Showing papers on "Line segment published in 2019"

PL-SLAM: A Stereo SLAM System Through the Combination of Points and Line Segments

Abstract: Traditional approaches to stereo visual simultaneous localization and mapping (SLAM) rely on point features to estimate the camera trajectory and build a map of the environment. In low-textured environments, though, it is often difficult to find a sufficient number of reliable point features and, as a consequence, the performance of such algorithms degrades. This paper proposes PL-SLAM, a stereo visual SLAM system that combines both points and line segments to work robustly in a wider variety of scenarios, particularly in those where point features are scarce or not well-distributed in the image. PL-SLAM leverages both points and line segments at all the instances of the process: visual odometry, keyframe selection, bundle adjustment, etc. We contribute also with a loop-closure procedure through a novel bag-of-words approach that exploits the combined descriptive power of the two kinds of features. Additionally, the resulting map is richer and more diverse in three-dimensional elements, which can be exploited to infer valuable, high-level scene structures, such as planes, empty spaces, ground plane, etc. (not addressed in this paper). Our proposal has been tested with several popular datasets (such as EuRoC or KITTI), and is compared with state-of-the-art methods such as ORB-SLAM2, revealing a more robust performance in most of the experiments while still running in real time. An open-source version of the PL-SLAM C++ code has been released for the benefit of the community.

PPGNet: Learning Point-Pair Graph for Line Segment Detection

Abstract: In this paper, we present a novel framework to detect line segments in man-made environments. Specifically, we propose to describe junctions, line segments and relationships between them with a simple graph, which is more structured and informative than end-point representation used in existing line segment detection methods. In order to extract a line segment graph from an image, we further introduce the PPGNet, a convolutional neural network that directly infers a graph from an image. We evaluate our method on published benchmarks including York Urban and Wireframe datasets. The results demonstrate that our method achieves satisfactory performance and generalizes well on all the benchmarks. The source code of our work is available at https://github.com/svip-lab/PPGNet.

Learning Attraction Field Representation for Robust Line Segment Detection

Abstract: This paper presents a region-partition based attraction field dual representation for line segment maps, and thus poses the problem of line segment detection (LSD) as the region coloring problem. The latter is then addressed by learning deep convolutional neural networks (ConvNets) for accuracy, robustness and efficiency. For a 2D line segment map, our dual representation consists of three components: (i) A region-partition map in which every pixel is assigned to one and only one line segment; (ii) An attraction field map in which every pixel in a partition region is encoded by its 2D projection vector w.r.t. the associated line segment; and (iii) A squeeze module which squashes the attraction field to a line segment map that almost perfectly recovers the input one. By leveraging the duality, we learn ConvNets to compute the attraction field maps for raw in-put images, followed by the squeeze module for LSD, in an end-to-end manner. Our method rigorously addresses several challenges in LSD such as local ambiguity and class imbalance. Our method also harnesses the best practices developed in ConvNets based semantic segmentation methods such as the encoder-decoder architecture and the a-trous convolution. In experiments, our method is tested on the WireFrame dataset and the YorkUrban dataset with state-of-the-art performance obtained. Especially, we advance the performance by 4.5 percents on the WireFramedataset. Our method is also fast with 6.6∼10.4 FPS, outperforming most of existing line segment detectors.

LiDAR-Based Real-Time Detection and Modeling of Power Lines for Unmanned Aerial Vehicles

Abstract: The effective monitoring and maintenance of power lines are becoming increasingly important due to a global growing dependence on electricity. The costs and risks associated with the traditional foot patrol and helicopter-based inspections can be reduced by using UAVs with the appropriate sensors. However, this implies developing algorithms to make the power line inspection process reliable and autonomous. In order to overcome the limitations of visual methods in the presence of poor light and noisy backgrounds, we propose to address the problem of power line detection and modeling based on LiDAR. The PL 2 DM, Power Line LiDAR-based Detection and Modeling, is a novel approach to detect power lines. Its basis is a scan-by-scan adaptive neighbor minimalist comparison for all the points in a point cloud. The power line final model is obtained by matching and grouping several line segments, using their collinearity properties. Horizontally, the power lines are modeled as a straight line, and vertically as a catenary curve. Using a real dataset, the algorithm showed promising results both in terms of outputs and processing time, adding real-time object-based perception capabilities for other layers of processing.

