Evolutionary Optimisation Methods for Template Based Image Registration

Transcription

1 Evolutionary Optimisation Methos for Template Base Image Registration Lukasz A Machowski, Tshilizi Marwala School of Electrical an Information Engineering University of Witwatersran, Johannesburg, South Africa. l.machowski@ee.wits.ac.za t.marwala@ee.wits.ac.za Abstract This paper investigates the use of evolutionary optimisation techniques to register a template with a scene image. An error function is create to measure the corresponence of the template to the image. The problem presente here is to optimise the horizontal, vertical an scaling parameters that register the template with the scene. The Genetic Algorithm, Simulate Annealing an Particle Swarm Optimisations are compare to a Neler-Mea Simplex optimisation with starting points chosen in a pre-processing stage. The paper investigates the precision an accuracy of each metho an shows that all four methos perform favourably for image registration. SA is the most precise, GA is the most accurate. PSO is a goo mix of both an the Simplex metho returns local minima the most. A pre-processing stage shoul be investigate for the evolutionary methos in orer to improve performance. Discrete versions of the optimisation methos shoul be investigate to further improve computational performance. 1. Introuction Image registration has great practical application in the fiel of computer vision, meicine, remote sensing an image watermarking [1][]. Being able to etermine how best the template image fits into the scene poses several problems that have to be overcome. The registration process may involve shifting, scaling, rotation, perspective projection or other non-linear transformations. The shear number of possible transformations makes it ifficult to automate the process an usually requires a person to verify the results manually. This paper presents finings on the use of evolutionary optimisation methos for automating the template matching of -imensional intensity images. 1.1 Image Registration Image registration is the process by which a template is oriente in such a way as to match an entire, or a portion of, a given scene [1][]. The template is transforme in such a way as to match the scene as closely as possible. There are four main steps require for registration of an image. These are feature etection, feature matching, transform moel estimation an image transformation [1]. Feature base etection makes it easier to etermine the orientation of the template with respect to the scene. Area base etection methos are much more computationally expensive ue to the amount of ata that nees to be processe. Since the area base etection methos epen on the appearance of the images, they are intolerant of changes in illumination an ambient conitions [1][]. The feature base etection methos o not suffer from this but it is more ifficult to automatically extract the features for any general image. It is common to combine the avantages from both methos to form a hybri approach to the registration process []. Correlation-like methos are typically use for area-base etection methos where a correlation surface is calculate for the template an the maximum point is foun an interprete as the best fit for the template [1]. This metho is aversely affecte by self similarity in the image an it is characterise by high computational complexity. It also oes not allow much variance in template rotation or other more complex transformations. This approach, is still however attractive for real-time object tracking [1][3]. An alternative to cross correlation is to use optimisation to fin the best fit for the template in the scene [4]. The avantage of this approach is that one can apply more complex transformations to the templates, an thus make the metho robust when compare to cross correlation. This metho also requires less computation because the entire correlation surface oes not have to be etermine. In this paper, we investigate the use of a Genetic Algorithm (GA), Simulate Annealing (SA) an Particle Swarm Optimisation (PSO) to register a template with a given scene. These methos are also compare to the Neler-Mea Simplex metho. For simplicity, only three transformation parameters are efine. These are horizontal translation, vertical translation an uniform scaling.

