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Robust Data Association for Multi-Object Detection in Maritime Environments Using Camera and Radar Measurements
Abstract
This letter presents a robust data association method for fusing camera and marine radar measurements in order to automatically detect surface ships and determine their locations with respect to the observing ship. In the preprocessing step for sensor fusion, convolutional neural networks are used to perform feature extraction from camera images and semantic segmentation from radar images. The correspondences between the camera and radar image features are optimized using a pair of geometric parameters, and the positions of all the matched object features are determined. The proposed method enables robust data association even without accurate calibration and alignment between the camera and radar measurements. The feasibility and performance of the proposed method are demonstrated using an experimental dataset obtained in a real coastal environment.
... Unmanned surface vehicles (USVs) play a significant role in water handling, such as water rescue [2], water quality monitoring [3], garbage collection [4] and geological exploration [5]. Based on extensive surveys, firstly, we find most models of water-surface perception can only execute a single task [6][7] [8], which is far to help USVs complete autonomous navigation. In addition, some researches show that multitasks can improve each other [9] [10]. ...
Current perception models for different tasks usually exist in modular forms on Unmanned Surface Vehicles (USVs), which infer extremely slowly in parallel on edge devices, causing the asynchrony between perception results and USV position, and leading to error decisions of autonomous navigation. Compared with Unmanned Ground Vehicles (UGVs), the robust perception of USVs develops relatively slowly. Moreover, most current multi-task perception models are huge in parameters, slow in inference and not scalable. Oriented on this, we propose Achelous, a low-cost and fast unified panoptic perception framework for water-surface perception based on the fusion of a monocular camera and 4D mmWave radar. Achelous can simultaneously perform five tasks, detection and segmentation of visual targets, drivable-area segmentation, waterline segmentation and radar point cloud segmentation. Besides, models in Achelous family, with less than around 5 million parameters, achieve about 18 FPS on an NVIDIA Jetson AGX Xavier, 11 FPS faster than HybridNets, and exceed YOLOX-Tiny and Segformer-B0 on our collected dataset about 5 mAP and 0.7 mIoU, especially under situations of adverse weather, dark environments and camera failure. To our knowledge, Achelous is the first comprehensive panoptic perception framework combining vision-level and point-cloud-level tasks for water-surface perception. To promote the development of the intelligent transportation community, we release our codes in \url{https://github.com/GuanRunwei/Achelous}.
The use of automatic safety-critical control for uncrewed surface vessel (USV) survey, inspection and intervention can provide a computationally lightweight controller which guarantees that a minimum safe standoff distance to a target of interest is always maintained. We propose a trajectory tracking safety-critical controller for the closest safe approach of an underactuated USV with nonholonomic dynamic (acceleration) motion constraints to a target. A backstepping-based control law is designed using a relaxed control barrier function and an analytical convex optimization method. The stability of the controller is proven. Simulations of a USV approaching both stationary and moving targets are used to demonstrate implementation of the method. The performance of the proposed controller is compared with that of a nonlinear model predictive control (MPC) controller in simulation. The simulation results demonstrate that, while the tracking error of the proposed controller is higher than that of an MPC controller, it requires lower computational resources, suggesting it is a good candidate for use on small USVs with low computational power.
Unmanned surface vehicles (USVs) have been widely used for a wide range of tasks in the past decades. Accurate perception of the surrounding environment on the water surface under complex conditions is crucial for USVs to conduct effective operations. This paper proposes a radar-vision fusion framework for USVs to accurately detect typical targets on the water surface. The modality difference between images and radar measurements, along with their perpendicular coordinates presents challenges in the fusion process. The swaying of USVs on water and the extensive areas of perception enhance the difficulties of multi-sensor data association. To address these problems, we propose two modules to enhance multi-sensor fusion performance: a movement-compensated projection module and a distance-aware probabilistic data association module. The former effectively reduces projection bias during the alignment process of radar and camera signals by compensating for sensor movement using measured roll and pitch angles from the inertial measurement unit (IMU). The latter module models target regions guided by each radar measurement as a bivariate Gaussian distribution, with its covariance matrix adaptively derived based on the distance between targets and the camera. Consequently, the association of radar points and images is robust to projection errors and works well for multi-scale objects. Features of radar points and images are subsequently extracted with two parallel backbones and fused at different levels to provide sufficient semantic information for robust object detection. The proposed framework achieves an AP of 0.501 on the challenging real-world dataset established by us, outperforming state-of-the-art vision-only and radar-vision fusion methods.
