Genetic Algorithms for Free-Form Surface Matching

Genetic Algorithms for Free-Form Surface MatchingK.Brunnstr¨o m and A.J.StoddartDept.of Electronic and Electrical EngineeringUniversity of SurreyGuildford,Surrey GU25XH,UKAbstractThe free-form surface matching problem is important inseveral practical applications,such as reverse engineering.An accurate,robust and fast solution is,therefore,of greatsignificance.Recently genetic algorithms have attractedgreat interest for their ability to robustly solve hard opti-mization problems.In this work we investigate the perfor-mance of such an approach forfinding the initial guess ofthe transformation,a translation and a rotation,between theobject and the model surface.This would be followed by alocal gradient descent method such as ICP to refine the es-timate.Promising results are demonstrated on accurate realdata.1.IntroductionThe free-form surface matching problem remains a diffi-cult problem.Apart from being interesting in its own right,there are a number of direct applications.These include in-spection and recognition of free-form surfaces from densedepth data.In these problems correspondences need to beestablished between models and the data.For example inreverse engineering processes range data is acquired by arange scanner.Separate views need to be registered andfused[11].The registration is usually based on gradient-descent e.g.the iterative closest point(ICP)algorithm.Tofind the right solution the user must input an initial guess.For these applications an accurate,robust and fast solutionis of vital importance.The problem of matching two free-form surfaces can becast as a search or an optimization problem.This leads toa6dimensional optimization problem with many local ex-trema.Hence,a gradient-descent method would most likelyThere are very few algorithms on the market that con-vincingly address the problem of free-form surface match-ing without an initial guess.One algorithm that apparently does solve the problem is based on a multi stage algorithm of some complexity[1].It requires in the region of30min-utes to compute a solution.Besl’s survey article[2]on free-form surface matching convincingly specifies the problem.It cites111other arti-cles almost none of which actually address the general free-form matching problem.Those that do,assume that reliable features(e.g.zero crossings of curvature)can be extracted. We make no such assumption.This paper,which is a shortened version of the technical report[4],is organized as follows.First we present the ge-netic algorithm for free-form surface matching,assuming that the reader is familiar with how a genetic algorithm is working in general.For such a description see e.g.Goldberg [6].We then describe our experiments and results.Finally we conclude with a discussion.2.The genetic algorithmTo use a genetic algorithm for solving our problem,it must be formulated in a suitable way.First we need to con-vert the continuous problem of free-form surface matching to discrete form.This is done by randomly sampling the model surface to a‘sufficient’density,and randomly sam-pling the scene surface to some other density.We now pose the problem as a point matching problem.We wish to la-bel(associate)each point in the scene with a point from the model,or a null label.Because the points are randomly chosen we do not expect an exact match.However,when sampled to a sufficient density we hope to match to a rela-tively close point.We also need a way to measurefitness of the chromosomes.The chromosomes represents an assign-ment of model points to scene points.Thefitness function used will be described in detail below.It is based on a func-tion that counts the number of good matches,by using the translational and rotational invariants such as relative ori-entation of surface normals and relative distances between points.Finally,we must decide on when a suitable result has been obtained.Thefitness function measures the num-ber of good matches.Ideally when all points are correctly matched,it will correspond to the number of scene points. However,due to the random sampling,noise and occlusion this cannot be achieved in practise.Furthermore,there is no way from within the genetic algorithm to check whether we have found the right solution or not.In thefield of GA’s the choice of termination is still an open question.We have investigated three different approaches:the optimization is stopped when a certain percentage of the total number of correct matches have been found;the maximumfitness has not increased for afixed number of generations;or it is ter-minated after a certain number of generations.There are a few free parameters that need to be selected. The number is small compared to many other methods1. The parameters are:population size i.e.the number of chro-mosomes;the crossover probability;the mutation probabil-ity;the stopping criteria;and one parameter in thefitness function,which we call the temperature.The problem of selecting parameters for the GA’s is an open question.They are usually chosen heuristically based on experience of the user community.This has guided us in our choice of parameters.2.1.Thefitness functionA genetic algorithm needs a cost function orfitness func-tion to be defined for a given problem.In this Section we develop afitness function appropriate for the free-form sur-face matching problem.Because thefitness must measure match quality in corre-spondence space it is designed to be invariant to translation or rotation of either the scene or model.It should be robust in the presence of clutter as