CN105303570A - Method for calibrating parabolic refraction-reflection camera via intersection image of two spheres and images of circular ring points - Google Patents
Method for calibrating parabolic refraction-reflection camera via intersection image of two spheres and images of circular ring points Download PDFInfo
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Abstract
The invention relates to a method for calibrating a parabolic refraction-reflection camera via an intersection image of two spheres and images of circular ring points. The parabolic refraction-reflection camera shoots a target to obtain one image, a main point is determined via four groups of antipodal image points corresponding to solid and virtual intersection points in a spherical image in the refraction-reflection image and solid and virtual intersection points of an antipodal spherical image, a disappearance line of a plane, where two small parallel projection circles are placed in a unit visual sphere, of the two spheres is solved, images of the circular ring points in the plane are determined, and finally, the circular ring points are used to solve other inner parameters of the camera according to limitations of the inner parameters. The method of the invention can be used to realize full-automatic calibration, and reduce errors caused by measurement in the calibration process. Due to the fact that the projection contours of the spheres can be completely extracted from the images, the calibration precision of the camera is improved.
Description
Technical field
The invention belongs to computer vision field, the picture relating to two ball that a kind of utilization blocks mutually and circular point solves the method for parabolic catadioptric camera intrinsic parameter.
Background technology
The central task of computer vision is understood image exactly, and its final goal makes computing machine have ability by the cognitive three-dimensional environment information of two dimensional image.The geological information that this ability will not only make the perception of machine energy comprise object in the three-dimensional environment of shape, attitude, motion etc., and can be described them, store, Understanding and reasoning.Camera calibration is exactly determine that it is that much computer vision applies requisite step from three dimensions point to the mapping relations its X-Y scheme picture point.In order to determine this mapping process, need the geometry imaging model setting up video camera, the parameter of geometric model is called camera parameters, and camera parameters can be divided into intrinsic parameter and outer parameter two class.Intrinsic parameter describes the imaging geometry characteristic of imaging system, and outer parameter describes imaging system about the direction of world coordinate system and position.Camera calibration can be divided into tradition demarcation, self-calibration and the demarcation based on geometry entity.No matter which kind of scaling method, be all intended to the restriction relation set up between two dimensional image and camera intrinsic parameter, particularly linear restriction relation, this is the target that current camera calibration is pursued, and is also one of focus of current computer vision field research.
Parabolic catadioptric video camera is made up of a parabolic mirror surface and an orthogonal camera, and its visual field is large, is one of focus of panoramic vision area research.Document " Catadioptricself-calibration ", (KangS.B., ProceedingsofIEEEConferenceonComputerVisionandPatternRec ognition, vol.1, pp.201-207,2000.) propose a kind of catadioptric camera self-calibration method, the advantage of these class methods does not need to use calibrating block, and shortcoming to obtain the corresponding point between image.And in computer vision, the method realizing highly effective searching corresponding point is very difficult.Document " Geometricpropertiesofcentralcatadioptriclineimagesandthe irapplicationincalibration ", (BarretoJ.P., AraujoH., IEEETransactionsonPatternAnalysisandMachineIntelligence, vol.27, no.8, pp.1327-1333,2005) have studied the geometric properties of the picture of straight line under central catadiotric video camera, and these character are applied to the demarcation of central catadiotric video camera.Document " Anewlinearalgorithmforcalibratingcentralcatadioptriccame ras ", (WuF., DuanF., HuZ.etal., PatternRecognition, vol.41, no.10, pp.3166-3172,2008) antipodal point is described and to opening up picture point, be derived the relation of a point between the projection looked on ball and its catadioptric picture point in space, use this relation to establish the linear restriction of central catadiotric camera intrinsic parameter, central catadiotric camera intrinsic parameter can be obtained by this linear restriction.Document " CalibrationofcentralcatadioptriccamerasusingaDLT-likeapp roach ", (PuigL., BastanlarY., SturmP., etal.InternationalJournalofComputerVision, vol.93, no.1, pp.101-114, 2011) a kind of scaling method based on Three dimensions control point is proposed, expand by using the coordinate of Veronese mapping pair three-dimensional point and its picture point, based on DLT (direct linear transformation)---similarity method achieves the demarcation of central catadiotric video camera on the basis of expansion coordinate, but these class methods need the position of known three-dimensional point, and easily from image, extract its picture point.
