the ground truth well. However, the HVS mechanisms behind them are difficult to explain [19] . In addition, one of the most challenging issues of end-to-end learning-based approaches is their requirement for large amounts of labeled data for training [20–22]. The learning model may have difficulties in handling various contents and distortions if the training set is not sufficiently large or fails to adequately represent real-world contents [23]. Besides, the shortage of data may probably cause serious over-fitting problems. In fact, only very size-limited, distortion-type unbalanced point cloud datasets are available in the PCQA field. In previous research, the perceptual quality of point clouds under the FR framework has been explored using multiple geometric features, textural features, or a combination of both. However, these studies typically compute geometric or textural similarity measurements from the referenced and distorted point clouds, independently [24–26]. This is achieved by identifying correspondences among points based on geometry. Considering the aforementioned limitations and drawing inspiration from PointPCA [25], our emphasis in this paper is on non-deep-learning-based FR PCQA metrics. To this aim, we describe https://doi.org/10.1016/j.image.2025.117262 Received 30 December 2023; Received in revised form 16 September 2024; Accepted 9 January 2025 Signal Processing: Image Communication 135 (2025) 117262 Available online 17 January 2025 0923-5965/© 2025 Published by Elsevier B.V. X. Zhou et al. Fig. 1. PointPCA+ architecture: both the reference and the distorted point cloud are passing from every stage to compute a quality score. Operations in the blue box are applied only to the geometry data of point clouds. Only the reference point cloud serves as a reference to identify neighborhoods. PointPCA+ and extend our prior analysis [27], incorporating results and discussions that take into account distortion types and feature importance. The contributions of the paper are threefold: •We extend the PointPCA framework by performing PCA on the geometry data of the reference point cloud and transforming both the reference and distorted point clouds onto the new basis. This way we can capture differences in their shape properties effectively. •We utilize 𝑘𝑛𝑛 algorithm to determine the neighborhood, which is faster and returns a consistent number of points, therefore further decreasing the computational cost of subsequent processing steps. •We perform extensive experimentation on four publicly available datasets, demonstrating that PointPCA+ consistently achieves superior performance across four distinct datasets. A thorough analysis investigates the effectiveness of different handcrafted features concerning specific distortion types, aiming to discern which features are more impactful for different distortion type categories. 2. Related works 2.1. Ground-truth labeling and subjective dataset construction In the development of objective PCQA metrics, subjective quality assessment datasets are the basis for their design and validation. Groundtruth ratings for visual impairments in stimuli, namely Mean Opinion Score (MOS) or Differential MOS (DMOS), are obtained through subjective quality assessment experiments [28,29]. These scores primarily indicate the degree of degradation affecting the contents with little information about the representations of distortion. Existing synthesized PCQA datasets, generated either by performing a test in a controlled laboratory environment or by mimicking it with point cloud processing algorithms, are size-limited and distortion-type and distortion-level unbalanced, often by design [30–33]. For example, when using test methods such as absolute category rating, the distortion level for a certain distortion type must be visually distinguishable to obtain meaningful MOS values and to avoid fatiguing subjects. Regarding the compression distortion of PCQA, recent research indicates a monotonic relationship with bit rate when evaluating compression quality [11,34]. However, it is crucial to emphasize that subjective quality evaluation sometimes does not follow a monotonic behavior. This is particularly evident in cases where the balance between geometry and color quality in point cloud content is challenging to establish [35]. This complexity adds difficulty to discerning between different point cloud coding modules. Therefore, it becomes imperative to integrate both geometry and texture distortions in assessing the perceptual quality of point clouds for PCQA. The utilization of effective geometryand texture-related features can enhance the optimization of various algorithms associated with perception. 