Linear Camera Calibration Methods

The motivation to study the geometry if conics arises from the fact that conics have more geometric information,and can be more robustly than points and lines.The the camera calibration method proposed in this paper,which based on the arbitrary distributed three(or more) coplanar circles,can deal with all the distributions of each two circles such as intersection,tangent, including,separation.What's more,we shouldn't know any of the properties of circles,and needn't handling matching.As the absolute conic has all the information needed for the camera's intrinsic parameters,we now just need to find the image of the absolute conic. Based on this,many people have proposed some calibration methods.[27][2] give the calibration using a circle and its diameters;the linear method introduced by[6]based on two rectangulars;which the calibration algorithms in [7][14]based are only circles,but the method is not linear.Aimed at computing the image of the absolute conic,which is also a conic,we should find at least five points which lay in the conic.Fortunately, the generally called circles p0ints are just such points.So the problem is now that how to find the images of at least three pairs of circle points.Realizing that circle points are the intersection of a circle and the infinite line,our task comes to finding the intersection of the images of a circle and the infinite line which is called the vanishing line.We can easily get the image of a circle which lav on the model plane,the coming difficulty is how to find the vanishing line.Suppose there are three circles which are distributed arbitrarily,we will show how to find the vanishing line by using these there circles.We should at first group them so that two circles are in a group,then the image of at least one infinite point(which lay in the infinite line) will be computed from the image of each group.So using the there image points lay in the vanishing line,we can be fitted(of course we can use more than three circles in the model plane so as to get more image points of the vanishing line to fit it). Suppose there is a point X=(x,y,t)^T in the projective plane of the circles O₁,O₂,the necessary and sufficient condition of X being the common pole is that O₂X=λO₁X,that is, (O₂-λO₁)X=0,(0-2-1) withλ∈R\{0}.We have pointed out the fact that:the common poles of O₁,O₂ must include at least one infinite point.Under the projective transformation H,circles O₁,O₂ have the image of quadratic curves C₁=λ₁H^-TO₁H^-1,C₂=λ₂H^-TO₂H^-1,withλ₁,λ₂ R\{0}.Substitute this into(0-2-1),we have:(C₂-λ(λ₂/λ₁)C₁)HX=0,that is, (C₂-λ′C₁)HX=0.(0-2-2) Therefore,HX is the common pole of C₁ and C₂.Because H is nonsingular, andλ₁,λ₂ are non-zero,so equation(0-2-1) and(0-2-2) are equivalent. This is just the algebra description of the invariance of pole and polar line about projective transformation.Therefore,the common poles of C₁,C₂ which are the images of the two circle must include at least one infinite point.The following is the method of finding the vanishing line base on the equation(0-2-3). det(C₂-λ′C₁)=0.(0-2-3)proposition 0.1(1) If the equation(0-2-3) haven't repeated root(in this situation the preimage of C₁,C₂ are neither concentric nor tangent circles),compute the independent real rootλ′,then substituteλ′into equation(0-2-2),the eigenvector HX is the image of infinite point d_m∞.(2) If equation(0-2-3) has repeated root,then it must be twofold root,compute this twofold rootλ′and substitute it into the matrix C₂-λ′C₁: (a) If the matrix C₂-λ′C₁ rank of 1(showing that the preimage of C₁,C₂ are concentric circles),substitute thisλ′into equation(0-2-2),the two independent eigenvectors HX⁽¹⁾ and HX² are the image of infinite points d_m∞⁽¹⁾ and d_m∞⁽²⁾;(b) If the matrix C₂-λ′C₁ rank of 2(showing that the preimage of C₁,C₂ are tangent circles),compute the independent real root of equation(0-2-3),then substitute it into equation(0-2-2),and the eigenvector HX is the image of infinite point d_m∞.Till now,we have find the image of a infinite point.If there are three circles in the model plane,they can be divided into three groups,and at least one image point of the infinite point can be detected by using the above method.So we now have at least three points which are the image of three infinite points.Using these image points,the vanishing line can be fitted.After getting the vanishing line,compute the intersections of the vanishing line and the image of a circle,so we can get two points of the absolute conic.In this way,if we take photos of the model plane from at least three different angles,there will be enough points for recovering the absolute conic which has the degree of freedom of 5.Now we get the image of the absolute conic,and the intrinsic parameters of the camera.One fact we should point out is that the position of the camera is nothing to do with the image of absolute conic.That to say when we are taking photos from different angles,the image of absolute conic is unchanged.In fact,proposition 0.1 has introduced a criterion of judging the relationship between two circles just by using the images of this two circles.