Far-range perception is essential for intelligent transportation systems. The main challenge of far-range perception is due to the difficulty of performing accurate object detection and tracking under far distances (e.g.,
) at low cost. To cope with such challenges, deploying millimeter wave Radars and high-definition (HD) cameras, and fusing their data for joint perception has become a common practice. The key to this solution is the precise association between two types of data captured from different perspectives. Towards this goal, the first question is which plane to conduct the association, i.e., the 2D image plane or the BEV plane. We argue that the former is more suitable because the location errors of the perspective projection points are smaller at far distances and can lead to more accurate associations. Thus, we project Radar-based target locations from the BEV to the 2D plane and then associate them with camera-based object locations. Subsequently, we map the camera-based object locations to the BEV plane through inverse projection mapping (IPM) with corresponding depth information from Radar data. Finally, we engage a BEV tracking module to generate target trajectories for traffic monitoring. We devise a transformation parameters refining approach based on the depth scaling technique. We have deployed an actual testbed on an urban expressway and conducted extensive experiments for evaluation. The results show that our system can improve
by 32%, and reduce the location error by
. Our system is capable of achieving an average location accuracy of
within the
range.
Autonomous ships are expected to extensively rely on perception sensors for situation awareness and safety during challenging operations, such as reactive collision avoidance. However, sensor noise is inevitable and its impact on end-to-end decision-making has not been addressed yet. This study aims to develop a methodology to enhance the robustness of decision-making for the reactive collision avoidance of autonomous ships against various perception sensor noise levels. A Gaussian-based noisy perception sensor is employed, where its noisy measurements and noise variance are incorporated into the decision-making as observations. A deep reinforcement learning agent is employed, which is trained in different noise variances. Robustness metrics that quantify the robustness of the agent’s decision-making are defined. A case study of a container ship using a LIDAR in a single static obstacle environment is investigated. Simulation results indicate sophisticated decision-making of the trained agent prioritising safety over efficiency when the noise variance is higher by conducting larger evasive manoeuvres. Sensitivity analysis indicates the criticality of the noise variance observation on the agent’s decision-making. Robustness is verified against noise variance up to 132% from its maximum trained value. Robustness is verified only up to 76% when the agent is trained without the noise variance observation with lack of its prior sophisticated decision-making. This study contributes towards the development of autonomous systems that can make safe and robust decisions under uncertainty.
This paper presents a multimodal maritime dataset and the data collection procedure used to gather it, which aims to facilitate autonomous navigation in restricted water environments. The dataset comprises measurements obtained using various perception and navigation sensors, including a stereo camera, an infrared camera, an omnidirectional camera, three LiDARs, a marine radar, a global positioning system, and an attitude heading reference system. The data were collected along a 7.5-km-long route that includes a narrow canal, inner and outer ports, and near-coastal areas in Pohang, South Korea. The collection was conducted under diverse weather and visual conditions. The dataset and its detailed description are available for free download at https://sites.google.com/view/pohang-canal-dataset .
In this paper, we present the development of autonomous navigation capabilities for small cruise boats, and their verification by field experiments in a canal and its surrounding waters. A cruise boat was converted to an autonomous surface vehicle (ASV) by installing various sensors and actuators to enable autonomous navigation. Navigation and perception sensors, such as global positioning system, attitude and heading reference system, radar, light detection and ranging (LiDAR), and cameras, were mounted on the ASV to estimate its motion and perceive the surrounding environment. Motors and potentiometers were installed for active control of the ASV. Software system components including navigation filters, object‐detection, path‐planning, and control algorithms were designed and implemented. In the narrow canal region, LiDARs were used to detect the side walls and boundaries of the canal. In open areas outside the canal, obstacles and object features were detected using various combinations of onboard sensors. A model‐based path‐planning algorithm was designed to avoid the detected obstacles, and the line‐of‐sight guidance was employed to control the vehicle. The performance of the developed system was verified through a field experiment in a real‐world maritime environment.
