Note: Descriptions are shown in the official language in which they were submitted.
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CAMERA CALIBRATING APPARATUS AND METHOD
Field of the invention
The present invention relates to the field of object imaging with electronic
cameras, and more particularly to camera calibrating targets, as well as
apparatus
and methods employing such targets for calibrating electronic cameras, in
combination with laser light source as part of detection/measurement
instrumentation.
Brief description of the background art
Laser and electronic imaging technologies have been used for many years
in many product manufacturing applications, particularly for quality control
involving numerous types of measurement such as dimension, shape, profile or
surface characteristics such as roughness and presence of defects. Typically,
measurements are based on the well known laser triangulation ranging principle
involving a direct relationship between the distance separating a reference
plane
and a given point of the surface of an object under inspection as measured
along
an axis extending in a direction perpendicular to the surface in one hand, and
the
reflected light being shifted from a corresponding reference position as
observed
at the imaging sensor or camera location in the other hand. Thus, following an
appropriate calibration step, profile data as defined by series of calculated
distance values for corresponding points on the surface can be directly
derived
from light beam shifts measurements. A known calibration approach consists of
currently establishing the correspondence between each pixel position
coordinates provided at the imaging sensor within an image reference system
and
the spatial position coordinates of any point located within an inspection
area
delimited by the optical field of view of the imaging sensor or camera and the
illumination plane defined by the laser beam and with respect to a world or
object
reference system, using a mathematical camera model such as proposed by
Roger Tsai in "A versatile camera calibration technique for High-Accuracy 3D
machine vision metrology using off-the-shelf TV cameras and lenses", IEEE
Journal of Robotics and Automation, Vol. RA-3, No. 4, August 1987, which model
is calibrated from position coordinates data obtained through initial
measurements
using a calibration target of either of the coplanar or non-coplanar type. A
non-
coplanar calibration target consists of a structure defining a three-
dimensional
arrangement of reference points having known position coordinates within a
three-
dimensional reference system associated with such structure. In use, the
structure
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is accurately disposed in a camera calibration position with respect to the
three
coordinates axis of the reference system and with respect to the illumination
plane
defined by the laser source beam. The use of non-coplanar calibration targets
may be required in certain cases where an unknown optical parameter such as
scale factor uncertainty, has to be estimated. However, they require
simultaneous
and precise alignment with respect to all three coordinates axis of the
reference
system, whereas coplanar calibration targets require precise alignment with
respect to only two coordinates axis of the same reference system. Known
coplanar and non-coplanar camera calibration targets are disclosed in U.S.
Patent
no. 6,621,921 B1, U.S. Patent no. 6,437,823 B1 and U.S. Patent no. 6,195,455
B1. Although coplanar camera calibration targets are less arduous to align
with
the illuminating plane defined by the laser source as compared with non-
coplanar
calibration targets, the alignment still remains a critical operation in order
to
achieve the measurement accuracy requirements. Therefore, there is still a
need
for improved coplanar calibration targets as well as apparatus and method
using
such improved targets exhibiting ease of operation while insuring high
position
coordinates measurement accuracy.
Summary of invention
It is therefore a main object of the present invention to provide a camera
calibrating target, a camera calibrating apparatus and method using such
target,
which accurate correspondence between actual position coordinates within
calibration plane defined by a laser beam and the image position coordinates
generated by the camera according to its specific optical characteristics
while
involving minimal target alignment accuracy requirements.
According to the above main object, from a broad aspect of the present
invention, there is provided a camera calibrating target for use with a laser
source
capable of generating a beam of coherent light defining a calibration plane,
the
camera being characterized by a field of view in the direction of an optical
axis
forming a predetermined angle with said calibration plane, the beam
intersecting
the field of view of the camera to define an object imaging area. The camera
calibrating target comprises a frame defining a coplanar arrangement of more
than five reference points having known position coordinates within an object
reference system associated with the frame and defining at least two
dimensions,
the frame being capable of being disposed in a camera calibration position
wherein the coplanar arrangement of reference points is substantially parallel
to
the calibration plane. The target further comprises an arrangement of light
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reflecting members protruding from the arrangement of reference points in a
direction substantially perpendicular thereto so as to extend within the
object
imaging area when the frame is disposed in the camera calibration position to
allow the camera to capture an image formed by illuminated portions of the
light
reflecting members.
