Patent Application: US-16285902-A

Abstract:
calibration for a camera is achieved by receiving images of a calibration object whose geometry is one - dimension in space . the received images show the calibration object in several distinct positions . calibration for the camera is then calculated based on the received images of the calibration object .

Description:
the following discussion is directed to camera calibration using one - dimensional objects . the subject matter is described with specificity to meet statutory requirements . however , the description itself is not intended to limit the scope of this patent . rather , the inventors have contemplated that the claimed subject matter might also be embodied in other ways , to include different elements or combinations of elements similar to the ones described in this document , in conjunction with other present or future technologies . fig1 illustrates an exemplary diagram of a camera calibration system 100 . calibration system 100 includes a computer system 102 , a camera 104 and a calibration object 106 . the computer system 102 is described in more detail with reference to fig9 and performs camera calibration calculations . based on the camera calibration calculations , the computer system 102 can send signals ( not is shown ) to the camera 104 ; the signals representing values that can be used to calibrate the camera 104 . the computer system 102 could also be distributed within a computing system having more than one computing device . see the description of “ exemplary computing system and environment ” below for specific examples and implementations of networks , computing systems , computing devices , and components that can be used to implement the calibration processes described herein . the camera 104 may be any type of object imaging device such as a digital camera and / or analog camera . as shall become more apparent , more than one camera can be used in camera calibration system 100 . for instance , suppose that camera 104 is mounted at the front of a room and another camera ( not shown ) is mounted at the back of the room . both cameras can then be calibrated by simultaneously imaging the same calibration object 106 . this is very difficult , if not impossible , to achieve with 3d or 2d calibration systems when two or more cameras are mounted at opposite ends of a room . this scenario , however , does not present a problem for calibration system 100 , because of the ability to calibrate the camera ( s ) 104 using calibration object 106 that is one - dimensional and is visible from virtually any point in the space . camera 104 sends images of the calibration object to computer system 102 though any type of connector , such as , for example through a 1394 link . the calibration object 106 is a one - dimensional ( 1d ) object . in the exemplary illustration , calibration object 106 consists of three collinear points a , b , and c . the collinear points a , b , and c should be measured to provide known relative positions between the points . moreover , one of the points , as shall be explained in more detail , should remain fixed ( i . e ., stationary ). the points a , b , and c can be implemented with any inexpensive objects such as a string of beads or balls hanging from a ceiling , or a stick with styrofoam spheres , etc . the calibration object 106 could also consist of four or more points ( not shown ), however , camera calibration is not recommended if less than three points are used due to having too many unknown parameters , making the calibration impossible . the fixed point does not need to be visible to the camera because it can be computed by intersecting image lines . fig2 is a flow chart illustrating a process 200 for calibrating a camera 104 in system 100 . process 200 includes various operations illustrated as blocks . the order in which the process is described is not intended to be construed as a limitation . furthermore , the process can be implemented in any suitable hardware , software , firmware , or combination thereof . in the exemplary implementation , the majority of operations are performed in software ( such as in the form of modules or programs ) running on computer system 102 . at block 202 , images of the calibration object 106 are received by the computer system 102 from camera 104 . based on the received images , the computer system 102 is able to provide a calibration solution that can be used to calibrate camera 104 accordingly ( block 204 ). fig3 is a flow chart illustrating the received images operation in more detail . at blocks 302 and 304 , calibration object 106 is moved around into different configurations while camera 104 images the calibration object 106 . during the movement of the calibration object 106 , at least one of the points remains fixed . for example , fig4 shows another view of calibration object 106 as points c and b move around point a , which remains fixed . images of the calibration object 106 are then sent to computer system 102 . to understand block 202 in more detail , reference is made to the notation used herein . a 2d point is denoted by m =[ u , v ] t . a 3d point is denoted by m =[ x , y , z ] t . the term { tilde over ( x )} is used to denote the augmented vector by adding 1 as the last element : { tilde over ( m )}=[ u , v , 1 ] t and { tilde over ( m )}=[ x , y , z , 1 ] t . a camera is modeled by the