WO2004092826A1 - Method and system for obtaining optical parameters of camera - Google Patents

Abstract

The present invention is a method and system for analyzing the mapping mechanism and accordingly obtaining the optical parameters of a camera. A specific mapping characteristic that one sight ray exclusively corresponds to one imaged point is utilized; with reference to a particular imaged point, absolute spatial coordinates conforming to the mapping characteristic are searched for analyzing the mapping mechanism of the camera. A planar target with a physical central-symmetric pattern (PCP) is employed to locate the principal point and absolutely orient the optical axis, with the aid of the both centers of the PCP and its corresponding image with the similar geometric feature, termed the imaged central-symmetric pattern (ICP). Then refer to the optical axis and actively adjust the relative distance between the camera and the target to enable the mapping traces, cast from different calibration marks on the target, to overlap on the image plane. Based on this phenomenon, the sight ray can be analyzed by the overlapping mechanism and a methodology developed thereby to obtain the optical parameters of the camera. Because the invention totally and solely employs the measured data to deduce the parameters, which means in other words that no postulation of a given mapping mechanism is necessary; it is most suitable for application to the kind of camera with an unknown optical model. Actually, the bigger the deformation of the image, the higher the sensitivity of the invention in its operations. Hence, the applications of wide-angle cameras can be widely expanded and, furthermore, the invention can evaluate or determine the specifications of the camera. The operating procedure of the invention is simple and low-cost, wherefore it has a major practicability commercially as well as industrially.

Description

METHOD AND SYSTEM FOR OBTAINING OPTICAL PARAMETERS OF

CAMERA

BACKGROUND OF THE INVENTION

Field of Invention

The present invention relates to a method and system for obtaining the optical

parameters of a camera. Particularly, it is a method and system for analyzing the camera

with a lens seriously diverging from the rectilinear projection mechanism, such as a

fisheye lens, to obtain the optical parameters comprising the principal point, the

viewpoint, the focal length constant and the projection function.

Related Art

The camera systems in the field of artificial vision have preferred using lenses with a

narrow field of view (FOV) in order to obtain images approaching an ideal perspective

projection mechanism for precise measurement and easy image processes. The pinhole

model is usually a basis to deduce the camera's parameters. The obtained intrinsic and

extrinsic parameters can be employed in visual applications in the quest for improved

precision, for instance in 3-D cubical inference, stereoscopy, automatic optical inspection,

etc. As for the image deformation, a polynomial function is used to describe the deviation

between original images and the ideal model or to conduct the job of calibration. These

applications, however, currently have a common limitation of narrow visual angles and

an insufficient depth of field. A fisheye camera (also termed a fisheye image sensor) mounted with a fisheye lens,

which focuses deeper and wider, can capture a clear image with a FOV of as much as over

180 degrees, but a severe barrel distortion develops. If the application is a surveillance

system and the request is only to monitor the movement of people or things, a partial

distortion in images can be tolerated. If the purpose is to take pictures for virtual reality

(VR), it is also acceptable that images "look like" normal ones. However, if the purpose

involves the measurement of an object's physical size or 3-D image metering, it has to be

admitted techniques for precisely obtaining the optical parameters of the fisheye camera

are still absent.

Because the optical geometry of the fisheye camera is far from the rectilinear

perspective projection model, the optical parameters are hard to be precisely deduced by

those methods employed in the related art for normal cameras. Therefore, technologies

developed for the usual visual disciplines have not resulted in the capability to process

images of the fisheye camera (simplified as "fisheye images" hereinafter).

R. Y. Tsai [1987] ( Tsai, "A versatile camera calibration technique for high-accuracy

3D machine vision metrology using off-the-shelf TV camera and lenses", IEEE Journal of

Robotics and Automation, Vol. RA-3, No 4, Aug, 1987, pp 323-344 ) brought up a radial

alignment constraint in the radial-symmetric projection mechanism to derive the

parameters of the camera. He employed five non-coplanar points of known absolute

coordinates in viewed space and the positions of their corresponding images in the image

plane, referring to the radial alignment of the optical axis constraining the vectors of the

image distance and absolute distance to deduce the coefficients of a rotation matrix and

translation matrix, which stand for the orientation, displacement and viewpoint of the camera. The focal length is obtained through the hypothesis of the rectilinear projection

geometry, but a non-linear function is taken to describe the distortion mechanism of the

image. Its chief merit is the ability to obtain the parameters of the camera with only

simple experimental devices. In cameras with little distortion, the results from Tsai's

model are quite accurate. But its demonstration is also based on the hypothesis of the

rectilinear projection; it will involve a large error under a severely nonlinear projection

system like the fisheye lens, and the results will be dependent on the arrangement of

calibration marks. Hence, Tsai's model cannot be directly quoted in the case of wide-

angle cameras, such as ones mounted with the fisheye lens.

However, if an artificial vision system has the advantages of wide-angle views, clear

images and the capability of handling a cubical projection mechanism, it will be

substantially more functional and competitive with a wider application field. Moreover,

the excellent advantages of a nearly infinite view depth, simple structure and tiny volume

are strengths vis-a-vis which other kinds of lenses are scarcely comparable to the fisheye

lens. However, the severe distortion is a vital disadvantage in some applications, so the

issue of identifying the features and irregular mapping mechanism of the fisheye lens and

accordingly developing the related methodology is extremely important. Further, the

applications depending on the accuracy of image calibration, for example in the

stereoscope or autonomous robotic vision, are difficult to accurately handle without the

precise optical parameters of the fisheye camera.

Owing to the poor deduced accuracy of the optical parameters of a camera based on

the rectilinear perspective projection model, some alternative solutions have been

advanced for handling the transformation of the fisheye image. Among these alternative approaches an image-based algorithm aims at a specific camera which mounts a specific

lens conforming to a specific projection mechanism so as to deduce the optical parameters

based simply on the images displayed. With reference to FIGs. 1 A and IB, wherein FIG.

1A expresses the imageable area 1 of a fisheye image in a framed oval/circular region and

FIG. IB is the hemispherical spatial projecting geometry corresponding to FIG. 1 A, both

figures note the zenithal distance of α, which is the angle defined by an incident ray and

the optical axis 21, and the azimuthal distance of β, which is the angular vector in the

polar coordinate system whose origin is set at the principal point. Quoting the positioning

concept of a globe, β is the angle referring to the mapping domain 13' of the prime

meridian 13 on the equatorial plane in the polar coordinate system, shown in FIG. IB.

Thus, π/2-α is regarded as latitude and β as longitude. Therefore, if several imaged points

are situated along the same radius of the imageable area 1, their corresponding spatial

incident rays would be on the same meridional plane (like the sector determined by the arc

C'E'G' and two spherical radii); namely, their azimuthal distances (β) are invariant, such

as points D, E, F, and G in FIG. 1 A corresponding to points D', E', F', and G' in FIG. IB.

(Note: the phenomenon utilized by the image-based algorithm is not only relevant to the

fisheye lens; actually, it is the radial alignment constraint in Tsai's model on condition of

using a rectilinear perspective projection lens.)

In addition to the specific projection mechanism, the image-based algorithm makes

several basic postulates: first, the imageable area 1 of the fisheye image is an analyzable

oval or circle, and the intersection of the major axis 11 and the minor axis 12 (or two

diameters instead) situates the principal point, which is cast by the optical axis 21 shown

in FIG. IB; secondly, the boundary of the image is projected by the light rays of α=π/2; third, α and p are linearly related, wherein p, termed a principal distance, is the length

between an imaged point (such as point E) and the principal point (point C). For example,

the value of α at point E is supposed to be π/4 since it is located in the middle of the radius

of the imageable area 1 and, therefore, the sight ray corresponding to point E is destined to

pass through point E' in the hemispherical sight space, as shown in FIG. IB. The same

occurs with points C and C\ points D and D', points F and F', and so on. An imaged point

on the image plane can be denoted as (u, v) in the Cartesian coordinate system or as (p, β)

in the polar coordinate system, both taking the principal point as their origin; the vector

coordinate of its corresponding sight ray in space is denoted as (α, β).