So as to maximize quality of the captured calibration image, the reference
points are located with one another within the arrangement and with respect to
the
coherent light beam so as to prevent shading of reflecting members which may
adversely affect the illuminated portions in forming the image.
Conveniently, the two-dimensional arrangement includes a plurality of
reference points series extending substantially transversely to the coherent
light
beam and being disposed in parallel spaced apart relationship in the direction
of
the light beam from a foremost position on the frame proximal to the laser
source
to a rearmost position on the frame distal to the laser source.
Preferably, the respective points of adjacent ones of said series are
transversely shifted with one another within the arrangement and with respect
to
the coherent light beam so as to prevent shading of reflecting members which
may adversely affect the illuminated portions in forming the image.
According to the above main object, from a further broad aspect of the
invention, there is provided a camera calibrating apparatus for use with a
laser
source capable of generating a beam of coherent light defining a calibration
plane,
the camera being characterized by intrinsic imaging parameters and a field of
view
in the direction of an optical axis forming a predetermined angle with the
calibration plane, the beam intersecting the field of view of the camera to
define an
object imaging area. The apparatus comprises a calibration target including a
frame defining a coplanar arrangement of more than five reference points
having
known position coordinates within an object reference system associated with
the
frame and defining at least two dimensions, the frame being capable of being
disposed in a camera calibration position wherein the coplanar arrangement of
reference points is substantially parallel to the calibration plane. The
target further
includes an arrangement of light reflecting members protruding from the
arrangement of reference points in a direction substantially perpendicular
thereto
so as to extend within the object imaging area when the frame is disposed in
the
camera calibration position to allow the camera to capture an image formed by
illuminated portions of the light reflecting members. The apparatus further
comprises image processor means for estimating position coordinates of the
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illuminated portions within an image reference system associated with the
camera,
for associating the estimated position coordinates of each illuminated portion
within the image reference system with the known position coordinates of a
corresponding one of said reference points in relation with a corresponding
one of
said calibration planes within the object reference system, for calibrating a
camera
model based on the intrinsic imaging parameters with the associated known
position coordinates and estimated position coordinates, and for using the
calibrated camera model to associate any position coordinates in the object
reference system with corresponding position coordinates in the image
reference
system.
According to the above main object, from another broad aspect of the
present invention, there is provided a method of calibrating a camera
characterized by intrinsic imaging parameters and a field of view in the
direction of
an optical axis. The method comprises the steps of: i) generating a beam of
coherent light defining a calibration plane; ii) disposing the camera so that
its field
of view in the direction of its optical axis forms a predetermined angle with
the
calibration plane and so that the beam intersects the field of view of the
camera to
define an object imaging area; iii) providing a calibration target including:
a) a
frame defining a coplanar arrangement of more than five reference points
having
known position coordinates within an object reference system associated with
the
frame and defining at least two dimensions; and b) an arrangement of light
reflecting members protruding from the arrangement of reference points in a
direction substantially perpendicular thereto; iv) disposing the frame in a
camera
calibration position wherein the coplanar arrangement of reference points is
substantially parallel to the calibration plane so that the light reflecting
members
extend within the object imaging area; v) generating with the camera an image
formed by illuminated portions of the light reflecting members; vi) estimating
position coordinates of the illuminated portions within an image reference
system
associated with the camera; vii) associating the estimated position
coordinates of
each illuminated portion within the image reference system with the known
position coordinates of a corresponding one of the reference points within the
object reference system; viii) calibrating a camera model based on the
intrinsic
imaging parameters with the associated known position coordinates and
estimated position coordinates; and ix) using the calibrated camera model to
associate any position coordinates in the object reference system with
corresponding position coordinates in the image reference system.
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Brief description of the drawings
Fig. 1 is an elevation view of the camera calibration apparatus according to
a first embodiment of the present invention, showing the calibration target
aligned
with the respective light beams generated by two laser sources associated with
5 respective electronic cameras subjected to a calibration procedure;
Fig. 2 is a front end view, along iines 2-2 shown in Fig. 1, of the camera
calibration target and laser source of the calibration set up of the first
embodiment,
showing the particular spatial arrangement of reference points from which
protrude the light reflecting members as part of the calibration target;
Fig. 3 is an enlarged partial view of the camera calibration target
illustrated
in Fig. 2, on which the trajectories of light rays are shown to illustrate
that the
particular arrangement of reference points and associated reflecting members
ensure efficient illumination thereof by laser beam to prevent shading effect;
Fig. 4 is a process flow diagram showing the main image processing steps
performed according to a coplanar calibration procedure using the calibration
target of the invention, to associate any position coordinates in the object
reference system with corresponding position coordinates in the image
reference
system; and
Fig. 5 is a detailed elevation view of the camera calibration target as part
of the embodiment shown in Fig. 1 and shown in two different positions, to
illustrate that the captured image is not significantly affected by a
deviation in
alignment of the calibration target with respect to the calibration plane.