Determining the dynamics of surface vehicles and marine robots is important for developing marine autopilot and autonomous navigation systems. However, this often requires extensive experimental data and intense effort because they are highly nonlinear and involve various uncertainties in real operating conditions. Herein, we propose an efficient data-driven approach for analyzing and predicting the motion of a surface vehicle in a real environment based on deep learning techniques. The proposed multistep model is robust to measurement uncertainty and overcomes compounding errors by eliminating the correlation between the prediction results. Additionally, latent state representation and mixup augmentation are introduced to make the model more consistent and accurate. The performance analysis reveals that the proposed method outperforms conventional methods and is robust against environmental disturbances.
This paper proposes an approach to estimate ship dynamic features (i.e., speed and heading) using radar sequential images, which can be used to assess the ship-ship collision risk in vessel traffic service (VTS) center. The proposed approach estimates the ship speed and heading in three steps, including ship detection, tracking, and data fusion. First, the ship target, together with its center coordinate, is detected by introducing the background subtraction detector in each radar image. Second, the ship position in one frame is mapped to the consecutive frame, and ship trajectory distribution is established using the Kalman filter and Hungarian algorithms. Third, fuzzy logic is introduced to fuse data from radar and AIS. Specifically, input variables include two parameters, which are the ship speed/heading difference in consecutive frames using radar and AIS, respectively. The variance of ship speed distribution is 1.30 at 10s sampling frequency, and the variance of ship heading distribution is 26.53, the experimental results indicate that the proposed approach can accurately estimate ship speed and heading in the radar sequential images of the Yangtze River. The strength of the proposed approach is its capability to estimate the ship speed and heading quantitatively from radar sequential images, which is interactive and interpretable for VTS operators.
This paper proposes a radar image segmentation and tracking method by learning radar images for autonomous surface vehicles. To identify marine objects from radar images, we propose a deep neural network named the dual path squeeze and excitation network (DPSE-Net). By learning the radar images, the proposed DPSE-Net is designed to segment every pixel of the radar images into four classes: marine objects, land, noise, and background. The proposed DPSE-Net shows the best performance in radar image segmentation while operating in real-time, compared to state-of-the-art real-time image segmentation network models. In addition, we design a real-time moving object tracking algorithm for estimating the position and velocity of marine objects based on deep simple online real-time tracking with a deep association metric (DeepSORT), a widely used tracking algorithm. The existing DeepSORT algorithm uses the intersection over union (IoU) metric and a deep appearance descriptor for data association, but since they are not suitable for radar images, successive tracking is difficult. To solve this problem, a new data association metric suitable for radar images is proposed. The field tests in ocean environments confirm that the proposed method performs better in marine object segmentation and tracking.
Ship detection plays a vital role in monitoring and managing maritime safety. Most recently proposed learning-based object detection methods have achieved marked progress in detection accuracy, but the size of these models is too large to be applied to mobile devices with limited resources. Although some compact models have been presented in the previous study, they achieve unsatisfactory results in ship detection, especially under extreme weather conditions. To address these challenges, this article presents a lightweight convolutional neural network (CNN) called Light-SDNet to perform an end-to-end ship detection under different weather conditions. In the proposed model, we introduce the improved CA-Ghost, C3Ghost, and DepthWise Convolution (DWConv) into the You Only Look Once version 5 (YOLOv5) to reduce the number of model parameters, while remaining its powerful feature expression ability. We use parallel attention to highlight the features that contribute to the ship detection in the marine surveillance. To enhance the adaptability of the proposed model, a hybrid training strategy with generating synthetically-degraded images is proposed to augment the volume and diversity of the original datasets. The proposed strategy enables Light-SDNet to improve the ship detection results under severe weather conditions such as haze, rain, and low illumination. We compare Light-SDNet with other competitive approaches on a large-scaled ship dataset called SeaShips. We show that Light-SDNet achieves a better balance between the detection accuracy and the model complexity. The ship detection results on degraded marine images have proven the superior performance of the proposed model in terms of detection accuracy, robustness and efficiency.
Maritime search and rescue (SAR) plays a very important role in emergency waterway traffic situations, which is supposed to trigger severe personal casualties and property loss in maritime traffic accidents. The study aims to exploit an optimal allocation strategy with limited SAR resources deployed at navigation-constrained coastal islands. The study formulates the problem of SAR resource allocation in coastal areas into a non-linear optimization model. We explore the optimal solution for the SAR resource allocation problem under constraints of different ship and aircraft base station settings with the help of an enhanced particle swarm optimization (EPSO) model. Experimental results suggest that the proposed EPSO model can reasonably allocate the maritime rescue resources with a large coverage area and low time cost. The particle swarm optimization and genetic algorithm are further implemented for the purpose of model performance comparison. The research findings can help maritime traffic regulation departments to make more reasonable decisions for establishing SAR base stations.
