Row Direction Research Articles

Two Years of Cotton (Gossypium hirsutum L.) Data from the Georgia Coastal Plain, USA

The sustainable management of Earth’s complex ecosystems requires an abundance of field data to support long term stewardship. Remotely sensed satellite data provide crucial supplements to field measurements and are essential for deriving key operational products for monitoring Earth systems. However, to accurately calibrate and validate the models used to develop monitoring datasets, coincident field measurements are required. In 2018 and 2019, data related to cotton (Gossypium hirsutum L.) crops were collected from five fields in two farms located in Georgia, USA. Collections were timed to coincide with satellite overpasses to support the development of remote sensing-based crop and soil data products. Data collected include soil moisture, plant water content, above ground biomass, crop height, plant phenology, and field management practices (row direction, row spacing, and plant density). The datasets include 512 records collected in 2018 and 303 records collected in 2019. The data are archived in the National Agricultural Library Ag Data Commons repository and are available for use by researchers seeking crop and soil validation data.

Effect of Row Orientation on Productivity Of Sugarcane With Potato as Intercrop

An experiment was carried out at the Regional Sugarcrop Research Station (RSRS) farm, Thakurgaon under the agro-ecological zone of Old Himalayan Piedmont Plain (AEZ-1) of Bangladesh, during 2016-2017 to 2017-2018 cropping seasons to evaluate the impact of row orientation on the productivity of sugarcane with potato as intercrop. Under the study eight treatments viz T1: Single row cane (SRC) only in north-south direction, T2: SRC + potato in North-South direction, T3: SRC only in east-west direction, T4: SRC + potato in east-west direction,T5: Paired row cane (PRC) only in North-South direction, T6: PRC + potato in north-south direction, T7: PRC only in east-west direction and T8: PRC + potato in east-west direction were tested following Randomized Complete Block Design with three replications Significant differences were found among the observed parameters except germination (%) of setts, number of internodes per stalk and unit stalk weight of sugarcane. The highest number of tiller and millable cane population, cane yield and brix reading were achieved from the treatment SRC + potato in the North-South direction. The highest intercrop yield, equivalent cane yield of intercrop, total adjusted yield and gross return were obtained from the treatment PRC + potato in north-south direction, but the highest net return and benefit cost ratio were obtained from the treatment SRC + potato in the North-South direction. Having the highest cane yield, net return and benefit cost ratio with less variable cost, sugarcane planting in a single row system with potato as intercrop in a North-South row direction could be considered the best package practice for successful sugarcane farming. Bangladesh Agron. J. 2024, 26(2): 7-15

A dual-domain cone beam computed tomography reconstruction framework with improved differentiable domain transform for cone-angle artifact correction

We propose a dual-domain cone beam computed tomography (CBCT) reconstruction framework DualCBR-Net based on improved differentiable domain transform for cone-angle artifact correction. The proposed CBCT dual-domain reconstruction framework DualCBR-Net consists of 3 individual modules: projection preprocessing, differentiable domain transform, and image post-processing. The projection preprocessing module first extends the original projection data in the row direction to ensure full coverage of the scanned object by X-ray. The differentiable domain transform introduces the FDK reconstruction and forward projection operators to complete the forward and gradient backpropagation processes, where the geometric parameters correspond to the extended data dimension to provide crucial prior information in the forward pass of the network and ensure the accuracy in the gradient backpropagation, thus enabling precise learning of cone-beam region data. The image post-processing module further fine-tunes the domain-transformed image to remove residual artifacts and noises. The results of validation experiments conducted on Mayo's public chest dataset showed that the proposed DualCBR-Net framework was superior to other comparison methods in terms of artifact removal and structural detail preservation. Compared with the latest methods, the DualCBR-Net framework improved the PSNR and SSIM by 0.6479 and 0.0074, respectively. The proposed DualCBR-Net framework for cone-angle artifact correction allows effective joint training of the CBCT dual-domain network and is especially effective for large cone-angle region.