Ball is as a kind of common solid, and its most important advantage is to block without self, the occluding contour of an a ball always circle space viewed from any one direction, and its contour projection can all extract.Because ball has abundant visual geometric characteristic, therefore utilizing ball to carry out camera calibration has become a focus in recent years.Document " Catadioptriccameracalibrationusinggeometricinvariants ", (YingX., HuZ., IEEETransactionsonPatternAnalysisandMachineIntelligence, vol.26, no.10, pp.1260-1271,2004) propose first and utilize ball to demarcate central catadiotric video camera.They demonstrate the picture of ball under the unit ball projection model of central catadiotric video camera for oval, and provide two invariants at the projection quafric curve of the next ball of non degenerate situation.In order to reduce the complexity solved, they propose a kind of substep scaling method, just the method at least needs the projection of 4 balls can complete the demarcation of video camera.But the scaling method that the document proposes is nonlinear, and the complexity of calculating is higher, and this scaling method can only demarcate the part intrinsic parameter of parabolic catadioptric video camera.Document " Identicalprojectivegeometricpropertiesofcentralcatadiopt riclineimagesandsphereimageswithapplicationstocalibratio n ", (YingX., ZhaH., InternationalJournalofComputerVision, vol.78, no.1, pp.89-105,2008) describe picture (MIAC) effect in central catadiotric camera calibration revising absolute conic.They propose two kinds of linear calibration's algorithms by the picture of research ball under central catadiotric video camera with the Analytic Geometry and Algebra relation of MIAC.The conclusion that they draw also is set up for dual form.But the theory in this section of document and scaling method are degenerate for the situation of parabolic catadioptric video camera.Document " Acalibrationmethodforparacatadioptriccamerafromsphereima ges ", (DuanH., WuY., PatternRecognitionLetters, vol.33, no.6, pp.677-684,2012) a kind of utilization is proposed to the linear method opening up ball image scale and determine parabolic catadioptric video camera based on circular point theory.But choose more complicated about the picture of circular point in this section of document.
Summary of the invention
The invention provides a kind of making simply, widely applicable, the target that utilizes of good stability solves the method for parabolic catadioptric camera intrinsic parameter, and this target is made up of two balls in space, and is crossing in the projection of parabolic catadioptric camera image plane.In the process solving parabolic catadioptric camera intrinsic parameter, the 1 width image that parabolic catadioptric video camera need be used to take target just can go out 5 intrinsic parameters of parabolic catadioptric video camera by linear solution.
The present invention adopts following technical scheme:
The image that 1 width contains the two balls mutually blocked is taken with parabolic catadioptric video camera.The present invention is the target be made up of the two balls in space, and target is crossing in the projection of imaging plane, utilizes target to solve the method for parabolic catadioptric camera intrinsic parameter, it is characterized in that only utilizing ball element.First, from this width image, extract marginal point and the target image marginal point of minute surface outline projection, use least square fitting to obtain the picture of minute surface outline projection and two ball.Secondly, according to picture point and its, relation of opening up picture point is obtained opening up picture point, thus estimate two ball picture to opening up ball picture, according to the intersection point (real and empty) of ball picture with to opening up intersection point (real and void) corresponding four groups of being formed of ball picture to opening up picture point determination principal point.On the basis obtaining principal point, solve two groups of parallel projection roundlet place planes on ball looked by two ball vanishing line in unit, thus determine the picture of circular point in this plane.Finally, picture all the other camera intrinsic parameters of constraint solving to camera intrinsic parameter of circular point are utilized.Concrete step comprises: matching minute surface outline projection equation and target projection equation, estimate two ball picture to opening up ball picture, determine the principal point of video camera, determine the picture of circular point, solve all the other intrinsic parameters of parabolic catadioptric video camera
.
1. matching minute surface outline projection equation and target projection equation
Utilize the function in Matlab program to extract the pixel coordinate of minute surface outline projection marginal point and target image marginal point, and obtain the equation of minute surface outline projection equation and two ball picture with least square fitting.
2. estimate two ball picture to opening up ball picture
A point in space has two points by unit depending on the projection centered by the centre of sphere of ball (also claim unit ball or depending on ball) on sphere, visiblely be called subpoint near the subpoint of spatial point on sphere, be called antipodal point away from spatial point being projected as on sphere is invisible, sphere projects 2 symmetrical relative to the centre of sphere; Take up an official post at sphere gets a bit simultaneously, subpoint and antipodal point is projected in a plane, is called the picture of subpoint corresponding to subpoint picture on a projection plane, is called the picture of antipodal point, i.e. sightless picture corresponding to antipodal point picture on a projection plane.Two balls in space
with
, and be crossing in the projection of imaging plane, be divided into two steps in the unit of Throwing thing catadioptric video camera depending on the projection under spherical model: the first step, by two ball
with
be projected as with
centered by unit look two groups of parallel roundlets on ball
with
, claim
for
to opening up circle, subscript
represent the picture corresponding with ball ,+representation unit looks perceptible projections on ball sphere, and-representation unit looks invisible projection on ball sphere.