Although the mapping mechanism was not really put on discussion in the image-

based algorithm, it is actually the equidistant projection (simplified as the EDP

hereinafter) with the postulation of a 180-degree visual angle (simplified totally as the

EDPπ hereinafter). The EDP's projection function is p=kα wherein k is a constant and,

actually, the focal length constant off. In order to fit the postulations described above, a

qualified camera body mounted with a qualified lens is utterly necessary. Generally it is a

special combination with no room for flexibility. Based on the EDPπ postulation, the

focal length constant (f) can be obtained by dividing the radius of the imageable area 1

with π/2; the spatial angle (α, β) of the corresponding incident ray can also be analyzed

from the planar coordinates (u, v) in the imageable area 1.

Therefore, in light of the known skills of image-analysis, an "ideal EDPπ image" can

be transformed into the image remapped by the rectilinear perspective projection referring

to any projection line as a datum axis. This image-based algorithm is easy and no extra

calibrating object is needed. The US patent 5,185,667 accordingly developed a method to transform fisheye

images conforming to the rectilinear perspective projection model along the projection

mechanism shown in FIGs. 1 A and IB so as to monitor a hemispherical field of view (180

degrees by 360 degrees). This patented technology has been applied in endoscopy,

surveillance and remote control as disclosed in US patents 5,313,306, 5,359,363 and

5,384,588. However, it is worth noting that these serial US patents did not concretely

demonstrate a general fitness toward average fisheye lenses. Thus, the image-transformed

accuracy of the patented technology is a big question when no specific fisheye lens is used.

Currently, in practice system application manufacturers ask for limited-specification

fisheye lenses combined with particular camera bodies and provide exclusive software,

and then the patented technology (US patent 5,185,667) will have practical and

commercial value.

Major parts of the image-based postulates mentioned, however, are unrealistic

because many essential factors or variations have not been taken into consideration. First,

the EDPπ might just be a special case among possible proj ection geometric models (note:

however, it is the most familiar projection model of the fisheye lens). Referring to FIG. 2,

three possible and typical projection curves of the fisheye lens are shown, implying

moreover that the natural projection mechanism of the fisheye lens might be the following:

the stereographic projection (or SGP, whose projection function is p = 2/xtan(α/2)) and

the orthographic projection ( or OGP, whose projection function is p =/ sin(α)).

Moreover, the coverage of the FOV is not constantly equal to π, perhaps being either

larger or smaller. From the curves in FIG. 2, the differences respectively between the

three projection models are obviously increasing along the growing zenithal distances (α). Thus, distortions will develop if all projection geometries are locked on the EDPπ to

transform images accordingly. Secondly, the FOV of π is hard to evaluate since the shape

of the imageable area 1 is always presented as a circle, irrespective of the angular scale of

the FOV. A third factor concerns the errors caused in locating the image border even

though the FOV is certainly equal to π. The radial decay caused by the radiometric

response is an unavoidable phenomenon in a lens, especially when dealing with a larger

FOV. This property will induce a radial decay on the image intensity, occurring especially

with some simple lenses, so that the actual boundary is extremely hard to set under that

bordering effect. Perhaps no real border feature even exists under the consideration of the

diffraction phenomenon of light. Finally, if the imageable area 1 of a camera is larger than

the sensitive zone of a CCD, only parts of the "boundary" of an image will show; hence

the image transformation cannot be effectively executed. Consequently, the image-based

algorithm depends extensively on the selected devices irrespective of whether the lens

conforms to the ideal EDPπ postulation or not. Alternatively, the method will result in

poor accuracy, modeling errors, a doubtful imageable area 1 extracted, an unstable

principal point situated, and practical limitations; these problems would keep the methods

in the related art from accurately solving the extrinsic and intrinsic parameters of the

camera in the interests of developing computer vision systems, not to mention the

viewpoint, in behalf of a camera's absolute position, which plays a key role in 3-D

metering.

Furthermore, Margaret M. Fleck [ Perspective Projection: The Wrong Image Model,

1994] has demonstrated that the projection mechanisms of lenses hardly fit a single ideal

model across the whole angular range in practice; otherwise, optics engineers could develop lenses with special projection functions, such as the fovea lens, in light of the

different requirements in applications. Thus, to obligate the postulation of the EDP on all

fisheye cameras is an extreme imposition.

On the other hand, although a lens is usually designed with a specific projective

mechanism, the refractivity of light limited by the properties of the material in question

keeps the lens from a perfect design. Moreover, following manufacture it is difficult to

verify whether they match the expected specifications or not. Further, when a fisheye lens

is installed in a real system (such as a camera), its focal length constant may vary

accordingly (depending on the precision of the mechanical installation). Consequently, if

a simple and common technology is developed which can verify the optical features of the

fabricated devices being produced in order to provide a guarantee of quality for the

products at their sale, it would significantly increase their value.

The Gaussian optics model is a convenient means for describing the imaging logic of

an optical system. It is usually the reference model in tracing a camera's errors. The

model regards an optical system (such as a camera) as a black box whose features have

been defined by several cardinal points. That is to say, the complicated projection

geometry is ignored and the projective behavior of light rays is logically analyzed directly

with the aid of the cardinal points. Referring to FIG. 3, the cardinal points defined by the

Gaussian optics model comprise the first and second focal points FI and F2, the first and

second principal points PI and P2, and the first and second nodal points. If the incident

medium of the optical system is air, the nodal points are regarded as the principal points;

at the same time, the first principal point PI is also termed the front nodal point (FNP),

and the second principal points P2 is called the back nodal point (BNP). Otherwise, two principal planes 141 and 142 are defined as the datum planes turning the proceeding

directions of light rays being projected into the optical system. The intersections

detennined by the two principal planes 141 and 142 and the optical axis 224 are simply

the two principal points PI and P2. In accordance with the cardinal points FI, F2, PI, and

P2 and the principal planes 141 and 142, the infinite light rays passing tlirough the first

focal point FI will turn to parallel the optical axis 224 at the first principal plane 141, like

the lines OC and CO'; conversely, if light rays are projected into the optical system in

parallel directions, they will turn to pass the second focal point F2 when meeting the

second principal plane 142, like the lines OB and BO'. A characteristic of this mapping

mechanism is that a light ray from the object point O projected toward the first principal

point PI (i.e. the line OP1) will turn in the direction along the optical axis 224 after

passing through PI, and turn again in the direction parallel to the line OP1 after passing

through P2 (i.e. the line P2O') until it's mapped on the sensitive element to form the

imaged point O'. In other words, the incident ray passing through PI is parallel to the

spatial traces of the light ray passing through P2. In the case of a single lens, the

phenomenon appears only in the paraxial zone of a thin lens. However, the Gaussian

optics model is an ideal imaging logic which average cameras seek to emulate. A wide-

angle lens has to attain this imaging mechanism and is quite different from the fisheye

lens.