2.2. Objective quality assessment of point cloud Objective PCQA metrics can be divided into point-based, projectionbased, and feature-based models based on the way to process the point clouds. Point-based metrics such as point-to-point, point-to-plane, and their variants [36–39] measure degradations between the original and distorted point clouds per point, mainly based on Euclidean or color space distances [40]. Alexiou et al. [41] propose the angular similarity of tangent planes among corresponding points, which considers neighborhood information. These metrics are computationally efficient but suffer from a crude correspondence of matching between points. Projection-based approaches adapt existing Image Quality Assessment (IQA) metrics to PCQA. The first attempt is reported in [42], which describes a framework for predicting point cloud quality by employing 2D IQA metrics on 6 orthographic projected views. This methodology was further extended in [43] by assigning non-uniform weighting on each view, the weights of a stimulus are computed as the ratio of the duration of inspection on corresponding views, divided by the total time. Liu et al. [44] provide a PCQA model based on the principle of information content weighted structural similarity (IWSSIM) [45]. Icosphere and a series of transformations are employed to generate viewpoints. PQA-Net [46] takes 6 orthographic projections of point clouds as inputs, features are extracted after convolution neural network blocks, and they share a distortion identification and a quality prediction module that assist in obtaining final quality scores. PQANet adopts a two-step strategy to train the multi-task neural network. However, the projection process and the number of viewpoints have a non-negligible impact on the final prediction accuracy; besides, how to combine the quality score on each viewpoint into a score is also not straightforward. Feature-based metrics consider perceptual loss from both geometry and texture properties. Viola et al. [47] extract the color statistics, histogram, and correlogram to assess the level of impairment and combine the color-based metrics with geometry-based metrics to form a global quality score. Alexiou et al. [26] employ the local distributions of point clouds to predict perceptual degradations from topology and color. Yang et al. [48] construct a local graph centered at resampled key points for both reference and distorted point clouds, with the color gradient on the local graph being used to measure distortions. Diniz et al. [49] introduced a texture descriptor based on perceptual color distance patterns, which is scale and rotation invariant [50]. Meynet et al. [24] utilize an optimally-weighted linear combination of curvature-based and color-based features to evaluate visual quality. Diniz et al. [51] adopt the statistical information of the extracted geometry/color features and feed them into a regression model. Deep learning-based modules have also been used to extract perceptual features. An extension using coarse-to-fine progressive knowledge transfer based on HVS is given in [52]. Zhang et al. [53] make use Signal Processing: Image Communication 135 (2025) 117262 2 X. Zhou et al. of multi-modal information to address the PCQA problem; the qualityaware encoder features are optimized with the assistance of symmetric cross-modality attention. They first split point clouds into two submodels, with a point cloud encoder to extract the geometry feature, and an image encoder to obtain the texture feature. Subsequently, the quality-aware encoder features are optimized with the assistance of symmetric cross-modality attention. Zhang et al. [54] utilize natural scene statistics and entropy on the quality-related geometry and color feature domains, which are projected from 3D space, and support vector regression is employed to obtain quality scores. IT-PCQA [55] utilizes the rich prior knowledge in images and builds a bridge between 2D and 3D perception in the field of quality assessment. Specifically, a hierarchical feature encoder and a conditional discriminative network is proposed to extract effective latent features and minimize the domain discrepancy. pmBQA [56] proposes a projection-based blind quality indicator via multimodal learning by using four homogeneous modalities (i.e., texture, normal, depth and roughness). Interested readers may refer to [40] for a more comprehensive review of the literature. In summary, point-based schemes may neglect the high-dimensional properties of point clouds and the interplay among these dimensions, thereby limiting their effectiveness. Projection-based methods often rely on 2D IQA, which may not adequately capture the intrinsic characteristics of point clouds. Feature-based schemes tend to have a high level of complexity, while the interpretability of deep learning-based methods is a drawback with training requiring a huge amount of data. 3. Proposed method In Fig. 1, the PointPCA+ framework is illustrated, which is split into three modules, namely, (a) pre-processing, (b) feature extraction, and (c) quality regression which are introduced in the following subsections. Note that a FR metric typically uses either the pristine or the impaired content as a reference, or both. In our metric design, only the pristine point cloud serves as a reference. 