macJNet: weakly-supervised multimodal image deformable registration using joint learning framework and multi-sampling cascaded MIND.

Deformable multimodal image registration plays a key role in medical image analysis. It remains a challenge to find accurate dense correspondences between multimodal images due to the significant intensity distortion and the large deformation. macJNet is proposed to align the multimodal medical images, which is a weakly-supervised multimodal image deformable registration method using a joint learning framework and multi-sampling cascaded modality independent neighborhood descriptor (macMIND). The joint learning framework consists of a multimodal image registration network and two segmentation networks. The proposed macMIND is a modality-independent image structure descriptor to provide dense correspondence for registration, which incorporates multi-orientation and multi-scale sampling patterns to build self-similarity context. It greatly enhances the representation ability of cross-modal features in the registration network. The semi-supervised segmentation networks generate anatomical labels to provide semantics correspondence for registration, and the registration network helps to improve the performance of multimodal image segmentation by providing the consistency of anatomical labels. 3D CT-MR liver image dataset with 118 samples is built for evaluation, and comprehensive experiments have been conducted to demonstrate that macJNet achieves superior performance over state-of-the-art multi-modality medical image registration methods.

[Automatic segmentation of kidney tumor based on cascaded multiscale convolutional neural networks].

The background of abdominal computed tomography (CT) images is complex, and kidney tumors have different shapes, sizes and unclear edges. Consequently, the segmentation methods applying to the whole CT images are often unable to effectively segment the kidney tumors. To solve these problems, this paper proposes a multi-scale network based on cascaded 3D U-Net and DeepLabV3+ for kidney tumor segmentation, which uses atrous convolution feature pyramid to adaptively control receptive field. Through the fusion of high-level and low-level features, the segmented edges of large tumors and the segmentation accuracies of small tumors are effectively improved. A total of 210 CT data published by Kits2019 were used for five-fold cross validation, and 30 CT volume data collected from Suzhou Science and Technology Town Hospital were independently tested by trained segmentation models. The results of five-fold cross validation experiments showed that the Dice coefficient, sensitivity and precision were 0.796 2 ± 0.274 1, 0.824 5 ± 0.276 3, and 0.805 1 ± 0.284 0, respectively. On the external test set, the Dice coefficient, sensitivity and precision were 0.817 2 ± 0.110 0, 0.829 6 ± 0.150 7, and 0.831 8 ± 0.116 8, respectively. The results show a great improvement in the segmentation accuracy compared with other semantic segmentation methods.

Cross-modal missing time-series imputation using dense spatio-temporal transformer nets.

Due to irregular sampling or device failure, the data collected from sensor network has missing value, that is, missing time-series data occurs. To address this issue, many methods have been proposed to impute random or non-random missing data. However, the imputation accuracy of these methods are not accurate enough to be applied, especially in the case of complete data missing (CDM). Thus, we propose a cross-modal method to impute time-series missing data by dense spatio-temporal transformer nets (DSTTN). This model embeds spatial modal data into time-series data by stacked spatio-temporal transformer blocks and deployment of dense connections. It adopts cross-modal constraints, a graph Laplacian regularization term, to optimize model parameters. When the model is trained, it recovers missing data finally by an end-to-end imputation pipeline. Various baseline models are compared by sufficient experiments. Based on the experimental results, it is verified that DSTTN achieves state-of-the-art imputation performance in the cases of random and non-random missing. Especially, the proposed method provides a new solution to the CDM problem.

Uncertain prediction of deformable image registration on lung CT using multi-category features and supervised learning.

The assessment of deformable registration uncertainty is an important task for the safety and reliability of registration methods in clinical applications. However, it is typically done by a manual and time-consuming procedure. We propose a novel automatic method to predict registration uncertainty based on multi-category features and supervised learning. Three types of features, including deformation field statistical features, deformation field physiologically realistic features, and image similarity features, are introduced and calculated to train the random forest regressor for local registration uncertain prediction. Deformation field statistical features represent the numerical stability of registration optimization, which are correlated to the uncertainty of deformation fields; deformation field physiologically realistic features represent the biomechanical properties of organ motions, which mathematically reflect the physiological reality of deformation; image similarity features reflect the similarity between the warped image and fixed image. The multi-category features comprehensively reflect the registration uncertainty. The strategy of spatial adaptive random perturbations is also introduced to accurately simulate spatial distribution of registration uncertainty, which makes deformation field statistical features more discriminative to the uncertainty of deformation fields. Experiments were conducted on three publicly available thoracic CT image datasets. Seventeen randomly selected image pairs are used to train the random forest model, and 9 image pairs are used to evaluate the prediction model. The quantitative experiments on lung CT images show that the proposed method outperforms the baseline method for uncertain prediction of classical iterative optimization-based registration and deep learning-based registration with different registration qualities. The proposed method achieves good performance for registration uncertain prediction, which has great potential in improving the accuracy of registration uncertain prediction.