Regarding a lens such as the fisheye lens, specialists skilled in the art think of no

"single viewpoint"; this is correct in the aspect of Gaussian optics. However, if the limits

of Gaussian optics could be overcome, the inherent mapping mechanism of the fisheye

lens might be analyzable so that the "single viewpoint" could be logically positioned and the optical parameters can be deduced thereby. At this point, not only the reliability of

analyzing fisheye images is raised but the applications can also be largely expanded in the

field of 3-D metering and so forth. Thus, the present invention will carefully look into

these issues and free the procedures of camera-parameterization from the ideal image-

based postulations, such as the EDPπ and the image boundary, so as to precisely obtain

the optical parameters of the fisheye camera.

SUMMARY OF THE INVENTION

In view of the foregoing, it is an object of this invention to provide a method and

system aiming at cameras with lenses of the non-linear perspective proj ection mechanism,

in order to analyze the natural optical projection properties of an optical system.

Another object of this invention is to provide a method and system for obtaining

optical parameters (comprising the viewpoint, the orientation of the optical axis, the focal

length constant and the projection mechanism of the camera) based simply on the natural

optical projection phenomena of a lens so as to extend the applications of the fisheye

camera to the fields of stereo graphic measuring and 3-D metering.

Another object of this invention is to provide a method and system for analyzing

image distortion according to the coordinates on the image plane, which can directly

quantify image distortion by the zenithal distances (α) deduced from the coordinates of

imaged points.

Another object of this invention is to provide a method and system capable of

examining a lens or the spatial mapping mechanism of the camera mounted with the lens

to act as a basis for detennining the specifications or testing the qualities of products. In accordance with the objects described above, the present invention refers to the

degree of distortion of an image projected from a target with a physical central-symmetric

pattern (PCP) so as to adjust the absolute coordinate of a camera in order to make the

features of the image similar to the PCP; namely, an imaged central-symmetric pattern

(ICP) appears on the image plane. Next, an object-to-image conjugate coordinate array,

composed of the spatial absolute coordinates of calibration marks on the target and the

corresponding image coordinates on the image plane, is sampled and utilized to describe

the projecting behavior between object space and the image plane. Accordingly, the

projection relationship between the object space (i.e. the sight rays) and the image (i.e. the

imaged coordinates) is deduced. Thus, the optical parameters of the camera system can be

obtained.

The present invention does not borrow any assumptions from existent ideal

projection functions. The deduction of the optical projection mechanism and the

quantification of the optical parameters of a camera utterly and totally with only the

assistance of the measured projection relationships between the given coordinates of the

calibration marks and the corresponding respective imaged coordinates comprise a

significant characteristic of the present invention. The invention makes a breakthrough

from the limitations and presumptions that those skilled in the related art have strongly

believed. The invention is suitable for application to the fisheye camera or the kind with

special projection functions and even, as reverse engineering to analyze camera devices

having unknown projection models.

Owing to the capability of the invention to precisely deduce the projection function

of the camera, its inverse projection function can calibrate the image distortions and further be applied in the fields of stereology and 3-D metering.

Further scope of applicability of the present invention will become apparent from the

detailed description given hereinafter. However, it should be understood that the detailed

description and specific examples, while indicating preferred embodiments of the

invention, are given by way of illustration only, since various changes and modifications

within the spirit and scope of the invention will become apparent to those skilled in the art

from this detailed description.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will become more fully understood from the detailed

description given herein below by illustration only, which illustrations are not limitative

of the present invention, and wherein:

FIGs. 1A and IB show the schematic view of a calibration method based on an

image-based algorithm aiming at the EDPπ of the fisheye images in the related art;

FIG. 2 sketches three typical projection functions of the fisheye lens;

FIG. 3 shows the schematic view of the mapping optical path of the Gaussian optics

model;

FIG. 4 cubically shows the 3-D optical paths between the PCP and the fisheye camera

in the invention;

FIG. 5 shows one embodiment of the PCP which is an octagonal symmetric pattern

defined by three concentric circles;

FIG. 6 shows the first embodiment of the theoretical model disclosed in the invention

where the specific sight ray is determined by three different calibration marks while the target is moved to three different positions;

FIG. 7 shows the first embodiment of the measuring system disclosed in the

invention and the related coordinate systems referred to;

FIG. 8 shows the second embodiment of the measuring system disclosed in the

invention and the related coordinate systems referred to;

FIG. 9 shows the second embodiment of the theoretical model disclosed in the

invention which takes the solid center of the PCP as the origin of the absolute coordinate

system and moves the camera to equivalently deduce the specific sight ray;

FIG. 10 statistically shows the moving traces of the camera in the platform

coordinate system measured in an experiment as the ICP is attained, the above traces

representing as well the spatial traces of the optical axis in the platform coordinate

system;

FIG. 11 statistically shows the pixel coordinates of the imaged center measured in the

experiment;

FIG. 12A statistically shows the profiles of the average image heights (p) defined by

the three concentric circles varied by the different locations (referring to FIG. 10) of the

camera in the experiment;

FIG. 12B shows the varying ranges and the overlapping situation of the average

image heights (p) corresponding to FIG. 12 A;

FIG. 13 shows the beautiful overlapping profiles of the zenithal distances (α) to the

image heights (p) as the viewpoint is exactly set in the experiment;

FIG. 14 shows the divergent profiles of the zenithal distances (α) to the image heights

(p) when the location of the viewpoint is shifted from its exact position; FIG. 15 shows the beautiful overlapping profiles of the zenithal focal length (zFL) to

the image heights (p) as the viewpoint is exactly set in the experiment;

FIG. 16 shows the divergent profiles of the zenithal focal length (zFL) to the image

heights (p) when the location of the viewpoint is shifted from its exact position; and

FIG. 17 shows the divergent length composed of multiple traces of the zFL, which is

used to evaluate the overlapping degree of the profiles, an example taken from FIGs. 15

and 16.

DETAILED DESCRIPTION OF THE INVENTION

Several coordinate systems are defined in advance of the detailed technical disclosure

necessary to a convenient analysis:

1. The absolute coordinate system of W(X,Y,Z) places its origin at the geometric

center of a target, and defines the direction perpendicularly away from the target

as the positive of the Z-axis.

2. The image-plane coordinate system of C'(x,y) or P'(p, β) represents the image

plane of the camera in the Cartesian coordinate system or the polar coordinate

system in which its origin is set at the principal point.

3. The pixel coordinate system of I(u,v) represents images which can be directly

observed on a computer screen with an unit of "pixel". The principal point is

imaged at the coordinate denoted as I(u_c,v_c) on the computer screen. Basically,

the dimensions on the image plane, C'(x', y') or P'(p', β'), can correspond to the

pixel coordinate system of I(u,v). Therefore, the Cartesian coordinate system of

C(u,v) or the polar coordinate system of P(p, β) can represent as well the pixel coordinate system of I(u,v) in which I(u_c, v_c) is the origin.

4. The camera outer-space coordinate system of N(α,β,h) describes the geometry of

the sight rays in the field of view (FOV) of the camera.

5. The camera inner-space coordinate system of S(α', β', / ) describes the

projection geometry inside the camera.

The serial numbers of sampled points will be identified at the subscript positions and

the sampling sequence is indicated by means of an array. For example, W_n(a,b,c)[k]

expresses that a calibration mark of n is located at (a, b, c) in the absolute coordinate

system during the k^th test. The other coordinates are denoted by a similar rule. Part of the

denotation could be omitted in the interests of fluent comprehension while readability is

not adversely affected. The coordinate denotations will be quoted in the invention

hereinafter.

The fisheye lens is severely diverged from the Gaussian optics model to be a non¬

linear perspective projection lens, which means its projective behavior cannot be

interpreted by the well-known pinhole model following the rectilinear perspective

projection mechanism. Compared with other lenses, an image captured by the fisheye

lens (simplified as the fisheye image) is possessed of a severe barrel distortion. The

fisheye lens is frequently employed to create dramatic or extraordinary effects but it is

found lacking in the accurate mapping of the original dimensions and features of objects.