Programmable thick-panel Miura-ori for ultra-large planar antennas with one-degree-of-freedom

Planar antennas are large spacecraft primarily used in various fields such as communication, Earth observation, astronomical observation, and scientific experiments. They require high surface accuracy, with the array needing to have uniform thickness and flat surfaces. An ultra-large planar antenna means higher receiving sensitivity and greater coverage area, which is the trend for the future. During launch, the planar antenna needs to be folded to fit into the payload capsule. In this study, a programmable thick-panel Miura-ori unit that has uniform thickness and satisfies the Bennett linkage condition is developed. After special design, this unit can be infinitely expanded in the row and column directions, allowing for the folding of planar antennas of any size (rows × columns). For each vertex inside the planar antenna, there are four panels around it, and their connection relationships all satisfy the Bennett linkage condition, resulting in the entire planar antenna having only one degree of freedom. The sector angle of each parallelogram can approach 90°, compacting the folding form of the planar antenna. Furthermore, a layout scheme for an ultra-large planar antenna is provided, the deployment ratio, envelope diameter and envelope height are derived, and the complete unfolding process of the ultra-large planar antenna is demonstrated. Finally, the effectiveness of the proposed method for satellite planar antennas is verified through an experiment.

47‐5: Analyzing and Enhancing Display Quality in FRC Algorithm

The FRC algorithm is a commonly employed technique in LCD displays, aimed at enhancing the data bit depth through pixel dithering. However, it can also give rise to certain display issues such as noise, scrolling stripes, and flickering [1]. To address these problems comprehensively, this paper conducts an analysis from both temporal and spatial dimensions. It reveals that screen noise is closely associated with the uniformity of pixel data distribution throughout the entire FRC cycle time, while rolling stripes are linked to the spatial distribution of pixel data in both row and column directions. Furthermore, flickering arises due to variations in the number of FRC pattern pixels actually displayed per frame. The validity of these theories has been verified on an FPGA platform.

An efficient full-size convolutional computing method based on memristor crossbar

Modern artificial intelligence systems based on neural networks need to perform a large number of repeated parallel operations quickly. Without hardware acceleration, they cannot achieve effectiveness and availability. Memristor-based neuromorphic computing systems are one of the promising hardware acceleration strategies. In this paper, we propose a full-size convolution algorithm (FSCA) for the memristor crossbar, which can store both the input matrix and the convolution kernel and map the convolution kernel to the entire input matrix in a full parallel method during the computation. This method dramatically increases the convolutional kernel computations in a single operation, and the number of operations no longer increases with the input matrix size. Then a bidirectional pulse control switch integrated with two extra memristors into CMOS devices is designed to effectively suppress the leakage current problem in the row and column directions of the existing memristor crossbar. The spice circuit simulation system is built to verify that the design convolutional computation algorithm can extract the feature map of the entire input matrix after only a few operations in the memristor crossbar-based computational circuit. System-level simulations based on the MNIST classification task verify that the designed algorithm and circuit can effectively implement Gabor filtering, allowing the multilayer neural network to improve the classification task recognition accuracy to 98.25% with a 26.2% reduction in network parameters. In comparison, the network can even effectively immunize various non-idealities of the memristive synaptic within 30%.

A texture feature extraction method considering spatial continuity and gray diversity

Texture features play an important role in the field of remote sensing classification. However, most existing methods lack a comprehensive consideration of spatial continuity, which makes them either destroy the spatial integrity of regular ground objects or fail to quantify the fragmentation degrees of irregular ground objects. These problems weak the ability of existing methods to distinguish ground objects with different fragmentation degrees. Therefore, this study proposed a new texture feature extraction method considering spatial continuity and gray diversity (SCGD). SCGD first connected all pixels in a neighborhood in series from end to end according to the row and column directions, and the diversities of the spatial continuity encoding in different directions were calculated by the Shannon index. Then, the Shannon index was used to calculate the gray diversity. Finally, SCGD calculated the weighted average of spatial continuity diversity and gray diversity to obtain the final texture feature values. Validation results indicated that SCGD can effectively distinguish ground objects with different fragmentation degrees, and its performance is better than that of traditional methods. As the spatial resolution decreases, its performance advantage becomes more obvious. Moreover, SCGD has great application potential in the field of ground object classification, and combining it with deep learning models will contribute to achieving the fine recognition of ground objects.