usual pinhole : the relationship between a 3d point m and its image projection m is given by where s is an arbitrary scale factor and ( r , t ), which is called the extrinsic parameters , is the rotation and translation which relates the world coordinate system to the camera coordinate system . without loss of generality , in one implementation , the camera coordinate system is used to define the calibration object 106 , therefore , r = i and t = 0 in equation 1 above . additionally , matrix a , called the camera intrinsic matrix , is given by with ( u 0 , ν 0 ) the coordinates of the principle point , α and β the scale factors in image u and ν axes , and γ the parameter describing the skewness of the two image axes . the task of camera calibration is to determine these five intrinsic parameters . finally , the abbreviation a − t will be used for ( a − 1 ) t or ( a t ) − 1 . as mentioned above calibration is not recommended with free moving 1d objects . however , if one of the points remains fixed as shown in fig4 , for instance point a is the fixed point , and a is the corresponding image point , then camera calibration can be realized . we need three parameters , which are unknown , to specify the coordinates of a in the camera coordinate system , while image point a provides two scalar equations according to equation 1 . by having at least three collinear points ( see fig1 and 4 ) comprising the calibration object 106 , the number of unknowns for the point positions is 8 + 2n , where n is the number of observations of the calibration object 106 . this is because given n observations of the calibration object , we have 5 camera intrinsic parameters , 3 parameters for the fixed point a , and 2n parameters for the n positions of the free point b . for the latter , more explanation is provided here : for each observation , we only need 2 parameters to define b because of known distance between a and b ; no additional parameters are necessary to define c once a and b are defined . for each observation , b provides two equations , but c only provides one additional equation , because of collinearity of a , b , c . thus , the total number of equations is 2 + 3n for n observations . by counting the numbers , if there are six or more observations , then it should be possible to solve camera calibration , to be described in more detail with reference to step 204 . if more than three collinear points are used with know distances , the number of unknowns and the number of impendent equations remains the same , because of cross - ratios . nevertheless , the more collinear points used , the more accurate camera calibration will be , because data redundancy tends to combat noise in the image data . after a number of observations of the calibration object have been received by computer system 102 ( i . e ., block 202 in fig2 and blocks 302 , 304 in fig3 ), it is possible to solve for camera calibration using a 1d calibration object ( block 204 in fig2 ). block 204 describes a process for calculating camera calibration , which will be described in more detail . there are two implementations for solving the calibration problem : one involves a closed - form solution and the other involves a non - linear optimization ( that is optional , but recommended for greater accuracy ). in this section , both implementations will be described in more detail with reference to the minimal configuration implementation of the calibration object ( three collinear points moving around a fixed point , e . g . a ). referring to fig4 , point a is the fixed point in space , and the calibration object ab moves around a . the length of the calibration object measured from a to b is known to be l , i . e ., the position of point c is know with respect to a and b , and therefore , where λ a and λ b are known . if c is the midpoint of ab , then λ a and λ b = 0 . 5 . points a , b , and c on the image plane 502 are a projection of space points a , b and c , respectively . again , without loss of generality , using the camera calibration system to define 1d objects , with r = i and t = 0 in equation 1 . let the unknown depths for a , b and c be z a , z b and z c , respectively . according to equation 1 , there are z c { tilde over ( c )}= z a λ a ã + z b λ b { tilde over ( b )} ( 8 ) after eliminating a − 1 from both sides . by performing a cross - product on both sides of the equation 8 with { tilde over ( c )}, the following equation is realized : z a λ a ( ã ×{ tilde over ( c )} )+ z b λ b ( { tilde over ( b )}×{ tilde over ( c )} )= 0 z b = - z a ⁢ λ a ⁡ ( a ~ × c ~ ) · ( b ~ × c ~ ) λ b ⁡ ( b ~ × c ~ ) · ( b ~ × c ~ ) ( 9 ) ∥ a − 1 ( z b { tilde over ( b )}− z a ã )∥= l z a ⁢  a - 1 ( a ~ + λ a ⁡ ( a ~ × c ~ ) · ( b ~ × c ~ ) λ b ⁡ ( b ~ × c ~ ) · ( b ~ × c ~ ) ⁢ b ~  = l h = a ~ + λ a ⁡ ( a ~ × c ~ ) · ( b ~ × c ~ ) λ b ⁡ ( b ~ × c ~ ) · ( b ~ × c ~ ) ⁢ b ~ ( 11 ) equation 10 contains the unknown intrinsic parameters a and the unknown depth , z a of the fixed point a . it is the basic constraint for camera calibration with 1d objects . vector h , give by equation 11 , can be computed from image point and known λ a and λ b . since the total number of unknowns is six , it is recommended that at least six imaging observations of the calibration object 106 be made ( that is , six different configurations of the calibration object ). note that a − t a actually describes the image