However, there are still a couple of rules to follow while mapping images. Rule 1, the

quantity of distortions throughout the fisheye image are distributed with a radial

symmetry whose point of origin is termed a principal point, and its optical projection

geometry in space symmetrically encircles the optical axis of the camera. Rule 2, all object points located on the same specific sight ray in object space are totally projected

onto a specific imaged point on the image plane. The projection mechanism in space can

be postulated as follows: incident rays (including active or inactive reflective light rays)

cast from an object in the FOV will logically converge on a unique spatial optical center

(or termed the viewpoint, simplified as VP) and then divergently map on the image plane

in light of a projection function. The rules and the postulation described above are well

known to specialists skilled in the related art of optical engineering.

The present invention designs a particular target according to the characteristic of the

radial symmetry of distortion across a fisheye image (rule 1) to locate the principal point

on the image plane and position the optical axis in space. Then, the specific projection

relationship between the sight ray and the imaged point (rule 2) is analyzed to obtain the

absolute coordinate of the VP on the optical axis as well as the absolute coordinate of the

sight ray in space; the focal length constant is deduced accordingly and the projection

model of the camera is induced consequently. The present invention needs no assumption

of any existent projection models, such as the equidistant projection (EDP), the

stereo graphic projection (SGP) or the orthographic projection (OGP), only if cameras

possessed of the mapping properties of fisheye images, or similar ones, all of which are

analyzable in the invention.

The spatial projecting symmetry of rule 1 is illustrated in FIG. 4, which shows the

optical projection paths between the fisheye camera and the planar target 30 being placed

in the FOV thereof; wherein take the fisheye lens 221 and the image plane 225 standing

equivalently for the fisheye camera. With a view of geometry, a planar drawing capable of

representing an axis-symmetric geometrical arrangement in space can image a center- symmetric image inside the camera. Therefore, referring to FIG. 5, a planar target 30 with

a physical central-symmetric pattern (PCP) 31 thereon is placed in the FOV of the camera.

The PCP 31 is composed of a central mark 38 located at the geometric center thereof and

a plurality of calibration marks 311-318, 321-328, 331-338 defined by a plurality of

center-symmetric geometric figures. The relative position of the target 30 and the camera

is adjusted in order to obtain an imaged central-symmetric pattern (ICP) 226 on the image

plane 225. Obtaining the ICP 230 means the optical axis 224 perpendicularly penetrates

both the principal point 227 on the image plane 225 and the central mark 38 of the PCP 31.

The position of the optical axis 224 in space can be determined absolutely by referring to

the target 30 because its absolute position is man-made and given in advance. The feature

coordinate of the blob imaged by the central mark 38 (or the center of gravity of the

imaged blob) is regarded as the principal point 227 on the image plane 225.

If the projection behavior of a camera conforms to any known circular- function

relationship (note: meaning the product of a circular function and a focal length), the

incident rays cast from the PCP 31 will certainly and essentially achieve a collimating

mechanism; namely, referring to FIG.4 again, all incident rays will converge at a logical

optical center of the fisheye lens 221, termed the front cardinal point (FCP) 222, and

divergently refract onto the image plane 225 (or the optical sensor) from the back cardinal

point (BCP) 223 according to the projection function so that two light cones whose

zeniths are separately at the FCP 222 and BCP 223 are fonned. The FCP 222 and BCP

223 are two referred points for the two distinct spaces delimiting the projecting behaviors

inside and outside the fisheye camera. Sight rays refer to the FCP 222 and the image plane

225 refers to the BCP 223 while analyzing the projection mechanism of the fisheye camera. The distance between the two cardinal points 222 and 223 is arbitrary because it

is not a parameter of the camera system. The present invention therefore merges the two

cardinal points 222 and 223 at a single viewpoint (VP) or picks the FCP 222 on behalf of

the VP in order to simplify the imaging logic. Such a technique of expression is often seen

in volumes on optics in discussing lenses.

The equivalent mapping mechanism of rule 2 is shown in FIG. 6. As far as an optical

model is concerned the different object points on the sight ray 80 (such as the three

calibration marks 313, 323, and 333 in FIG. 5 whose absolute coordinates are W₃₁₃[p],

W₃₂₃[q], and W₃₃₃[r] respectively while the target 30 is passed through at the three

different locations of p, q, and r) can hardly be told apart simply by a single image

message (such as the imaged point 91) on the image plane 225. Another aspect is that if at

least two different object points simultaneously map at the same imaged blob, the sight

ray 80 defined by these object points can be determined by the spatial absolute

coordinates thereof. The intersection of the sight ray 80 and the optical axis 224 situates

the FCP 222 or, instead, the VP.

The projection mechanism of any sight ray 80 (also called the incident ray) of the

fisheye lens can be explained by the Gaussian optics model. The sight ray 80 is assumed

to be refracted at the FCP 222 (that is the FNP 222' in the Gaussian optics model,

referring to FIG. 3) and then mapped on the image plane 225 to form an imaged point 91

whose coordinate is C'(u,v) after the sight ray 80 meets the optical axis 224; therefore a

track parallel to the sight ray 80 from the imaged point 91 can be inferred inversely to

obtain the conesponding BNP 223'. If the projection behavior of the sight ray 80

conforms to the Gaussian optics model, the BNP 223' matches the BCP 223, and the focal length constant (/) can be derived by an object distance, an object height and an image

height only by employing simple mathematic geometry. Only the lenses following the

Gaussian optics model can obtain the same focal length as a constant wherever the

imaged point 91 is located.

If the sight ray 80 corresponding to any coordinate on the image plane 225 is

analyzable, the mapping geometry of the camera is totally describable without the need

for the camera's projection function. This is a vital subject and basis of the invention and

will be disclosed hereinafter.

Due to the severe distortions caused by the fisheye lens, it is impossible to enable all

sight rays 80 to pass through a single BNP 223'. That is to say, there is no unique focal

length constant in view of the Gaussian optics model. However, the geometric projection

mechanism between a specific sight ray 80 and its corresponding imaged point 91 is still

separately describable by a Gaussian model. The invention terms the individual focal

length attained by this method as a zenithal focal length (simplified as the zFL

hereinafter), that is, the distance between the BNP 223 ' and the principal point 227 shown

in FIG. 6; wherein the location of the BNP 223' is determined by the line parallel to the

sight ray 80 and passing the imaged point 91 C'(u,v) observable on the image plane 225.

The zFL can also be called the image-height focal length because image heights of

equivalent values will correspond to the same zFL and each image height is determined

by an imaged point 91. Thus, the existence of a different but unique zFL corresponding to

every single imaged point 91 is definitely inferred in view of the Gaussian optics model,

but its values will be decreasing while the image heights are increasing. Based on the

one-to-one correspondence, the mapping/distortion mechanism of the fisheye camera can also be described by the zFL, one of the parameters of the camera.

If the projection function of the fisheye lens is describable by a circular function, the

relationship between the image height (p) and the zenithal distance (α) is deducible as

well; wherein the zenithal distance (α) is the angular distance of an incident ray away

from the optical axis 224 in space. Taking the EDP as an example, the image height (p) is

determined by the product of the zenithal distance (α) and the focal length constant (f),

namely p=/*α; the value of α is derivable when both p and/ are given.

