Full Camera Calihration from a Single View of Planar Scene 819 a unique focal length corresponding to specified p.v.and v.which can be computed from the following cquation [3]. f=V-(x-0j(x'-0)-(y-o(-0) (7) Different orthogonal vanishing point pairs lead to different estimated focal lengths. Therefore,for each guessed principal point p and a set of orthogonal vanishing points V=1,{2,n,,we can estimate a corresponding set of focal lengths F =f1,f2....Our basic idea is to employ the set F containing large amounts of estimated values to statistically put constraint on the principal point. We reasonably expect that,if the principal point is correctly estimated,then the f values in the F set surely form a densely distributed cluster.On the contrary.if the guessed principal point is far from the correct position,then the distribution of the focal lengths computed by equation(7)is more likely quite sparse.Therefore,the distribution of the entries of the F set provides a confidence measure of the guess about the principal point ()Naturally,the variance of the distribution,D(F),is a good candidate to measure such confidence and evaluate the goodness of the guess.In other words, although the probability density function P()is hidden from us it can he measured through the observable focal length distribution D(F).Smaller D(F)corresponds to higher confidence of (o,yo).Under this formulation we can use D(F)to characterize the probability density function of the principal point and perform calibration through maximum likelihood estimate.Note that is determined by (zo.and should be more strictly written as F(ro,0).We take D(F)as the cost function and try to solve the following optimization problem: Minimize()(D(F(ro:)) (8) Under this formulation,from every guess about the principal point a confidence value can be estimated and the corresponding focal length can be computed.An optimization routine is required to seek the minimum of equation (8).which corresponds to the max- imum likelihood estimate of the intrinsic parameters (o,2o,f).In our study the above statistical function is not easily differentiated analytically.So a derivative-free opti- mizer is preferred.The downhill simplex method is a good candidate for this type of optimization [16].In addition,Experiments show that a lot of local minimums exist in the solution space.We solve this problem by employing multiple initial points.Namely. the optimization is repcated several times with multiple rundomly chosen starting points and the best result produced is adopted as the final solution.This strategy ensures the reliability and robustness.After the principal point (u,v)and the focal length f are de- termined,the conic based pose estimate algorithm in 4,8]is employed to calculate the extrinsic parameters.This completes a full single-view hased calibration framework. 3.3 Taking Aspect Ratio into Account Having made the above calibration algorithm work,adding an extra intrinsic parameter, i.e.,the seale factor a,becomes straightforward.All we need to do is just introduce a as a fourth unknown variable into the optimization routine.During optimization,for each guessed value of the scale factor,the image is first scaled horizontally to give a corrected Full Camera Calibration from a Single View of Planar Scene 819 a unique focal length corresponding to specified p, v, and v , which can be computed from the following equation [3]. f = −(x − x0)(x − x0) − (y − y0)(y − y0) (7) Different orthogonal vanishing point pairs lead to different estimated focal lengths. Therefore, for each guessed principal point p and a set of orthogonal vanishing points V = {{v1, v 1}, {v2, v 2}, ...{vn, v n}} , we can estimate a corresponding set of focal lengths F = {f1, f2, ...fn}. Our basic idea is to employ the set F containing large amounts of estimated f values to statistically put constraint on the principal point. We reasonably expect that, if the principal point is correctly estimated, then the f values in the F set surely form a densely distributed cluster. On the contrary, if the guessed principal point is far from the correct position, then the distribution of the focal lengths computed by equation (7) is more likely quite sparse. Therefore, the distribution of the entries of the F set provides a confidence measure of the guess about the principal point (x0, y0). Naturally, the variance of the distribution, D(F), is a good candidate to measure such confidence and evaluate the goodness of the guess. In other words, although the probability density function P(x0, y0)is hidden from us it can be measured through the observable focal length distribution D(F). Smaller D(F) corresponds to higher confidence of (x0, y0). Under this formulation we can use D(F) to characterize the probability density function of the principal point and perform calibration through maximum likelihood estimate. Note that F is determined by (x0, y0) and should be more strictly written as F(x0, y0). We take D(F) as the cost function and try to solve the following optimization problem: Minimize(x0,y0)(D(F(x0, y0))) (8) Under this formulation, from every guess about the principal point a confidence value can be estimated and the corresponding focal length can be computed. An optimization routine is required to seek the minimum of equation (8), which corresponds to the max￾imum likelihood estimate of the intrinsic parameters (x0, y0, f). In our study the above statistical function is not easily differentiated analytically. So a derivative-free opti￾mizer is preferred. The downhill simplex method is a good candidate for this type of optimization [16]. In addition, Experiments show that a lot of local minimums exist in the solution space. We solve this problem by employing multiple initial points. Namely, the optimization is repeated several times with multiple randomly chosen starting points and the best result produced is adopted as the final solution. This strategy ensures the reliability and robustness. After the principal point (u,v) and the focal length f are de￾termined, the conic based pose estimate algorithm in [4,8] is employed to calculate the extrinsic parameters. This completes a full single-view based calibration framework. 3.3 Taking Aspect Ratio into Account Having made the above calibration algorithm work, adding an extra intrinsic parameter, i.e., the scale factor α, becomes straightforward. All we need to do is just introduce α as a fourth unknown variable into the optimization routine. During optimization, for each guessed value of the scale factor, the image is first scaled horizontally to give a corrected