This study addresses the development of algorithms for multiple target detection and tracking in the framework of sensor fusion and its application to autonomous navigation and collision avoidance systems for the unmanned surface vehicle (USV) Aragon. To provide autonomous navigation capabilities, various perception sensors such as radar, lidar, and cameras have been mounted on the USV platform and automatic ship detection algorithms are applied to the sensor measurements. The relative position information between the USV and nearby objects is obtained to estimate the motion of the target objects in a sensor‐level tracking filter. The estimated motion information from the individual tracking filters is then combined in a central‐level fusion tracker to achieve persistent and reliable target tracking performance. For automatic ship collision avoidance, the combined track data are used as obstacle information, and appropriate collision avoidance maneuvers are designed and executed in accordance with the international regulations for preventing collisions at sea (COLREGs). In this paper, the development processes of the vehicle platform and the autonomous navigation algorithms are described, and the results of field experiments are presented and discussed.
With the development of robot technology, the expectation of autonomous mission operations has increased, and the research on robot control architectures and mission planners has continued. A scalable and robust control architecture is required for unmanned surface vehicles (USVs) to perform a variety of tasks, such as surveillance, reconnaissance, and search and rescue operations, in unstructured and time-varying maritime environments. In this paper, we propose a robot control architecture along with a new utility function that can be extended to various applications for USVs. Also, an additional structure is proposed to reflect the operator's command and improve the performance of the autonomous mission. The proposed architecture was developed using a robot operating system (ROS), and the performance and feasibility of the architecture were verified through simulations.
In the scenario where an underwater vehicle tracks an underwater target, reliable estimation of the target position is required. While USBL measurements provide target position measurements at low but regular update rate, multibeam sonar imagery gives high precision measurements but in a limited field of view. This paper describes the development of the tracking filter that fuses USBL and processed sonar image measurements for tracking underwater targets for the purpose of obtaining reliable tracking estimates at steady rate, even in cases when either sonar or USBL measurements are not available or are faulty. The proposed algorithms significantly increase safety in scenarios where underwater vehicle has to maneuver in close vicinity to human diver who emits air bubbles that can deteriorate tracking performance. In addition to the tracking filter development, special attention is devoted to adaptation of the region of interest within the sonar image by using tracking filter covariance transformation for the purpose of improving detection and avoiding false sonar measurements. Developed algorithms are tested on real experimental data obtained in field conditions. Statistical analysis shows superior performance of the proposed filter compared to conventional tracking using pure USBL or sonar measurements.
A new technique is proposed to carry out the nearest neighbor (NN) association in a large set of different class points in an automated, optimized, and speedy manner. The algorithm makes use of the K-dimensions tree to, mutually, organize the examined set of points and initiate the different stages of NN search algorithm. Our algorithm assumes no prior knowledge about the spatial distribution of the examined set of points, which means it has the potential to be applied to many applications in signal processing, wireless communications modeling, image processing, computer vision, biochemical, and feature extraction. It will be very useful for many applications that require real-time output. Our simulations show that we can get an optimal solution of associations while saving more than 80% of the processing time when compared with the exhaustive sorted list-based search method.
This article proposes a vision-based target motion analysis and collision avoidance method for unmanned surface vehicles. To estimate the relative position of the target, the calibrated camera model and camera height constraint are used. In addition, the optical flow equation is adopted to estimate the relative velocity of the target. The estimated relative position and velocity are then integrated using the Kalman filter in order to reduce the effect of measurement noise. Once the kinematic information of the target is determined by the vision-based target motion analysis, the collision risk is calculated by a fuzzy estimator. Based on the collision risk, vision-based collision avoidance control is performed. To validate the effectiveness of the suggested method in various encounter situations, a collision avoidance simulation is conducted.
There is large consent that successful training of deep networks requires
many thousand annotated training samples. In this paper, we present a network
and training strategy that relies on the strong use of data augmentation to use
the available annotated samples more efficiently. The architecture consists of
a contracting path to capture context and a symmetric expanding path that
enables precise localization. We show that such a network can be trained
end-to-end from very few images and outperforms the prior best method (a
sliding-window convolutional network) on the ISBI challenge for segmentation of
neuronal structures in electron microscopic stacks. Using the same network
trained on transmitted light microscopy images (phase contrast and DIC) we won
the ISBI cell tracking challenge 2015 in these categories by a large margin.