Narrow-band loss - a novel loss function focused on target boundary.

Loss functions widely employed in medical image segmentation, e.g., Dice or Generalized Dice, treat each pixel of segmentation target(s) equally. These region-based loss functions are concerned with the overall segmentation accuracy. However, in clinical applications, the focus of attention is often the boundary area of the target organ(s). Existing region-based loss functions lack attention to boundary areas. We designed narrow-band loss, which computes the integration of the predicted probability within the narrow-band around the target boundary. From the aspect of how it's derived, Narrow-band loss belongs to the region-based loss function. The difference from normal region-based loss is that Narrow-band loss calculates based on the degree of coincidence of the region surrounding the organ boundary. The advantage is that narrow-band loss can guide networks to focus on the target's boundary and neighborhood. We also generalize narrow-band loss to multi-target segmentation. We tested narrow-band loss on two datasets of different parts of the human body: the brain dataset with 416 cases, each case with 35 labels, and the abdominal dataset with 50 cases, each case with 12 labels. Narrow-band loss has improved greatly in hd95 metric and dice metric compared with baseline, which is dice loss only.

An artificial intelligence-assisted diagnosis modeling software (AIMS) platform based on medical images and machine learning: a development and validation study.

Background: Supervised machine learning methods [both radiomics and convolutional neural network (CNN)-based deep learning] are usually employed to develop artificial intelligence models with medical images for computer-assisted diagnosis and prognosis of diseases. A classical machine learning-based modeling workflow involves a series of interconnected components and various algorithms, but this makes it challenging, tedious, and labor intensive for radiologists and researchers to build customized models for specific clinical applications if they lack expertise in machine learning methods. Methods: We developed a user-friendly artificial intelligence-assisted diagnosis modeling software (AIMS) platform, which supplies standardized machine learning-based modeling workflows for computer-assisted diagnosis and prognosis systems with medical images. In contrast to other existing software platforms, AIMS contains both radiomics and CNN-based deep learning workflows, making it an all-in-one software platform for machine learning-based medical image analysis. The modular design of AIMS allows users to build machine learning models easily, test models comprehensively, and fairly compare the performance of different models in a specific application. The graphical user interface (GUI) enables users to process large numbers of medical images without programming or script addition. Furthermore, AIMS also provides a flexible image processing toolkit (e.g., semiautomatic segmentation, registration, morphological operations) to rapidly create lesion labels for multiphase analysis, multiregion analysis of an individual tumor (e.g., tumor mass and peritumor), and multimodality analysis. Results: The functionality and efficiency of AIMS were demonstrated in 3 independent experiments in radiation oncology, where multiphase, multiregion, and multimodality analyses were performed, respectively. For clear cell renal cell carcinoma (ccRCC) Fuhrman grading with multiphase analysis (sample size =187), the area under the curve (AUC) value of the AIMS was 0.776; for ccRCC Fuhrman grading with multiregion analysis (sample size =177), the AUC value of the AIMS was 0.848; for prostate cancer Gleason grading with multimodality analysis (sample size =206), the AUC value of the AIMS was 0.980. Conclusions: AIMS provides a user-friendly infrastructure for radiologists and researchers, lowering the barrier to building customized machine learning-based computer-assisted diagnosis models for medical image analysis.

Dual attention guided multiscale neural network trained with curriculum learning for noninvasive prediction of Gleason Grade Group from MRI.

