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PubMed2006

Estimating vaccine efficacy from outbreak size household data in the presence of heterogeneous transmission probabilities.

We develop a Bayesian approach for estimating vaccine efficacy for susceptibility (VEs) and infectiousness (VEI) using outbreak size household data. Our method allows for heterogeneity in transmission probabilities due to factors that are related to individuals' characteristics, such as age, in addition to vaccination status. It also allows for between-household heterogeneity in transmission probabilities due to random effects associated with households, such as genetic or environmental effects. Using age as a potential covariate causing heterogeneity in individuals' transmission probabilities in households consisting of adults and children, we present the results of a simulation study designed to evaluate the performance of the proposed estimators of VEs and VE(I). We found that estimates of VE(I) have larger bias and variance compared to those of VEs. We also use the approach to compare two vaccination designs: one vaccinating both adults and children, the other only children. Simulations reveal that the design that vaccinates both adults and children provides better estimates of VEs. There is no obvious difference between the two designs in the performance of the estimates of VE(I). In regard to random effects between households and the scenarios considered, models that do not account for between-households heterogeneity produce fairly robust estimates even when household-level random effects are present.

Adult↗
PubMed2002

Using Bayesian tissue classification to improve the accuracy of vestibular schwannoma volume and growth measurement.

BACKGROUND AND PURPOSE: True 3D measurements of tumor volume are time-consuming and subject to errors that are particularly pronounced in cases of small tumors. These problems complicate the routine clinical assessment of tumor growth rates. We examined the accuracy of currently available methods of size and growth measurement of vestibular schwannomas compared with that of a novel fast partial volume tissue classification algorithm. METHODS: Sixty-three patients with unilateral sporadic vestibular schwannomas underwent imaging. Thirty-eight of these patients underwent imaging two or more times at approximately 12-month intervals. Contrast-enhanced 3D T1-weighted images were used for all measurements. An experienced radiologist performed standard size estimations, including maximal diameter, elliptical area, perimeter, manually segmented area, intensity thresholded seeding volume, and manually segmented volume. A method for calculating volume was also used, incorporating Bayesian probability statistics to estimate partial volume effects. Manually segmented volume was obtained as a baseline standard measure. A computer-generated phantom exhibiting the intensity and partial volume characteristics of brain tissue, CSF, and intracanalicular vestibular schwannoma tissue was used to measure absolute accuracy of the standard technique and Bayesian partial volume segmentation. RESULTS: The Bayesian partial volume segmentation method showed the highest correlation (R(2) = 0.994) with the standard method, whereas the commonly used method of maximal diameter measurement showed poor correlation (R(2) = 0.732). Accuracy of Bayesian segmentation was shown to be more than twice that of manual segmentation, with an absolute accuracy of 5% (cf, 13%) and a remeasurement accuracy of 70 mm(3) (cf, 150 mm(3)). For the 38 patients who underwent imaging twice, definite tumor growth was shown for 12, potential growth for seven, no growth for 17, and definite shrinkage for two. CONCLUSION: Commonly used methods such as maximal diameter measurements do not provide adequate statistical accuracy with which to monitor tumor growth in patients with small vestibular schwannomas. Bayesian partial volume segmentation provides a more accurate and rapid method of volume and growth estimation. These differences in measurement accuracy translated into a significant improvement in clinical assessment, allowing identification of tumor growth in 10 of 12 cases that appeared to be static in size when manual segmentation techniques are used. The technique is quick to perform and suitable for use in routine clinical practice.

Adult↗
PubMed2004

Adaptive BCI based on variational Bayesian Kalman filtering: an empirical evaluation.

This paper proposes the use of variational Kalman filtering as an inference technique for adaptive classification in a brain computer interface (BCI). The proposed algorithm translates electroencephalogram segments adaptively into probabilities of cognitive states. It, thus, allows for nonstationarities in the joint process over cognitive state and generated EEG which may occur during a consecutive number of trials. Nonstationarities may have technical reasons (e.g., changes in impedance between scalp and electrodes) or be caused by learning effects in subjects. We compare the performance of the proposed method against an equivalent static classifier by estimating the generalization accuracy and the bit rate of the BCI. Using data from two studies with healthy subjects, we conclude that adaptive classification significantly improves BCI performance. Averaging over all subjects that participated in the respective study, we obtain, depending on the cognitive task pairing, an increase both in generalization accuracy and bit rate of up to 8%. We may, thus, conclude that adaptive inference can play a significant contribution in the quest of increasing bit rates and robustness of current BCI technology. This is especially true since the proposed algorithm can be applied in real time.