Fast 3D Line Segment Detection From Unorganized Point Cloud

Abstract: This paper presents a very simple but efficient algorithm for 3D line segment detection from large scale unorganized point cloud. Unlike traditional methods which usually extract 3D edge points first and then link them to fit for 3D line segments, we propose a very simple 3D line segment detection algorithm based on point cloud segmentation and 2D line detection. Given the input unorganized point cloud, three steps are performed to detect 3D line segments. Firstly, the point cloud is segmented into 3D planes via region growing and region merging. Secondly, for each 3D plane, all the points belonging to it are projected onto the plane itself to form a 2D image, which is followed by 2D contour extraction and Least Square Fitting to get the 2D line segments. Those 2D line segments are then re-projected onto the 3D plane to get the corresponding 3D line segments. Finally, a post-processing procedure is proposed to eliminate outliers and merge adjacent 3D line segments. Experiments on several public datasets demonstrate the efficiency and robustness of our method. More results and the C++ source code of the proposed algorithm are publicly available at this https URL.

The Value Function Polytope in Reinforcement Learning

Abstract: We establish geometric and topological properties of the space of value functions in finite state-action Markov decision processes. Our main contribution is the characterization of the nature of its shape: a general polytope (Aigner et al., 2010). To demonstrate this result, we exhibit several properties of the structural relationship between policies and value functions including the line theorem, which shows that the value functions of policies constrained on all but one state describe a line segment. Finally, we use this novel perspective to introduce visualizations to enhance the understanding of the dynamics of reinforcement learning algorithms.

Visual-Inertial Odometry with Point and Line Features

Abstract: In this paper, we present a tightly-coupled monocular visual-inertial navigation system (VINS) using points and lines with degenerate motion analysis for 3D line triangulation. Based on line segment measurements from images, we propose two sliding window based 3D line triangulation algorithms and compare their performance. Analysis of the proposed algorithms reveals 3 degenerate camera motions that cause triangulation failures. Both geometrical interpretation and Monte-Carlo simulations are provided to verify these degenerate motions which prevent triangulation. In addition, commonly used line representations are compared through a monocular visual SLAM Monte-Carlo simulation. Finally, real-world experiments are conducted to validate the implementation of the proposed VINS system using the “closest point” line representation.

Splitting method for elliptic equations with line sources

Abstract: In this paper, we study the mathematical structure and numerical approximation of elliptic problems posed in a (3D) domain Ω when the right-hand side is a (1D) line source Λ. The analysis and approximation of such problems is known to be non-standard as the line source causes the solution to be singular. Our main result is a splitting theorem for the solution; we show that the solution admits a split into an explicit, low regularity term capturing the singularity, and a high-regularity correction term w being the solution of a suitable elliptic equation. The splitting theorem states the mathematical structure of the solution; in particular, we find that the solution has anisotropic regularity. More precisely, the solution fails to belong to H 1 in the neighbourhood of Λ, but exhibits piecewise H 2 -regularity parallel to Λ. The splitting theorem can further be used to formulate a numerical method in which the solution is approximated via its correction function w . This recasts the problem as a 3D elliptic problem with a 3D right-hand side belonging to L 2 , a problem for which the discretizations and solvers are readily available. Moreover, as w enjoys higher regularity than the full solution, this improves the approximation properties of the numerical method. We consider here the Galerkin finite element method, and show that the singularity subtraction then recovers optimal convergence rates on uniform meshes, i.e. , without needing to refine the mesh around each line segment. The numerical method presented in this paper is therefore well-suited for applications involving a large number of line segments. We illustrate this by treating a dataset (consisting of ~3000 line segments) describing the vascular system of the brain.

Learnable Line Segment Descriptor for Visual SLAM

Abstract: Traditionally, the indirect visual motion estimation and simultaneous localization and mapping (SLAM) systems were based on point features. In recent years, several SLAM systems that use lines as primitives were suggested. Despite the extra robustness and accuracy brought by the line segment matching, the line segment descriptors used in such systems were hand-crafted, and therefore sub-optimal. In this paper, we suggest applying descriptor learning to construct line segment descriptors optimized for matching tasks. We show how such descriptors can be constructed on top of a deep yet lightweight fully-convolutional neural network. The coefficients of this network are trained using an automatically collected dataset of matching and non-matching line segments. The use of the fully-convolutional network ensures that the bulk of the computations needed to compute descriptors is shared among the multiple line segments in the same image, enabling efficient implementation. We show that the learned line segment descriptors outperform the previously suggested hand-crafted line segment descriptors both in isolation (i.e., for the subtask of distinguishing matching and non-matching line segments), but also when built into the SLAM system. We construct a new line based SLAM pipeline built upon a state-of-the-art point-only system. We demonstrate generalization of the learned parameters of the descriptor network between two well-known datasets for autonomous driving and indoor micro aerial vehicle navigation.