of the absolute conic [ 12 ]. b =[ b 11 , b 12 , b 22 , b 13 , b 23 , b 33 ] t . ( 13 ) let h =[ h 1 , h 2 , h 3 ] t , and x = z a 2 b , then equation ( 10 ) becomes ν =[ 1 2 , 2h 1 h 2 , h 2 2 , 2h 1 h 3 , 2h 2 h 3 , h 3 2 ] t . when n images of the 1d object are observed , by stacking n such equations as ( 14 ) we have where v =[ v 1 , . . . , v n ] t and 1 =[ 1 , . . . , 1 ] t . the least - squares solution is then given by once x is estimated , computer system 102 can compute all the unknowns based on x = z a 2 b . let x =[ x 1 , x 2 , . . . , x 6 ] t . without difficulty , it is possible to uniquely extract the intrinsic parameters for the camera and the depth z a as : ν 0 =( x 2 x 4 − x 1 x 5 )/( x 1 x 3 − x 2 2 ) z a =√{ square root over ( x 6 −[ x 4 2 + ν 0 ( x 2 x 4 − x 1 x 5 )]/ x 1 )} β =√{ square root over ( z a x 1 /( x 1 x 3 − x 2 2 ))} u 0 = γν 0 / α − x 4 α 2 / z a . at this point , it is possible for the computer system 102 to compute z b according to equation ( 9 ), so points a and b can be computed from equations ( 5 ) and ( 6 ), while point c can be computed according to equations ( 3 ). in block 204 , it is possible to refine the above calibration solutions through a maximum likelihood inference . supposing there are n images of the calibration object 106 and there are three points on the comprising the object . point a is fixed , and points b and c move around a . assume that the images points are corrupted by independent and identically distributed noise . the maximum likelihood estimate can be obtained by minimizing the following functional : ∑ i = 1 n ⁢ ⁢ (  a i - ϕ ⁡ ( a , a )  2 +  b i - ϕ ⁡ ( a , b i )  2 +  c i - ϕ ⁡ ( a , c i )  2 ) ( 17 ) where φ ( a , m )( m ∈{ a , b i , c i }) is the projection of point m onto the image , according to equations ( 5 ) to ( 6 ). more precisely , ϕ ⁡ ( a , m ) = 1 z m ⁢ a ⁢ ⁢ m , the unknowns to be estimated are : five camera intrinsic parameters α , β , γ , u 0 and ν 0 that define matrix a ; three parameters for the coordinates of the fixed point a ; and two n additional parameters to defined points b i and c 1 at each instant . therefore there are a total of 8 + 2n unknowns . regarding the parameterization for b and c , we use the spherical coordinates φ and θ to define the direction of the calibration object 106 , and point b is then given by : b = a + l ⁡ [ sin ⁢ ⁢ θ ⁢ ⁢ cos ⁢ ⁢ ϕ sin ⁢ ⁢ θ ⁢ ⁢ sin ⁢ ⁢ ϕ cos ⁢ ⁢ θ ] where l is the known distance between a and b . in turn , point c is computed according to equation ( 4 ). therefore , only two additional parameters are needed for each observation . minimizing equation ( 17 ) is a nonlinear minimization problem , which is solved with the levenberg - marquardt algorithm as implemented in minpack [ 15 ]. this algorithm requires an initial guess of a , a , { b 1 , c 1 | i = 1 . . . n }, which can be obtained using the closed - form solution described earlier . using computer simulations suppose that a simulated camera 104 has the following properties : α = 1000 , β = 1000 , γ = 0 , u 0 = 320 and ν 0 = 240 . the image resolution is 640 × 480 . a calibration object 106 in the form of a stick is 70 cm long and is also simulated . the object has a fixed point a at [ 0 , 35 , 150 ] t . the other endpoint of the object is b , and c is located at the half way point between a and b . one hundred random orientations of the calibration object 106 are generated by sampling θ in [ π / 6 , 5π / 6 ] and φ in [ π , 2π ] according to uniform distribution . points a , b , and c are then projected onto the image 402 . gaussian noise with 0 mean and σ standard deviation is added to the projected image points a , b and c . the estimated camera parameters are compared with the ground truth , and their relative errors are measured with respect to focal length α . note that the relative errors in ( u 0 , ν 0 ) with respect to α , are as proposed by triggs in [ 18 ]. triggs pointed out that the absolute errors in ( u 0 , ν 0 ) is not geometrically meaningful , while computing the relative error is equivalent to measuring the angle between the true optical axis and the estimated one . the noise level is varied from 0 . 1 pixels to 1 pixel . for each noise level , the calibration system 100 performs 120 independent trials , and the results as shown in fig5 and 6 , are the average . fig5 is a graph illustrating relative errors for calibration results using the closed - form solution . fig6 is a graph illustrating the relative errors for calibration results using the nonlinear minimization result . errors increase almost linearly with the noise level . the nonlinear minimization refines the closed - form solution , and produces significantly better results ( with 50 % less errors ). at 1 pixel noise level , the errors for the closed - form solution are about 12 % while those for the nonlinear minimization are about 6 %. using actual real data , three toy beads were strung together with a stick to form the calibration object 106 . the beads are approximately 14 cm apart ( i . e ., l = 28 ). the stick was then moved around while fixing one end with the aid of a book . a video of 150 frames was recorded . a bead in the image is modeled as gaussian blob in the rgb space , and the centroid of each detected blob is the image point we use for camera calibration . the proposed algorithm is therefore applied to the 150 observations of the beads , and the estimated parameters are provided in a table shown in fig7 . the first row is the estimation from the closed - form solution , while the second row is the refined result after nonlinear minimization . for the image skew parameter γ , the angle between the image axes are also provided in parenthesis ( it should be close to 90 degrees ). for comparison , a plane - based calibration technique described in [ 22 ] was used to calibrate the same camera . five images of a planar pattern were taken . the calibration result is shown in the third row of the fig7 . the fourth row displays the relative difference between the plane - based result and the nonlinear solution with respect to the focal length ( 828 . 