Algorithms↗
PubMed2004

Pathways of urothelial cancer progression suggested by Bayesian network analysis of allelotyping data.

Urothelial cancers of the bladder (UC) comprise biologically heterogeneous group of tumors and display complex genetic alterations. Several genetic changes have been analyzed in detail and some of them are associated with the development and progression of UCs. Only a few studies, however, are focused on identifying the order in which the aberrations may appear during UC tumorigenesis. We have analyzed 123 papillary UCs of the bladder by microsatellites for each of the chromosomal regions that have been suggested to be specifically involved in this type of tumor. We used Bayesian network modeling that enables to uncover multivariate probabilistic dependencies between variables. This methodology applied to LOH data allowed us to discover patterns of losses in UCs. Exploiting the mechanism of probabilistic reasoning in Bayesian networks we suggest primary and secondary events in tumor pathogenesis and reconstruct the possible flow of progression of allelic changes. Losses of chromosome 9p and 9q were found to be the primary events. Losses of 8p and 17p are important events leading to progression of tumor cell clones. The loss of 17p occurs when both abnormalities of chromosome 9 and 8p are already present. There are chromosomal losses related to 8p (1q, 18q, 10q) and some losses like 5q/5p were associated with 17p, leading to the hypothesis of different genetic pathways of UC progression. The abnormalities of chromosome regions 13q, 16q, 6q, 14q, 3p are suggested to be late events being accumulated during the progression of cancer. Although some genetic changes were associated only with the 8p pathway, most secondary genetic changes appear in both pathways. Supplementary material for this article can be found on the International Journal of Cancer website at http://www.interscience.wiley.com/jpages/0020-7136/suppmat/index.html.

Alleles↗
PubMed2005

A Markov chain Monte Carlo technique for identification of combinations of allelic variants underlying complex diseases in humans.

In recent years, the number of studies focusing on the genetic basis of common disorders with a complex mode of inheritance, in which multiple genes of small effect are involved, has been steadily increasing. An improved methodology to identify the cumulative contribution of several polymorphous genes would accelerate our understanding of their importance in disease susceptibility and our ability to develop new treatments. A critical bottleneck is the inability of standard statistical approaches, developed for relatively modest predictor sets, to achieve power in the face of the enormous growth in our knowledge of genomics. The inability is due to the combinatorial complexity arising in searches for multiple interacting genes. Similar "curse of dimensionality" problems have arisen in other fields, and Bayesian statistical approaches coupled to Markov chain Monte Carlo (MCMC) techniques have led to significant improvements in understanding. We present here an algorithm, APSampler, for the exploration of potential combinations of allelic variations positively or negatively associated with a disease or with a phenotype. The algorithm relies on the rank comparison of phenotype for individuals with and without specific patterns (i.e., combinations of allelic variants) isolated in genetic backgrounds matched for the remaining significant patterns. It constructs a Markov chain to sample only potentially significant variants, minimizing the potential of large data sets to overwhelm the search. We tested APSampler on a simulated data set and on a case-control MS (multiple sclerosis) study for ethnic Russians. For the simulated data, the algorithm identified all the phenotype-associated allele combinations coded into the data and, for the MS data, it replicated the previously known findings.

Algorithms↗
PubMed1999

Feature selection for computerized mass detection in digitized mammograms by using a genetic algorithm.

RATIONALE AND OBJECTIVES: To investigate optimization of feature selection for computerized mass detection in digitized mammograms, and to compare the effectiveness of a genetic algorithm (GA) in such optimization with that of an "exhaustive" search of all feature permutations. MATERIALS AND METHODS: A Bayesian belief network (BBN) was used to classify positive and negative regions for masses depicted in digitized mammograms; 20 features were computed for each of 592 positive and 3,790 negative regions in two databases. Conditional probabilities for the BBN were computed by using a "training" database of 288 positive and 2,204 negative regions. Performance was measured by the area under the receiver operating characteristic curve (A) by using the remainder database (304 positive and 1,586 negative regions). The optimal set was first found by using an "exhaustive" (complete permutation) searching method. A GA-based search for the optimal set then was applied, and the results of the two approaches were compared. RESULTS: As the number of features in the classifier increased, the A value increased until it reached a maximum performance for 11 features of 0.876 +/- 0.008. The A value then decreased monotonically as the number of features increased from 11 to 20. Using 100 random chromosomes (seeds) in the first generation, the GA identified the same optimal set of features but reduced the total computation time by a factor of 65. CONCLUSION: A GA-based search might be an efficient and effective approach to selecting an optimal feature set.