3.1. Pre-processing To ensure coherent geometry and color information without redundancies, points with identical coordinates that belong to the same point cloud are merged [37]. The color of a merged point is obtained by averaging the color of corresponding points sharing the same coordinates. For an FR PCQA metric, identifying matches between reference and distorted point clouds is crucial for comparing corresponding local properties. In our method, we use the 𝑘𝑛𝑛 algorithm to identify neighborhood pairs between two point clouds. In particular, for each point that belongs to a reference point cloud , we find its 𝑁nearest reference points, and its 𝑁nearest distorted points from the distorted point cloud , in terms of Euclidean distance. 3.2. Feature extraction To capture local perceptual quality degradations of a distorted point cloud, we compute geometry and texture descriptors based on the identified neighborhoods. Statistics based on these descriptors are subsequently calculated and serve as predictors of visual quality. Features are finally obtained via pooling over these predictors. As mentioned earlier, our method uses only the pristine point cloud as a reference to find the matches in the distorted point cloud. Table 1 Definition of descriptors. Descriptor Definition Distance Geometric Error vector 𝒆= (𝝎 𝑖−𝝎 𝑖)𝑟𝛼 Error along axes 𝜖𝑚= (𝝎 𝑖−𝝎 𝑖)𝑇⋅𝐮𝑚𝑟𝛽 Error from origin 𝜺=𝝎 𝑖𝑟𝛼,𝑟𝛽 Mean 𝝁=1 𝑁∑𝑛𝝎 𝑛𝑟𝛼,𝑟𝛽 Variance 𝝀=1 𝑁∑𝑛(𝝎 𝑛−𝝁)2𝑟𝛿 Sum of variance 𝛴=∑𝑚𝜆 𝑚𝑟𝛿 Covariance 𝜮=1 𝑁∑𝑛(𝝎 𝑛−𝝁)⋅(𝝎 𝑛−𝝁)T𝑟𝛾 Omnivariance O=3 √∏𝑚𝜆 𝑚𝑟𝜆 Eigenentropy E= −∑𝑚𝜆 𝑚 ⋅log 𝜆 𝑚𝑟𝛿 Anisotropy A= (𝜆 1−𝜆 3)∕𝜆 1𝑟𝛿 Planarity P= (𝜆 2−𝜆 3)∕𝜆 1𝑟𝛿 Linearity L= (𝜆 1−𝜆 2)∕𝜆 1𝑟𝛿 Scattering S=𝜆 3∕𝜆 1𝑟𝛿 Change of curvature C=𝜆 3/∑𝑚𝜆 𝑚𝑟𝛿 Parallelity P𝑚= 1 −𝐮𝑚⋅𝐯 𝑚– Angular similarity 𝜃= 1 −2⋅ar ccos(cos(𝐮𝑚,𝐯 𝑚)) 𝜋– Textural Mean  𝝁=1 𝑁∑𝑛𝒑𝑡, 𝑛𝑟𝛿 Variance  𝒔=1 𝑁∑𝑛(𝒑𝑡, 𝑛− 𝝁)2𝑟𝛿 Sum of variance  𝛴=∑𝑚𝑠 𝑚𝑟𝛿 Covariance  𝜮=1 𝑁∑𝑛(𝒑𝑡, 𝑛− 𝝁)⋅(𝒑𝑡, 𝑛− 𝝁)T𝑟𝛾 Omnivariance  O=3 √∏𝑚𝑠 𝑚𝑟𝛿 Entropy  H= −∑𝑚𝑠 𝑚 ⋅log 𝑠 𝑚𝑟𝛿 3.2.1. Geometry descriptors Given a query point 𝐩𝑖of , the subscript 𝑖denotes the point index, 1≤𝑖≤||, and ||is the cardinality. The coordinates of 𝐩𝑖’s 𝑁nearest neighbors in are indicated as 𝒑𝑔 , 𝑛= (𝑥𝑛, 𝑦𝑛, 𝑧𝑛)T, with 1≤𝑛≤𝑁and ∈ {,}. The geometry of 𝐩𝑖is denoted as 𝒑𝑔 , 𝑖, and the geometry of its closest neighbor in is denoted as 𝒑𝑔 , 𝑖. Initially, the covariance matrix 𝜮 𝑖is computed as 𝜮 𝑖=1 𝑁∑𝑁 𝑛=1 (𝒑𝑔 , 𝑛− 𝒑𝑔 , 𝑖)⋅(𝒑𝑔 , 𝑛− 𝒑𝑔 , 𝑖)T ,(1) where  𝒑𝑔 , 𝑖indicates the centroid, given as  𝒑𝑔 , 𝑖=1 𝑁∑𝑁 𝑛=1 𝒑𝑔 , 𝑛.(2) Then, eigen-decomposition is applied to 𝜮 𝑖, to obtain the eigenvectors which form an orthonormal basis 𝐕composed of eigenvectors 𝐯 𝑚, where 𝑚= 1,2,3. Next, we map the reference and distorted neighborhoods to the new orthonormal basis, denoted as 𝝎 𝑛= (𝒑𝑔 , 𝑛−  𝒑𝑔 𝑖)⋅𝐕. Finally, we apply PCA to the covariance matrix of 𝝎 𝑛and compute the eigenvectors 𝐯 𝑚. This process is visually demonstrated in Fig. 2, showcasing the distinction between the two bases. The merit inherent in projecting the geometry of both the reference and distorted point clouds onto a shared orthonormal basis, established by the reference point cloud, lies in the capacity to unify the representation of geometry degradation within a common space. This enables a more precise measurement of geometry similarity within the framework of the FR PCQA paradigm. The mapped coordinates of the reference and distorted points 𝝎 𝑛, the eigenvectors 𝐯 𝑚and the unit vectors 𝐮𝑚, with 𝐮1= [1,0,0]T, 𝐮2= [0,1,0]Tand 𝐮3= [0,0,1]T, are used to construct the geometric descriptors defined in Table 1. 3.2.2. Texture descriptors The color space is first converted from RGB to YCbCr [57]. This conversion is motivated by the fact that the human eye is more sensitive to changes in brightness than changes in color according to HVS. We denote the texture information of 𝐩𝑖’s 𝑁nearest neighbors in as 𝒑𝑡, 𝑛= (𝑌𝑛, 𝐶 𝑏𝑛, 𝐶 𝑟𝑛)T. The proposed 6 texture descriptors are defined in Table 1. Signal Processing: Image Communication 135 (2025) 117262 3 X. Zhou et al. Fig. 2. The orthonormal basis formed by both reference (longdress) and distorted (longdress_Octree-Lifting_R04) point clouds. The orange color represents the geometry around one point (2130th point) of the reference point cloud and the corresponding basis after PCA operation; the purple color represents the geometry around the matched point of the distorted point cloud and the corresponding basis after PCA operation. Fig. 3. The point cloud longdress and statistical features using mean of linearity (Figs. 3(b)–3(c)), planarity (Figs. 3(d)–3(e)). The amplitudes of statistical features are color-mapped, with red indicating higher and blue lower values. It can be noticed that the mean of linearity 3(b) and planarity 3(d) of PointPCA/PointPCA+ capture highand low-frequency geometric regions, respectively. Additionally, PointPCA+ has lower complexity as the neighborhood is determined through knn. 3.2.3. Explanation of descriptors Each geometry descriptor represents an interpretable shape property inside the neighborhood. Specifically, 𝒆denotes the error vector between the mapped coordinates of the reference query point and its nearest neighbor, and 𝜖𝑚is the projected distance of the error vector across the 𝑚th axis. The 𝜺is used to capture the Euclidean and projected distances of the mapped reference query point or its nearest distorted neighbor from the centroid and principal axes, respectively. 