with
meet at two real intersection points,
with
also two real intersection points are met at.Second step, looks a bit on ball surface with unit
for projection centre, i.e. principal point, here
the photocentre of a video camera can be regarded as, by these two groups of parallel roundlets
with
be projected as two groups of quafric curves on the parabolic catadioptric plane of delineation respectively
with
, wherein claim visible quafric curve
for ball
with
picture, sightless quafric curve
for ball picture
to opening up ball picture, the parabolic catadioptric plane of delineation and straight line
vertically.
with
there are two real intersection points and a pair imaginary intersection point;
with
also two real intersection points and a pair imaginary intersection point is had.Make with
for the Intrinsic Matrix of the video camera of photocentre is
, wherein
aspect ratio,
effective focal length,
inclination factor,
it is video camera principal point
homogeneous coordinates.Utilize the function in Matlab to extract the pixel coordinate of minute surface outline projection marginal point in this width image and target image marginal point, obtain corresponding quadratic curve equation by least square fitting, use here
represent the matrix of coefficients of minute surface outline projection curve,
represent the matrix of coefficients (in order to simplify statement, representing curve and its matrix of coefficients by same letter) of two ball pictures in this width image.Pass through
camera intrinsic parameter matrix can be obtained
an initial value
, thus obtain the picture (IAC) of absolute conic
initial value
.Here
,
the initial value of aspect ratio,
the initial value of effective focal length,
the initial value of inclination factor,
the initial homogeneous coordinates of video camera principal point, note
.Get
on one group of point
, represent by homogeneous coordinates,
represent which point got, upper subscript
represent which ball, then corresponding with it one group to opening up picture point
, represent by homogeneous coordinates, can by relational expression
determine.According to the definition of opening up picture point, point
at ball picture
to opening up ball picture
on, therefore can obtain opening up ball picture with least square fitting
original equation, finally by minimization function
obtain
an estimation, wherein
lagrange multiplier factor,
,
, here
representing matrix
?
row
column element.
3. determine the principal point of video camera
Two balls in space
with
the crossing projection roundlet on ball is looked in unit
two real intersection points
with projection roundlet
two real intersection points
corresponding formation two pairs of antipodal points.According to the definition of antipodal point, line segment
and line segment
for two diameters of ball are looked by unit, their intersection point is the center that ball is looked by unit
.With
with
represent respectively
with
picture, here
.According to the definition of opening up picture point,
be two to opening up picture point.Meanwhile, according to the associativity of projective transformation,
with
it is ball picture
with
two real intersection points,
with
be they to opening up ball picture
with
two real intersection points, and line segment
and line segment
intersection point be the principal point of video camera
.With
with
represent ball picture
with
two imaginary intersection points,
with
represent they to opening up ball picture
with
two imaginary intersection points.According to the definition of opening up picture point,
be that two pairs of virtual pairs open up picture point, they can regard two as to empty antipodal point
picture, and phantom line segments
with
intersection point be also the principal point of video camera
.By analysis above, use 1 width image of target can determine the principal point of video camera
, namely obtain
for 2 parameters of video camera.
4. determine the picture of circular point
At ball
projection roundlet
on get the point of two inequalities
with
, use
with
represent respectively
with
antipodal point, then point
with
?
to opening up circle
on.Straight line
for roundlet
point
the tangent line at place, straight line
for roundlet
point
the tangent line at place, wherein
.According to the definition of antipodal point and the geometric properties of circle, have
so,
with
there is identical infinity point, use here
represent
on infinity point, wherein
.Pass through
with
roundlet can be determined
two infinity points in the plane, so the line at infinity in this plane can be determined, use here
represent the line at infinity in this plane, wherein subscript
.According to the definition of circular point in projective geometry,
with roundlet
and roundlet
intersection point identical, be the circular point in this plane
.With
represent respectively
picture, then
be two to opening up picture point.Note ball picture
point
the tangent line at place is
, it to opening up ball picture
point
the tangent line at place is
, then according to the character of projective transformation, straight line
for straight line
picture, straight line
for straight line
picture, so by straight line
and straight line
determine roundlet
two end points in the plane
(
picture), thus the vanishing line of this plane can be determined
(
picture), wherein subscript
.Finally, straight line is asked
with ball picture
intersection point obtain a pair conjugate imaginary points, ask straight line simultaneously
with ball picture
to opening up ball picture
intersection point also can obtain a pair conjugate imaginary points.These two pairs of conjugate imaginary pointses are identical in theory, are roundlet
the picture of circular point in the plane, due to noise effect, their may be different, here by asking the mean value of corresponding point coordinate as net result, thus obtain the picture of the circular point in this plane
; For roundlet
the picture of circular point in the plane
also can obtain; Subscript
represent and correspond to circular point
.