Robust two-dimensional PCANet with F-norm distance metric

The principal component analysis network (PCANet) is a simplified convolutional neural network (CNN) implemented using a deep subspace learning model. Since its typical PCA filters utilize the squared Frobenius-norm (F-norm) as the distance metric, it exhibits extreme sensitivity to outliers hidden within the data. Furthermore, PCANet faces challenges in extracting structural information in the row and column directions of images as the filters need to be pre-vectorized. To address these issues, we propose a robust bi-directional two-dimensional PCANet (RBDPCANet), wherein the filters utilize the F-norm instead of the squared F-norm to mitigate the impact of outliers. Additionally, the relationship between reconstruction errors and projection distances is simultaneously considered in the objective function. After a concise analytical description of the PCANet concept and related work, we present the methodology for generating three convolutional kernels in the form of RBDPCANet-1, RBDPCANet-2, and RBDPCANet-3 algorithms. These convolution kernels are iteratively obtained and are suitable for numerical implementation. Each of the three algorithms uses another optimization objective function. It has been shown that in image recognition tasks, the developed methods are characterized by better resistance to noise compared to the classic PCANet approach. The rotation invariance and convergence of RBDPCANet are demonstrated according to theoretical principles. The efficacy of the proposed method is validated using images from publicly available datasets. Extensive experimental results on these datasets demonstrate that the proposed RBDPCANet not only significantly boosts the efficiency of image classification but also has strong resistance to occlusions, changes in lighting, changes in orientation, etc. Our implementation code is available at https://github.com/Wesley-li1/RBDPCANet-.

Analysis of the Directional Characteristics of the Clumping Index (CI) Based on RAMI-V Canopy Scenes

The vegetation canopy clumping index (CI) is an important indicator for understanding radiative transport processes, radiation interception, and the photosynthesis of vegetation canopies. However, most studies consider CI only in the nadir or specific direction. In this study, we analyze the directional characteristics of the CI based on RAMI-V (radiation transfer model intercomparison) activity, which represents most typical canopies. The directional gap fraction and CI of these scenes are accurately calculated based on the LESS (large-scale remote sensing data and image simulation framework) model. According to our results, the directional characteristics of the CI are affected by many factors, such as vegetation type, season, and canopy structure. Generally, the CI of a coniferous forest varies little with zenith angle, while the CI of a broad-leaf forest demonstrates the different trend. In winter, the CI is smaller than that in summer, and the variation in the CI at the zenith angle is less. The row structure scenes exhibit different directional characteristics along and perpendicular to the row direction, and their CIs tend to increase with zenith angle. To accurately model the directional CI, we propose a modified Gompertz function model. Compared with other directional CI models, this model has the advantages of high precision and strong applicability ( R 2 = 0.975). By studying the directional characteristics of CI, we can enhance the usability of radiative transfer modeling and the accuracy of canopy biophysical parameter retrieval for vegetation with different structures.

Three-Dimensional Modeling of Maize Canopies Based on Computational Intelligence.