2 1. Evolutionary Optimisation The term evolutionary refers to the fact that the optimum solution graually evolves from a population of iniviuals that share information an have group ynamics [5]. This is in contrast to the non-evolutionary or classical optimisation methos which always try to travel in the best irection. Typically, the evolutionary concept is linke with GA alone but in this paper, we group GA, SA an PSO into the subset. All evolutionary optimisation methos have the following operations [5]: Evaluation Selection Alteration An initial population of iniviuals is initialise, covering the parameter space an the objective function is evaluate for each iniviual. From this ata, a subset of iniviuals is selecte an altere to form new iniviuals. The egree to which each of these operations is performe in GA, SA an PSO varies from algorithm to algorithm. To fin the optimal registration parameters for template matching, it is important to construct a multivariate cost function that represents how well the template matches the scene [6]. The traitional techniques for optimisation make use of the objective function value, first erivative or its secon erivative [6][7]. The general approach for all non-evolutionary optimisation methos is to select an initial guess for the registration parameters an travel in a irection as to improve the objective function. Once a suitable irection is foun, it is possible to make either fixe or varying successive steps towars the local optimum. Evolutionary Optimisation methos, however, o not make use of any other information but the objective function values themselves. This eliminates the evaluation of graients which may be expensive an misleaing for image registration. Evolutionary methos typically sample the search space significantly more than the non-evolutionary techniques but this improves the probability of the algorithm fining the global optimum point. The evolutionary algorithms are typically base on the processes which occur in the natural worl, such as genetics, the swarming behaviour of bees an the annealing of metals. Various ata structures are use to simulate these [5]. A brief review of the methos suitable for image registration is given below: 1..1 Simplex Metho This is a non-evolutionary (classical) metho. A simplex is a geometric figure that has one more vertex than the number of imensions in the parameter space (a triangle in two imensions, as shown in Figure 1). The objective function is sample at each vertex an the one that has the worst value gets remove from the simplex. A new vertex is then create by reflecting the simplex about the remaining points. Depening on whether the fitness of the new point improves or not, the simplex is expane or contracte to look for a more precise solution. In this manner, the algorithm steps its way towars the local optimum point. This metho is relatively robust when use for iscontinuous objective functions [8]. x Figure 1 A simplex in two imensions, showing a reflection. 1.. Simulate Annealing Simulate Annealing is a Monte Carlo Technique [5] an is base on the analogy of metals cooling slowly to form a crystalline structure with low energy. The SA introuces a probability of acceptance of a new sample point for a hill-climbing process. The probability of accepting the new point is base on a control variable calle the temperature. The higher the temperature, the more likely it is for a worse point to be accepte. This allows the algorithm to escape from local optima [5][9]. This algorithm is commonly referre to as Metropolis after its founer. The SA technique use in this paper moifies the simplex metho escribe above to allow the simplex to accept a worse vertex with a probability istribution that is base on the temperature. If the change in energy is negative (we have a better point) then the new point will always be chosen. If the change in energy is positive (the point is worse) then the probability of accepting it is given by: E kb. T Reflecte Vertex p = e (1) where E is the change in energy, k B is Boltzmann s constant an T is the current temperature. The cooling scheule (how many iterations to spen at each temperature) is an important factor in the success of the algorithm. x 1

3 1..3 Genetic Algorithm Base on the theory of genetics, the GA encoes each iniviual in the population with a chromosome [5][]. This encoing represents the parameters for the objective function being optimise. There are several ifferent techniques for encoing parameters, performing the selection, an the alteration stages of the algorithm. The alteration stage is separate into Crossover an Mutation. The metho use in this paper selects a ranom sample of parents from the population with a specifie probability. An arithmetic crossover is then performe on these iniviuals which creates chilren base on a linear interpolation of the two parents. This is shown in Figure. where v (t+1) is the next velocity for particle, α is an acceleration constant, β i is an attraction constant for the iniviual best position, β g is an attraction constant for the group best position, p i, is the best iniviual position for particle, p i,g is the best group position for particle s neighbourhoo an x (t) is the current position for particle. The position of each particle is then calculate for each iteration as: x ( t + 1) = x ( t) + v ( t + 1) (3) The various components that make up the velocity for each particle are shown in Figure 3. x c 1 = p 1.(mix) + p.(1-mix) c = p 1.(1-mix) + p.