Automatic Building Outline Extraction from ALS Point Clouds by Ordered Points Aided Hough Transform

Abstract: Many urban applications require building polygons as input. However, manual extraction from point cloud data is time- and labor-intensive. Hough transform is a well-known procedure to extract line features. Unfortunately, current Hough-based approaches lack flexibility to effectively extract outlines from arbitrary buildings. We found that available point order information is actually never used. Using ordered building edge points allows us to present a novel ordered points–aided Hough Transform (OHT) for extracting high quality building outlines from an airborne LiDAR point cloud. First, a Hough accumulator matrix is constructed based on a voting scheme in parametric line space (θ, r). The variance of angles in each column is used to determine dominant building directions. We propose a hierarchical filtering and clustering approach to obtain accurate line based on detected hotspots and ordered points. An Ordered Point List matrix consisting of ordered building edge points enables the detection of line segments of arbitrary direction, resulting in high-quality building roof polygons. We tested our method on three different datasets of different characteristics: one new dataset in Makassar, Indonesia, and two benchmark datasets in Vaihingen, Germany. To the best of our knowledge, our algorithm is the first Hough method that is highly adaptable since it works for buildings with edges of different lengths and arbitrary relative orientations. The results prove that our method delivers high completeness (between 90.1% and 96.4%) and correctness percentages (all over 96%). The positional accuracy of the building corners is between 0.2–0.57 m RMSE. The quality rate (89.6%) for the Vaihingen-B benchmark outperforms all existing state of the art methods. Other solutions for the challenging Vaihingen-A dataset are not yet available, while we achieve a quality score of 93.2%. Results with arbitrary directions are demonstrated on the complex buildings around the EYE museum in Amsterdam.

Certified space curve fitting and trajectory planning for CNC machining with cubic B-splines

Abstract: In CNC machining, the tool path following G01 codes generally introduces large computations and inherent discontinuities. A common way to avoid these shortcomings is fitting the G01 polyline with a parametric curve and then scheduling the feedrate along the fitted curve. However, curve fitting with confined error in three dimensional space is nontrivial since the Hausdorff distance between a space G01 segment and a rational parametric curve segment is difficult to formulate. In this paper, we derive the explicit expression for the Hausdorff distance between a line segment and a curve segment, and then propose a curve fitting algorithm for G01 polylines. Instead of the traditional two-stage model, we present a combined trajectory planning model with error constraints as well as dynamic constraints. Moreover, an effective and efficient algorithm is designed to solve this model. Experimental results are provided to illustrate and clarify our methods.

DLD: A Deep Learning Based Line Descriptor for Line Feature Matching

Abstract: In this paper, we present an appearance based line descriptor which was developed with the help of machine learning. Our descriptor uses a ResNet which we modified in its size to improve the performance. We utilized the Unreal Engine and multiple scenes provided for it to create training data. The training was performed using a triplet loss, where the loss of the network is calculated with triplets each consisting of three lines including a matching pair and another non-matching line. During learning, the goal of the minimization function is to calculate descriptors with minimal descriptor distance to matching lines’ descriptors and maximal descriptor distance to other lines’ descriptors. We evaluate the performance of our descriptor on our synthetic datasets, on real-world stereo images from the Middlebury Stereo Dataset and on a benchmark for line segment matching. The results show that in comparison to state-of-the-art line descriptors our method achieves a greatly improved line matching accuracy.

Power Line Extraction From Aerial Images Using Object-Based Markov Random Field With Anisotropic Weighted Penalty

Abstract: The extraction of power line plays a key role in power line inspection by Unmanned Aerial Vehicles (UAVs). While it is challenging to extract power lines in aerial images because of the weak targets and the complex background. In this paper, a novel power line extraction method is proposed. First of all, we create a line segment candidate pool which contains power line segments and large amount of other line segments. Secondly, we construct the irregular graph model with these line segments as nodes. Then a novel object-based Markov random field with anisotropic weighted penalty (OMRF-AWP) method is proposed. It defines a new neighborhood system based on the irregular graph model and builds a new potential function by considering the region angle information. With the OMRF-AWP method, we can distinguish between the power line segments and other line segments. Finally, an envelope-based piecewise fitting (EPF) method is proposed to fit the power lines. Experimental results show that the proposed method has good performance in multiple scenes with complex background.

