Commission III, Working Group III3

MODEL PARAMETER ESTIMATION USING SIMULATED ANNEALINGOlaf HellwichChair for Photogrammetry and Remote SensingTechnische Universit¨at M¨unchen,D-80290Munich,GermanyE-mail:olaf@photo.verm.tu-muenchen.deURL:http://www.photo.verm.tu-muenchen.deCommission III,Working Group III/3KEY WORDS:Model Parameter Estimation,Image Analysis,Markov Random FieldsABSTRACTIn image analysis and image processing most methods require the selection of model parameters such as thresholds. As this task can be rather complex,it is often conducted by a human operator.Here a generally applicable automatic estimation scheme for model parameters is proposed.The parameter space is explored tofind a global optimum by simulated annealing controlled by a divergence measure resulting from a comparison of processing results and a training data set,i.e.“ground truth”.In comparison with an interactive selection of parameter values the automatic estimation has advantages when parameters are to be determined from many data sets,or in a parameter space which is not intuitively accessible.Furthermore,interdependencies between parameters can be investigated,e.g.to avoid overparameterization. The parameter estimation method can be interpreted as a means of learning in order to adapt the analysis method to different data sets.The estimation method is applied to two Markov randomfield models for line ing examples, its convergence properties are demonstrated.1INTRODUCTIONGiven is the following scenario in image analy-sis/processing:An image is to be processed with aparticular method,for instance for line extraction.Certainlythe processing results should be as good as possible.Theresult of the method depends on a model parameter vector,e.g.thresholds.Therefore,has to be set to optimumparameter values.Frequently,the optimum values arefound by trial and error which may be acceptable as longas the dimension of is low and the influence of theparameters is intuitively estimable.Y et,it is not difficult to imagine a case where an imageanalysis method depends on several model parameters,or where there is a large amount of data such that theselection of optimum parameter values might be rather acomplex procedure.In those cases a more systematic ap-proach than trial and error would be very helpful.Here amethod using simulated annealing is proposed.It was in-spired by methods for continuous minimization using simu-lated annealing(Vanderbilt and Louie,1984,Lakshmananand Derin,1989,Press et al.,1992).It requires a trainingdata set which should represent an optimum output the im-age analysis method is supposed to give,i.e.ground truth.At the beginning of the estimation the model parametersare set to initial values.These values should be reasonablycorrect,but the selection can be rather arbitrary and doesnot require special care.Then the method is applied tothe given data.The output is compared with the trainingdata.The comparison results in a divergence measure.It should be low when output and training data agree andhigh when they do not.Such a divergence measure couldfor instance be the number of misclassified pixels,e.g.inthe state of minimum energy is reached.The metal then has certain mechanical properties,like an enhanced stiff-ness.Simulated annealing is a stochastic optimization method.Its purpose is the computation of the value of a variable where a function has a global minimum.Gener-ally,is a vector,i.e.multidimensional.An initial valueof is randomly changed over several,often many iter-ations.The change of the variable from to in itera-tion can result in an increase or decrease of the function value.It is essential for simulated annealing thatis transformed using a temperature parameter such that the transformed function isflat at the beginning and steeply modulated at the end of the iterations.Such a transfor-mation is frequently conducted according to the Boltzmann probability distribution(Press et al.,1992)(2) where C is a cooling constant.2.1Continuous MinimizationSimulated annealing is applied to various optimization taskssuch as the travelling salesman problem(Press et al.,1992)or the maximum a posteriori(MAP)estimation in Markov randomfield models(Geman and Geman,1984,Winkler,1995,Hellwich,1997).These tasks are combinatorial prob-lems where functions of discrete variables are involved.The application of simulated annealing proposed here isbelonging to the alternative class of minimization of func-tions of continuous variables(Press et al.,1992).Its main problem is the proposal of“meaningful”random changes of between two iterations.Several approaches have been proposed,e.g.(Vanderbilt and Louie,1984,Lakshmananand Derin,1989,Press et al.,1992).In this work two ofthem,(Press et al.,1992,Lakshmanan and Derin,1989),are used.