The 3-dimensional (3D) modeling of crop canopies is fundamental for studying functional-structural plant models. Existing studies often fail to capture the structural characteristics of crop canopies, such as organ overlapping and resource competition. To address this issue, we propose a 3D maize modeling method based on computational intelligence. An initial 3D maize canopy is created using the t-distribution method to reflect characteristics of the plant architecture. The subsequent model considers the 3D phytomers of maize as intelligent agents. The aim is to maximize the ratio of sunlit leaf area, and by iteratively modifying the azimuth angle of the 3D phytomers, a 3D maize canopy model that maximizes light resource interception can be constructed. Additionally, the method incorporates a reflective approach to optimize the canopy and utilizes a mesh deformation technique for detecting and responding to leaf collisions within the canopy. Six canopy models of 2 varieties plus 3 planting densities was constructed for validation. The average R2 of the difference in azimuth angle between adjacent leaves is 0.71, with a canopy coverage error range of 7% to 17%. Another 3D maize canopy model constructed using 12 distinct density gradients demonstrates the proportion of leaves perpendicular to the row direction increases along with the density. The proportion of these leaves steadily increased after 9 × 104 plants ha-1. This study presents a 3D modeling method for the maize canopy. It is a beneficial exploration of swarm intelligence on crops and generates a new way for exploring efficient resources utilization of crop canopies.

The balance between spectral and spatial information to estimate straw cereal plant density at early growth stages from optical sensors

Estimating straw cereal plant density at early stages is important for field crop management and phenotyping. Usual plant density estimation methods include manual counting and image-based counting, both of which have limited throughput, due to the need for high spatial resolution images. In this study, we explored the potential of high-throughput estimations with spectral information. A large and diverse dataset was collected on micro plot field experiments, encompassing six sites, three leaf stages, and four species of straw cereals. Canopy spectral reflectance was acquired with a spectrometer, both in 0° or 45° view zenith angle, perpendicularly to the row direction. Two reflectance-based approaches were then tested. In the direct approach, density was directly estimated from reflectance using Gaussian process regression (GPR) and spectral bands selected based on Akaike’s information criterion. In the indirect approach, the green fraction derived from high spatial resolution RGB images (GF_rgb) was first estimated from reflectance using GPR and selected bands, and then linearly related to density. These reflectance-based methods were compared to a classical image-based baseline method, which estimates density directly from GF_rgb.An ablation study firstly showed the superiority of 45° observations, and the necessity to calibrate one model for each site, growth stage, and species. The band selection process recommended using no more than four bands as inputs to the GPR models. The resulting direct and indirect estimations had an overall relative error of 30 %. The image-based baseline method had a lower error of 22 % for submillimeter spatial resolutions, but it performed worse than reflectance-based methods when degrading the spatial resolution to more than 1 to 2 mm to mimic an increase in sensor altitude. These results showed that spectral information can compensate for spatial information and that spectral methods can potentially provide high-throughput and reasonably accurate estimates of straw cereal plant density.

Navigation path extraction for inter-row robots in Panax notoginseng shade house based on Im-YOLOv5s.

The accurate extraction of navigation paths is crucial for the automated navigation of agricultural robots. Navigation line extraction in complex environments such as Panax notoginseng shade house can be challenging due to factors including similar colors between the fork rows and soil, and the shadows cast by shade nets. In this paper, we propose a new method for navigation line extraction based on deep learning and least squares (DL-LS) algorithms. We improve the YOLOv5s algorithm by introducing MobileNetv3 and ECANet. The trained model detects the seven-fork roots in the effective area between rows and uses the root point substitution method to determine the coordinates of the localization base points of the seven-fork root points. The seven-fork column lines on both sides of the plant monopoly are fitted using the least squares method. The experimental results indicate that Im-YOLOv5s achieves higher detection performance than other detection models. Through these improvements, Im-YOLOv5s achieves a mAP (mean Average Precision) of 94.9%. Compared to YOLOv5s, Im-YOLOv5s improves the average accuracy and frame rate by 1.9% and 27.7%, respectively, and the weight size is reduced by 47.9%. The results also reveal the ability of DL-LS to accurately extract seven-fork row lines, with a maximum deviation of the navigation baseline row direction of 1.64°, meeting the requirements of robot navigation line extraction. The results shows that compared to existing models, this model is more effective in detecting the seven-fork roots in images, and the computational complexity of the model is smaller. Our proposed method provides a basis for the intelligent mechanization of Panax notoginseng planting.