(mix) x p i p g p 1 c 1 x V (t) x 1 eg: Mix =.6 Figure One Dimensional Arithmetic crossover operator. A multi-non-uniform mutation is performe which moifies the parent parameters with a binomial istribution which narrows as the number of generations gets larger. More etails of the iniviual techniques are given in [5] Particle Swarm Optimisation PSO is base on the swarming behaviour of bees, flocking of birs, schooling of fish an social relations of humans [11]. A population of particles is ranomly initialise within the parameter space an each one is given an initial velocity. At each iteration of the algorithm, the particle position is upate an a new velocity is calculate, taking the best position for the particle an the group into account. There are ifferent ways of grouping the particles together. The metho use in this paper creates a social grouping where each particle has n logical neighbours (reference by ajacent inex numbers). This means that particles can be neighbours even though they are not close to each other spatially. This metho also tens to prouce better global exploration by the particles since there are many more attractors. The velocity of each particle is calculate as: α. v( t) v + = ( t 1) + βi.( pi, + β g.( p g, c x ( t)) x ( t)) p () Figure 3 Various components making up the new velocity.. Metho.1 Objective Function In orer to perform optimisation, it is necessary to efine an objective function that captures the essence of the problem at han. In image registration, one wants to maximise corresponence between the scene image an the template pose at its current position. The corresponence can be measure as the sum-square ifference between the intensities of overlapping pixels. This can be expresse as an error function where a value of zero represents a perfect match. The parameters to be optimise in this problem are horizontal translation (x), vertical translation (y) an uniform scaling (s). The objective function use when there are overlapping pixels between the template an the scene image is given by: error = ( A T) numel( A) (3) where A is the scene image, T is the template, the refers to element-wise squaring an the summation is over each element of the resultant matrix. This error is then normalise with the number of pixels that are overlapping between both images. It is also important to interpolate sub-pixel values for the optimisation algorithm to be able to function correctly. This allows the traitional algorithms to be run unchange an also allows it to be compare to other general optimisation algorithms.

4 It is necessary to penalise the error function when there are template pixels that o not lie within the image. This error component is ae to the existing value calculate above. The penalise error then becomes: error p = error + OutPixels c (4) where error is from equation (3), OutPixels is the number of template pixels that o not lie in the image, an c is a penalisation constant which shoul be large (~). This penalisation has the effect of constraining the x an y parameters back into range when the images no longer overlap. It is necessary to har-limit the scale parameter because the optimisation algorithms might try riiculously high values which require extremely large amounts of memory. Very selom oes a template match a scene at very high scaling values. Similarly, if the template gets scale to one pixel in size, then a fit can be foun nearly anywhere in the scene.. Test Image The test image use is the familiar picture of Lena, which has a goo mix of various image features an provies several local minima for registration. The template is a cut-out of Lena s face an is shown in Figure 4. The image scene is 56x56 pixels an the template is 17x138 pixels taken from the coorinates (151.5, 151.5) with a scale of.. This means that the global optimum for our objective function is at the coorinates (151.5, 151.5,.5) with an error value of. (a) Figure 4 a) Test Image b) Template for registration..3 Optimisation This section escribes implementation etails for each of the optimisation algorithms. All routines are implemente in Matlab R13. The implementation etails for each algorithm are given below:.3.1 Simplex Metho Matlab s Optimisation Toolbox is use to perform the Neler-Mea simplex optimisation [8]. The function use is fminsearch.m. A simple pre-processing algorithm is use to choose initial starting points. (b).3. Simulate Annealing The SA algorithm use in this paper is base on the coe given in [1] an is extene to inclue restarts. The coe is abstracte into a higher level in orer to take avantage of Matlab s matrix arithmetic capabilities. The following is a high level escription of the algorithm use: x = RanomStartingSimplex(); y = EvaluateSimplex(S); for each temperature in cooling scheule: for number of iterations: yfluc = AFluctuation(y); sort(yfluc); ReflectSimplex; If better than best then ExpanSimplex(x) Else if worse than n highest: ContractSimplex(x); If still ba then: ContractOtherVerices(x); En En If SimplexIsStuck() then: % Restart, keeping best point: x = RanomStartingSimplex(); KeepBestPoint(x); En En for each iteration En for each temperature The restart allows the algorithm to oscillate at a local minimum for only a limite number of iterations. After this, it is restarte with the best point as one of the vertices. This behaviour is justifie because it is recommene in [1] that the algorithm be re-run in the same fashion once a solution is foun. This merely allows the simplex to further explore the parameter space..3.3 Genetic Algorithm The Genetic Optimization Toolbox (GAOT) [13] is use for the implementation of the GA. This is an extensive toolbox with many functions for the encoing, selection, crossover an mutation operators. The following operators were use for the optimisation: normgeomselect.m, with the probability of selecting the best, set to.6. arithxover.m, with crossovers per generation multinonunifmutation.m, with mutations per generation. The particular selection operator use gives a goo mix of exploration an precision. The arithmetic crossover is very useful in this problem since it assists in fining more precise parameters for the objective function. It is important to use the given mutation operator so that a sufficient amount of exploration occurs. The nature of the image registration problem creates an objective function with many local minima so it is important to mutate out of these valleys. Since the GAOT maximises an objective function, we merely multiply our original objective function by a factor of -1 to perform the minimisation.