BACKGROUND: The Gleason Grade Group (GG) is essential in assessing the malignancy of prostate cancer (PCa) and is typically obtained by invasive biopsy procedures in which sampling errors could lead to inaccurately scored GGs. With the gradually recognized value of bi-parametric magnetic resonance imaging (bpMRI) in PCa, it is beneficial to noninvasively predict GGs from bpMRI for early diagnosis and treatment planning of PCa. However, it is challenging to establish the connection between bpMRI features and GGs. PURPOSE: In this study, we propose a dual attention-guided multiscale neural network (DAMS-Net) to predict the 5-scored GG from bpMRI and design a training curriculum to further improve the prediction performance. METHODS: The proposed DAMS-Net incorporates a feature pyramid network (FPN) to fully extract the multiscale features for lesions of varying sizes and a dual attention module to focus on lesion and surrounding regions while avoiding the influence of irrelevant ones. Furthermore, to enhance the differential ability for lesions with the inter-grade similarity and intra-grade variation in bpMRI, the training process employs a specially designed curriculum based on the differences between the radiological evaluations and the ground truth GGs. RESULTS: Extensive experiments were conducted on a private dataset of 382 patients and the public PROSTATEx-2 dataset. For the private dataset, the experimental results showed that the proposed network performed better than the plain baseline model for GG prediction, achieving a mean quadratic weighted Kappa (Kw ) of 0.4902 and a mean positive predictive value of 0.9098 for predicting clinically significant cancer (PPVGG>1 ). With the application of curriculum learning, the mean Kw and PPVGG>1 further increased to 0.5144 and 0.9118, respectively. For the public dataset, the proposed method achieved state-of-the-art results of 0.5413 Kw and 0.9747 PPVGG>1 . CONCLUSION: The proposed DAMS-Net trained with curriculum learning can effectively predict GGs from bpMRI, which may assist clinicians in early diagnosis and treatment planning for PCa patients.

Using radiomics based on multicenter magnetic resonance images to predict isocitrate dehydrogenase mutation status of gliomas.

Background: Isocitrate dehydrogenase (IDH) mutation status is an important biomarker for the treatment strategy selection and prognosis evaluation of glioma. The purpose of this study is to predict the IDH mutation status of gliomas based on multicenter magnetic resonance (MR) images using radiomic models, which were composed from the selected radiomics features and logistic regression (LR), support vector machine (SVM), and LR least absolute shrinkage and selection operator (LASSO) classifiers. Methods: We retrospectively reviewed the medical records of 205 patients with gliomas. We enrolled 78 patients from Shandong Provincial Hospital from January 2018 to December 2019 as testing sets and 127 patients from The Cancer Genome Atlas (TCGA) as training sets. Preoperative MR images were stratified according to their IDH status, and the participants formed a consecutive and random series. Four MR modalities, including T1C, T2, T1 fluid-attenuated inversion recovery (FLAIR), and T2 FLAIR, were used for analysis. Five-fold cross-validation was adopted to train the models, and the models' performances were verified through the testing set. Tumor volumes of interest (VOI) were delineated on the 4 MR modalities. A total of 428 radiomics features were extracted. Two feature selection algorithms, Pearson correlation coefficient (PCC) and recursive feature elimination (RFE), were used to select radiomics features. These features were fed into 3 machine learning classifiers, which were LR, SVM, and LR LASSO, to construct prediction models. The accuracy (ACC), sensitivity (SEN), specificity (SPEC), and area under the curve (AUC) were applied to measure the predictive performance of the radiomics models. Results: The LR (SVM and LR LASSO) classifier predicted IDH mutation status with an average testing set ACC of 80.77% (80.64% and 80.41%), a SEN of 73.68% (84.21% and 89.47%), a SPEC of 87.50% (67.50% and 62.50%), and an AUC of 0.8572 (0.8217 and 0.8164). Conclusions: The radiomics models based on MR modalities demonstrated the potential to be used as tools across different data sets for the noninvasive prediction of the IDH mutation status in glioma.

Unsupervised registration for liver CT-MR images based on the multiscale integrated spatial-weight module and dual similarity guidance.