Detailed description of the preferred embodiments
Preferred embodiments of camera calibration apparatus and method
making use of a calibration target according to the invention will now be
described
in detail. Referring now to Fig. 1, a basic camera calibration apparatus set
up
using the calibration target of the invention and generally designated at 10,
includes a laser source 12, such as a LasirisTM SNF laser supplied by Stocker
Yale
(Salem, NH) for generating a beam of coherent light represented at 14 being
preferably of a fan-shaped type as better shown in Fig. 2 so as to define a
calibration plane generally designated at 16 and delimited by axis lines 18,
18' and
28 shown in Fig. 2 as determined by a preset scanning range characterizing
laser
source 12. As shown in Fig.1, a camera 20 such as model A501 K supplied
by Basler Vision Technologies (Exton, PA), which is subjected to the
calibration
procedure, is disposed with respect to the calibration plane 16 defined by
laser
source 12 so that its field of view in the direction of optical axis 22 forms
a
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predetermined angle B with the calibration plane defined by laser beam 14,
which
angle 0 is formed at an intersection points PB of the calibration plane as
better
shown in Fig. 2. Angle 0 should be preferably of at least 30 for providing
depth
to reference points defined by the calibration target, as will be explained
later in
more detail. It can be seen from Figs. 1 and 2 that the laser beam 14
intersects
the field of view of camera 20 so as to define and object imaging area
generally
designated at 24 forming a projection within an object reference system
generally
designated at 26 defining at least two dimensions, namely represented by X.
axis 28 and Y. axis 30, delimited by X axis 28 between coordinates X=X, and
X=XZ and by Y axis 30 between coordinates Y=0 and Y=Y, . According to a
preferred embodiment, the camera calibrating target 32 includes a frame 34 ,
defining a coplanar arrangement of reference points 37, 38, 39, 40, 41 as
better
shown in Fig. 2, which reference points have known position coordinates within
the object reference system 26 associated with frame 34. It can be seen from
Fig.
1 in view of Fig. 2 that the frame 34 is capable of being disposed in a camera
calibration position wherein the coplanar arrangement of reference points 37,
38,
39, 40, 41 is substantially parallel to the calibration plane 16. The
calibration target
frame 34 may be adjustably maintained in the camera calibration position using
any appropriate support means (not shown). In the example shown in Figs. 1 and
2, camera and laser source 12 are angularly oriented with one another so that
the
object imaging area 24 corresponding to camera calibration position of target
frame 34 substantially coincides with an object scanning location intersected
by an
object travelling path represented by axis 42 as defined by conveyors 44, 44'
of a
transport system adapted to be used in combination with laser source 12 and
camera 20 following the calibration procedure. It can be seen from Fig. 1 that
the
calibration target 32 further includes an arrangement of light reflecting
members
46, 47, 48, 49, 50, 51 protruding from the arrangement of reference points 36,
37,
38, 39, 40, 41, in a direction substantially perpendicular thereto so as to
extend
within the object imaging area 24 when the frame 34 is disposed in the camera
calibration position allowing camera 20 to capture an image formed by
illuminated
portions of the light reflecting members 46, 47, 48, 49, 50, 51. Although the
frame
34 is of a rectangular shape in the example shown, it is to be understood that
any
other appropriate shape may be used, provided that a sufficient number of
adequately distributed reflecting members extend within the object imaging
area.