Algorithms↗
PubMed2000

Mapping a quantitative trait locus via the EM algorithm and Bayesian classification.

Mapping a locus controlling a quantitative genetic trait (e.g., blood pressure) to a specific genomic region is of considerable interest. Data on the quantitative trait under consideration and several codominant genetic markers with known genomic locations are collected from members of families and statistically analyzed to draw inferences on the genomic position of the trait locus. The vector of parameters of interest comprises the pairwise recombination fractions, theta, between the putative quantitative trait locus and the marker loci. One of the major complications in estimating theta for a quantitative trait in humans is the lack of haplotype information on members of families. The purpose of this study was to devise a computationally simple and efficient method of estimation of theta in the absence of haplotype information. We have proposed a two-stage estimation procedure using the expectation-maximization (EM) algorithm. In the first stage, parameters of the QTL are estimated based on data of a sample of unrelated individuals. From estimates thus obtained, we have used a Bayes' rule to infer QTL genotypes of parents in families. Finally, in the second stage of the procedure, we have proposed an EM algorithm for obtaining the maximum likelihood estimate of theta based on data of informative families (which are identified upon inferring parental QTL genotypes performed in the first stage). We have shown, using simulated data, that the proposed procedure is cost-effective, computationally simple, and statistically efficient. As expected, analysis of data on multiple markers jointly is more efficient than the analysis based on single markers.

Algorithms↗
PubMed1996

Bayesian imputation of predictive values when covariate information is available and gold standard diagnosis is unavailable.

We suggest a conceptually simple Bayesian approach to inferences about the conditional probability of a specimen being infection-free given the outcome of a diagnostic test and covariate information. The approach assumes that the infection state of a specimen is not observable but uses the outcomes of a second test in conjunction with those of the first, that is, dual testing data. Dual testing procedures are often employed in clinical laboratories to assure that samples are not contaminated or to increase the likelihood of correct diagnoses. Using the CD4 count and a proxy for risk behavior as covariates, we apply the method to obtain inferences about the conditional probability of an individual being HIV-1 infection-free given the individual's covariates and a negative outcome with the standard enzyme-linked immunoad-sorbent assay/Western blotting test for HIV-1 detection. Inferences combine data from two studies where specimens were tested with the standard and with the more sensitive polymerase chain reaction test.

Algorithms↗
PubMed2003

Comparing bootstrap and posterior probability values in the four-taxon case.

Assessment of the reliability of a given phylogenetic hypothesis is an important step in phylogenetic analysis. Historically, the nonparametric bootstrap procedure has been the most frequently used method for assessing the support for specific phylogenetic relationships. The recent employment of Bayesian methods for phylogenetic inference problems has resulted in clade support being expressed in terms of posterior probabilities. We used simulated data and the four-taxon case to explore the relationship between nonparametric bootstrap values (as inferred by maximum likelihood) and posterior probabilities (as inferred by Bayesian analysis). The results suggest a complex association between the two measures. Three general regions of tree space can be identified: (1) the neutral zone, where differences between mean bootstrap and mean posterior probability values are not significant, (2) near the two-branch corner, and (3) deep in the two-branch corner. In the last two regions, significant differences occur between mean bootstrap and mean posterior probability values. Whether bootstrap or posterior probability values are higher depends on the data in support of alternative topologies. Examination of star topologies revealed that both bootstrap and posterior probability values differ significantly from theoretical expectations; in particular, there are more posterior probability values in the range 0.85-1 than expected by theory. Therefore, our results corroborate the findings of others that posterior probability values are excessively high. Our results also suggest that extrapolations from single topology branch-length studies are unlikely to provide any general conclusions regarding the relationship between bootstrap and posterior probability values.

Computer Simulation↗
PubMed1994

Bayesian restoration of chest radiographs. Scatter compensation with improved signal-to-noise ratio.