PURPOSE: Multimodal registration is a key task in medical image analysis. Due to the large differences of multimodal images in intensity scale and texture pattern, it is a great challenge to design distinctive similarity metrics to guide deep learning-based multimodal image registration. Besides, since the limitation of the small receptive field, existing deep learning-based methods are mainly suitable for small deformation, but helpless for large deformation. To address the above issues, we present an unsupervised multimodal image registration method based on the multiscale integrated spatial-weight module and dual similarity guidance. METHODS: In this method, a U-shape network with our multiscale integrated spatial-weight module is embedded into a multi-resolution image registration architecture to achieve end-to-end large deformation registration, where the spatial-weight module can effectively highlight the regions with large deformation and aggregate discriminative features, and the multi-resolution architecture further helps to solve the optimization problem of the network in a coarse-to-fine pattern. Furthermore, we introduce a special loss function based on dual similarity, which represents both global gray-scale similarity and local feature similarity, to optimize the unsupervised multimodal registration network. RESULTS: We verified the effectiveness of the proposed method on liver CT-MR images. Experimental results indicate that the proposed method achieves the optimal DSC value and TRE value of 92.70 ± 1.75(%) and 6.52 ± 2.94(mm), compared with other state-of-the-art registration algorithms. CONCLUSION: The proposed method can accurately estimate the large deformation field by aggregating multiscale features, and achieve higher registration accuracy and fast registration speed. Comparative experiments also demonstrate the effectiveness and generalization ability of the algorithm.

Multi-phase-combined CECT radiomics models for Fuhrman grade prediction of clear cell renal cell carcinoma.

Objective: This study aimed to evaluate the effectiveness of multi-phase-combined contrast-enhanced CT (CECT) radiomics methods for noninvasive Fuhrman grade prediction of clear cell renal cell carcinoma (ccRCC). Methods: A total of 187 patients with four-phase CECT images were retrospectively enrolled and then were categorized into training cohort (n=126) and testing cohort (n=61). All patients were confirmed as ccRCC by histopathological reports. A total of 110 3D classical radiomics features were extracted from each phase of CECT for individual ccRCC lesion, and contrast-enhanced variation features were also calculated as derived radiomics features. These features were concatenated together, and redundant features were removed by Pearson correlation analysis. The discriminative features were selected by minimum redundancy maximum relevance method (mRMR) and then input into a C-support vector classifier to build multi-phase-combined CECT radiomics models. The prediction performance was evaluated by the area under the curve (AUC) of receiver operating characteristic (ROC). Results: The multi-phase-combined CECT radiomics model showed the best prediction performance (AUC=0.777) than the single-phase CECT radiomics model (AUC=0.711) in the testing cohort (p value=0.039). Conclusion: The multi-phase-combined CECT radiomics model is a potential effective way to noninvasively predict Fuhrman grade of ccRCC. The concatenation of first-order features and texture features extracted from corticomedullary phase and nephrographic phase are discriminative feature representations.

Automated quantitative assessment of pediatric blunt hepatic trauma by deep learning-based CT volumetry.

BACKGROUND: To develop an end-to-end deep learning method for automated quantitative assessment of pediatric blunt hepatic trauma based on contrast-enhanced computed tomography (CT). METHODS: This retrospective study included 170 children with blunt hepatic trauma between May 1, 2015, and August 30, 2021, who had undergone contrast-enhanced CT. Both liver parenchyma and liver trauma regions were manually segmented from CT images. Two deep convolutional neural networks (CNNs) were trained on 118 cases between May 1, 2015, and December 31, 2019, for liver segmentation and liver trauma segmentation. Liver volume and trauma volume were automatically calculated based on the segmentation results, and the liver parenchymal disruption index (LPDI) was computed as the ratio of liver trauma volume to liver volume. The segmentation performance was tested on 52 cases between January 1, 2020, and August 30, 2021. Correlation analysis among the LPDI, trauma volume, and the American Association for the Surgery of Trauma (AAST) liver injury grade was performed using the Spearman rank correlation. The performance of severity assessment of pediatric blunt hepatic trauma based on the LPDI and trauma volume was evaluated using receiver operating characteristic (ROC) analysis. RESULTS: The Dice, precision, and recall of the developed deep learning framework were 94.75, 94.11, and 95.46% in segmenting the liver and 72.91, 72.40, and 76.80% in segmenting the trauma regions. The LPDI and trauma volume were significantly correlated with AAST grade (rho = 0.823 and rho = 0.831, respectively; p < 0.001 for both). The area under the ROC curve (AUC) values for the LPDI and trauma volume to distinguish between high-grade and low-grade pediatric blunt hepatic trauma were 0.942 (95% CI, 0.882-1.000) and 0.952 (95% CI, 0.895-1.000), respectively. CONCLUSIONS: The developed end-to-end deep learning method is able to automatically and accurately segment the liver and trauma regions from contrast-enhanced CT images. The automated LDPI and liver trauma volume can act as objective and quantitative indexes to supplement the current AAST grading of pediatric blunt hepatic trauma.