92 ). there is about a two percent difference between the calibration techniques , which can be attributed to noise , imprecision of the extracted data points and a rudimentary experimental setup . nevertheless , despite these non - ideal factors , the calibration results using a one - dimensional object are very encouraging . fig8 illustrates an example of a computing environment 800 within which the computer , network , and system architectures ( such as camera calibration system 100 ) described herein can be either fully or partially implemented . exemplary computing environment 800 is only one example of a computing system and is not intended to suggest any limitation as to the scope of use or functionality of the network architectures . neither should the computing environment 800 be interpreted as having any dependency or requirement relating to any one or combination of components illustrated in the exemplary computing environment 800 . the computer and network architectures can be implemented with numerous other general purpose or special purpose computing system environments or configurations . examples of well known computing systems , environments , and / or configurations that may be suitable for use include , but are not limited to , personal computers , server computers , thin clients , thick clients , hand - held or laptop devices , multiprocessor systems , microprocessor - based systems , set top boxes , programmable consumer electronics , network pcs , minicomputers , mainframe computers , gaming consoles , distributed computing environments that include any of the above systems or devices , and the like . the camera calibration methodologies may be described in the general context of computer - executable instructions , such as program modules , being executed by a computer . generally , program modules include routines , programs , objects , components , data structures , etc . that perform particular tasks or implement particular abstract data types . the camera calibration methodologies may also be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network . in a distributed computing environment , program modules may be located in both local and remote computer storage media including memory storage devices . the computing environment 800 includes a general - purpose computing system in the form of a computer 802 . the components of computer 802 can include , by are not limited to , one or more processors or processing units 804 , a system memory 806 , and a system bus 808 that couples various system components including the processor 804 to the system memory 806 . the system bus 808 represents one or more of any of several types of bus structures , including a memory bus or memory controller , a peripheral bus , an accelerated graphics port , and a processor or local bus using any of a variety of bus architectures . by way of example , such architectures can include an industry standard architecture ( isa ) bus , a micro channel architecture ( mca ) bus , an enhanced isa ( eisa ) bus , a video electronics standards association ( vesa ) local bus , and a peripheral component interconnects ( pci ) bus also known as a mezzanine bus . computer system 802 typically includes a variety of computer readable media . such media can be any available media that is accessible by computer 802 and includes both volatile and non - volatile media , removable and non - removable media . the system memory 806 includes computer readable media in the form of volatile memory , such as random access memory ( ram ) 810 , and / or non - volatile memory , such as read only memory ( rom ) 812 . a basic input / output system ( bios ) 814 , containing the basic routines that help to transfer information between elements within computer 802 , such as during start - up , is stored in rom 812 . ram 810 typically contains data and / or program modules that are immediately accessible to and / or presently operated on by the processing unit 804 . computer 802 can also include other removable / non - removable , volatile / non - volatile computer storage media . by way of example , fig8 illustrates a hard disk drive 816 for reading from and writing to a non - removable , non - volatile magnetic media ( not shown ), a magnetic disk drive 818 for reading from and writing to a removable , non - volatile magnetic disk 820 ( e . g ., a “ floppy disk ”), and an optical disk drive 822 for reading from and / or writing to a removable , non - volatile optical disk 824 such as a cd - rom , dvd - rom , or other optical media . the hard disk drive 816 , magnetic disk drive 818 , and optical disk drive 822 are each connected to the system bus 808 by one or more data media interfaces 