5. solve all the other intrinsic parameters of parabolic catadioptric video camera
Owing to obtaining the principal point of video camera, therefore remaining three intrinsic parameter of a demand solution video camera.Use 1 width image of parabolic catadioptric video camera shooting target.First, by a translation transformation matrix
the true origin of image coordinate system is moved to principal point
form new coordinate system, wherein
.Intrinsic Matrix under new coordinate system
can be reduced to
, here
; So,
can be reduced to
,
expression is equivalent to, now
only there are 3 degree of freedom.Roundlet under new coordinate system
the picture of a pair circular point in the plane
be transformed to
, wherein,
for homogeneous coordinates; Secondly, the picture of circular point under new coordinate system is solved by SVD method
to the picture of the absolute conic simplified
linear restriction obtain
, that is:
, wherein
represent real and imaginary part respectively; Finally, right
carry out Cholesky and decompose the Intrinsic Matrix of inverting again and just can obtain under new coordinate system
, namely obtain in its excess-three intrinsic parameter of parabolic catadioptric video camera
.
Advantage of the present invention:
(1) this target makes simple, only two balls need be fixed on a support, make it have occlusion effect.
(2) to the physical size not requirement of this target, without the need to knowing the coordinate of the centre of sphere under world coordinate system.
(3) sharp point of this target almost can all extract, and can improve the degree of accuracy of curve like this, thus improves stated accuracy.
Accompanying drawing explanation
Fig. 1 is the target schematic diagram for solving parabolic catadioptric camera intrinsic parameter.
Fig. 2 is the projection of target on unit ball.
Fig. 3 is the projection of target on the parabolic catadioptric plane of delineation.
Embodiment
The invention provides a kind of target for solving parabolic catadioptric camera intrinsic parameter, it is made up of the two balls in space, and be crossing in the projection of imaging plane, as Fig. 1.Completing solving of parabolic catadioptric camera intrinsic parameter with this modulation needs through following steps: from catadioptric image, extract minute surface outline projection marginal point and target image marginal point, uses least square fitting to obtain the picture of minute surface outline projection and two ball.According to picture point and its, relation of opening up picture point is obtained opening up picture point, thus estimate two ball picture to opening up ball picture.According to the intersection point of ball picture (real and empty) with to opening up intersection point (real and empty) corresponding four groups of being formed of ball picture to opening up picture point determination principal point.On the basis obtaining principal point, solve two groups of parallel projection roundlet place planes on ball looked by two ball vanishing line in unit, thus determine the picture of circular point in this plane.Utilize picture all the other camera intrinsic parameters of constraint solving to camera intrinsic parameter of circular point.Concrete step comprises: matching minute surface outline projection equation and target projection equation, estimate two ball picture to opening up ball picture, determine the principal point of video camera, determine the picture of circular point, solve all the other intrinsic parameters of parabolic catadioptric video camera
.。Utilize the method in the present invention to demarcate the Throwing thing catadioptric video camera for testing, concrete steps are as follows:
1. matching minute surface outline projection equation and target projection equation
Utilize the function in Matlab program to extract the pixel coordinate of minute surface outline projection marginal point and target image marginal point, and obtain the equation of minute surface outline projection equation and two ball picture with least square fitting.
2. estimate two ball picture to opening up ball picture
Two balls in space
with
(as Fig. 1), and be crossing in the projection of imaging plane, the projection of two ball under the unit ball model of Throwing thing catadioptric video camera is divided into two steps: the first step, by two ball
with
be projected as with
two groups of parallel roundlets on ball are looked by the unit at center
with
, claim
for
to opening up circle.Point
for
with
two real intersection points, point
for
with
two real intersection points.This process as shown in Figure 2.Second step, by the photocentre of video camera
by these two groups of parallel roundlets
with
be projected as two groups of quafric curves on the parabolic catadioptric plane of delineation respectively
with
, claim visible quafric curve here
for ball
with
picture, sightless quafric curve
for ball picture
to opening up ball picture.Point
for
with
two real intersection points, point
for
with
two imaginary intersection points; Point
for
with
two real intersection points, point
for
with
two imaginary intersection points.This process as shown in Figure 3.Utilize the function in Matlab to extract the pixel coordinate of minute surface outline projection in this width image and target image marginal point, obtain corresponding quadratic curve equation by least square fitting, use here
represent the matrix of coefficients of minute surface outline projection curve,
represent the matrix of coefficients of two ball pictures in this width image.Pass through
camera intrinsic parameter matrix can be obtained
an initial value
.