𝝁,𝝀, 𝛴and 𝜮reveal local statistics. Eprovides an estimation of the space uncertainty on the projected surfaces. Additionally, P𝑚and 𝜃𝑚 assess the parallelity and the angular dispersion of the distorted plane. The remaining geometry descriptors explore the topology of a local region from different aspects, relying on the spatial dispersion along different principal axes.  𝝁, 𝒔and  𝛴of the YCbCr channel express the intrinsic distribution of luminance and chromatic components.  𝜮 and  Oshow the variability of color information.  𝐇provides an estimation of color uncertainty of the local region. Every descriptor is computed per point 𝐩𝑖. 3.2.4. Predictors Predictors are defined as the error samples obtained by computing a distance over descriptor values. We define different distance functions for different descriptors. We use the Euclidean distance to measure the Fig. 4. The predictors of linearity (Fig. 4(b)) and planarity (Fig. 4(c)) of point cloud longdress-Octree-Lifting-R02 (Fig. 4(a)). The amplitudes of predictors are color-mapped, with red indicating higher and blue lower values. Notably, the relative differences in linearity and planarity of PointPCA+ effectively highlight regions that can form linear structures and detect bulges, such as wrinkles in clothing, in the longdress. point-to-point distances between query point pairs under the new basis 𝑟𝛼=√∑𝑚𝒅𝟏2,(3) Signal Processing: Image Communication 135 (2025) 117262 4 X. Zhou et al. where 𝒅𝟏is the difference between two points. We use the absolute value to measure the point-to-plane distance, as 𝑟𝛽=|𝒅𝟐|,(4) where 𝒅𝟐indicates the projected distance between a point and the reference axes. We use the following definition of relative difference for the covariance features 𝑟𝛾=|𝒒⊙𝒒−𝑸| 𝒒⊙𝒒,(5) where {𝒒=𝝀,𝐐=𝜮}and {𝒒= 𝒔,𝐐= 𝜮}, for geometry and texture attributes, respectively, ⊙is for element-wise product. We use the relative difference formula [26], for the remaining descriptors 𝑟𝛿=|𝜙−𝜙| ||𝜙||+||𝜙||+𝜀,(6) where 𝜀is a small constant to avoid undefined operations. Finally, the definitions of parallelity and angular similarity descriptors incorporate a distance function. For notational purposes only, we define distances 𝑟𝜌and 𝑟𝜃to be identical to the definitions of P𝑚and 𝜃𝑚, respectively. Table 1enlists distance function(s) used per descriptor. 3.2.5. Features Features are defined by pooling over predictor values. Specifically, predictors 𝜓𝑖,𝑗 ,𝑘 are obtained per point 𝐩𝑖, descriptor 𝑗, and distance function 𝑟𝑘,𝑘∈ {𝛼 , 𝛽 , 𝛾 , 𝛿 , 𝜌, 𝜃}. This is done for all descriptors 𝑗in Table 1, using the corresponding distances 𝑟𝑘. Through pooling, we obtain a feature 𝑓𝑗 ,𝑘 for every predictor: 𝑓𝑗 ,𝑘 =1 ||∑|| 𝑖=1 𝜓𝑖,𝑗 ,𝑘.(7) 3.3. Quality regression To obtain a quality score that is well-aligned with the HVS, the Recursive Feature Elimination (RFE) algorithm is used to select the most relevant predictor set among all the proposed predictors. RFE [58] improves model accuracy, and efficiency, and reduces overfitting. Machine learning-based regression models have been extensively used to tackle the quality regression problem in the domain of quality assessment, we then use the random forest algorithm to regress the selected predictors to the final quality score. 3.4. Differences with PointPCA In comparison to PointPCA, our pre-processing methodology involves solely utilizing the pristine point cloud as a reference, dispensing with the need for both pristine and distorted point clouds. Regarding geometric features, we transform the 𝑥𝑦𝑧 values of both the pristine and distorted point clouds into the basis formed by the pristine point cloud, eliminating the necessity for separate bases for distinct point cloud sets. For textural features, we directly apply statistical functions to the color values in the YCbCr space without resorting to PCA decomposition. The decision to forego PCA on texture stems from the lack of physical significance post-decomposition of YCbCr channel values, as observed in geometry. Simultaneously, this approach aids in reducing computational complexity. Additionally, PointPCA+ adopts distinct distance functions for various descriptors, eliminating constraints on the computation of mean and standard deviation values. We illustrate the disparities between PointPCA and PointPCA+ for a specific point cloud, longdress, in Fig. 3. Notably, we visualize the linearity and planarity geometric descriptors before comparison. From Fig. 3, the geometric dissimilarity measurements differ when using 𝑘𝑛𝑛 and 𝑟-𝑠𝑒𝑎𝑟𝑐 ℎmethods. For PointPCA, we employed the 𝑟-𝑠𝑒𝑎𝑟𝑐 ℎmethod with 𝑟= 0.008 ×𝐵, where 𝐵represents the maximum length of the bounding box of the reference point cloud. In contrast, for PointPCA+, we used the 𝑘𝑛𝑛 algorithm with 𝑘= 81. While for the statistical features 𝜇, the 𝑘𝑛𝑛 algorithm with 𝑘= 9was applied. Both the linearity and planarity are decreased compared with PointPCA. We also visualize the predictions corresponding to linearity and planarity in Fig. 4. We can see that the predictors exactly describe the line and the uneven region of point cloud longdress. 