Referring to FIG. 4 again, the relationship between the zenithal distance (α) defined

by the outer light cone and the image height (p) which is the radius at the bottom of the

inner light cone is described by the projection function. Inversely, if the relationship can

be measured, the projection function can be inferred as well. This mechanism is not

limited in one single function with a closed form, such as trigonometric functions. The

present invention terms as an ideal lens the kind whose mapping mechanism throughout

the entire FOV can be described by a single circular function. Logically, once the natural

projection function of the lens is obtained, there does exist a BCP 223 in a modal.

However, as far as the outer light cone is concerned, it is correct to say that there is only

one FCP 222 because the origin of the sight ray 80 utilized to describe the absolute

projection space is modally infinite, and it is reasonable to regard the camera as a point.

Referring to FIG. 4 again, if there is a camera mounting an ideal lens, the value of α

corresponding to a physical object point in the FOV can be obtained by a simple tangent

function if the absolute coordinate of the FCP 222 is given. Furthermore, the point of

intersection of the incident sight ray 80 and the optical axis 224 situates the FCP 222;

taking the image plane 225 as a base and referring to the focal length constant (f) as the height, the unique image height (p) corresponding to the sight ray 80 can infer the BCP

223, the zenith of the inner light cone.

The absolute coordinate of the FCP 222 and the orientation of the optical axis 224,

both of which are the extrinsic parameters of the camera, can represent the position of the

camera; the focal length constant and the projection function are regarded as the intrinsic

parameters of the camera. The invention develops a measuring system and an analyzing

methodology to verify that these parameters are deducible in logic without knowledge of

the camera's projection model.

The realization of the mapping mechanism described above is a key basis to design

the measuring system in the invention. The arrangement of the measuring system is

shown in FIG. 7, in which the movement of the target 30 refers to the one in FIG. 6. The

measuring system employs a computer program executed automatically for the

automatized measuring procedures comprising the capture of images, the extraction of

feature coordinates of imaged blobs, and the deductions for the intrinsic and extrinsic

parameters of the camera 22.

Speaking generally, the measuring system is a composition of hardware devices and

software elements used to perform the mapping mechanism described above. Apart from

the devices and elements in operation, the qualities of measurement are also greatly

influenced by the surrounding factors of the laboratory such as the relative positions of

the devices, and both the specification and installation of lamps. The theoretical model in

FIG. 6 coupled with the measuring system in FIG. 7 present the first embodiment of the

invention. However, in practice, the first embodiment will cause irregular illumination on

the surface of the target 30 cast from the illuminant 24 while the target 30 is moving at different locations; this can certainly affect the experimental accuracy. A second

embodiment of the invention is therefore introduced with the benefits of simplified

calculation and unifonn illumination, as shown in FIG. 8, in which the target 30 is fixed

as the origin of reference of the absolute coordinate system 28 and the camera 22 is

moved instead. The corresponding theoretical model of the second embodiment is shown

in FIG. 9. The description hereinafter will take the second embodiment as a representative

example to disclose the details of the invention. However, it does not imply a limitation in

the invention; any variations or modifications following the same spirit are not to be

regarded as a departure from the spirit and scope of the invention.

The present invention defines four coordinate systems depending on each other in the

measuring system, referring to FIG. 8 to know their inset positions: (1) the absolute

coordinate system 28 (denoted as W=(X,Y,Z)) defined by the target 30; (2) the platform

coordinate system 29 (denoted as W'=(X',Y',Z')) defined by the adjusting platform 23

driving the orientation and location of the camera 22; (3) the pixel coordinate system 27

(denoted as I=(u,v)) displayed on a computer screen and corresponding to the image-

plane coordinate system 27' ( denoted as C'(x', y') or P'(p', β')) on the image plane 225;

and (4) the camera coordinate system 26 (denoted as N( α, β, h) and S( α', β',/)) utilized

to describe the imaging geometry of the camera 22.

The camera coordinate system 26 is composed of N(α, β, h) and S(α', β',/), in which

α and β have been defined hereinbefore while α' and β' are the corresponding angular

distances determined by virtual rays with reference to the image plane 225. Referring to

FIG. 4 again, S(α', β',. ) defines the refractive light rays bounded on the inner light cone

placing the zenith at the BCP 223 while N( α, β, h) defines the corresponding sight ray 80 bounded on the outer light cone whose zenith is at the FCP 222. Owing to the irregular

refraction from the outer to the inner space of the camera 22, α' is not equal to α but β' is

usually the equivalent of β (or β+π). The functional relationship between α and α' can

represent the mapping mechanism of the camera 22; however, α' cannot be directly

observed.

The image-plane coordinate system 27' defines the dimensions of images on the

image plane 225 separately in the Cartesian coordinate system (C'(x\ y')) or the polar

coordinate system (P'(p'_? β'))_> placing the origin at the principal point 227.

The pixel coordinate system 27 expresses the dimensions of images displayed on the

computer screen individually in the Cartesian coordinate system (C(x, y)) or the polar

coordinate system (P(p, β)), placing the origin at the feature coordinate imaged by the

principal point 227 (denoted as I(u_c, v_c)=C(0, 0)=(0, β)) with a unit of a pixel.

Referring to FIG. 5 again, the absolute coordinate system 28 regards the center of the

PCP 31 (i.e. the barycentric coordinate of the central mark 38) as the origin, and defines

the X-axis by the feature coordinates of the horizontal calibration marks 335, 325, 315, 38,

311, 321 and 331 and the Y-axis by the feature coordinates of the vertical calibration

marks 333, 323, 313, 38, 317,327 and 337; accordingly W₃₈=W(0,0,0).

During the experiment, the position of the target 30 is keeping fixed so that the

absolute coordinates of all the calibration marks 38, 311-318, 321-328, and 331-338 are

consequently ensured. Then, the camera 22 is moved within a particular object space and

the image changes enable the mapping mechanism of the sight ray 80 defined by α and β

in the camera coordinate system 26 to be analyzable. The details of the analysis will be

disclosed hereinafter. Referring to FIG. 8 again, the camera 22 is fixed on the adjusting platform 23 which

is composed of three rigid axes perpendicular to one another, that is the X' rigid axis 231,

the Y' rigid axis 232 and the Z' rigid axis 233 correspondingly representing the X'-, Y'-

and Z'-axis in the platform coordinate system 29; wherein the positive of the Z'-axis is

the direction departing from the target 30. Ideally, the three axes (X', Y', Z') of the

platform coordinate system 29 have to be parallel with the ones (X, Y, Z) of the absolute

coordinate system 28. However, in practice, initially there is a six-dimensional difference

between the two coordinate systems 28 and 29. Therefore, in addition to freely driving the

three rigid axes 231, 232, and 233 of the adjusting platform 23 to position the camera 22

fixed thereon, an omnidirectional base 70 is installed on the Y' -axis (under the camera 22)

for pamiing, tilting or rotating the camera 22. The mechanical arrangement can collimate

the optical axis 224 to the Z-axis. The details of the operation will be disclosed

hereinafter.

The pixel coordinate system 27 is utilized to express the two-dimensional memory

coordinates of digital video signals captured by the camera 22, digitized by a frame

grabber 252 and then provided to a CPU 251 or a digital image processor 253. Logically,

the value in the pixel coordinate system 27 can represent the dimensions of images on the

image plane 225; however, the proportion of both their units reveals a transformed

relationship, termed an aspect ratio. A square image displayed on the screen might not

have the aspect ratio equal to 1 ; it will turn an original circular image into an elliptic one.

In practice, the image mapped on the image plane 225 is displayed on the screen for

observers who can only read the values in the pixel coordinate system 27 to indirectly

represent the dimensions of images. The value of the aspect ratio can also be ensured by the invention. The details will be disclosed hereinafter.