Correlation between meso-structures and magnetoelectric properties in 0-3 magnetoelectric composites

The quantitative characterization of macroscopic properties of 0-3 polymer-based magnetoelectric (ME) composites is the basis of their application, and the meso-structure is the source that affects their macroscopic properties. In this study, theoretical models consistent with the meso-structures of the composites are developed to address scenarios involving random, chain, and agglomeration particle distributions. The focus is on investigating the nonlinear mechanical behavior of multi-field coupling. The validity and rationality of the theoretical model are confirmed by comparing the theoretical results with existing experimental data. Based on this, the distribution characteristics of elastic field, electric field and magnetic field of 0-3 polymer-based ME composites with different particle arrangements are analyzed. Additionally, the influence of several factors on the ME properties, such as magnetic field strength and direction, particle proportion and frequency, is discussed. The results indicate that the ME performance is optimized when the direction of the densest chain row of particles aligns with the direction of the applied magnetic field. Furthermore, the phenomenon of current leakage is examined when the direction of the chain row of ferromagnetic particles coincides with the direction of polarization. To mitigate this leakage effect, a solution is proposed by adding thin layers of piezoelectric polymer to the surface of the composite, resulting in a 62.6% increase in ME performance. The research findings reveal the response mechanisms and essence of 0-3 polymer-based ME composites under applied magnetic fields, establishing a connection between meso-structures and macro properties. This provides guidance for the design and optimization of mesomorphic structures and performance, and serves as a foundation for the design, fabrication, and development of a new generation of ME devices.

A Modified 2D-Checksum Error Detecting Method for Data Transmission in Noisy Media

In data transmission a change in single bit in the received data may lead to miss understanding or a disaster. Each bit in the sent information has high priority especially with information such as the address of the receiver. The importance of error detection with each single change is a key issue in data transmission field.The ordinary single parity detection method can detect odd number of errors efficiently, but fails with even number of errors. Other detection methods such as two-dimensional and checksum showed better results and failed to cope with the increasing number of errors.Two novel methods were suggested to detect the binary bit change errors when transmitting data in a noisy media.Those methods were: 2D-Checksum method and Modified 2D-Checksum. In 2D-checksum method, summing process was done for 7×7 patterns in row direction and then in column direction to result 8×8 patterns. While in modified method, an additional parity diagonal vector was added to the pattern to be 8×9. By combining the benefits of using single parity (detecting odd number of error bits) and the benefits of checksum (reducing the effect of 4-bit errors) and combining them in 2D shape, the detection process was improved. By contaminating any sample of data with up to 33% of noise (change 0 to 1 and vice versa), the detecting process in first method was improved by approximately 50% compared to the ordinary traditional two dimensional-parity method and gives best detection results in second novel method

Effect of Weeding Regime and Row Direction on Growth and Yield of Upland Rice in Uganda