4. Experimental results In this Section, we report the evaluation results of the proposed PointPCA+ metric under three public datasets in Section 4.2 with other 8 state-of-the-art metrics. Moreover, we report the performance achieved in the ICIP 2023 Point Cloud Visual Quality Assessment (PCVQA) grand challenge1in Section 4.3. Specifically, the challenge consists of 5 tracks, which correspond to different use cases in which quality metrics are typically used. The first two tracks aim to assess the perceptual fidelity of distorted contents with/without respect to the originals for any level of distortion, respectively. This is the most generic and traditional set-up for quality metrics. The next two tracks focus on metrics for high-end quality with/without access to the original content. These are desirable in applications such as content production, high-quality streaming, digital twins, etc. The last track should be sensible to quality differences within different processed versions of the same point cloud content, which is suitable for optimization scenarios. We participated in Track#1 FR broad-range quality estimation, Track#3 FR high-range quality estimation, and Track#5 FR intra-reference quality estimation. Additional analysis related to crossdataset validation and feature importance is carried out across all the aforementioned datasets, as detailed in Section 4.4 and Section 4.5. These sections aim to demonstrate the generalizability of the proposed PointPCA+. 4.1. Setup 4.1.1. Datasets Three publicly available datasets were recruited for performance evaluation, namely, M-PCCD, SJTU, and WPC. The M-PCCD [60] consists of 8 point clouds whose geometry and color are encoded using V-PCC and G-PCC variants, resulting in 232 distorted stimuli. Detailed distortion types include Octree-Lifting, Octree-RAHT, TriSoup-Lifting, TriSoup-RAHT and V-PCC. The contents in M-PCCD depict either human figures or objects. The SJTU [61] includes 9 reference point clouds with each point cloud corrupted by seven types of distortions under six levels, generating 378 distorted stimuli. Detailed distortion types include Octree-based compression, Color Noise (CN), Geometric Gaussian Noise (GGN), downsampling, and combinations of the CN, GGN and downsampling. SJTU includes 5 human body models and 4 inanimate objects. The WPC [44] contains 20 reference point clouds with each point cloud degraded under five types of distortions and different levels, leading to 740 distorted stimuli. Detailed distortion types include Octree-LPCC, TriSoup-SPCC, V-PCC, Gaussian noise and downsampling. WPC dataset only collects objects including snacks, fruits and vegetables, etc. The Broad Quality Assessment of Static Point Clouds (BASICS) [34] is used in the ICIP 2023 PCVQA grand challenge, and comprises 75 point clouds from 3 different semantic categories: (i) Humans & Animals, (ii) Inanimate Objects, and (iii) Buildings & Landscapes. Each point cloud is compressed with 3 compression methods from the MPEG standardization field, i.e., OctreeRAHT, Octree-Predlift and V-PCC; 1 learning-based algorithm, i.e., GeoCNN, at varying compression levels, resulting in 1494 processed point clouds. BASICS dataset is aimed to provide a foundation for research that supports the telepresence applications, in terms of compression and quality assessment. Table 3summarized the characteristics of the four datasets. 1https://sites.google.com/view/icip2023-pcvqa-grand-challenge Signal Processing: Image Communication 135 (2025) 117262 5 X. Zhou et al. Table 2 SROCC performance on M-PCCD, SJTU and WPC datasets. Metric PointPCA+ PointPCA [25] PCQM [24] PointSSIM [26] BitDance [59] Plane2Plane [41] P2Plane_MSE [40] P2P_MSE [40] PSNR Y [40] M-PCCD 0.943±0.022 0.941±0.032 0.940 ±0.032 0.925 ±0.024 0.859 ±0.061 0.847 ±0.076 0.901 ±0.025 0.896 ±0.042 0.798 ±0.162 SJTU 0.865±0.064 0.890±0.056 0.862 ±0.030 0.708 ±0.070 0.748 ±0.077 0.761 ±0.039 0.578 ±0.155 0.612 ±0.157 0.743 ±0.083 WPC 0.857±0.040 0.866±0.036 0.749 ±0.036 0.465 ±0.059 0.451 ±0.054 0.454 ±0.069 0.452 ±0.065 0.563 ±0.071 0.614 ±0.061 Table 3 Overview of M-PCCD, SJTU, WPC and BASICS datasets. Datasets Contents Distortion types Distortion levels Total M-PCCD Humans & Inanimate Objects 8 Octree-Lifting, Octree-RAHT, TriSoup-Lifting, TriSoup-RAHT, V-PCC 5 G-PCC: 6 V-PCC: 5 232 SJTU Humans & Inanimate Objects 9 Octree-based compression, CN, GGN, downsampling, CN + GGN, CN + downsamplin, GGN + downsampling 7 6 378 WPC Inanimate Objects 20 Octree-LPCC, TriSoup-SPCC, V-PCC, Gaussian noise, downsampling 5 Geometry: 3 Texture:1/3/4 740 BASICS 1. Humans & Animals 2. Inanimate Objects 3. Buildings & Landscapes 75 Octree-RAHT, Octree-Predlift, V-PCC, GeoCNN 4 G-PCC: 5 V-PCC: 6 1494 4.1.2. Evaluation metrics For the PCVQA Grand Challenge, all submissions undergo testing on a designated test set curated by the organizers. The evaluation of performance relies on five standard criteria provided by the organizers: including Pearson Linear Correlation Coefficient (PLCC), Spearman Rank Order Correlation Coefficient (SROCC), Difference/Similar Analysis quantified by Area Under the Curve (D/S𝐴𝑈 𝐶), Better/Worse Analysis quantified by Correct Classification percentage (B/W𝐶 𝐶) [62], and the Runtime Complexity (RC). Notably, no function is employed for score mapping. Additionally, for the assessment of performance metrics on three other commonly used datasets, the criteria include SROCC, PLCC, Kendall Rank Order Correlation Coefficient (KROCC), and Mean Square Error (MSE) are chosen. Higher values of PLCC, SROCC, and KROCC indicate better performance in terms of correlation with human opinion while lower RMSE indicates better consistency. 