，（1）
Here,
representing matrix
?
row
column element,
,
for the half at camera field of view angle,
for paraboloidal mirror outline projection is oval
major semi-axis long.In acquisition
basis on can obtain
initial value
；（2）
Get
on one group of point
, then corresponding with it one group to opening up picture point
can be determined by relational expression (3)
；（3）
According to the definition of opening up picture point, point
at ball picture
to opening up ball picture
on, therefore can obtain opening up ball picture with least square fitting
original equation, finally by minimization function
obtain
an estimation, wherein
lagrange multiplier factor,
,
, here
representing matrix
?
row
column element.
3. determine the principal point of video camera
As shown in Figure 2, the two balls in space
with
the crossing projection roundlet on ball is looked in unit
two real intersection points
with projection roundlet
two real intersection points
corresponding formation two pairs of antipodal points.According to the definition of antipodal point, line segment
and line segment
for looking two diameters of ball, their intersection point is the center looking ball
.As shown in Figure 3, use
with
represent respectively
with
picture, here
.According to the definition of opening up picture point,
be two to opening up picture point.Meanwhile, according to the associativity of projective transformation,
with
it is ball picture
with
two real intersection points,
with
be they to opening up ball picture
with
two real intersection points, and line segment
and line segment
intersection point be the principal point of video camera
.With
with
represent ball picture
with
two imaginary intersection points,
with
represent they to opening up ball picture
with
two imaginary intersection points.According to the definition of opening up picture point,
be that two pairs of virtual pairs open up picture point, they can regard two as to empty antipodal point
picture, and phantom line segments
and line segment
intersection point be also the principal point of video camera
.If
homogeneous coordinates be
, wherein
.Here system of equations (4), (5), (6) can be obtained
，（4）
，（5）
，（6）
Wherein
for the pixel coordinate on the parabolic catadioptric plane of delineation.By analysis above,
4 groups of solutions of system of equations (4),
4 groups of solutions of system of equations (5), wherein
.
it is the solution of system of equations (6).Because system of equations (6) is overdetermination, SVD method is used to solve.Thus use 1 width image of target can determine the principal point of video camera
.
4. determine the picture of circular point
As shown in Figure 2, at ball
projection roundlet
on get the point of two inequalities
with
, use
with
represent respectively
with
antipodal point, then point
with
?
to opening up circle
on.Straight line
for roundlet
point
the tangent line at place, straight line
for roundlet
point
the tangent line at place, wherein
.According to the definition of antipodal point and the geometric properties of circle, have
so,
with
there is identical infinity point, use here
represent
on infinity point, wherein
.Pass through
with
roundlet can be determined
two infinity points in the plane, so the line at infinity in this plane can be determined, use here
represent the line at infinity in this plane, wherein
.According to the definition of circular point,
with roundlet
and roundlet
intersection point identical, be the circular point in this plane
.As shown in Figure 3, use
represent respectively
picture, then
be two to opening up picture point.Note ball picture
point
the tangent line at place is
, it to opening up ball picture
point
the tangent line at place is
.If
homogeneous coordinates be
,
homogeneous line coordinates be
, then
，（7）
，（8）
Wherein
non-zero constant factor,
.According to the character of projective transformation, straight line
for straight line
picture, straight line
for straight line
picture, so by straight line
and straight line
determine roundlet
two end points in the plane
.Pass through simultaneous
with
equation can obtain
，（9）
In like manner, simultaneous is passed through
with
equation can obtain
，（10）
If
with
homogeneous coordinates be respectively
with
, then
the solution of system of equations (9),
it is the solution of system of equations (10).Pass through roundlet
two end points in the plane
with
the vanishing line of this plane can be determined
.If
homogeneous line coordinates be
, then
，（11）
Wherein
it is non-zero constant factor.Finally, simultaneous is passed through
with
equation can obtain
，（12）
In like manner, simultaneous is passed through
with
equation can obtain
；（13）
There is identical solution system of equations (12) and (13) in theory, are roundlet
the picture of circular point in the plane
but due to the impact of noise, their solution may be different.The solution of note system of equations (12) is
with
, the solution of system of equations (13) is
with
, then
；（14）
Roundlet can be obtained by (14) formula
the picture of circular point in the plane
.For roundlet
the picture of circular point in the plane
available similar method obtains.