A study determined the effects of row direction and integrated weed management on the growth, productivity and economics of upland rice at Ikulwe Research Station using a completely randomized block design, replicated thrice with NAMCHE 5 upland rice in Uganda (2021b & 2022b). 12 treatments with 6 weeding regimes namely Pre- emergence Butanil (PREB) + 1hand hoeing (HH), 2HH, 3HH, PREB + Post emergence Butanil (POEB), weekly weeding (42 days) and control (EW & NS orientation) were adopted. Nitrogen (100 kg) Phosphorus (60 kg) and Potash (40 kg) were applied ha-1. Collected data on plant height, tillers, leaves, panicles; and grains were subjected to ANOVA (13th edition Genstat). Higher growth and yield parameters were in EW than NS direction. Leaves (27-28 leaves), height (65-70 cm), tillers (5-6 tillers and panicles (4-5 panicles) plant-1 were high under 2HH, PREB +1HH, PREB + POEB and 3HH (EW) compared to similar treatments in NS direction during 2021b and 2022b. Grain yield (2.34 t ha-1; HH), (2.26 t ha-1; PREB + 1 HH), (2.01 t ha-1; PREB + POEB), (1.89 t ha-1; 3HH) was significantly high in 2021b. During 2022b increased panicles (4-5 panicles), panicle length (21 cm), grains panicle-1 (110-117 grains) and grain yield (3.8 - 4.4 t ha-1) were recorded relative to 2021b. Weekly weeding (EW) developed high grains panicle-1 (102 grains) and grain yield (3.0 t ha-1) in 2022b relative to reduced tillers (4.5 tillers), filled panicles (3.6 panicles) and grain yield (1.9 t ha-1) in 2021b. Row directions had no effect on leaf width and grass biomass was numerically higher under EW than NS direction. Benefit Cost Ratios and Marginal Rate of Returns were optimum under 2HH and PREB + 1HH (EW). 2HH and PREB + 1HH (EW) may be recommended in Uganda having been the most economic technologies.

Estimation of Muscle Fiber Direction by Multi-channel Surface EMG Conductiong Wave Analysis Using Grid Surface Electrodes

Skeletal muscle is a set of motor units (MU). Muscle contractile activity is carried out by regulating the firing frequency of MU and the types of muscle fibers to be mobilized. Multi-channel surface electromyogram (sEMG) contains many conducting waves that represent a single motor unit action potentials (MUAP). In previous study, we proposed a method to extract conducting waves quantitatively and automatically. This method is useful for elucidating mobilized MUs and can be applied to kinematic analysis and diagnosis of muscle diseases. In multi-channel sEMG measurement, rows of electrodes need to be applied along the direction of the muscle fibers. However, the electrodes are difficult to set because the direction of the muscle fibers cannot be visually confirmed on the skin. This study investigated a method for estimating muscle fiber direction by analyzing conducting waves from two-dimensional multichannel surface electromyograms using grid-shaped surface electrodes. By examining the number of conducting waves acquired in each direction of the electrode row, it was suggested that the direction of muscle fiber may be estimated from the direction with the largest number of propagating wave acquisitions. It was also found that the method of acquiring differential potential signals has a significant impact on the analysis of propagation direction.

A Low-Complexity Direction-of-Arrival Estimation Algorithm for Noncircular Signals via Subspace Rotation Technique

In this paper, we investigate the problem of the heavy computational burden of the direction-of-arrival (DoA) estimation for the noncircular (NC) signals. A novel low-complexity direction-of-arrival estimation algorithm for NC signals via subspace rotation technique (SRT) is proposed. The proposed algorithm divides the noise subspace matrix along its row direction into two submatrices, and the SRT is performed to get a new reduced-dimension noise subspace. Then, utilizing the separation of variables and the orthogonality between the reduced-dimension noise subspace and the space spanned by the columns of the extended manifold matrix, a new one-dimensional spectral search function is derived to estimate DoAs. As the size of the block matrices of the noise subspace matrix has a great impact on the computational complexity of the spectral search, the optimal number of rows of the block matrices is determined. The proposed algorithm not only avoids the two-dimensional spectral search but also efficiently removes the redundancy computations in the one-dimensional spectral search. Theoretical analysis and simulation results show that the proposed algorithm can significantly improve the computational efficiency on the premise of ensuring the accuracy of DoA estimation for the NC signals, especially in scenarios where large numbers of sensors are applied.