4.1.3. Implementation details We use RFE to select the best feature set among all the predictors, with the best SROCC performance on the PCVQA grand challenge test set. In the inference stage, the default configuration of scikit-learn (version 1.2.2) in Python is used. Regarding the neighborhood size for the computation of descriptors, 𝐾= 81 is chosen considering complexity and performance, after experimenting with 𝐾∈ {9,25,49,81,121}. 4.2. Performance evaluation on M-PCCD, SJTU and WPC We compare PointPCA+ with existing FR point-based quality metrics, the results are shown in Table 2. The best performance among these metrics is highlighted in boldface, with the second best underlined. Specifically, each dataset is split into two partitions that contain 80% and 20% of the contents for training and testing, respectively, with all the distorted versions of a specific content placed in one partition. For M-PCCD, SJTU, and WPC, we use 6/2, 7/2, and 16/4 contents for training/testing, respectively. Then, a quality prediction model is trained on the training data and tested on the corresponding testing data of the same dataset, for within-dataset validation. this process is repeated for all possible 80%–20% splits of each dataset, leading to 28, 36, and 4845 testing partitions and an equal number of corresponding quality prediction models for M-PCCD, SJTU, and WPC respectively. Finally, the average and the standard deviation of SROCC index computed across all testing splits of each dataset, are reported. From Table 2we can see that PCA-based metrics are competitive with the highest SROCC on the three datasets, especially the performance of PointPCA on WPC is increased by 15.62% in terms of SROCC though the performance of PointPCA+ is a slightly lower than PointPCA (0.866 VS 0.857). Table 4 Track#1 (FR broad range): top 4 performance comparison on the official PCVQA grand challenge test set, evaluated by the challenge organizers. Best in bold and second best underlined. Our submission is ranked in 2nd place. Submission SROCC PLCC D/S𝐴𝑈 𝐶B/W𝐶 𝐶RC(s) KDDIUSCJoint 0.875 0.917 0.888 0.970 42.80 PointPCA+ 0.874 0.909 0.871 0.961 1000.00 SJTU MMLAB 0.871 0.896 0.832 0.955 8.60 SlowHand 0.791 0.825 0.805 0.924 130.47 Table 5 Track#3 (FR high range): top 4 performance comparison on the official PCVQA grand challenge test set, evaluated by the challenge organizers. Best in bold and second best underlined. Our submission is ranked in 3rd place. Submission SROCC PLCC D/S𝐴𝑈 𝐶B/W𝐶 𝐶RC(s) SJTU MMLAB 0.630 0.592 0.665 0.909 8.60 KDDIUSCJoint 0.551 0.516 0.642 0.872 42.80 PointPCA+ 0.603 0.479 0.625 0.886 1000.00 SlowHand 0.377 0.423 0.565 0.780 130.47 Table 6 Track#5 (FR intra-reference): top 4 performance comparison on the official PCVQA grand challenge test set, evaluated by the challenge organizers. Best in bold and second best underlined. Our submission is ranked in 3rd place. Submission D/S𝐴𝑈 𝐶B/W𝐶 𝐶RC(s) SJTU MMLAB 0.808 0.947 8.60 KDDIUSCJoint 0.822 0.933 42.80 PointPCA+ 0.811 0.938 1000.00 SlowHand 0.753 0.854 130.47 4.3. Performance evaluation on BASICS We split BASICS into training–validation-test with 60%-20%–20% following the rules from the PCVQA grand challenge [63]. Table 4to Table 6show the official evaluation results of Track#1, Track#3 and Track#5, respectively. Referencing Tables 5–6, several notable observations emerge from the competition results across all three FR tracks. (1) Despite strong performances in Track 1 and Track 5, none of the teams attained satisfactory results in Track 3 of in PCVQA, highlighting the challenges associated with fine-grained PCQA. (2) Examining PointPCA+, it is evident that the extracted features within a neighborhood size of 81, combined with the application of statistical functions (e.g., mean and variance) on geometry, may fail to capture subtle differences between two point clouds. This highlights the limitations of using statistical features to capture subtle differences. Signal Processing: Image Communication 135 (2025) 117262 6 X. Zhou et al. Table 7 Cross-dataset validation among M-PCCD, SJTU, WPC and BASICS datasets. Both the training and testing used all the content among the datasets. Best in bold. Test M-PCCD SJTU WPC BASICS Train PLCC SROCC KROCC RMSE PLCC SROCC KROCC RMSE PLCC SROCC KROCC RMSE PLCC SROCC KROCC RMSE M-PCCD – – – – 0.726 0.725 0.541 3.148 0.500 0.469 0.328 4.003 0.847 0.777 0.589 0.899 SJTU 0.855 0.881 0.708 2.578 – – – – 0.578 0.608 0.442 2.575 0.732 0.717 0.523 1.946 WPC 0.731 0.878 0.703 4.767 0.610 0.604 0.435 2.049 – – – – 0.848 0.726 0.536 3.675 BASICS 0.832 0.880 0.692 1.057 0.579 0.645 0.472 2.810 0.500 0.490 0.347 3.202 – – – – 4.4. Cross-dataset validation To verify the generalization and robustness of the proposed PointPCA+, we conduct cross-dataset experiments among all 4 datasets. We train the model using the entire content of one dataset and then test it separately