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Turning now to Fig. 3, reference points 36, 37, 38, 39, 40, 41 are located
with one another within the arrangement and with respect to light beam 14 so
as
to prevent shading of reflecting members 47, 48, 49, 50, 51 as shown in Fig. I
which may adversely affect the illuminated portions thereof in forming the
image
by camera 20. The external diameter of the light reflecting members 46, 47,
48,
49, 50, 51 as well as their location on frame 34 as defined by corresponding
reference points 36, 37, 38, 39, 40, 41 are determined considering the camera
resolution, width and height dimensions of the imaging area along Xw axis 28
and
Y. axis 30, as well as the relative position of the laser source 12, so as to
ensure
that all light reflecting members, which are optimally distributed within the
imaging
area, receive appropriate illumination. More particularly, the coplanar
arrangement
is designed to include a plurality of reference point series respectively
corresponding to points 36, 37, 38, 39, 40, 41 which extend substantially
transversely to coherent light beam 14 as represented by a set of rays in Fig.
3
originating from laser source 12, and being disposed in parallel spaced apart
relationship in the direction of light beam 14 from a foremost position on the
frame
34 proximal to laser source 12, corresponding to series of reference points
36, to
a rearmost position on frame 34 distal to laser source 12, corresponding to
series
of reference points 41 in the example shown. It can be seen from Fig. 3 that
the
shadow area extending along any ray axis 52 aligned with any corresponding ray
53 as a result of the illumination of any corresponding reflecting member such
at
46 in the example shown, is offset to any other ray axis associated with any
other
incident light ray that illuminates it. It can also be seen from Fig. 3 that
the
minimum transverse clearance along X axis 28 is set to a value "c" that is
sufficient to prevent adverse shading while complying with camera image
resolution to ensure reliable position coordinates determination of each point
from
the captured image. In other words, the respective points of adjacent ones of
series 36, 37, 38, 39, 40 and 41 are transversely shifted with one another
within
the arrangement and with respect to light beam 14 so as to prevent shading of
deflecting members 47, 48, 49, 50 and 51 which may adversely affect the
illuminated portion thereof in forming the image captured by camera 20. It can
be
appreciated that the spacing between adjacent reflecting members 46 of the
foremost series must be set at a higher value as the imaging area height is
increased. Preferably, an appropriate coating is applied onto the surface
subjected
to illumination on each light-reflecting member, to provide a resulting
reflectance
that is comparable with the reflectance obtained with the real object to be
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inspected. Since the calibration procedure requires position identification of
each
light-reflecting member on a captured image, each reflecting member must
receive a minimal illumination level from the laser beam. A minimum
identification
threshold corresponding to a minimum image grey level, allowing segmentation
of
the image portion corresponding to reflecting member as opposed to image
background, may be estimated. With the particular camera used in the present
example, such identification threshold has been set to at least 10/255 to
prevent
segmentation error. The calibration procedure requires establishing a
correspondence between the actual world object coordinates within the object
reference system 26 with image coordinates within the image reference system
derived from the calibration target image. For so doing, two sorting steps are
preferably performed successively on the basis of estimated image coordinates.
Assuming that the actual world, object coordinates are generated line by line,
from
top to bottom and from left to right, knowing the exact number of points on
each
line, the image coordinates are first sorted from top to bottom and then from
left to
right within each line. Hence, exact correspondence between image coordinates
and object coordinates can be obtained, provided the calibration target is
adequately positioned. The calibration target 32 must be positioned in such a
manner that none of the reference points within any series, for instance
series 36,
located along Yw axis 30 is over any point of a following reference points
series,
such as series 37 for the instant example. It is to be understood that any
other
appropriate manner to establish a correspondence between the object
coordinates
and the image coordinates, such as using a predetermined matching data file
stored in the computer, can be employed.