well.Because of the random sampling we can never achieve an exact match but there will usually be reasonably“good”approximate matches.The quality of the best match will be a function of the average spacing between the randomly selected points on the surface.In genetic algorithms the choice of afitness function needs careful consideration.The convergence of the algo-rithm depends crucially on relative sizes of this over solu-tion space.A choice we make is to build up a cost that has some meaning,in particular we aim tofind an interme-diate quality measure which counts the number of good matches.This will then be transformed into which will be thefitness passed to the GA.We suppose that the model points are labeled by and the scene points are labeled by or .If scene point takes on label this is denotedby.We denote a joint labeling of all objects by the set.As a shorthand we define the multi-index.Now suppose that we choose2points from the scene surface.They will contain4vectors(2positions and2nor-mal vectors)of information between them,i.e.. There are10independent parameters and4invariants under rigid translation and rotation.We are interested in features that remain invariant to rotation and translation.The most obvious is the length of the vector from point to point ,i.e..Pairwise relative orientation is also an invariant.Finally,there will be a twist angle between thetwo normals representing the extent to which they are out of plane.Thus the four invariants are(1)(2)(3)(4)Now consider two model points and two scene points. We wish to define a pairwise match quality which will be a product over four terms.Thefirst termcan then be expressed as.Where is the density of points on the sur-face,which we estimate to be the area of the surface divided by the number of randomly sampled points.This agrees well with the empirical average closest distance.The angular quality measure is defined as(10)The form of the expression and the parameter(tempera-ture)are intended to bear a close resemblance to the formal-ism of the simulated annealing method of optimization.We believe that this is a good form for thefitness func-tion.Evolutionary pressure in GA’s is exerted by the ratio of costs,and the exponential form of thefitness ensures that each gain of correct matches is accompanied by an in-crease infitness of a factor.A few values of were tested,with decreasing performance for and no real improvement for values,so we have used the value throughout this work.2.2.Finding the transformationThe result of the GA is a good labeling,but we actually want a transformation,a translation and a rotation,which takes the object data set into the model.When the genetic algorithm are stopped,the chromosome of the best scoring individual will contain both good and bad matches.If we can pick out the subset that just contain good matches we could use a standard method tofind the transformation.This could indeed be done by looking at the point-wise matching 3score,Equation8.If this is close to we can use it for calculating the transformation.In this work we have simply been using a threshold,typically set to about30%of.The transformation is then found by least-squaresfitting, which is solved using a SVD-based based method[8].plexityThe core computation is thefitness function which is evaluated for each individual in the population,to deter-mine its relative strength.As can be seen by equation9, this computation will be of.Consider the situation that we have a population of in-dividuals and the algorithm is stopped after generations. This will give an overall complexity of .(g)(h)(i)(j)(k)(l)Figure2.High precision3D data of a bust of Beethoven produced by Viewpoint Animation Engineering.3.Experiments and ResultsWe will here present a few experiments illustrating the strengths and weaknesses of the presented approach to free-form surface matching.It should be stressed that our ob-jective is not accurate registration,rather it is to develop a technique tofind an approximate pose.The parameters of the algorithm have been kept constant throughout the exper-iments.In one case the number of points has been increased to obtain a better match.Both results are presented.In the standard case the population has consists of individu-als,each having points i.e.(see Section2.1). The genetic algorithm searches for matches among la-bel points i.e..The crossover probability was and the mutation rate per label.The temperature has beenfixed at.In all cases the GA was run for1000gen-erations.The GA specific parameters have been obtained by testing a few values around those given in Goldberg’s book about GA’s and selecting the set with the fastest con-vergence for a particular example.The computation time in the examples with these parameters is about2minutes on a AlphaStation 250.