A robust lane detection method based on hyperbolic model

Abstract: Lane detection is an essential part of safety assurance in intelligent vehicle and advanced driver assistance systems. Despite many methods having been proposed, there still remain challenges such as complex road surface and large curvature. In this paper, we present a robust lane detection method under structured roads to solve these issues. The method contains two parts: straight line detection in near field and curve matching in far field. Instead of generating top-view image by inverse perspective mapping (IPM), we propose a new form of IPM application to reduce noise that we only take advantage of sub-pixel-level spatial relations and project line segments obtained by line segments detector to top-view image. Then, we apply density-based spatial clustering of applications with noise to clustering segments and design a fusion method to extract the optimal lines combination from clusters. Finally, a weighted hyperbolic model is proposed to finish curve fitting. The results of experiment indicate that the method has robust performance in complex environment.

Feasible Self-Calibration of Larger Field-of-View (FOV) Camera Sensors for the Advanced Driver-Assistance System (ADAS)

Abstract: This paper proposes a self-calibration method that can be applied for multiple larger field-of-view (FOV) camera models on an advanced driver-assistance system (ADAS). Firstly, the proposed method performs a series of pre-processing steps such as edge detection, length thresholding, and edge grouping for the segregation of robust line candidates from the pool of initial distortion line segments. A novel straightness cost constraint with a cross-entropy loss was imposed on the selected line candidates, thereby exploiting that novel loss to optimize the lens-distortion parameters using the Levenberg-Marquardt (LM) optimization approach. The best-fit distortion parameters are used for the undistortion of an image frame, thereby employing various high-end vision-based tasks on the distortion-rectified frame. In this study, an investigation was carried out on experimental approaches such as parameter sharing between multiple camera systems and model-specific empirical γ -residual rectification factor. The quantitative comparisons were carried out between the proposed method and traditional OpenCV method as well as contemporary state-of-the-art self-calibration techniques on KITTI dataset with synthetically generated distortion ranges. Famous image consistency metrics such as peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), and position error in salient points estimation were employed for the performance evaluations. Finally, for a better performance validation of the proposed system on a real-time ADAS platform, a pragmatic approach of qualitative analysis has been conducted through streamlining high-end vision-based tasks such as object detection, localization, and mapping, and auto-parking on undistorted frames.

Packing circles into perimeter-minimizing convex hulls

Abstract: We present and solve a new computational geometry optimization problem in which a set of circles with given radii is to be arranged in unspecified area such that the length of the boundary, i.e., the perimeter, of the convex hull enclosing the non-overlapping circles is minimized. The convex hull boundary is established by line segments and circular arcs. To tackle the problem, we derive a non-convex mixed-integer non-linear programming formulation for this circle arrangement or packing problem. Moreover, we present some theoretical insights presenting a relaxed objective function for circles with equal radius leading to the same circle arrangement as for the original objective function. If we minimize only the sum of lengths of the line segments, for selected cases of up to 10 circles we obtain gaps smaller than $$10^{-4}$$ using BARON or LINDO embedded in GAMS, while for up to 75 circles we are able to approximate the optimal solution with a gap of at most $$14\%$$ .

Scale invariant line-based co-registration of multimodal aerial data using L1 minimization of spatial and angular deviations