Conveniently, in cases where more than one surface or side of an
object has to be scanned simultaneously, a same camera calibration target 32
can
be provided with complementary sets of reference points series designated at
36',
37', 38', 39', 40', 41' in Fig. 2 and corresponding light reflecting members
46', 47',
48', 49', 50', 51' as shown in Fig. I which are disposed symmetrically with
respect
to X axis 28 in the example shown, as part of a complementary section 54
provided on the calibration target 32 It is to be understood that a different,
non-
symmetrical configuration may also be used depending on the specific
application
contemplated. For example, camera 20' may be chosen to have different imaging
resolution as compared to camera 20, and the spatial distribution and diameter
of
the reflecting members provided on target complementary section 32 may be set
accordingly. Furthermore, in a case where articles to be inspected while
carried by
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conveyers 44, 44' shown in Fig. 1 exhibit thickness variations so that the
position
of their respective top surfaces varies with reference to axis Yw 30, the
field of
view characterizing camera 20 must be different to the field of view of camera
20'
which scans article bottom surfaces whose position with reference to axis Yw
30 is
substantially maintained at the level of conveyers 44,44'. The complementary
section 54 of calibration target 32 can be used with a further laser source
12' for
generating a second laser beam 14', associated with a further camera 20'
characterized with a complementary field of view in the direction of its
optical axis
22' forming a predetermined angle B' with a complementary calibration plane
16'
delimited bv axis lines 19, 19' and associated with further camera 12'. The
calibration plane 16' is aligned with calibration plane 16 when a single
calibration
frame for two cameras is used. Although axis lines 18, 19 and 18', 19'
intersect
axis XW 28 respectively at same points x3, x4 since laser sources 12, 12' use
equal fan angles in the example shown in Fig. 2, it is to be understood that
different fan angles may be set for laser sources 12, 12', provided that the
desired
target areas are illuminated. As part of a camera calibrating apparatus using
the
calibration target of the present invention, there is provided a processor
means
including an image signal acquisition module 55 for receiving image signals
from
either camera 20 or 20' through lines 59, 59' using commercially available
hardware such as model Odyssey X-ProT"" supplied by Matrox Electronic System
(Dorval, Quebec,Canada) and a computer 56 having the capability to run the
required image analysis software that is especially programmed to carry out
image
processing steps, as will be explained in detail below with reference to Figs.
4
and 7, using a well-known camera calibrating algorithm based on a camera model
proposed by Roger Y. Tsai in "A Versatile Camera Calibration Technique for
High-
Accuracy 3D Machine Vision Metrology Using Off-the-Shelf TV Cameras and
Lenses, IEEE Journal of Robotics and Automation, Vol. RA-3, NO. 4, August
1987, pages 323-344.
A first preferred mode of operation of the camera calibrating target as
described above according to a preferred embodiment of the invention will now
be
explained in more detail. The same calibration procedure can be performed in a
same manner for each camera 20, 20'. Referring to Fig. 1, the computer 56
sends
a control signal to the acquisition module 55 through data line 58 for causing
thereof to transfer through input data line 60 the data representing camera
image
formed by illuminated portions of light reflecting members 46, 47, 48, 49, 50,
51 or
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46', 47', 48', 49', 50', 51' as received from either camera 20 or 20'
depending on
the computer command sent.
Referring now to Fig. 4, the image data represented at 63 in the process
flow diagram can be obtained as follows. In order to prevent image
contamination
5 due to background pixels in the identification of image points corresponding
to
reference points provided on the calibration target, a preliminary image
processing
algorithm is preferably performed on the raw image captured by the camera 20.
Such algorithm typically requires that all reference points of the calibration
target
are made visible in the image, and when background contamination areas exist
in
10 the image, such areas are individually smaller than the illuminated portion
of each
reflecting member, or their respective grey levels are lower than a preset
detection
threshold. According to this algorithm, a current threshold is first
initialized to the
value of the preset minimum threshold. Then the raw image is binarized using
the
current threshold, followed by a detection of all image blobs which respective
area
is higher than the minimum area corresponding to the level of reflection of
the
laser beam onto a given reflecting member. If the detected number of blobs
corresponds with the number of reflection members provided on the target, the
resulting image containing only the calculated blobs is generated so as to
eliminate any background contamination. Otherwise, the current threshold is
incremented and the algorithm is repeated until the above condition is
satisfied.