CT-based peritumoral radiomics signatures for malignancy grading of clear cell renal cell carcinoma.

OBJECTIVE: To evaluate the efficiency of CT-based peritumoral radiomics signatures of clear cell renal cell carcinoma (ccRCC) for malignancy grading in preoperative prediction. MATERIALS AND METHODS: 203 patients with pathologically confirmed as ccRCC were retrospectively enrolled in this study. All patients were categorized into training set (n = 122) and validation set (n = 81). For each patient, two types of volumes of interest (VOI) were masked on CT images. One type of VOIs was defined as the tumor mass volume (TMV), which was masked by radiologists delineating the outline of all contiguous slices of the entire tumor, while the other type defined as the peritumoral tumor volume (PTV), which was automatically created by an image morphological method. 1760 radiomics features were calculated from each VOI, and then the discriminative radiomics features were selected by Pearson correlation analysis for reproducibility and redundancy. These selected features were investigated their validity for building radiomics signatures by mRMR feature ranking method. Finally, the top ranked features, which were used as radiomics signatures, were input into a classifier for malignancy grading. The prediction performance was evaluated by receiver operating characteristic (ROC) curve in an independent validation cohort. RESULTS: The radiomics signatures of PTV showed a better performance on malignancy grade prediction of ccRCC with AUC of 0.807 (95% CI 0.800-0.834) in train data and 0.848 (95% CI 0.760-0.936) in validation data, while the radiomics signatures of TMV with AUC of 0.773 (95% CI 0.744-0.802) in train data and 0.810 (95% CI 0.706-0.914) in validation data. CONCLUSION: The CT-based peritumoral radiomics signature is a potential way to be used as a noninvasive tool to preoperatively predict the malignancy grades of ccRCC.

A new machine learning based user-friendly software platform for automatic radiomics modeling and analysis.

Supervised machine learning methods are usually used to build a custom model for disease diagnosis and auxiliary prognosis in radiomics studies. A classical machine learning pipeline involves a series of steps and multiple algorithms, which leads to a great challenge to find an appropriate combination of algorithms and an optimal hyper-parameter set for radiomics model building. We developed a freely available software package for radiomics model building. It can be used to lesion labeling, feature extraction, feature selection, classifier training and statistic result visualization. This software provides a user-friendly graphic interface and flexible IOs for radiologists and researchers to automatically develop radiomics models. Moreover, this software can extract features from corresponding lesion regions in multi-modality images, which is labeled by semi-automatic or full-automatic segmentation algorithms. It is designed in a loosely coupled architecture, programmed with Qt, VTK, and Python. In order to evaluate the availability and effectiveness of the software, we utilized it to build a CT-based radiomics model containing peritumoral features for malignancy grading of cell renal cell carcinoma. The final model got a good performance of grading study with AUC=0.848 on independent validation dataset.Clinical Relevance-the developed provides convenient and powerful toolboxes to build radiomics models for radiologists and researchers on clinical studies.

Machine learning-based quantitative analysis of barium enema and clinical features for early diagnosis of short-segment Hirschsprung disease in neonate.

OBJECTIVE: To develop a mathematical model based on a combination of clinical and radiologic features (barium enema) for early diagnosis of short-segment Hirschsprung disease (SHSCR) in neonate. METHODS: The analysis included 54 neonates with biopsy-confirmed SHSCR (the cases) and 59 neonates undergoing barium enema for abdominal symptoms but no Hirschsprung disease (the control). Colon shape features extracted from barium enema images and clinical features were used to develop diagnostic models using support vector machine (SVM) and L2-regularized logistic regression (LR). The training cohort included 32 cases and 37 controls; testing cohort consisted 22 cases and 22 controls. Results were compared to interpretation by 2 radiologists. RESULTS: In the analysis by radiologists, 87 out of 113 cases were correctly classified. Six SHSCR cases were mis-classified into the non-HSCR group. In the remaining 20 cases, radiologists were unable to make a decision. Both the SVM and LR classifiers contained five clinical features and four shape features. The performance of the two classifiers was similar. The best model had 86.36% accuracy, 81.82% sensitivity, and 90.91% specificity. The AUC was 0.9132 for the best-performing SVM classifier and 0.9318 for the best-performing LR classifier. CONCLUSION: A combination of clinical features and colon shape features extracted from barium enemas can be used to improve early diagnosis of SHSCR in neonate.