The measuring system not only builds a mechanical structure for the coordinate

systems described above but also functionally plays as a device for capturing images,

calculating feature coordinates, and adjusting coordinate systems. The details of the rest

of the major devices are described as follows:

1. The camera 22: a BW camera applied to surveillance, which is equipped with a

1 /2-inch CCD (charge coupled device) and a fisheye lens (with a vendor's

specification of 2.8mm focal length). It has the capability of focusing infinitely

and outputs video signals along the standards of the NTSC (National

Television System Committee) and further transmits the video signals to the

frame grabber 252. In addition to the CCD camera, it is another embodiment to

adopt a CMOS (Complementary Metal Oxide Semiconductor) camera or

cameras mounting image sensors.

2. An illuminant 24: an important element in the invention. The category and

arrangement of the illuminant 24 totally affect the distribution of the

illumination and cause different results in the experiment. The invention takes

two lamps with high frequency conversions as the illuminant 24 to light up the

target 30. The relative position of the illuminant 24 and the target 30 is fixed

for the full duration of the experiment in order to keep the illumination stable.

3. A platform controller 21: utilized to control the movement of the adjusting

platform 23 through the commands of software and provide power to and limit

the moving range of the adjusting platform 23. If necessary, users can

manually adjust the orientation of the camera 22 as well. 4. A processing unit 25: a normal personal computer (PC), which is employed to

retrieve, process and calculate the images of the camera 22 and command the

platform controller 21 to adjust the position of the camera 22. Wherein, the

CPU 251 is utilized to execute the software, handle the entire operation and

manage data; the digital image processor 253, which is connected with the

frame grabber 252, is employed to process digital signals in order to extract

pixel coordinates; the frame grabber 252 is utilized to turn analogical signals

into digital ones and store them in a memory in order to supply the digital

image processor 253 and the CPU 251 in calculating the imaged feature

coordinates corresponding to the calibration marks 38, 311-318, 321-328, and

331-338 in real time. The frame grabber 252, the digital image processor 253

and the CPU 251 are integrated together in the PC with the operation system of

MS Windows. The software developed for the experimental operation will be

disclosed hereinafter.

5. The target 30: fixed in the FOV of the camera 22 as a reference for analyzing

the sight ray 80. A physical central-symmetric pattern (PCP) 31 is illustrated

on the target 30. The PCP 31 is composed of a central origin located at the

geometric center thereof and a plurality of calibration marks is defined by a

plurality of center-symmetric geometric figures. The embodiment of the PCP

31 shown in FIG. 5 takes the central mark 38 as its central origin to define three

concentric circles as the plurality of center-symmetric geometric figures. Eight

individual calibration marks 311-318, 321-328, and 331-338 are

symmetrically placed on the three concentric circles to form three symmetric regular octagons. The radii of the three concentric circles are 20mm, 40mm

and 60mm respectively. The locations of the calibration marks 311-318, 321-

328, and 331-338 begin at 0 degree and mark every π/4 shift, totaling 24 marks.

They are black squares each of 8mm width and 8mm length. Otherwise, take

the four extreme external calibration marks 331, 333, 335, and 337 as the

tangent points to form a square whose four vertexes are the test marks 341-344.

The PCP 31 is drafted by a computer-aided designer (CAD) and printed on a

piece of high-quality photo paper by an ink-jet printer to form the target 30.

Besides, there is another embodiment to form the marks 38, 311-318, 321-328,

331-338, and 341-344 by using LEDs (light emitting diodes) as active lighting

elements in order to attain better image quality; meanwhile, the illuminant 24

could be absent in the measuring system. During the experiment, the target 30

is fixed at a proper location on an experimental table and its absolute

coordinate can be precisely defined.

The embodiment of the PCP 31 is not limited in FIG. 5, depicting three regular

octagons defined by three concentric circles. It performs well as long as the PCP 31 fits a

concentric and symmetric design. Hence triangles, rectangles, squares or any other

polygons shaped by a number of calibration marks are all possible forms for the PCP 31.

The better choice, however, is if each of the geometric figures is composed of even

calibration marks. It has the advantage of easy calculation. The extremeness of a polygon

is a circle, such as the PCP 31 shown in FIG. 4. Besides, a 3-D calibration target 30 might

have the same function if it can symmetrically surround the optical axis 224.

Before entering into the details of implementing the invention, the issues that the invention intends to solve are listed as follows:

1. deducing the principal point 227 on the image plane 225 and situating the

absolute position of the optical axis 224 in space;

2. deducing the absolute coordinate of the FCP 222 (also called the viewpoint);

3. deducing a length profile of the zFL ( also termed the length profile of the

image-height focal length);

4. deducing the projection function from the absolute coordinate system 28 to the

camera coordinate system 26; and

5. deducing the distortive degree of the image and the calibration mechanism.

Relative to the above issues the invention discloses an experimental procedure and a

deductive method described as follows:

A. Locate the principal point 2271(u_c,v_c) on the pixel coordinate system and collimate the

optical axis 224 to W(0,0,z) by regularizing the image of the PCP 31 to achieve an

ICP 226.

According to the axial-symmetric projection geometry of the fisheye lens, the

radial-symmetric distortion of the fisheye image and the centric-symmetric arrangement

of the PCP 31, if and only if the optical axis 224 is collimated to the Z-axis in the absolute

coordinate system 28, a concentric and symmetric image, i.e. the ICP 226, can be

achieved. The spatial disposition of the measuring system is adjusted in light of the

symmetry of the imaged blobs (actually the symmetry of their barycentric coordinates,

termed the feature coordinates) mapped by the calibration marks displayed on the

computer screen. A computer program, which controls the procedure to dynamically

adjust the absolute coordinate of the camera 22, perfonns the work for adjusting camera's orientation with manual assistance. The camera coordinate system 26 is therefore

collimated to the absolute coordinate system 28 once the adjustment is completed. At the

time, the geometric center of the ICP 226 (i.e. the feature coordinate of the imaged blob

mapped by the central mark 38 if the PCP 31 is similar to the one shown in FIG. 5) is the

principal point 227; meanwhile, the optical axis 224 peφendicularly penetrates both the

principal point 227 and the geometric center of the PCP 31 (i.e. the feature coordinate of

the central mark 38). The details are described as follows:

1. Use one's eyesight to properly set the relative position between the adjusting

platform 23 and the target 30 in order to make the three rigid axes 231 -233 of the

adjusting platform 23 as parallel as possible to the axes of the absolute

coordinate system 28.

2. Properly place the illuminant 24 to distribute unifonn illumination on the target

30 and regard the center of the PCP 31 (the barycenter of the central mark 38) as

the origin W(0,0,0) of the absolute coordinate system 28.

3. Install the camera 22 on the Y' rigid axis 232 of the adjusting platform 23. An

omnidirectional base 70 is mounted at the bottom of the camera 22 to manually

pan, tilt or rotate the camera 22. The optical axis 224 denoted as S(0,0,f) in the

camera coordinate system 26 has to coincide with the Z'-axis denoted as

W'(0,0,z) in the platform coordinate system 29 so that the movement of the

camera 22 along the Z' rigid axis 233 can be regarded as the equivalent of the

one along the optical axis 224. Therefore, in practice, the utmost care is exerted

to collimate the Z-axis in the absolute coordinate system 28, the Z'-axis in the

platform coordinate system 29 and the optical axis 224 in the camera coordinate system 26 in order to align them along the same straight line.

4. Vary the coordinate of the camera 22 on the adjusting platform 23 to locate the

four test marks 341-344 beside the four corners of the computer screen in order

to maximize the calibrated range.