OBJECTIVES: The authors introduce a Bayesian algorithm for digital chest radiography that increases the signal-to-noise ratio, and thus detectability, for low-contrast objects. METHOD: The improved images are formed as a maximum a posteriori probability estimation of a scatter-reduced (contrast-enhanced) image with decreased noise. Noise is constrained by including prior knowledge of image smoothness. Variations between neighboring pixels are penalized for small variations (to suppress Poisson noise), but not for larger variations (to avoid affecting anatomical structure). The technique was optimized to reduce residual scatter in digital radiographs of an anatomical chest phantom. RESULTS: The contrast in the lung was improved by a factor of two, whereas signal-to-noise ratio was improved by a factor of 1.8. Image resolution was unaffected for objects with a contrast greater than 2%. CONCLUSION: This statistical estimation technique shows promise for improving object detectability in radiographs by simultaneously increasing contrast, while constraining noise.

Algorithms↗
PubMed2004

Layered motion segmentation and depth ordering by tracking edges.

This paper presents a new Bayesian framework for motion segmentation--dividing a frame from an image sequence into layers representing different moving objects--by tracking edges between frames. Edges are found using the Canny edge detector, and the Expectation-Maximization algorithm is then used to fit motion models to these edges and also to calculate the probabilities of the edges obeying each motion model. The edges are also used to segment the image into regions of similar color. The most likely labeling for these regions is then calculated by using the edge probabilities, in association with a Markov Random Field-style prior. The identification of the relative depth ordering of the different motion layers is also determined, as an integral part of the process. An efficient implementation of this framework is presented for segmenting two motions (foreground and background) using two frames. It is then demonstrated how, by tracking the edges into further frames, the probabilities may be accumulated to provide an even more accurate and robust estimate, and segment an entire sequence. Further extensions are then presented to address the segmentation of more than two motions. Here, a hierarchical method of initializing the Expectation-Maximization algorithm is described, and it is demonstrated that the Minimum Description Length principle may be used to automatically select the best number of motion layers. The results from over 30 sequences (demonstrating both two and three motions) are presented and discussed.

Algorithms↗
PubMed2004

Performance of maximum parsimony and likelihood phylogenetics when evolution is heterogeneous.

All inferences in comparative biology depend on accurate estimates of evolutionary relationships. Recent phylogenetic analyses have turned away from maximum parsimony towards the probabilistic techniques of maximum likelihood and bayesian Markov chain Monte Carlo (BMCMC). These probabilistic techniques represent a parametric approach to statistical phylogenetics, because their criterion for evaluating a topology--the probability of the data, given the tree--is calculated with reference to an explicit evolutionary model from which the data are assumed to be identically distributed. Maximum parsimony can be considered nonparametric, because trees are evaluated on the basis of a general metric--the minimum number of character state changes required to generate the data on a given tree--without assuming a specific distribution. The shift to parametric methods was spurred, in large part, by studies showing that although both approaches perform well most of the time, maximum parsimony is strongly biased towards recovering an incorrect tree under certain combinations of branch lengths, whereas maximum likelihood is not. All these evaluations simulated sequences by a largely homogeneous evolutionary process in which data are identically distributed. There is ample evidence, however, that real-world gene sequences evolve heterogeneously and are not identically distributed. Here we show that maximum likelihood and BMCMC can become strongly biased and statistically inconsistent when the rates at which sequence sites evolve change non-identically over time. Maximum parsimony performs substantially better than current parametric methods over a wide range of conditions tested, including moderate heterogeneity and phylogenetic problems not normally considered difficult.

Bayes Theorem↗
PubMed2026

NExON-Bayes: a Bayesian approach to network estimation informed by ordinal covariates.