using the entire content of the other three datasets. The experimental results are shown in Table 7. From Table 7, we can draw the following observations: 1. PointPCA+ performs well in generalization and robustness, particularly when trained on the small M-PCCD dataset and tested on the large BASICS dataset. Combined with Table 2, the crossdataset evaluation performance is even higher than certain FR PCQA metrics, for example BitDance [59], PSNR_Y [40], etc. 2. Compared with M-PCCD, SJTU and BASICS datasets, the WPC dataset has the worst performance among all the evaluation metrics. SROCC and PLCC of PointPCA+ on WPC trained on M-PCCD and BASICS achieve the same accuracy as random guessing, this may be because the contents in WPC only contain objects, and the distortion types of WPC are more complex compared with the other three datasets. 3. PointPCA+ shows better SROCC performance on the SJTU dataset compared to the WPC dataset. This is likely because SJTU shares specific human figures with the M-PCCD dataset and includes both human and object categories for M-PCCD and BASICS, while SJTU and WPC have no overlapping content. In conclusion, the cross-dataset performance of PointPCA+ is promising but its generalization depends on dataset composition. Training on large, diverse datasets and testing on smaller ones normally yields better results, while the reverse leads to poor generalization. However, this cannot hold if there exists a domain shift (i.e., in our case, contents and distortion type) between the testing set and the training set. 4.5. Performance on individual distortion type To further explore the effectiveness of the designed geometry and texture features for a specific distortion type, we test the performance per distortion type per dataset, with the results listed in Table 8. We can draw the following observations. 1. When assessing compression distortion, V-PCC emerges as the most challenging distortion type of which to evaluate perceptual quality, aligning with findings from a prior study [46]. Predicting the perceptual quality of compression distortions from G-PCC and learning-based methods proves to be more manageable, with Octree-Lifting exhibiting a slight advantage over Octree-RAHT on M-PCCD and BASICS datasets. 2. For CN and Gaussian noise distortions, CN has the poorest performance. However, prediction accuracy improves by 16.06% for CN+GGN in terms of SROCC on the SJTU dataset, indicating that geometry-related features help capture these distortions. Gaussian noise performs worst on WPC, as it affects both geometry and texture, creating a compounded distortion. 3. For downsampling distortion types, the performance of PointPCA+ is notably high on the SJTU dataset but relatively lower on the WPC dataset. This implies that downsampling on objects presents a challenge for the HVS to discern, as it may exert a masking effect on objects more prominently than on humans. Table 8 Performance comparison of PointPCA+ metrics for different distortion types per dataset. DT refers to distortion type and NC denotes the number of contents. Dataset DT NC PLCC SROCC KROCC RMSE M-PCCD Octree-Lifting 48 0.938 0.991 0.962 0.826 Octree-RAHT 48 0.962 0.984 0.931 0.704 TriSoup-Lifting 48 0.951 0.965 0.879 0.597 TriSoup_RAHT 48 0.970 0.963 0.870 0.561 V-PCC 40 0.823 0.815 0.674 1.266 SJTU CN 54 0.621 0.741 0.545 2.594 CN + GGN 54 0.894 0.860 0.697 0.825 Downsampling 54 0.969 0.944 0.848 0.562 Downsampling+CN 54 0.946 0.937 0.788 1.371 Downsampling+GGN 54 0.981 0.965 0.879 0.736 GGN 54 0.955 0.958 0.848 0.810 Octree (PCL) 54 0.975 0.965 0.879 0.797 WPC Downsampling 60 0.802 0.795 0.594 1.177 Octree (LPCC) 80 0.916 0.881 0.708 0.850 Trisoup (SPCC) 240 0.987 0.923 0.818 0.594 Gaussian noise 180 0.628 0.650 0.417 1.826 V-PCC 180 0.737 0.756 0.566 2.054 BASICS GeoCNN 294 0.971 0.941 0.787 0.305 Octree-Predlift 375 0.975 0.937 0.792 0.289 Octree-RAHT 375 0.895 0.876 0.693 0.441 V-PCC 450 0.787 0.690 0.514 0.371 In summary, PointPCA+ exhibits proficiency in predicting compression distortions. However, its effectiveness diminishes when confronted with distortion instances primarily manifesting in color values, as observed for CN distortions on the SJTU dataset. Additionally, when the equilibrium between geometric and color is disrupted, as exemplified by Gaussian noise on the WPC dataset, PointPCA+ struggles to accurately gauge the extent of degradation. 4.6. Feature importance of PointPCA+ for different distortion types per dataset We evaluated the effectiveness of the 40 selected features after RFE. The model was trained on 80% of the dataset, and feature importance was calculated on the remaining 20% for each distortion type. This was done using the permutation importance technique [64] based on MSE. Permutation importance shows how crucial a feature is for a specific model, rather than its standalone predictive value, helping to assess the generalization ability of the features across different distortion types. The feature importance of PointPCA+ across the four datasets is shown in Fig. 5. The feature importance for each distortion type per dataset is illustrated in Fig. 6. Combining Fig. 5and Fig. 6, we can draw the following conclusions: 1. The feature importance ranking varies across different datasets, primarily due to distinctions in content and distortion types among the four datasets. The variance on the 𝑧axis (𝑓𝑧 𝜆,𝛿) within the geometric features obtained the lowest importance ranking on both M-PCCD and BASICS, ranked 29th and 38th on SJTU and WPC datasets, respectively. 