Moreover, the network is fast. Segmentation of a 512x512 image takes less than
a second on a recent GPU. The full implementation (based on Caffe) and the
trained networks are available at
http://lmb.informatik.uni-freiburg.de/people/ronneber/u-net .
Maritime surveillance systems can be employed to increase the security of ports, airports, merchant and war ships against pirates, terrorists or any hostile vessel attacks, to avoid collisions, to control maritime traffic at ports and channels and for coastal and oil platforms defense. Cameras are one of the main sensors of these systems. They are cheap and complement other types of sensors. This survey was motivated by the importance of the subject, to motivate new researches and because there are no surveys about video detection and tracking of marine vehicles or they are not widespread. It is an insignificant quantity compared with other kinds of video surveillance systems. The paper presents the state of the art algorithms.
Data association is a fundamental problem in multitarget-multisensor tracking. It entails selecting the most probable association between sensor measurements and target tracks from a very large set of possibilities. With N sensors and n targets in the detection range of each sensor, even with perfect detection there are (n!) N different configurations which renders infeasible a solution by direct computation even in modestly-sized applications. We describe an iterative method for solving the optimal data association problem in a distributed fashion; the work exploits the framework of graphical models, which are a powerful tool for encoding the statistical dependencies of a set of random variables and are widely used in many applications (e.g., computer vision, error-correcting codes). Our basic idea is to treat the measurement assignment for each sensor as a random variable, which is in turn represented as a node in an underlying graph. Neighboring nodes are coupled by the targets visible to both sensors. Thus we transform the data association problem to that of computing the maximum a posteriori (MAP) configuration in a graphical model to which efficient techniques (e.g., the max-product/min-sum algorithm) can be applied. We use a tree-reweighted version of the usual max-product algorithm that either outputs the MAP data association, or acknowledges failure. For acyclic graphs, this message-passing algorithm can solve the data association problem directly and recursively with complexity O (n!) 2 N . On graphs with cycles, the algorithm may require more iterations to converge, and need not output an unambiguous assignment. However, for the data association problems considered here, the coupling matrices involved in computations are inherently of low rank, and experiments show that the algorithm converges very fast and finds the MAP configurations in this case.
The Pascal Visual Object Classes (VOC) challenge is a benchmark in visual object category recognition and detection, providing the vision
and machine learning communities with a standard dataset of images and annotation, and standard evaluation procedures. Organised
annually from 2005 to present, the challenge and its associated dataset has become accepted as the benchmark for object detection.
This paper describes the dataset and evaluation procedure. We review the state-of-the-art in evaluated methods for both classification
and detection, analyse whether the methods are statistically different, what they are learning from the images (e.g. the object
or its context), and what the methods find easy or confuse. The paper concludes with lessons learnt in the three year history
of the challenge, and proposes directions for future improvement and extension.
The design of a robust perception method is a substantial component towards achieving underwater human–robot collaboration. However, in complex environments such as the oceans, perception is still a challenging issue. Data‐fusion of different sensing modalities can improve perception in dynamic and unstructured ocean environments. This study addresses the control of a highly maneuverable autonomous underwater vehicle for diver tracking based on visual and acoustic signals data fusion measured by low‐cost sensors. The underwater vehicle U‐CAT tracks a diver using a 3‐degree‐of‐freedom fuzzy logic Mamdani controller. The proposed tracking approach was validated through open water real‐time experiments. Combining acoustic and visual signals for underwater target tracking provides several advantages compared to previously done related research. The obtained results suggest that the proposed solution ensures effective detection and tracking in poor visibility operating conditions.