5. solve all the other intrinsic parameters of parabolic catadioptric video camera
Owing to obtaining the principal point of video camera, therefore remaining three intrinsic parameter of a demand solution video camera.Use 1 width image of parabolic catadioptric video camera shooting target.First, by a translation transformation
the true origin of image coordinate system is moved to principal point
, wherein
；（15）
Intrinsic Matrix under new coordinate system
can be reduced to
；（16）
So,
can be reduced to
，（17）
Now
only there are 3 degree of freedom.Roundlet under new coordinate system
the picture of a pair circular point in the plane
be transformed to
，（18）
Secondly, had the constraint of IAC by the picture of circular point
，（19）
Wherein
represent real and imaginary part respectively.Obtain by SVD method solving equation group (19)
.Finally, right
carry out Cholesky to decompose and invert again and just can obtain
, namely obtain in its excess-three intrinsic parameter of parabolic catadioptric video camera
.
Embodiment
The present invention proposes a kind of two balls in space that utilize as target, and target is crossing in the projection as plane of parabolic catadioptric video camera, utilizes target linearly to determine the method for parabolic catadioptric camera intrinsic parameter.The experiment pattern structural representation that the present invention adopts as shown in Figure 1.With an example, description is specifically made to embodiment of the present invention below.
Based on the two balls in space, the experiment pattern that demarcation parabolic catadioptric video camera adopts is the two balls in space, and is crossing in the projection as plane, and as shown in Figure 1, two balls are respectively
with
.Utilize the method in the present invention to demarcate the parabolic catadioptric video camera for testing, concrete steps are as follows:
1. matching image boundary and target curvilinear equation
The image size that the present invention adopts is
, the field angle of parabolic catadioptric video camera is 212 degree.By 1 width experimental image of parabolic catadioptric video camera shooting target, read in image, utilize the Edge sum functions in Matlab to extract the pixel coordinate of minute surface outline projection marginal point and target image marginal point, and obtain the equation of minute surface outline projection equation and two ball picture with least square fitting.The matrix of coefficients of minute surface outline projection equation is
, the matrix of coefficients of the equation of two ball picture is
, result is as follows
，（20）
，（21）
；（22）
2. estimate two ball picture to opening up ball picture
(20) are substituted into (1) and (2) can obtain
, result is as follows
，（23）
First (23) are substituted into (3) to obtain opening up the initial coordinate that ball picture is put, then obtained the original equation opening up ball picture, finally by minimization function by least square fitting
obtain opening up ball picture
estimation, result is as follows
，（24）
；（25）
3. determine the principal point of video camera
(21) and (22) are substituted into two real intersection points and a pair imaginary intersection point that (4) can obtain ball picture, and result is as follows
，（26）
；（27）
，（28）
；（29）
In like manner, (24) and (25) are substituted into (5) can to two real intersection points and a pair imaginary intersection point opening up ball picture, and result is as follows
，（30）
；（31）
，（32）
；（33）
First (26) and (33) are substituted into system of equations (6), re-use SVD method solving equation group (6), the principal point of video camera can be obtained
, result is as follows
；（34）
4. determine the picture of circular point
We are at ball picture
take up an official post and get the point of two inequalities, making at these 2 is exactly
, their homogeneous coordinates are respectively
，（35）
；（36）
According to the character of opening up picture point, can obtain and point
corresponding to opening up picture point
, result is as follows
，（37）
；（38）
(35), (36) are substituted into (7) and can obtain ball picture
point
with
the tangent line at place
with
homogeneous line coordinates, result is as follows
，（39）
；（40）
By (37), (38) substitute into (8) can to opening up ball picture
point
with
the tangent line at place
with
homogeneous line coordinates, result is as follows
，（41）
；（42）
(39) and (41) are substituted into (9) and can end point be obtained
, result is as follows
；（43）
(40) and (42) are substituted into (10) and can end point be obtained
, result is as follows
；（44）
(43) and (44) are substituted into (11) and can roundlet be determined
vanishing line in the plane
, its homogeneous line coordinates is
；（45）
First (21) are substituted into (12), (24) substitute into (13) and obtain result, then its result substitution (14) can be obtained roundlet
the picture of circular point in the plane
, result is as follows
，（46）
；（47）
For roundlet
the picture of circular point in the plane
available similar method obtains.Result is as follows
，（48）
；（49）
5. solve all the other intrinsic parameters of parabolic catadioptric video camera
First, (34) are substituted into (15) and obtain translation transformation
, result is as follows
，（50）
Secondly, (46) ~ (50) substitution (18) can be obtained roundlet under new coordinate system
the picture of a pair circular point in the plane
, wherein
.Result is as follows
，（51）
；（52）
，（53）
; (54) following, (51) and (53) are substituted into (19) and obtains
the system of linear equations of middle element, uses SVD method to solve this system of linear equations and obtains
.Result is as follows
。（55）
Finally, in (55)
carry out Cholesky to decompose and invert again and just can obtain
, have
，（56）
Wherein aspect ratio
(
representing matrix
the 1st row the 1st arrange element,
representing matrix
the element that arranges of the 2nd row the 2nd), therefore all the other 3 intrinsic parameters of parabolic catadioptric video camera are respectively:
,
,
.