2. Geometric features consistently exhibit superior feature importance rankings when compared to textural features, as observed in the top-5 features across all four datasets. In M-PCCD, SJTU, Signal Processing: Image Communication 135 (2025) 117262 7 X. Zhou et al. Fig. 5. The feature importance of all the extracted geometry and texture features within pointPCA+ metric for M-PCCD, SJTU, WPC and BASICS datasets. The numbers are obtained by ranking the 40 features based on the importance score per dataset. Fig. 6. Feature importance of PointPCA+ per distortion type for M-PCCD, SJTU, WPC, and BASICS datasets, separately. The numbers are obtained by ranking the 40 features based on the permutation feature importance score per distortion type. and WPC, the top-3 rankings comprise a combination of texture and geometry features. However, in BASICS, only geometry features are represented in the top-3 ranking. 3. The point-to-point distance (𝑓𝒆,𝛼) and the mean value of 𝑣channel (𝑓𝑣  𝝁,𝛿) demonstrate strong performance across various distortion types in M-PCCD dataset. Notably, in TriSoup-Lifting, the mean value of 𝑦channel (𝑓𝑦  𝝁,𝛿) excels but performs less optimally in the case of V-PCC distortion. In SJTU dataset, the cosine similarity of the 𝑦axis (𝑓𝑦 𝜃) outperforms other features for all distortion types, except for CN+GGN distortion. Similarly, the covariance of the 𝑢channel (𝑓𝑢  𝜮) excels for all distortion types, with the exception of Gaussian noise in WPC. Meanwhile, in BASICS, the projected distances of the distorted centroid from reference planes on the 𝑧axis (𝑓𝑧 𝝎𝐵,𝛽 ) consistently yield high rankings across all distortion types. 4.7. Discussion At the dataset level, performance is significantly influenced by the total number of point clouds. In extensive datasets encompassing diverse content, prediction accuracy tends to be lower compared to smaller datasets with fewer variations, even when their distortion types are similar. Furthermore, distortion levels play a crucial role in impacting prediction accuracy, with fine-grained distortion proving more challenging than coarse division. In the context of point clouds, compound distortion does not necessarily result from the mixture of multiple distortion types, even a single distortion type can concurrently compromise texture and geometry. Hand-crafted geometry and texture features exhibit distinct strengths and weaknesses across various distortion types. Combining both types of features adaptively with distortions may enhance prediction accuracy. Signal Processing: Image Communication 135 (2025) 117262 8 X. Zhou et al. 5. Conclusion and future work This paper proposes a PCA-based FR PCQA metric, namely PointPCA+, which relies on an enriched set of lower complexity descriptors with respect to its PointPCA predecessor. After a pre-processing step, features are extracted from both geometric and textural domains. A subset of features is selected to enhance the stability of the model, and a learning-based feature fusion based on ensemble learning is applied to the feature subset, to provide a quality score for a distorted point cloud. Our experimental results demonstrate that PointPCA+ outperforms the majority of existing PCQA metrics, reaching second place in Track#1 and third place in Track#3 and Track#5 of the ICIP 2023 PCVQA grand challenge. Compared to other teams in the PCVQA grand challenge, PointPCA+ has the highest computational complexity, with a processing time of 1000 ms, while the lowest reported is 8.6 ms. This inefficiency stems from the point-based FR PCQA framework, especially the PCA decomposition applied to each point. One solution to reduce the computational load is to downsample the point cloud before processing, while maintaining its overall structure. Additionally, both PointPCA and PointPCA+ mainly focus on local features, neglecting the broader geometric and texture context of the point cloud. To improve, incorporating global features that capture more comprehensive information is essential. This will lead to a more efficient and holistic characterization of point clouds, meeting the demands of the PCQA field. These additions are poised to contribute to a more holistic and efficient characterization of the point cloud, aligning with the rigorous demands of the PCQA field. CRediT authorship contribution statement Xuemei Zhou: Writing – review & editing, Writing – original draft, Methodology, Investigation, Data curation. Evangelos Alexiou: Writing – review & editing, Validation, Supervision, Methodology. Irene Viola: Writing – review & editing, Visualization, Validation, Supervision, Funding acquisition. Pablo Cesar: Writing – review & editing, Supervision, Funding acquisition. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 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