Abstract: In this work, we investigate the coregistration of multimodal data, such as photogrammetric/LiDAR point clouds, digital surface models, orthoimages, or 3D CAD city models, using corresponding line segments. The lines are analytically derived as intersections of adjacent planar surfaces, which can be determined more robustly and are deemed more accurate compared to single point based features. We propose a two-stage approach, which first focuses on finding optimal line correspondences between the datasets using a scale-invariant graph matching method, and then utilizes the found matching as a basis for calculating the optimal coregistration transform. By decoupling the correspondence search from the transform calculation, our approach can use more line pairs for determining the optimal transform than would be practicable with a combined, sampling-style approach. As opposed to competing methods, our transform computation is based on explicitly minimizing the average L1 distance on the matched line set. The assumed model accounts for an isotropic scaling factor, three translations and three rotation angles. We conducted experiments on two publicly available ISPRS datasets: Vaihingen and Dortmund, and compared the performance of several variations of our approach with three competing methods. The results indicate that the L1 methods decreased the median matched line distance by up to one third in case of pre-aligned Z axes. Moreover, when coregistering two photogrammetric datasets acquired from distinct viewing perspectives, our method was able to triple the number of matched lines (under a strict proximity-based criterion) compared to its competitor. Our results show that it is worthwhile to base the transform calculation on significantly more line pairs than is customary for sample consensus-based approaches. Our established validation dataset for line-based coregistration has been published and made available online ( https://doi.org/10.17632/dmp7tkn8kc.2 ).

No-Bend Orthogonal Drawings and No-Bend Orthogonally Convex Drawings of Planar Graphs (Extended Abstract)

Abstract: A plane graph is a planar graph with a fixed planar embedding in the plane. In an orthogonal drawing of a plane graph each vertex is drawn as a point and each edge is drawn as a sequence of vertical and horizontal line segments. A bend is a point at which the drawing of an edge changes its direction. A necessary and sufficient condition for a plane graph of maximum degree 3 to have a no-bend orthogonal drawing is known which leads to a linear-time algorithm to find such a drawing of a plane graph, if it exists. A planar graph G has a no-bend orthogonal drawing if any of the plane embeddings of G has a no-bend orthogonal drawing. Since a planar graph G of maximum degree 3 may have an exponential number of planar embeddings, determining whether G has a no-bend orthogonal drawing or not using the known algorithm for plane graphs takes exponential time. The best known algorithm takes \(O(n^{2})\) time for finding a no-bend orthogonal drawing of a biconnected planar graph of maximum degree 3. In this paper we give a linear-time algorithm to determine whether a biconnected planar graph G of maximum degree 3 has a no-bend orthogonal drawing or not and to find such a drawing of G, if it exists. We also give a necessary and sufficient condition for a biconnected planar graph G of maximum degree 3 to have a no-bend “orthogonally convex” drawing D; where any horizontal and vertical line segment connecting two points in a facial polygon P in D lies totally within P. Our condition leads to a linear-time algorithm for finding such a drawing, if it exists.

The Value Function Polytope in Reinforcement Learning

Abstract: We establish geometric and topological properties of the space of value functions in finite state-action Markov decision processes. Our main contribution is the characterization of the nature of its shape: a general polytope (Aigner et al., 2010). To demonstrate this result, we exhibit several properties of the structural relationship between policies and value functions including the line theorem, which shows that the value functions of policies constrained on all but one state describe a line segment. Finally, we use this novel perspective to introduce visualizations to enhance the understanding of the dynamics of reinforcement learning algorithms.

Computing Linear Restrictions of Neural Networks

Abstract: A linear restriction of a function is the same function with its domain restricted to points on a given line. This paper addresses the problem of computing a succinct representation for a linear restriction of a piecewise-linear neural network. This primitive, which we call ExactLine, allows us to exactly characterize the result of applying the network to all of the infinitely many points on a line. In particular, ExactLine computes a partitioning of the given input line segment such that the network is affine on each partition. We present an efficient algorithm for computing ExactLine for networks that use ReLU, MaxPool, batch normalization, fully-connected, convolutional, and other layers, along with several applications. First, we show how to exactly determine decision boundaries of an ACAS Xu neural network, providing significantly improved confidence in the results compared to prior work that sampled finitely many points in the input space. Next, we demonstrate how to exactly compute integrated gradients, which are commonly used for neural network attributions, allowing us to show that the prior heuristic-based methods had relative errors of 25-45% and show that a better sampling method can achieve higher accuracy with less computation. Finally, we use ExactLine to empirically falsify the core assumption behind a well-known hypothesis about adversarial examples, and in the process identify interesting properties of adversarially-trained networks.