(a)(b)(c)(d)(e)(f)(g)(h)(i)Figure3.High precision3D data of a cow produced by Viewpoint Animation Engineering.d)-f)Shows the re-sult when using the above described set of parameters, except that the number of labels for the GA have been doubled to reflect that the model surface has double the area.g)-i)The result when usingand.In thefirst example,Figure1,an image of a rabbit ob-tained by a laser rangefinder,Cyberware,have been used. The scene,1(a),and the model,1(b),have been taken from different directions,with a more than50%overlap,1(c). The resulting transformation applied to the scene is shown in Figure1(d),and in1(e)and1(f)it is shown together with the model from two different viewing directions.The data was used in Turk and Levoy[11]and made available by ftp. It has been registered with an ICP based algorithm starting from a user supplied initial position.The ICP transforma-tion is given as comparison to the GA computed transfor-mation in the table of Figure5(a).As can be seen there are good agreement between the values computed by the two methods.In the next example,Figure2,we have been using3D data of a bust of Beethoven.The object or the scene is shown in Figure2(a).The model is a translated and rotated version of the object,Figure2(b).This data was produced by Viewpoint Animation Engineering using point measure-ments of high accuracy.The two data sets are shown to-gether in Figure2(c).Applying the computed transforma-tion to the object,the result can be seen in Figures2(d)-(f). The table in Figure5(b)gives the computed transformation and the errors in these parameters.In Figure3the model is a cow2,3(b).The scene,3(a), has been generated by cutting out half the cow from head to tail and then adding a rotation and a translation to its posi-tion.The match in thefirst example has been obtained byusing the same parameters as above,except that we have increased the number of labels to reflect the fact that the size of the model is double that of the object. The transformation found was not very good,Figure3(d)-(f)and5(c)(Estimate1).This is improved considerably, Figure3(g)-(i)and5(c)(Estimate2),by increasing the num-ber of points on the object surface as well as the number of labels points on the model surface.In thefinal example,Figure4and5(d),the front legs of the cow above have been cutout and rotated relative to each other.A rather surprising solution is found,even though we expected the algorithm to have problems in this case.The points around the cylindrical part of the body will dominate the solution,which make the algorithm stuck in a local op-timum.(a)(b)(c)(d)(e)(f)Figure4.High precision3D data of a cow produced by Viewpoint Animation Engineering.The scene and model are cut out pieces from the front leg region of the cow.Note the near cylindrical shape of most of the data.4.DiscussionWe have in this paper presented a simple approach,with promising results,based on genetic algorithm,forfinding the initial guess for the free-form matching problem.The basic steps of the algorithm are:1)Randomly sampling the object and model surface.2)Search for a solution with a genetic algorithm.3)Calculating the transformation from the matched points.Our goal is tofind a pose in the close vicinity of the global ually thefitness function has many lo-cal optima.We use a stochastic search technique that can in principle locate the global optimum to any desired prob-ability.In this sense it resembles Simulated Annealing.Naturally there is a tradeoff between the success rate and the resources allocated.On the datasets we studied we ob-tain good results in2-10minutes on a workstation.The results fall short of the performance that would be desired in practical applications such as multi view fusion of range images.However we believe that there is consider-able scope for refinement of the algorithm.Recent advancesin thefield of Genetic Algorithms suggest that small modi-fications can significantly improve their performance.On the other hand almost all optimization algorithms benefit from the inclusion of domain-specific knowledge into the basic algorithm.This we have not yet done,but seems likely to lead to a much more efficient algorithm.Work is in progress on these points.ParameterICP estimate0.000.019-0.0130.0380.039-1.0031.4b) 4.00-1.830.00180.0Estimate8.95-0.99-0.091True0.71 1.000.00-0.91-2.43-0.08521.2 Estimate2 1.15 1.000.010True 2.00-1.000.003.40 2.74-0.91171.5Figure5.The computed transformation for the example shown in Figure1,2,3and4.Thefirst three columns represent the computed translation.The next three the rotation axis and the last the rotation angle.a)Result for the example in Figure1b)Figure2c)Figure3d) Figure4References[1]R.Bergevin,urendeau,and D.Poussart.Registeringrange view of multipart objects.IEEE Trans.Pattern Analysisand Machine Intell.,61(1):1–16,Jan.1995.[2]P.Besl.The free-form surface matching problem.In H.Free-man,editor,Machine Vision for Three-Dimensional Scenes,pages25–69.Academic,New York,1990.[3]P.Besl and N.McKay.A method for registration of3-dshapes.IEEE Trans.Pattern Analysis and Machine Intell.,14(2):239–256,1992.[4]K.Brunnstr¨o m and A.J.Stoddart.Genetic algorithmsfor free-form surface matching.Technical Report ISRNKTH/NA/P--95/19--SE,Dept.of Numerical Analysis andComputing Science,KTH(Royal Institute of Technology),Oct.1995.[5] A.Cross and E.Hancock.Genetic search for structuralmatching.In R.Cipolla,editor,Proc.4th European Confer-ence on Computer Vision,volume1064of Lecture Notes inComputer Science,pages514–525,Cambridge,United King-dom,Apr.1996.Springer Verlag,New York.[6] D.Goldberg.Genetic algorithms in search,optimization andmachine learning.Addison-Wesley,Reading,Mass.,1988.[7]J.-J.Jacq and C.Roux.Registration of3-d images by geneticoptimization.Pattern Recognition,16(8):823–856,1995.[8]K.Kanatani.Analysis of3-d rotationfitting.IEEE Trans.Pattern Analysis and Machine Intell.,16(5):543–549,1994.[9] A.Papoulis.Probability,Random Variables,and StochasticProcesses.McGraw-Hill,Inc.,Singapore,1984.[10] A.J.Stoddart,J.Illingworth,and T.Windeatt.Optimal pa-rameter selection for derivative estimation in range images.Image and Vision Computing,13(8):629–635,1995.[11]G.Turk and M.Levoy.Zippered polygon meshes from rangeimages.In Proc.SIGGRAPH’94,In Computer GraphicsProceedings,Annual Conference Series,pages311–318,Or-lando,Florida,1994.5。