(Press et al.,1992)is based on a simplex method related to(Nelder and Mead,1965).The simplex,i.e.a geometricalfigure in an-dimensional space consisting of vertices and their connections and faces,is used to determine the change of.The method gains its ef-ficiency from changing all elements of a vectorial variable at the same time.This is different in the approach of (Lakshmanan and Derin,1989)where only one element of is changed between two epochs.As the simpler,latter approach clarifies the principle sufficiently,it is described in greater detail.According to(Lakshmanan and Derin,1989),each iterationconsists of epochs treating the elements of the modelparameter vector in random order.In an epoch a new value is proposed for an element of where is the epoch index continuously increasing over the iterations and is the element index.It is equally distributed in a small intervall around the previous value.This means that is selected from the intervalwhere is the size of the interval for element.With the model parameter vector changed in one el-ement the method is executed on the data.Then,the di-vergence between its results and the training data is computed.The Metropolis algorithm is used to decide about accep-tance or rejection of the parameter change:A value is computed fromtriangle in2dimensions,tetrahedron in3dimensionswhere is the difference of the divergence measures of the present and the previous epoch.is computed ac-cording to the cooling schedule.If the divergenceof the changed model parameter vector is smaller than the divergence of the previous epoch,the parameter change is accepted.If the change resulted in an increase of the divergence,it is only accepted with a probability. The decision results in a model parameter vector of epochifif with probabilityotherwise(4)The decision in case is conducted with the help of a random number equally distributed in the interval. If,the changed model parameter vector is accepted, otherwise the previous one is used.In the following epoch the next element of the model parameter vector is treated.2.2ImplementationFor the implementation of the proposed model parameter estimation scheme the following algorithm was used:1.Selection of initial model parameters2.Execution of the method on the dataputation of divergence4.Optimization of the model parameters by simulated an-nealing(a)Proposal of changed model parameters(b)Execution of the method on the data(c)Computation of divergence(d)Decision about acceptance or rejection of modelparameter change(e)If stopping criterion is not fulfilled,go to4a.3RESULTSThe model parameter estimation scheme has been applied to two Markov randomfield(MRF)models for line extrac-tion(Hellwich,1997).In both cases the MRF expresses prior knowledge about lines.Thefirst model aims at the ex-traction of one pixel wide horizontal lines demonstrating the characteristics of MRF models.To optimize its parameters the simulated annealing method of(Press et al.,1992)has been used.For the second,more sophisticated model for the extraction of thin lines with arbitrary directions,the sim-ulated annealing method of(Lakshmanan and Derin,1989) has been applied.As object parameters,both methods are determining line and no-line labels for the image pixels.3.1MRF Model for Horizontal Line ExtractionIn thefirst MRF model,prior knowledge for more or less ar-tificial cases is used.It states that there are primarily con-tinuous,horizontal,one pixel wide lines.Consequently,it penalizes end points of horizontal lines and vertically neigh-bouring line pixels using energy parameters and,re-spectively.For the grey value data,a stationary normal dis-tribution is assumed for both line pixels and no-line pixels. They are characterized by their mean values and standard deviations.The data evaluation is a pixel-wise classification using the normal distributions as likelihood functions.The line extraction method has been applied to a simulateddata set.Figure1a)shows the true object parameters:linepixels in black and no-line pixels in white.Figure1b)dis-plays the simulated image data.The normal distributionsused have parameters and for line pixels, and and for no-line pixels.Figure1c) contains the results of a maximum likelihood(ML)estima-tion based on the known normal distributions.In this casethe prior knowledge in form of the MRF model has not beenapplied.The number of misclassified pixels is45.Figure1d)contains the results of a maximum a posteriori(MAP) estimation using the MRF model for a priori probability den-sity function.This reduces the number of misclassified pix-els to15.The MAP estimation is based on a Gibbs sampler in combination with simulated annealing(Geman and Ge-man,1984,Koch and Schmidt,1994,Winkler,1995).The Gibbs sampling has been conducted over10000iterations. In the MRF model the parameters and have been used;simulated annealing has been conducted with a cooling constant of the logarithmic cooling sched-ule according to(2).These three parameters have been estimated using the model parameter estimationscheme.a)“True”objectparameters b)Observed histogram-equalizedgreyvaluesc)MLestimation d)MAP estimation Figure1:MRF model for horizontal line extraction.The results are generated without model parameter estimation.For the implementation of the simulated annealing method (Press et al.,1992)the example given in(Vetterling et al., 1992)has been adapted.No algorithmic changes were re-quired;the temperature parameter was set to and the iteration parameter to.The number of misclassified pix-els was used as divergence measure.For the model pa-rameter estimation the number of Gibbs sampling iterations was reduced ing the initial values,and the optimized values and were found.They give a result with misclas-sified pixels which means that they describe a global mini-mum.With the initial values,andanother global minimum is found at,and .Applying spontaneous small changes to the opti-mized parameters it was found that they are not particularly robust.This behaviour will be commented in Section4.3.2General MRF Model for Line ExtractionTo a line extraction method for Synthetic Aperture Radar (SAR)data(Hellwich,1997,Hellwich et al.,1996),the sim-ulated annealing method proposed by(Lakshmanan and Derin,1989)was applied.Here,an MRF model is used to express continuity and narrowness of lines.The SAR intensity data is evaluated using rotating templates