Claims (1)
1. the method using two crossing picture of ball and the image scale of circular point to determine Throwing thing catadioptric video camera, it is characterized in that by the two balls in space as target, and two ball is crossing in the projection as plane of parabolic catadioptric video camera, the concrete steps of described method comprise: first, the image that 1 width contains the two balls mutually blocked is taken with parabolic catadioptric video camera, from this width image, extract marginal point and the target image marginal point of minute surface outline projection, use least square fitting to obtain the image of minute surface outline projection and two ball; Secondly, picture point according to two ball obtains opening up picture point the relation of opening up picture point with it, thus estimate two ball picture to opening up ball picture, according to the intersection point of ball picture, with to the intersection point opening up ball picture, corresponding four groups of being formed to opening up picture point determination principal point, the intersection point of ball picture and real intersection point and imaginary intersection point are comprised to the intersection point opening up ball picture; On the basis obtaining principal point, solve two groups of parallel projection roundlet place planes on ball looked by two ball vanishing line in unit, thus determine the picture of circular point in this plane; Finally, picture all the other camera intrinsic parameters of constraint solving to camera intrinsic parameter of circular point are utilized;
(1) matching minute surface outline projection equation and target projection equation
Utilize the function in Matlab program to extract the pixel coordinate of minute surface outline projection marginal point and target image marginal point, and obtain the equation of minute surface outline projection equation and two ball picture with least square fitting;
(2) estimate two ball picture to opening up ball picture
A point in space has two points by unit depending on the projection on sphere centered by the centre of sphere of ball, the visible subpoint that is called at the subpoint of close spatial point on sphere, away from spatial point on sphere, be projected as the invisible antipodal point that is called, on sphere, 2 of projections are with respect to centre of sphere symmetry; Take up an official post get a bit at sphere, subpoint and antipodal point are projected in a plane, correspond to the picture that subpoint picture is on a projection plane called subpoint, correspond to the picture that antipodal point picture is on a projection plane called antipodal point, i.e. sightless picture simultaneously; Two balls in space
With
, and be crossing in the projection of imaging plane, the projection of two balls under the unit ball model of Throwing thing catadioptric video camera is divided into two steps: the first step, by two balls
With
Be projected as with
Centered by unit look two groups of parallel roundlets on ball
With
, claim
For
To opening up circle, subscript
Represent the picture corresponding with ball,+ representation unit is looked perceptible projections on ball sphere, and-representation unit is looked invisible projection on ball sphere;
With
Meet at two real intersection points,
With
Also two real intersection points are met at; Second step, looks on the surface a bit of ball with unit
For projection centre, i.e. principal point,Here
Regard the photocentre of a video camera as, by these two groups of parallel roundlets
With
Be projected as respectively two groups of conic sections on the parabolic catadioptric plane of delineation
With
,Wherein claim visible conic section
For ball
With
Picture, sightless conic section
For ball picture
To opening up ball picture, the parabolic catadioptric plane of delineation and straight line
Vertically;
With
There are two real intersection points and two imaginary intersection points;
With
Also two real intersection points and two imaginary intersection points are had; Order with
For the Intrinsic Matrix of the video camera of photocentre is
,Wherein
Aspect ratio,
Effective focal length,
Obliquity factor,
It is video camera principal point
Homogeneous coordinates; Utilize function in Matlab to extract minute surface outline projection marginal point in this width image and the pixel coordinate of target image marginal point, obtained corresponding quadratic curve equation by least square fitting,Here use
Represent the coefficient matrix of minute surface outline projection curve,
Represent the coefficient matrix of two ball pictures in this width image, in order to simplify statement, represent the coefficient matrix of curve and it by same letter; Pass through
Obtain camera intrinsic parameter matrix
An initial value
,Thus obtain the picture of absolute conic
Initial value
, here:
,
, wherein
,
The initial value of aspect ratio,
The initial value of effective focal length,
The initial value of obliquity factor,
The initial homogeneous coordinates of video camera principal point, note
; Get
On one group of point
,
Represent which point of getting, upper subscript
Represent which ball, then corresponding with it one group to opening up picture point,
By relational expression
Determine,
,
Represent by homogeneous coordinates ,-represent sightless picture; According to the definition of opening up picture point, point
At ball picture
To opening up ball picture
Upper, therefore obtain opening up ball picture with least square fitting
Original equation, finally by minimization function
Obtain
An estimation, wherein
Lagrange multiplier factor,
,
, here
Representing matrix
?