We present some updates to YOLO! We made a bunch of little design changes to make it better. We also trained this new network that's pretty swell. It's a little bigger than last time but more accurate. It's still fast though, don't worry. At 320x320 YOLOv3 runs in 22 ms at 28.2 mAP, as accurate as SSD but three times faster. When we look at the old .5 IOU mAP detection metric YOLOv3 is quite good. It achieves 57.9 mAP@50 in 51 ms on a Titan X, compared to 57.5 mAP@50 in 198 ms by RetinaNet, similar performance but 3.8x faster. As always, all the code is online at https://pjreddie.com/yolo/
We present a survey on maritime object detection and tracking approaches, which are essential for the development of a navigational system for autonomous ships. The electro-optical (EO) sensor considered here is a video camera that operates in the visible or the infrared spectra, which conventionally complement radar and sonar and have demonstrated effectiveness for situational awareness at sea has demonstrated its effectiveness over the last few years. This paper provides a comprehensive overview of various approaches of video processing for object detection and tracking in the maritime environment. We follow an approach-based taxonomy wherein the advantages and limitations of each approach are compared. The object detection system consists of the following modules: horizon detection, static background subtraction and foreground segmentation. Each of these has been studied extensively in maritime situations and has been shown to be challenging due to the presence of background motion especially due to waves and wakes. The main processes involved in object tracking include video frame registration, dynamic background subtraction, and the object tracking algorithm itself. The challenges for robust tracking arise due to camera motion, dynamic background and low contrast of tracked object, possibly due to environmental degradation. The survey also discusses multisensor approaches and commercial maritime systems that use EO sensors. The survey also highlights methods from computer vision research which hold promise to perform well in maritime EO data processing. Performance of several maritime and computer vision techniques is evaluated on newly proposed Singapore Maritime Dataset.
There is large consent that successful training of deep networks requires many thousand annotated training samples. In this paper, we present a network and training strategy that relies on the strong use of data augmentation to use the available annotated samples more efficiently. The architecture consists of a contracting path to capture context and a symmetric expanding path that enables precise localization. We show that such a network can be trained end-to-end from very few images and outperforms the prior best method (a sliding-window convolutional network) on the ISBI challenge for segmentation of neuronal structures in electron microscopic stacks. Using the same network trained on transmitted light microscopy images (phase contrast and DIC) we won the ISBI cell tracking challenge 2015 in these categories by a large margin. Moreover, the network is fast. Segmentation of a 512x512 image takes less than a second on a recent GPU. The full implementation (based on Caffe) and the trained networks are available at
http://lmb.informatik.uni-freiburg.de/people/ronneber/u-net
.
Enhancing the performance of passive target tracking and trajectory estimation of marine traffic ships is focused using a monocular camera mounted on an unmanned surface vessel. To accurately estimate the trajectory of a target traffic ship, the relative bearing and range information between the observing ship and the target ship is required. Monocular vision provides bearing information with reasonable accuracy but with no explicit range information. The relative range information can be extracted from the bearing changes induced by the relative ship motion in the framework of bearings-only tracking (BOT). BOT can be effective in crossing situations with large bearing angle changes. However, it often fails in head-on or overtaking situations due to small bearing angle changes and the resulting low observability of the tracking filter. To deal with the lack of observability, the vertical pixel distance between the horizon and the target ship in the image is used, which improves the overall target tracking performance. The feasibility and performance of the proposed tracking approach were validated through field experiments at sea.
Visual surveillance in the maritime domain has been explored for more than a decade. Although it has produced a number of working systems and resulted in a mature technology, surveillance has been restricted to the port facilities or areas close to the coast line assuming a fixed-camera scenario. This paper presents a novel algorithm for open-sea visual maritime surveillance. We explore a challenging situation with a forward-looking camera mounted on a buoy or other floating platform. The proposed algorithm detects, localizes, and tracks ships in the field of view of the camera. Specifically, developed algorithm is uniquely designed to handle rapidly moving camera. Its performance is robust in the presence of a random relatively-large camera motion. In the context of ship detection we developed a new horizon detection scheme for a complex maritime domain. The performance of our algorithm and its comprising elements is evaluated. Ship detection precision of 88% is achieved on a large dataset collected from a prototype system.
We consider a generalization of the assignment problem in which an integer k is given and one wants to assign k rows to k columns so that the sum of the corresponding costs is a minimum. The problem can be seen as a 2-matroid intersection, hence is solvable in polynomial time; immediate algorithms for it can be obtained from transformation to min-cost flow or from classical shortest augmenting path techniques. We introduce original preprocessing techniques for finding optimal solutions in which g⩽k rows are assigned, for determining rows and columns which must be assigned in an optimal solution and for reducing the cost matrix. A specialized primal algorithm is finally presented. The average computational efficiency of the different approaches is evaluated through computational experiments.