Then, the mass centre of each image blob is calculated using a well-known
image
processing subroutine, assuming that the calculated mass centre substantially
corresponds to the actual mass centre position. It is to be understood that
many
alternate image pre-processing algorithms may be used to generate reliable
image data 63. Hence, image data 63 is used by computer 56 at step 70 to
estimate position coordinates of the illuminated portions within an image
reference
system generally designated at 57, 57' in Figs. 1 and 2 associated with the
selected camera 20, 20', respectively formed by X axis 62 in Fig. 1 and Y axis
64 in Fig. 2 for camera 20, and X axis 66 in Fig. I and Y axis 68 in Fig. 2
for
camera 20' wherein axis Z is colinear with optical axis 22'. Since the
reference
points 36, 37, 38, 39, 40, 41 provided on the upper portion of calibration
target 32
shown in Fig. 2 are coplanar, the ( Xw , Y. , Zw ) object reference system at
26 may
be chosen so that Z,=O, while the origin at 31 is distant from optical Z axis
22 at
point P. as well as from Y axis 64 of the image system 57, to ensure that
T, component of the translation vector be different from zero, in order to
simplify
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the computation procedure that will be explained later in detail. At step 70,
the
position coordinates (Xd , Yd ) of the illuminated portion within image
reference
system 57 are computed using predetermined intrinsic imaging parameters
characterizing the camera 20. A whole image frame stored in memory of computer
56 is fetched to detect row and column number of each reference point i as
expressed by ( Xr , Yr ) for i=1, ..., N, wherein N is the total number of
reference
points. Then the position coordinates (Xd , Yd ) representing the distorted or
true
image coordinates are derived using the following equations:
Xd;=sX'dx(X f-Cx) (1)
Ydi=dy(Yr-Cv) (2)
dx=dx N~x (3)
Nfx
wherein ( Cx , Cy ) are row and column numbers of the centre of computer image
frame memory;
dx is the centre-to-centre distance between adjacent sensor elements in
X scanning direction;
dy is centre-to-centre distance between adjacent CCD sensors provided
on the camera in the Y direction;
N, is the number of sensor elements in the X direction;
Nfx is the number of pixels in a line as sampled by the computer; and
sx is an image scale factor used to compensate uncertainty about
estimation of d,r and dy.
When unknown, the scale factor sx can be estimated using a simple technique as
proposed by R. K. et al. in "R. K. Lenz and R. Y. Tsai, "Techniques for
calibration
of the scale factor and image centre for high accuracy 3D machine vision
metrology", Proc. IEEE Int. Conf. Robotics and Automation, Raleigh, NC, March
31-April 3, 1987. Then, at a following step 72 shown in Fig. 4, the estimated
position coordinates ( Xd, , Ydi ) of each illuminated portion within image
reference
system 57 is associated with the known position coordinates ( XH,i , Y;, Z; )
of a
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corresponding one of the reference points 36, 37, 38, 39, 40, 41 within the
object
reference system 26. At a following step 74, a camera model based on intrinsic
imaging parameters characterizing camera 20 is calibrated with known position
coordinates ( Xwi , Ywi , Zwi ) and estimated position coordinates ( Xdi , Ydi
) that have
been associated at prior step 72. For each pair of position coordinates
( Xwi , Ywi, Zwi ), ( Xdi , Ydi ), and using a number of reference points N
being much
larger than 5, the following linear equation is used to derive the values of
external
parameter related components Ty'r, , Ty'rZ , Ty'Tx , Ty'r4 , Ty'r5
Ty'r
Tyz
[Ydixwi Ydi.ywi Ydi - Xdixwi - Xdi.ywi ] Ty'Tx =Xdi (4)
Tv'ra
Ty lr5
,~ xw
y =R yw +T (5)
Z zw
r r2 r3
R r4 rs r6 (6)
r7 r8 r9
TX
T T,. (7)
TZ
(8)
Xu=f z
Yu=f~ (9)
Xd+Dx =Xu (10)
Yd +Dy=Y (11)
Dx=Xd1C, r2 (12)
Dy=Yd K, r2 (13)
r= Xd+Y~ (14)
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wherein:
(x,y,z) represent the coordinates of any point within a three-dimensional
reference system (not shown) associated with the camera;
R is the 3 x 3 rotation matrix as an external parameter;
r, ,...r9 are the rotation coefficient of the rotation matrix R;
T is the translation vector as an external parameter;
Tx,TY,TZ are the three components of the translator vector T;
(Xu,Yõ ), are ideal, undistorted image coordinates using perspective
projection within the camera three-dimensional reference system according to
known pinhole camera geometry, which corresponds to the image reference
system 57;
f is the focal length as an intrinsic parameter characterizing the camera,
which is the distance between the image plane defined by axis X and Y of the
image reference system 57 and the optical centre corresponding to the origin
of
the camera three-dimensional coordinates system;
Dx,Dy represent radial distortion respectively along axis X and Y of image
reference system 57, as estimated by the first term of a corresponding
infinite
series;
(Xd,Yd ) are actual, distorted position coordinates in the image reference
system 57; and
ic, is the radial distortion coefficient.