5 .3.4 Particle Swarm Optimisation After evaluating the performance of a free PSO toolbox, an getting poor results, it was ecie to write a custom PSO routine base on the metho escribe in the previous section. The high level escription of the algorithm is given below: Swarm = CreateRanomSwarm(); While we have more iterations to go: EvaluateObjectiveFunction(Swarm); If Swarm.BestValue<GlobalBestValue then GlobalBestValue = Swarm.BestValue; GlobalBestPosition=Swarm.BestPosition; En UpateIniviualAnGroupBestValues(Swarm); CalculateParticleVelocities(Swarm); UpateParticlePositions(Swarm); En While loop Output GlobalBestPosition; The initial swarm is create with ranom particles between the bouns of the parameter space. A recor is kept of each particle s best value that it has sample. Similarly, a group-best is maintaine for a social neighbourhoo size of 3 particles to either sie of the current particle. Moulo inexing is use. The α parameter (escribe in the previous section) is set to.99 so that the swarm oes not become unstable an iverge. Both β parameters are set to.1 in orer that the particles approach the best locations graually. This samples the objective function many times along the trajectory of the particle. 3. Results 3.1 Optimisation In orer to be able to compare the algorithms, the maximum number of function evaluations for each metho is set to. The algorithms are stoppe as close to this value as possible (since the number of function evaluations may vary from run to run). The amount of time taken by each algorithm is not a goo measure of its performance in this case because the amount of processing in the objective function is highly epenent on the parameter values being sample. All four algorithms escribe in this paper ultimately approach the global optimum so a suitable measure for their performance is to investigate their precision an accuracy. Each algorithm is run 5 times an an accuracy histogram is calculate for how close an how consistently the algorithm reache the global optimum of (151.5, 151.5,.5). The istance is measure geometrically by the following equation: i = ( xg xi ) + ( yg yi ) + ( sg si ) (5) where i is the istance for the i th run, (x G, y G, s G ) are the global optimum parameters an (x i, y i, s i ) are the optimum parameters as calculate in the i th run. The histogram is ivie into equal bins so that the last bin has a istance which is units away from the global optimum. The accuracy histograms for the various methos are given in Figure 5 an the results are analyse below. Higher counts towars bin zero are better Simplex GA Figure 5 Accuracy Histograms Simplex Metho This metho is especially robust for functions that have peculiar graients or that are iscontinuous, as is common in image registration. The metho of travel by the simplex acts as a pseuo-graient that plays a similar role as in the graient methos. The success of this algorithm is attribute to the pre-processing stage of the algorithm which selects suitable starting points for the Neler-Mea optimisation. The histogram shows that when the algorithm is near the global optimum, it is reasonably precise. It oes however, fin local minima quiet often Simulate Annealing It is expecte that this algorithm performs better than the Simplex Metho escribe above ue to the Metropolis metho that it employs. It was foun that the performance of the algorithm is significantly improve by restarting when the simplex becomes stuck. Without this behaviour, the algorithm tries oscillatory values which waste precious computation time since the samples o not avance the simplex at all. The histogram shows that the algorithm is the least accurate (repeatable) algorithm of the four but is precise (fins values near to the global optimum many times) Genetic Algorithm The performance of the GA can be attribute to the number of mutations that take place. This is important for image registration because the objective function has many local optima that vary in shallowness. It is more SA PSO