which consist of a line zone and two side zones.The SAR in-tensity ratio is used to determine the line strength for the center pixel of a template.Three model parameters have to be set by the user:a factor enforcing line continuity, a factor enforcing line narrowness,and a threshold balancing intensity ratios of line and no-line pixels.For a simulated data set,the three model parameters have been estimated.Figure2a)shows the simulated data,and Figure2b)the“true”line and no-line pixels on which the simulation is based.Two test runs have been conducted: thefirst one started with a model parameter vector which favours no-line pixels(Fig.2c)),whereas the second one’s initial parameters strongly support line pixels(Fig.2e)).In both cases thefinal results after simulated annealing are very similar(Figs.2d)and2f)).Table1shows the param-eter values before and after simulated annealing.The dif-ferences between thefinal parameter values of both test runs are acceptable,as the percentages of misclassified pixels are approximately equal.This means that both re-sults’divergence values are equivalently close to the global minimum of the divergence function.A careful interactive selection of parameter values has shown that the values found by simulated annealing are good representations of the parameter vector at the global minimum.The converg-ing of the parameter vectors during thefirst and the second test run is shown in Figure3.test run2parameter initial valuesfinal values0.000130.0002050.0150.005550.20.299iterations0250misclassified9.70 3.95pixels%Table1:Estimation of model parameters,,and for a simulated SAR scene by simulated annealingThe model parameter estimation scheme has also been ap-plied to real SAR data.Figures4and5show the magnitude (square root of the intensity)and coherence,respectively, of the evaluated TOPSAR data set.For the image section displayed in Figure6a)a training data set was generated by interactive digitization(Figure6b)).The training data was used in two model parameter estimations by simulated annealing.In thefirst case,the model parameters,, and were estimated based on the intensity data.In the second case,intensity and coherence were combined us-ing a Bayesian approach.In addition to the previous model parameters,two further model parameters were estimated for the evaluation of coherence:balancing lineversusa)Simulated SARdata b)“True”line and no-linepixelsc)Initial result of test run1d)Final result of test run1e)Initial result of test run2f)Final result of test run2 Figure2:Model parameter estimation by simulated anneal-ing for an MRF model for line extraction from SAR data. Estimated parameters:,,no-line evidence from coherence,and weighting coher-ence with respect to intensity.Table2gives the estimated parameter values.A comparison of the values reveals that the parameters,and remain unchanged after the introduction of coherence to the evaluation process,which eases the handling of the line extraction method.Further investigations showed that the model parameters are suffi-ciently robust,as the results don’t change when the model parameters are varied slightly.Figures7and8display the extracted lines for the complete data set using the esti-mated model parameters.parameter evaluation ofintensity and coherence0.0005470.0001450.6071iterations82misclassified pixels%15.13Table2:Estimation of model parameters,,,,and for a TOPSAR scene by simulated annealing0.0050.010.0150.10.150.20.250.30.350.40.450.55e-050.00010.000150.00020.000250.00030.000350.00040.000450.00050.000550.0006Start 1End 1Start 2End 2Figure 3:Model parameter estimation by simulated annealing:sequence of parameters ,,andduring the first(continuous line)and the second test run (dashed line)Figure 4:TOPSAR scene:histogram-equalized magnitude Prior to the investigations described here,during the de-sign phase of the MRF-based lineextraction method,the model parameter estimation scheme was used to discover interdependencies between model parameters which led to a reduction of the number of necessary parameters thus avoiding overparameterization.4CONCLUSIONSIt has been shown that model parameters can be success-fully estimated using simulated annealing when a trainingFigure 5:TOPSAR scene:histogram-equalized coherencedata set and a divergence measure are available.The use of the optimized parameters leads to improved results if (1)the training data is representative for the image data to be evaluated,(2)the divergence measure describes the prob-lem properly and robustly,and (3)the results depend suf-ficiently robustly on the parameters.The third condition is only fulfilled when the global minima of the divergence func-tion are not located in narrow valleys or troughs.Therefore,it is recommended that the user of the proposed model pa-rameter estimation scheme is sufficiently aware of the prop-erties of the divergence function.It may also be possible toa)Subsection from Figure4b)Training dataFigure6:TOPSAR scene:interactivly digitized trainingdataFigure7:Line pixels extracted from TOPSAR magnitude include curvature properties of an intermediate divergence function into afinal divergence function such that global minima of the intermediate function inflat areas are pre-ferred in comparison with global minima in narrow valleys.REFERENCESGeman,D.and Geman,S.,1984.Stochastic Relaxation, Gibbs Distribution,and the Bayesian Restoration of Im-ages.IEEE Transactions on Pattern Analysis and Machine Intelligence P AMI-6(6),pp.721–741.Heipke,C.,Mayer,H.,Wiedemann,C.and Jamet,O.,1997. 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