826 . alternatively , the hard disk drive 816 , magnetic disk drive 818 , and optical disk drive 822 can be connected to the system bus 808 by a scsi interface ( not shown ). the disk drives and their associated computer - readable media provide non - volatile storage of computer readable instructions , data structures , program modules , and other data for computer 802 . although the example illustrates a hard disk 816 , a removable magnetic disk 820 , and a removable optical disk 824 , it is to be appreciated that other types of computer readable media which can store data that is accessible by a computer , such as magnetic cassettes or other magnetic storage devices , flash memory cards , cd - rom , digital versatile disks ( dvd ) or other optical storage , random access memories ( ram ), read only memories ( rom ), electrically erasable programmable read - only memory ( eeprom ), and the like , can also be utilized to implement the exemplary computing system and environment . any number of program modules can be stored on the hard disk 816 , magnetic disk 820 , optical disk 824 , rom 812 , and / or ram 810 , including by way of example , an operating system 826 , one or more application programs 828 , other program modules 830 , and program data 832 . each of such operating system 826 , one or more application programs 828 , other program modules 830 , and program data 832 ( or some combination thereof ) may include an embodiment of the camera calibration methodologies . computer system 802 can include a variety of computer readable media identified as communication media . communication media typically embodies computer readable instructions , data structures , program modules , or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media . the term “ modulated data signal ” means a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal . by way of example , and not limitation , communication media includes wired media such as a wired network or direct - wired connection , and wireless media such as acoustic , rf , infrared , and other wireless media . combinations of any of the above are also included within the scope of computer readable media . a user can enter commands and information into computer system 802 via input devices such as a keyboard 834 and a pointing device 836 ( e . g ., a “ mouse ”). other input devices 838 ( not shown specifically ) may include a microphone , joystick , game pad , satellite dish , serial port , scanner , and / or the like . these and other input devices are connected to the processing unit 804 via input / output interfaces 840 that are coupled to the system bus 808 , but may be connected by other interface and bus structures , such as a parallel port , game port , or a universal serial bus ( usb ). a monitor 842 or other type of display device can also be connected to the system bus 808 via an interface , such as a video adapter 844 . in addition to the monitor 842 , other output peripheral devices can include components such as speakers ( not shown ) and a printer 846 which can be connected to computer 802 via the input / output interfaces 840 . computer 802 can operate in a networked environment using logical connections to one or more remote computers , such as a remote computing device 848 . by way of example , the remote computing device 848 can be a personal computer , portable computer , a server , a router , a network computer , a peer device or other common network node , and the like . the remote computing device 848 is illustrated as a portable computer that can include many or all of the elements and features described herein relative to computer system 802 . logical connections between computer 802 and the remote computer 848 are depicted as a local area network ( lan ) 850 and a general wide area network ( wan ) 852 . such networking environments are commonplace in offices , enterprise - wide computer networks , intranets , and the internet . when implemented in a lan networking environment , the computer 802 is connected to a local network 950 via a network interface or adapter 854 . when implemented in a wan networking environment , the computer 802 typically includes a modem 856 or other means for establishing communications over the wide network 852 . the modem 856 , which can be internal or external to computer 802 , can be connected to the system bus 808 via the input / output interfaces 840 or other appropriate mechanisms . it is to be appreciated that the illustrated network connections are exemplary and that other means of establishing communication link ( s ) between the computers 802 and 848 can be employed . in a networked environment , such as that illustrated with computing environment 800 , program modules depicted relative to the computer 802 , or portions thereof , may be stored in a remote memory storage device . by way of example , remote application programs 858 reside on a memory device of remote computer 848 . for purposes of illustration , application programs and other executable program components , such as the operating system , are illustrated herein as discrete blocks , although it is recognized that such programs and components reside at various times in different storage components of the computer system 802 , and are executed by the data processor ( s ) of the computer . duane c . brown . close - 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