Row
Column element;
(3) principal point of video camera is determined
Two balls in space
with
the crossing projection roundlet on ball is looked in unit
two real intersection points
with projection roundlet
two real intersection points
corresponding formation two pairs of antipodal points; According to the definition of step (2) antipodal point, line segment
and line segment
for two diameters of ball are looked by unit, intersection point is the center that ball is looked by unit
; With
with
represent respectively
with
picture, here
; According to the definition of opening up picture point,
each other to opening up ,+represent picture point ,-represent opening up picture point; Meanwhile, according to the associativity of projective transformation,
with
it is ball picture
with
two real intersection points,
with
to opening up ball picture
with
two real intersection points, and line segment
and line segment
intersection point be the principal point of video camera
; With
with
represent ball picture
with
two imaginary intersection points,
with
represent opening up ball picture
with
two imaginary intersection points; According to the definition of opening up picture point,
be that two pairs of virtual pairs open up picture point, regard two as to empty antipodal point
picture, and phantom line segments
with
intersection point be also the principal point of video camera
; By analysis above, 1 width image of target is used namely to determine the principal point of video camera
, namely obtain
for 2 parameters of video camera;
(4) picture of circular point is determined
At ball
projection roundlet
on get the point of two inequalities
with
, use
with
represent respectively
with
antipodal point, then point
with
?
to opening up circle
on; Straight line
for roundlet
point
the tangent line at place, straight line
for roundlet
point
the tangent line at place, wherein subscript
; According to the definition of antipodal point and the geometric properties of circle, have
so,
with
there is identical infinity point, use here
represent
on infinity point, wherein subscript
; Pass through
with
determine roundlet
two infinity points in the plane, so determine the line at infinity in this plane, use here
represent the line at infinity in this plane, wherein subscript
; According to the definition of circular point in projective geometry,
with roundlet
and roundlet
intersection point identical, be the circular point in this plane
; With
represent respectively
picture, then
be two to opening up picture point, subscript
; Note ball picture
point
the tangent line at place is
, it to opening up ball picture
point
the tangent line at place is
, then according to the character of projective transformation, straight line
for straight line
picture, straight line
for straight line
picture, so pass through straight line
and straight line
determine roundlet
two end points in the plane
, namely
picture, thus determine the vanishing line of this plane
, namely
picture, wherein subscript
; Finally, straight line is asked
with ball picture
intersection point obtain a pair conjugate imaginary points, ask straight line simultaneously
with ball picture
to opening up ball picture
intersection point also obtain a pair conjugate imaginary points; According to projective theory, these two pairs of conjugate imaginary pointses are identical, are roundlet
the picture of circular point in the plane, but due to noise effect, here by asking the mean value of corresponding point coordinate as net result, thus obtain the picture of the circular point in this plane
; For roundlet
the picture of circular point in the plane
also obtain; Subscript
represent and correspond to circular point
;
(5) all the other intrinsic parameters of parabolic catadioptric video camera are solved
Owing to obtaining the principal point of video camera, therefore remaining three intrinsic parameter of a demand solution video camera; Use 1 width image of parabolic catadioptric video camera shooting target; First, by a translation transformation matrix
the true origin of image coordinate system is moved to principal point
form new coordinate system, wherein
; Intrinsic Matrix under new coordinate system
be reduced to
, here
; So,
be reduced to
,
expression is equivalent to, now
only there are 3 degree of freedom; Roundlet under new coordinate system
the picture of a pair circular point in the plane
be transformed to
, wherein,
for homogeneous coordinates; Secondly, the picture of circular point under new coordinate system is solved by SVD method
to the picture of the absolute conic simplified
linear restriction obtain
, that is:
,
, wherein
represent real and imaginary part respectively; Finally, right
carry out Cholesky and decompose the Intrinsic Matrix of inverting again and just obtaining under new coordinate system
, namely obtain in its excess-three intrinsic parameter of parabolic catadioptric video camera
, wherein
aspect ratio,
effective focal length,
it is inclination factor.
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