5. A symmetry-analyzing background program is employed to keep tracing the

geometric center of the ICP 226 (i.e. the feature coordinate of the imaged blob

mapped by the central mark 38), and by referring to the center I(u_3S,v₃₈), to

calculate the "image-distortion indexes" and "horizontal/vertical deviation

indexes" of the imaged blobs mapped by the calibration marks 311-318, 321-328

and 331-338. These indexes are displayed on the computer screen and as a

feedback to the program to command the platform controller 21 driving the

adjusting platform 23 to vary the coordinate of the camera 22 in the platform

coordinate system 29, that is W'(x', y', z'), until these indexes approach optimal

values. If these indexes displayed on the screen have reached satisfactory

standards, the program will go on to the next step, or repeat this step.

6. Record the "imaged-distortion indexes", the "horizontal/vertical deviation

indexes" and the "object-to-image conjugate coordinates", denoted as (W_c'

(x',y',z')[0], l_n(u,v)[0]), obtained during the procedure. Wherein, W_c'(x',y',z')[0]

means the platform coordinate of the camera 22 while l_n(u,v)[0] is the pixel

coordinate of the calibration mark of n; n may be equal to 38, 311-318, 321-328

or 331-338, representing any calibration mark of the PCP 31 in FIG. 5; k=0

means an initial position of the camera 22, increasing by 1 with each movement,

such as the p^th, q^th and r^tb measured in FIGs. 6 and 9. As shown here the collimation procedure for the camera coordinate system 26 and

the absolute coordinate system 28 has been achieved. The "imaged-distortion indexes"

and the "horizontal/vertical deviation indexes" are going to be inteφreted before the

advanced discussion. The symmetry-analyzing background program is kept running

during the whole experiment. In order to enable software to guide the task of adjustment,

not only the image of the PCP 31 but these indexes representing the symmetry of the

image are also displayed on the screen. The measuring system actively adjusts the

position of the camera 22 in light of these symmetric indexes, sometimes with the

assistance of hands. The imaged-distortion indexes and the horizontal/vertical deviation

indexes are defined as follows:

a. The imaged-distortion indexes (su[m][k], sv[m][k]) are the summation of

imaged differences between the calibration marks 311-318, 321-328, 331-

338 and the central mark 38 individually in the u- vector and v- vector of the

pixel coordinate system 27. Referring to the serial numbers shown in FIG. 5,

the formulae of the indexes are given as follows:

(1)

sv[m][k] = ∑ (v_{(300+w*10+fl)} [k] - v₃₈ [/c]) a=\

(2)

wherein l ≤m≤ 3 ; 1 ≤a≤ 8 and k=0. u_(300+m,_10+a) actually is u„, standing for

the u-vector of I_n(u,v), and the same rule applies to v_(300+m,_10+a). The imaged-distortion indexes, denoted as (su[m][k], sv[m][k]), should both

approach zero if an ideal ICP 226 is obtained by reason of the symmetric

distribution of the calibration marks 311-318, 321-328, and 331-338.

b. The horizontal deviation index is the standard deviation of the v- vectors of

L/u,,, v_n)[k], which are the feature coordinates of all horizontal imaged

blobs in the pixel coordinate system 27. Referring to the PCP 31 shown in

FIG. 5, n is equal to 335, 325, 315, 38, 311, 321 and 331, so the horizontal

deviation index is the standard deviation of the series composed of v₃₃₅[k],

V₃₂₅M, v₃₁₅[k], v₃₈[k], v₃₁₁[k], v₃₂₁[k] and v₃₃₁[k].

c. The vertical deviation index is the standard deviation of the u- vectors of

I yL_n, v_n)[k], which are the feature coordinates of all vertical imaged blobs in

the pixel coordinate system 27. Referring to the PCP 31 shown in FIG. 5, n

is equal to 333, 323, 313, 38, 317, 327 and 337, so the vertical deviation

index is the standard deviation of the series composed of u₃₃₃[k], u₃₂₃[k],

u₃₁₃[k], u₃₈[k], u₃₁₇[k], u₃₂₇[k] and u₃₃₇[k].

Minimizing the symmetric indexes described above can help collimate the optical

axis 224 (S(0,0,f)) to the Z-axis of the absolute coordinate 28. This implies that the Z-axis

also peφendicularly passes through the principal point 227 (I(u_c, v_c)) on the image plane

225, and the optical axis 224 is traceable by referring to the given absolute coordinate of

the PCP 31. Nevertheless the absolute coordinate of the camera 22 (i.e. the absolute

coordinate of the viewpoint) is unknown till now.

The aspect ratio is also a parameter in the field of camera calibration. The invention

can easily attain the parameter because the horizontal vectors and the vertical vectors of [k] have reflected it directly. If the aspect ratio is equal to one, in an ideal

situation the image heights (p) of the vertexes of a regular polygon will be exactly the

same after calibration. This is the case in practice.

B. Deduce the absolute coordinate of an identical sight ray 80 by realizing an overlapping

mechanism of different calibration marks located in the same radial direction and

locate the viewpoint of the camera 22.

From the analysis that different absolute coordinates map on the same imaged point

91 to deduce the mapping mechanism of the camera 22 is a significant innovation of the

invention. These different coordinates construct a sight ray 80, termed the identical sight

ray 80. Any single imaged point on the image plane 225 can be modally analyzed to

obtain its corresponding identical sight ray 80.

Referring to the measuring system in FIG. 8, the camera 22 is moved further along

the optical axis 224 which is locked on the normal passing through the central mark 38.

The imaged blobs mapped by the calibration marks 311-318, 321-328, and 331-338 are

getting closer toward the principal point 227 while the obj ect distances are getting bigger.

During the period of time different calibration marks may map at the same location of one

imaged blob. The relative offsets of the camera 22 (actively driven by the program) are

measurable. This offset data couples with the feature coordinate of the particular imaged

blob overlapped by different calibration marks, and the given absolute coordinates of the

calibration marks 311-318, 321-328, 331-338 can infer the absolute location of the

identical sight ray 80 in space corresponding to the particular imaged blob.

Referring to FIG. 6 again, the present invention takes the first embodiment as an

example to explain how to locate the identical sight ray 80 in theory; the camera 22 is fixed but the target 30 is being moved in this embodiment. If at least two different

calibration marks (like the three calibration marks 313,323, and 333 in the vertical

direction on the target 30) jointly map at the same imaged point 91 (I(u,v)) while they

move to at least two different absolute coordinates in space (such as W₃₁₃[p], W₃₂₃[q], and

W₃₃₃[r] in the figure), the identical sight ray 80 corresponding to the imaged point 91 can

be defined thereby. Because the line composed of the calibration marks lying on the same

diameter of the PCP 31 is constantly peφendicular to the optical axis 224, the particular

imaged point 91 overlapped is obtainable, that is I₃₁₃(u, v)[p]= I₃₂₃(u, v)[q]= I₃₃₃(u, v)[r],

while driving the target 30 to move along the optical axis 224. Actually, it is identical to

Tsai's radial alignment constraint and is a characteristic of the radial-symmetric mapping

mechanism; basically it is fully identified by those skilled in the art. The intersection

point of the identical sight ray 80 and the optical axis 224 is exactly the FCP 222, also

termed the viewpoint (VP), representing the absolute coordinate of the camera 22 in

space.

In addition to marking the imaged point 91 1(u, v) distorted by the fisheye lens, FIG.

6 also shows the imaged point 921(u', v') calibrated to conform to the rectilinear mapping

mechanism. The difference between the two imaged points 91 and 92 is customarily

called the distortion value of the imaged point 91 1(u, v).