Feature Tracking Based on Line Segments With the Dynamic and Active-Pixel Vision Sensor (DAVIS)

Abstract: As a novel vision sensor, the dynamic and active-pixel vision sensor (DAVIS) combines a standard camera and an asynchronous event-based sensor in the same pixel array. In this paper, we propose a novel asynchronous feature tracking method based on line segments with the DAVIS. The proposed method takes asynchronous events, synchronous image frames, and IMU data as the input. We first use the Harris detector to extract feature points and the Canny detector to extract line segment templates from image frames. Then we select spatio-temporal windows from asynchronous events and perform registration to estimate the optical flow. The registration is achieved by associating the extracted line segments with the events inside the window. Expectation maximization-iterative closest point (EM-ICP) is adopted for the registration. Afterward, we use the estimated optical flow and the IMU data to update the position of line segments, and take them as the new templates. We evaluate our method on the public event camera datasets. The results show that our method can achieve comparable performance to other methods in terms of accuracy and tracking time.

Line-Based Stereo SLAM by Junction Matching and Vanishing Point Alignment

Abstract: In this paper, we propose a stereo simultaneous localization and mapping (SLAM) method based on line segments. For the front-end module of SLAM, we designed a novel method based on the coplanar junction detection, description, and matching. Then the junctions along with their multi-scale rotated BRIEF descriptors are used in other SLAM modules, including line tracking, mapping, and loop closure. The line extraction and matching thread runs at 20 ~ 40Hz for stereo image sequences on a laptop, making it a practical front-end for line-based SLAM system. For the back-end module, a cost function is designed to minimize both of the reprojection error of line segments and alignment error of the vanishing points. The experimental results demonstrate that the proposed method exhibits more accurate localization and reconstruction than state-of-the-art line-based SLAM systems in line-rich environments.

A Single-Line Single-Channel Method With Closed-Form Formulas for the Characterization of Dielectric Liquids

Abstract: This paper presents a method for the measurement of liquid permittivity without using liquid reference materials or calibration standards. The method uses a single transmission line and a single microfluidic channel which intercept the line twice. As a result, two transmission line segments are formed with channel sections to measure liquid samples. In single-connection measurements, the scattering parameters of three transmission line states can be obtained. The three states are both segments filled with air, one segment filled with a liquid sample while the other with air, and both segments filled with liquid. By choosing a 2:1 ratio for the two line segment lengths, closed-form formulas are provided to calculate line propagation constants directly from measured S-parameters. Then, sample permittivity values are obtained. A coplanar waveguide is built and tested with deionized water, methanol, ethanol, and 2-propanol from 0.1 to 9 GHz. The obtained line performance agrees with the simulation results. The obtained sample permittivity values agree with commonly accepted values.

CAE-RLSM: Consistent and Efficient Redundant Line Segment Merging for Online Feature Map Building

Abstract: In order to obtain a compact line segment-based map representation for localization and planning of mobile robots, it is necessary to merge redundant line segments which physically represent the same part of the environment in different scans. In this paper, a consistent and efficient redundant line segment merging approach (CAE-RLSM) is proposed for online feature map building. The proposed CAE-RLSM is composed of two newly proposed modules: one-to-many incremental line segment merging (OTM-ILSM) and multi-processing global map adjustment (MP-GMA). Different from state-of-the-art offline merging approaches, the proposed CAE-RLSM can achieve real-time mapping performance, which not only reduces the redundancy of incremental merging with high efficiency, but also solves the problem of global map adjustment after loop closing to guarantee global consistency. Furthermore, a new correlation-based evaluation metric is proposed for the quality evaluation of line segment maps. This evaluation metric does not require manual measurement of the environmental metric information, instead it makes full use of globally consistent laser scans obtained by simultaneous localization and mapping (SLAM) systems to compare the performance of different line segment-based mapping approaches in an objective and fair manner. Comparative experimental results with respect to a mean shift-based offline redundant line segment merging approach (MS-RLSM) and an offline version of one-to-one incremental line segment merging approach (O$^2$TO-ILSM) on both public data sets and self-recorded data set are presented to show the superior performance of CAE-RLSM in terms of efficiency and map quality in different scenarios.

On Bi-objective Convex-Quadratic Problems

Abstract: In this paper, we analyze theoretical properties of bi-objective convex-quadratic problems. We give a complete description of their Pareto set and prove the convexity of their Pareto front. We show that the Pareto set is a line segment when both Hessian matrices are proportional. We then propose a novel set of convex-quadratic test problems, describe their theoretical properties and the algorithm abilities required by those test problems. This includes in particular testing the sensitivity with respect to separability, ill-conditioned problems, rotational invariance, and whether the Pareto set is aligned with the coordinate axis.