6 important for the algorithm to explore the parameter space than it is to improve the precision (using the arithmetic crossover). It was also observe that the mutation rate is more important than the initial population size (which ranomly explores the parameter space), since sufficient mutations will explore the space more wisely. The histogram shows a clear binomial istribution near the global optimum. This shows that the GA prouces reasonably accurate (consistent) results Particle Swarm Optimisation The PSO prouces acceptable results because of the group ynamics in the system. The social groups promote exploration of the search space while the iniviual best position lets each particle improve its precision. Goo parameter values were foun by investigating what effect they have on the swarm behaviour an then tweaking the values to suite the problem omain. The behaviour of the swarm is preictably base on the algorithm parameters so it is relatively easy to infer goo parameter values by watching how the particles swarm in the objective function. The histogram shows that the algorithm fins precise results but not always accurately. 3. Other Sample Data The algorithms manage to register the templates of istorte an noisy images of Lena s face to the image. The various moifications to the templates that were mae inclue, Gaussian blurring, the aition of Gaussian noise, vortex rotation, smuging an non-uniform stretching. The characteristics an accuracy histograms for each algorithm remain relatively consistent with the results escribe in the previous section. 4. Recommenations The current objective function makes use of sub-pixel sampling to obtain a continuous parameter space. This is very computationally intensive an unnecessary for certain parameters in image registration such as x an y coorinates. The algorithms presente above shoul be moifie so that the user can specify that certain parameters may vary iscretely whilst others continuously. This shoul reuce the amount of time taken for each function evaluation. Another improvement to the algorithms woul be to select regions of interest that are likely to contain the template. This requires looking for higher level features in the image first an then creating bouns for the optimisation algorithms. 5. Conclusion This paper introuce evolutionary optimisation methos an contextualise them in the image registration fiel. It was foun that the algorithms perform well in ifferent aspects of the image registration process. The parameters for each metho nee to be tune to suite the given image but the behaviour of the methos presente in this paper are intuitive an insight into how to moify parameters can easily be gaine by simulating a few initial runs. These evolutionary optimisation methos were compare to the non-evolutionary Neler-Mea simplex metho with starting points selecte by oing higher level preprocessing. SA returns the most precise results, while GA returns the most accurate results. PSO is in between these two an it is followe by the Simplex metho. Preprocessing shoul be investigate for the evolutionary methos. 6. References [1] B. Zitova, J. Flusser. Image registration methos: a survey ; Elsevier B.V.; Image an Vision Computing 1 (3) 977 ; June 3. [] X. Peng, M. Ding, C. Zhou, Q. Ma. A practical twostep image registration metho for two-imensional images ; Elsevier B.V.; Information Fusion; Article in press; 4. [3] S. Chien, S. Sung. Aaptive winow metho with sizing vectors for reliable correlation-base target tracking ; Elsevier B.V.; Pattern Recognition 33 () 37}49; [4] H. Lim, S. Hossein Cheraghi. An optimization approach to shape matching an recognition ; Computers & Electrical Engineering 4 (1998) 183-; [5] Z. Michalewicz. Genetic Algorithms + Data Structures = Evolution Programs 3r E. ; Springer; New York; [6] E. Chong, S. Zak. An Introuction to Optimization, n E. ; Wiley; 1. [7] I. T. Nabney. Netlab: Algorithms for Pattern Recognition ; Springer; UK;. [8] Optimization Toolbox User s Guie V ; The Mathworks Inc.;. [9] C. Skiscim. Optimisation by Simulate Annealing ; Proceeings of the 1983 Winter Simulation Conference; pp ; [] C. Houck; J. Joines; M. Kay. "A Genetic Algorithm for Function Optimization: A Matlab Implementation"; NCSU-IE TR 95-9, [11] J. Kenney, R. C. Eberhart. Particle swarm optimization, Proceeings of the 1995 IEEE International Conference on Neural Networks, vol. 4, pp IEEE Press; [1] W. Press, W. Vetterling, S. Teukolsky, B. Flannery. Numerical Recipes in C++, n E. ; Cambrige University Press;. [13] Genetic Optimization Toolbox ; Last Accesse: 11/5/4.