MRI Based Radiomics Compared With the PI-RADS V2.1 in the Prediction of Clinically Significant Prostate Cancer: Biparametric vs Multiparametric MRI.

PURPOSE: To compare the performance of radiomics to that of the Prostate Imaging Reporting and Data System (PI-RADS) v2.1 scoring system in the detection of clinically significant prostate cancer (csPCa) based on biparametric magnetic resonance imaging (bpMRI) vs. multiparametric MRI (mpMRI). METHODS: A total of 204 patients with pathological results were enrolled between January 2018 and December 2019, with 142 patients in the training cohort and 62 patients in the testing cohort. The radiomics model was compared with the PI-RADS v2.1 for the diagnosis of csPCa based on bpMRI and mpMRI by using receiver operating characteristic (ROC) curve analysis. RESULTS: The radiomics model based on bpMRI and mpMRI signatures showed high predictive efficiency but with no significant differences (AUC = 0.975 vs 0.981, p=0.687 in the training cohort, and 0.953 vs 0.968, p=0.287 in the testing cohort, respectively). In addition, the radiomics model outperformed the PI-RADS v2.1 in the diagnosis of csPCa regardless of whether bpMRI (AUC = 0.975 vs. 0.871, p= 0.030 for the training cohort and AUC = 0.953 vs. 0.853, P = 0.024 for the testing cohort) or mpMRI (AUC = 0.981 vs. 0.880, p= 0.030 for the training cohort and AUC = 0.968 vs. 0.863, P = 0.016 for the testing cohort) was incorporated. CONCLUSIONS: Our study suggests the performance of bpMRI- and mpMRI-based radiomics models show no significant difference, which indicates that omitting DCE imaging in radiomics can simplify the process of analysis. Adding radiomics to PI-RADS v2.1 may improve the performance to predict csPCa.

Efficient construction of sparse radial basis function neural networks using L1-regularization.

This paper investigates the construction of sparse radial basis function neural networks (RBFNNs) for classification problems. An efficient two-phase construction algorithm (which is abbreviated as TPCLR1 for simplicity) is proposed by using L1 regularization. In the first phase, an improved maximum data coverage (IMDC) algorithm is presented for the initialization of RBF centers and widths. Then a specialized Orthant-Wise Limited-memory Quasi-Newton (sOWL-QN) method is employed to perform simultaneous network pruning and parameter optimization in the second phase. The advantages of TPCLR1 lie in that better generalization performance is guaranteed with higher model sparsity, and the required storage space and testing time are much reduced. Besides these, only the regularization parameter and the maximum number of function evaluations are required to be prescribed, then the entire construction procedure becomes automatic. The learning algorithm is verified by several classification benchmarks with different levels of complexity. The experimental results show that an appropriate value of the regularization parameter is easy to find without using costly cross validation, and the proposed TPCLR1 offers an efficient procedure to construct sparse RBFNN classifiers with good generalization performance.

Generalized Hybrid Constructive Learning Algorithm for Multioutput RBF Networks.

An efficient generalized hybrid constructive (GHC) learning algorithm for multioutput radial basis function (RBF) networks is proposed to obtain a compact network with good generalization capability. By this algorithm, one can train the adjustable parameters and determine the optimal network structure simultaneously. First, an initialization method based on the growing and pruning algorithm is utilized to select the important initial hidden neurons and candidate ones. Then, by introducing a generalized hidden matrix, a structured parameter optimization algorithm is presented to train multioutput RBF network with fixed size, which combines Levenberg-Marquardt (LM) algorithm with least-square method together. Beginning from an appropriate number of hidden neurons, new neurons chosen from the candidates are added one by one each time when the training entraps into local minima. By incorporating an improved incremental constructive scheme, the training is built on previous results after adding new neurons such that the GHC learning algorithm avoids a trial-and-error procedure. Furthermore, based on the improved computation for LM training, the memory limitation problem is solved. The computational complexity analysis and experimental results demonstrate that better performance is efficiently achieved by this algorithm.