MOTIVATION: In heterogeneous disease settings, accounting for intrinsic sample variability is crucial for obtaining reliable and interpretable omic network estimates. However, most graphical model analyses of biomedical data assume homogeneous conditional dependence structures, potentially leading to misleading conclusions. To address this, we propose a joint Gaussian graphical model that leverages sample-level ordinal covariates (e.g. disease stage) to account for heterogeneity and improve the estimation of partial correlation structures. RESULTS: Our modelling framework, called NExON-Bayes, extends the graphical spike-and-slab framework to account for ordinal covariates, jointly estimating their relevance to the graph structure and leveraging them to improve the accuracy of network estimation. To scale to high-dimensional omic settings, we develop an efficient variational inference algorithm tailored to our model. Through simulations, we demonstrate that our method outperforms the vanilla graphical spike-and-slab (with no covariate information), as well as other state-of-the-art network approaches which exploit covariate information. Applying our method to reverse phase protein array data from patients diagnosed with stage I, II or III breast carcinoma, we estimate the behaviour of proteomic networks as cancer progresses. Our model provides insights not only through inspection of the estimated proteomic networks, but also of the estimated ordinal covariate dependencies of key groups of proteins within those networks, offering a comprehensive understanding of how biological pathways shift across disease stages. AVAILABILITY AND IMPLEMENTATION: A user-friendly R package for NExON-Bayes with tutorials is available on Github at github.com/jf687/NExON, and archived at https://doi.org/10.5281/zenodo.20312938. The source of the dataset used is cited in the relevant section.

Bayes Theorem↗
PubMed2006

S. pneumoniae transmission according to inclusion in conjugate vaccines: Bayesian analysis of a longitudinal follow-up in schools.

BACKGROUND: Recent trends of pneumococcal colonization in the United States, following the introduction of conjugate vaccination, indicate that non-vaccine serotypes tend to replace vaccine serotypes. The eventual extent of this replacement is however unknown and depends on serotype-specific carriage and transmission characteristics. METHODS: Here, some of these characteristics were estimated for vaccine and non-vaccine serotypes from the follow-up of 4,488 schoolchildren in France in 2000. A Bayesian approach using Markov chain Monte Carlo data augmentation techniques was used for estimation. RESULTS: Vaccine and non-vaccine serotypes were found to have similar characteristics: the mean duration of carriage was 23 days (95% credible interval (CI): 21, 25 days) for vaccine serotypes and 22 days (95% CI: 20, 24 days) for non-vaccine serotypes; within a school of size 100, the Secondary Attack Rate was 1.1% (95% CI: 1.0%, 1.2%) for both vaccine and non-vaccine serotypes. CONCLUSION: This study supports that, in 3-6 years old children, no competitive advantage exists for vaccine serotypes compared to non-vaccine serotypes. This is an argument in favour of important serotype replacement. It would be important to validate the result for infants, who are known to be the main reservoir in maintaining transmission. Overall reduction in pathogenicity should also be taken into account in forecasting the future burden of pneumococcal colonization in vaccinated populations.

Bayes Theorem↗
PubMed2004

Comparison of methods for handling censored records in beef fertility data: simulation study.

A simulation study was conducted to compare methods for handling censored records for days to calving in beef cattle data. Days to calving was defined as the time, in days, between when a bull is turned out in the pasture and the subsequent parturition. Simulated data were generated to have data structure and genetic relationships similar to an available field data set. Records were simulated for 33,176 daughters of 4,238 sires. Data were simulated using a mixed linear model that included the fixed effects of contemporary group and sex of calf, linear and quadratic covariates for age at mating, and random effects of animal and residual error. Two methods for handling censored records were evaluated, and two censoring rates of 12 and 20% were applied to assess the influence of higher censoring rates on inferences. Censored records were assigned penalty values on a within-contemporary group basis under the first method (DCPEN). Under the second method (DCSIM), censored records were drawn from their respective predictive distributions. A Bayesian approach via Gibbs sampling was used to estimate variance components and predict breeding values. Posterior means (PM) and standard deviations (SD) of additive genetic variance for DCPEN at 12 and 20% censoring were 23.2 (3.7) and 21.0 (3.6), respectively, whereas the same estimates for DCSIM at 12 and 20% censoring were 23.7(3.3) and 21.9 (3.4), respectively. In all cases, the true value of the genetic variance was within the high posterior density (HPD) interval (95%). The PM (SD) of residual variance for DCPEN at 12 and 20% censoring were 415.7 (4.7) and 440.0 (4.8) respectively, whereas the same estimates for DCSIM at 12 and 20% censoring were 371.0 (4.3) and 365.4 (4.4), respectively. The true value of the residual variance was within the HPD (95%) for DCSIM, but it was outside this interval for DCPEN at both censoring rates, indicating a systematic bias for this parameter. Bayes Factor and Deviance Information Criteria were used for model comparisons, and both criteria indicated the superiority of the DCSIM method. However, little difference was observed between the two methods for correlations between true breeding values and posterior means of animal effects for sires, indicating that no major reranking of sires would be expected. This finding suggests that either censored data handling technique can be successfully used in a genetic evaluation for days to calving.