Having solved equation (4) to obtain values of TY'r , Ty'rZ , Ty'Tx , TY'r4 ,
TY'r5
the values of rotation coefficient r, ... r9 as well as the translation
components
TX,Ty can be computed using the following 2 x 2 submatrix:
rl rz - Lr ITY r2 /Ty J ()
C[r4 rs ] ra ITY r5 ITY
if none of whole rows or columns of C vanishes, the component Ty can be
computed from
CA 02508595 2008-06-27
14
iiz
Sr -4M+r rs-r4r2 ~ ]
TY S. [
z(16)
- ~z
2~r rs-r4rz 1
wherein Sr = r,z+rZ+r4z+r52
if a given row or column of C vanishes, the component TY can be obtained from:
TY = (r,?+rz) (17)
wherein r-, rj are the elements in the row or column of C that do not vanish.
Having extracted the square root of component TY from either equation (16) or
(17), the sign of TY can be determined by first choosing an object reference
point
i associated with computer image coordinates ( X f, Yf, ) away from image
centre
( Cx , C, ) corresponding to object coordinates Then by assuming
that the sign of TY is "+1 ",the following expression are computed:
r =(T)T 'r~y (18)
r2 = (Ty'rz YY (19)
r4 = (TY Ir4 Y (20)
rs=i,Ty'rsYy (21)
Tx = (TY'Tx YY (22)
x=r, xw+rZ yw+Tx (23)
y=r4 xw+rs yw+TY (24)
assuming that Ty'r, , Ty'rz , Ty'Tx , TY'r4 and Ty Irs have been previously
determined in a manner explained above. Hence, if the values for x and X have
a same sign as well as the sign of y and Yvalues, then the sign of TY is "+1
",
otherwise the sign of TY is "-1". As to the rotation coefficient r, ,...r9 and
Tx , they
can be derived as follows:
CA 02508595 2005-05-27
r, =
(Ty'r 1TY (25)
r = (Tv'r2 ~y (26)
r4 = (TYIr4~Y (27)
5 rs = (7')T 'r5 ky (28)
Tx = (Ty-'Tx Yy (29)
Then, an approximation of focal length f is computed from the following linear
equations:
[y, -dyY ~ ~~ ]=wj dY, (30)
Y =r4 xN,i +r5 yW,+r6 =O+TY (31)
w; =r7 x;+r8 ywi +r9 =0 (32)
if the obtained value forf is positive, the rotation matrix R can be
calculated
using the following expression:
,iz
r, r2 (1-r,z-r2~
R= r4 r5 s(1-r4 -rs Y/z (33)
r7 r8 r9
wherein s is the inverted sign of the resulting sign of (r, r4 +r2 rs with r7
, r8
r9 being determined from the outer product of the first two rows using the
orthonormal and right-handed property of rotation matrix R. If the approximate
value of f obtained with equation (30), (31) and (32) is negative, the
rotation
matrix R can be calculated from the following expression:
r r, -(1-r12-r2 2Y12
R= r4 r5 -s(1-r2-r2y'2 (34)
-r7 -r8 r9
CA 02508595 2005-05-27
16
Then, equations (30), (31) and (32) may be used to derive an approximate value
for translation components T. which will be used along with approximate values
obtained for f and x, an exact solution for these parameters using the
following
equations:
dyY+dyYK, rz= f r4xw+r5 yw+r6 Zw+Ty (35)
r7 Xw+rg yw+r9 Zw+Tz
with:
r= (sx'dxXY+(dyYY (36)
The above equations are solved using a standard optimization scheme such as
well known steepest descent, wherein approximate values for f and TZ are used
as initial values with x, =0. All intrinsic parameters in the camera model as
generally defined in equation (5) having been determined, the calibrated
camera
model so obtained can be used, in final step 76 shown in Fig. 4 to associate
any
position coordinates ( Xw , Yw , Zw ) in the object reference system 26 with
corresponding position coordinates ( Xd , Yd ) in the image reference system
57.
Turning now to Fig. 5, it can be seen that a misalignment of the calibration
target 32, in rotation within the plane defined by axis Yw at 30 and axis Zw
at 42
and related to the object reference system 26, which rotation is indicated in
phantom lines at 78, does not have any significant effect on the position of
the
illuminated portions 78, 78' on each reflecting members 46-51 and 46'-51',
even if
a relatively important shift "s" may be measured along axis Zw 42 for
reflecting
members 46 and 46'.