A Novel Lane Line Detection Algorithm for Driverless Geographic Information Perception Using Mixed-Attention Mechanism ResNet and Row Anchor Classification

Lane line detection is a fundamental and critical task for geographic information perception of driverless and advanced assisted driving. However, the traditional lane line detection method relies on manual adjustment of parameters, and has poor universality, a heavy workload, and poor robustness. Most deep learning-based methods make it difficult to effectively balance accuracy and efficiency. To improve the comprehensive perception ability of lane line geographic information in a natural traffic environment, a lane line detection algorithm based on a mixed-attention mechanism residual network (ResNet) and row anchor classification is proposed. A mixed-attention mechanism is added after the backbone network convolution, normalization and activation layers, respectively, so that the model can focus more on important lane line features to improve the pertinence and efficiency of feature extraction. In addition, to achieve faster detection speed and solve the problem of no vision, the method of lane line location selection and classification based on the row direction is used to detect whether there are lane lines in each candidate point according to the row anchor, reducing the high computational complexity caused by segmentation on a pixel-by-pixel basis of traditional semantic segmentation. Based on TuSimple and CurveLane datasets, multi-scene, multi-environment, multi-linear road image datasets and video sequences are integrated and self-built, and several experiments are designed and tested to verify the effectiveness of the proposed method. The test accuracy of the mixed-attention mechanism network model reached 95.96%, and the average time efficiency is nearly 180 FPS, which can achieve a high level of accuracy and real-time detection process. Therefore, the proposed method can meet the safety perception effect of lane line geographic information in natural traffic environments, and achieve an effective balance between the accuracy and efficiency of actual road application scenarios.

Selection of Optimum Land Leveling Design Method for Surface Irrigation

Land leveling or land grading of surface irrigated fields improve irrigation water distribution and application efficiencies, conserve water and increases crop productivity. Land formation for irrigation face many constraints (ensuring proper slopes, prevention of flood, ensuring canal water command over the field, optimizing earthwork, minimizing truck travel distances, proper equipment utilization). Design engineers traditionally, perform site formation design manually by plane shape, least squire or linear programming methods. Such methods are with different characteristics. The main objective this study is to select and compare performance of these three design methods for proper land leveling design. Consequently, the basic theory of these alternative design methods are reviewed and their performance using data surveyed from five fields in Khartoum North-Sudan, each with different soil surface topographic configurations, is analyzed. The statistical analysis revealed that the linear programming method is the most appropriate design method. Employing the linear programming design method revealed that design slopes in row and cross row directions are within the acceptable range (0.1 to 0.5, the ratio of Cut/fill volumes is within the recommended range (1.1 to 1.3), uniformity of distribution of design elevations of grid points are acceptable and within the target limits (80%), while their deviation is at 80% of grid points around the mean before leveling.

Kaczmarz's anomaly: A surprising feature of Kaczmarz's method

This paper exposes one of the more fascinating features of Kaczmarz's method. Let Ax=b denote the linear system to solve, where A∈Rm×n and b=(b1,b2,…,bm)T∈Rm. Let the rows of A be denoted as aiT,i=1,…,m. Then the ith step of the basic iteration achieves projection on the hyperplane {x|aiTx=bi}. Normalizing the equations allows us to assume that aiTai=1, so the vectors ai,i=1,…,m, are points on the unit sphere in Rn. The “Kaczmarz anomaly” occurs when these points are randomly scattered over a large area of the unit sphere. Assume first that the number of equations, m, is considerably smaller than the number of unknowns, n. In this case Kaczmarz's method enjoys a fast rate of convergence. Then, as m increases toward n, the rate of convergence slows down. That is, the more equations we have more iterations are needed. In particular, as m approaches n there is dramatic increase in the number of iterations. The closer are m and n the slower is the convergence. However, as m passes n the situation is reversed. From now on, the more equations we have less iterations are needed. Finally, when solving overdetermined linear systems in which m is considerably larger than n the method returns to enjoy rapid convergence.The paper illustrates this phenomenon and explains its reasons. Linear systems with randomly scattered row's directions are common in practical applications. The exposed behavior helps to see the reasons behind success (or failure) of Kaczmarz's method when applied to solve such systems.