Animals↗
PubMed2004

Predicting survival causes after out of hospital cardiac arrest using data mining method.

BACKGROUND: The prognosis of life for patients with heart failure remains poor. By using data mining methods, the purpose of this study was to evaluate the most important criteria for predicting patient survival and to profile patients to estimate their survival chances together with the most appropriate technique for health care. METHODS: Five hundred and thirty three patients who had suffered from cardiac arrest were included in the analysis. We performed classical statistical analysis and data mining analysis using mainly Bayesian networks. RESULTS: The mean age of the 533 patients was 63 (+/- 17) and the sample was composed of 390 (73 %) men and 143 (27 %) women. Cardiac arrest was observed at home for 411 (77 %) patients, in a public place for 62 (12 %) patients and on a public highway for 60 (11 %) patients. The belief network of the variables showed that the probability of remaining alive after heart failure is directly associated to five variables: age, sex, the initial cardiac rhythm, the origin of the heart failure and specialized resuscitation techniques employed. CONCLUSIONS: Data mining methods could help clinicians to predict the survival of patients and then adapt their practices accordingly. This work could be carried out for each medical procedure or medical problem and it would become possible to build a decision tree rapidly with the data of a service or a physician. The comparison between classic analysis and data mining analysis showed us the contribution of the data mining method for sorting variables and quickly conclude on the importance or the impact of the data and variables on the criterion of the study. The main limit of the method is knowledge acquisition and the necessity to gather sufficient data to produce a relevant model.

Bayes Theorem↗
PubMed2005

An empirical Bayesian solution to the source reconstruction problem in EEG.

Distributed linear solutions of the EEG source localisation problem are used routinely. In contrast to discrete dipole equivalent models, distributed linear solutions do not assume a fixed number of active sources and rest on a discretised fully 3D representation of the electrical activity of the brain. The ensuing inverse problem is underdetermined and constraints or priors are required to ensure the uniqueness of the solution. In a Bayesian framework, the conditional expectation of the source distribution, given the data, is attained by carefully balancing the minimisation of the residuals induced by noise and the improbability of the estimates as determined by their priors. This balance is specified by hyperparameters that control the relative importance of fitting and conforming to various constraints. Here we formulate the conventional "Weighted Minimum Norm" (WMN) solution in terms of hierarchical linear models. An "Expectation-Maximisation" (EM) algorithm is used to obtain a "Restricted Maximum Likelihood" (ReML) estimate of the hyperparameters, before estimating the "Maximum a Posteriori" solution itself. This procedure can be considered a generalisation of previous work that encompasses multiple constraints. Our approach was compared with the "classic" WMN and Maximum Smoothness solutions, using a simplified 2D source model with synthetic noisy data. The ReML solution was assessed with four types of source location priors: no priors, accurate priors, inaccurate priors, and both accurate and inaccurate priors. The ReML approach proved useful as: (1) The regularisation (or influence of the a priori source covariance) increased as the noise level increased. (2) The localisation error (LE) was negligible when accurate location priors were used. (3) When accurate and inaccurate location priors were used simultaneously, the solution was not influenced by the inaccurate priors. The ReML solution was then applied to real somatosensory-evoked responses to illustrate the application in an empirical setting.

Algorithms↗
PubMed2005

Bayesian algorithms for simultaneous structure from motion estimation of multiple independently moving objects.

In this paper, the problem of simultaneous structure from motion estimation for multiple independently moving objects from a monocular image sequence is addressed. Two Bayesian algorithms are presented for solving this problem using the sequential importance sampling (SIS) technique. The empirical posterior distribution of object motion and feature separation parameters is approximated by weighted samples. The first algorithm addresses the problem when only two moving objects are present. A singular value decomposition (SVD)-based sample clustering algorithm is shown to be capable of separating samples related to different objects. A pair of SIS procedures is used to track the posterior distribution of the motion parameters. In the second algorithm, a balancing step is added into the SIS procedure to preserve samples of low weights so that all objects have enough samples to propagate empirical motion distributions. By using the proposed algorithms, the relative motions of all the moving objects with respect to the camera can be simultaneously estimated. Both algorithms have been tested on synthetic and real-image sequences. Improved results have been achieved.

Algorithms↗