Nevertheless, in practice, in order to keep the illumination uniform and simplify the

calculation, the invention adopts the second embodiment and implements it in an

experiment; wherein, on the contrary, the camera 22 is moved but the target 30 is fixed

during the experiment; it is also able to achieve the same mapping mechanism as in FIG.

6. Referring to FIG. 9, move the camera 22 (represented by the FCP 222) away from the target 30 along the optical axis 224, which is already collimated to the Z' rigid axis 233, in

order to change the relative offsets between the target 30, possessed of three calibration

marks 313, 323, and 333, and the camera 22 (represented by the FCP 222). Further,

enable the three calibration marks 313, 323, and 333, whose coordinates are separately

W₃₁₃, W₃₂₃ and W₃₃₃, to map at the same coordinate of the imaged point 91 (i.e. I₃ι₃[p] ⁼

Ϊ₃₂₃[^£l] ^=:l₃₃₃[^r]) ^on the image plane 225 in three different tests numbered as p, q, and r

( which means the FCP 222 of the camera 22 is individually located at W_c[p], W_c[q] and

W_c[r]).

Note the initial offset of the FCP 222 (W_c[p]) from the central mark 38 (W₃₈(0,0,0))

as D. With unvarying direction the camera 22 is driven along the optical axis 224 with

several movements; meanwhile, the feature coordinates of imaged points (I_n[k]) mapped

by the calibration marks 38, 311-318, 321-328, and 331-338 are extracted and separately

coupled with the corresponding locations (W_c'[k]) of the camera 22 in the platform

coordinate system 29 to form an object-to-image conjugate-coordinate pair (W_c' [k], I_n[k]),

in which k is the sampled sequence.

The procedure for data extraction is divided into two parts: (1) each time after the

camera 22 moves a distance of dZ' along the Z' rigid axis 233, actively and finely adjust

the position but keep the direction of the camera 22 unvaried through the symmetry-

analyzing program in order to keep an optimal symmetry on the ICP 226; (2) extract the

data of the object-to-image conjugate-coordinate pair (W₀'(x',y',z')[k], I_n(u,v)[k]) in each

test, and pool it in each of a sequence of tests to form an object-to-image conjugate-

coordinate array. Continuing after the steps in section A, the details are described as

follows: 7. Keep the arrangement of the measuring system unchanged from the former

section A, while the optical axis 224 has been collimated to the Z-axis in the

absolute coordinate 28 and the initial object-to-image conjugate-coordinate pair

of (W_c'[0], I_n[0]) is already obtained. Set the initial offset of the FCP 222 from

the target 30 as D — the aim of the calculation (note: generally the orientation of

the camera 22 needn't be adjusted in this procedure);

8. Raise the location index of k and actively control the camera 22 to increase an

offset of dZ' along the Z' rigid axis 233;

9. In light of the symmetric indexes (the imaged-distortion indexes and the

horizontal/vertical deviation indexes) displayed on the screen, the position of the

camera 22 (i.e. W'(X',Y'), while the z-vector is fixed) is finely adjusted until the

symmetry of the ICP 226 reaches a preset standard; then, record the object-to-

image conjugate-coordinate pair (W_c'[k], I_n [k]);

10. If the location of the camera 22 is still within the default experimental range, the

program will back up to the previous step 8, or go on to the next step;

11. Close the symmetry-analyzing background program;

12. Finish acquiring the data regarding the object-to-image conjugate-coordinate

array provided for calculation; and

13. Deduce the related coefficients of the parameters of the camera 22 (the details

will be described hereinafter).

The following will introduce the data obtained in a practical experiment to verify the

practicability of the invention. In the experiment, each offset of the camera 22 (i.e. the dZ')

moving along the Z' rigid axis 233 is increased by 10 mm. There are totally 19 offsets, plus the initial one, constructing an object-to-image conjugate-coordinate arcay

(W_c'[0..19], L 0..19]) composed of 20 object-to-image conjugate-coordinate pairs. After

the procedure described above, the data for deriving the parameters of the camera 22 is

already obtained and will be analyzed by the following steps:

1. The position profiles of the camera 22 (W_c'[0..19]) in the platform coordinate

system 29: FIG. 10 illustrates the distribution of the series of positions of the

camera 22 from W_c'[0]=W'(-7.5mm,-15mm,0mm) to W_c'[19]= W'(-8mm,-

19mm, 190mm), while the ICP 226 is reached in each test; the profiles also

suggest the traces of the optical axis 224 in the platform coordinate system 29.

W„'[0]= W'(-7.7mm,-15.0mm,0mm) indicates the initial position in the platform

coordinate system 29 in which the x'-vector is -7.7 mm and the y' -vector -15.0

mm, and the z'-vector here is treated as the reference point during the experiment.

Although the profiles of X_c'[0..19] and Y_c'[0..19] show slight deviations, they still

hold linearity. This reveals that the optical axis 224 can be efficiently traced by

means of the symmetry of the ICP 226. Nevertheless, it also emerges that the

platform coordinate system 29 is not perfectly collimated to the absolute

coordinate system 28 in the experiment, but the deviation is quite small; that is

0.3% in the x'-vector and 2% in the y'-vector. The result suggests that the

camera's offsets in the absolute coordinate system 28 can be replaced by the ones

just along the Z'-axis and it is reliable, because the percentage of the error is only

0.002% calculated by the formula of x 100% . Therefore, d the absolute offsets of the camera 22 (Z_c[l during the experiment are regarded as

Z₀'[k] +D. 2. The position profiles of the imaged blob mapped by the central mark 38 (I₃₈(u,v)[0..19]) in the pixel coordinate system 27: FIG.11 illustrates the feature

coordinates of the imaged blobs mapped by the central mark 38; each data pair also stands for the principal point 227 practically measured in the pixel coordinate

system 27. In accordance with the spatial mapping symmetry of the camera 22

shown in FIG. 4, I₃₈(u,v)[k] should be a constant and does not vary with the

position of the camera 22 while it is moving along the Z'-axis (Z_c'[0..19]) in the

platform coordinate system 29. Based on these measured data, the standard

deviations of the principal point 227 are separately 0.25 pixel in the u- vector and

0.18 pixel in the v- vector. Further, the result in linear matching indicates the

location of the principal point 227 at I(u_c,v_c)= 1(318.1 ,236.1) pixels. In conclusion,

the slight values of the standard deviations attest to the reliability of the

experimental result, and verify that the postulation that the coordinate of the

principal point 227 is a constant.

3. The featured image-height profiles (ρ_m[0..19]; m=[1..3]) of the ICP 226 in the

pixel coordinate system 27: FIG. 12 A illustrates the profiles of three average

image heights (individually pi, p2 and p3, each defined by the calibration marks

located on the same circle from the inside to the outside; for example, pi

determined by calibration marks 311-318) in the pixel coordinate system 27,

which are varied with Z_c'[0..19] while the camera 22 is moving along the Z'-axis.

The formula is as follows:

PM , 1 ≤m≤3

(3) wherein k is the serial number of samplings, m the layer of the concentric circles

from the inside to the outside, n the number of the calibration marks 311-318,

321-328 and 331-338 shown in FIG. 5, and p_m[k] the average image-height array

corresponding to the concentric circles in each test. FIG. 12B, redrawn from FIG.

12 A, shows a clear overlapping phenomenon between each pair of the three

average image heights (p^O.,19], p₂[0..19] and p₃[0..19]) corresponding to the

three layers of the concentric circles. This supports the postulation described

above that measured image-height ranges hold the information for positioning the

identical sight ray 80. In an ideal model, the image heights defined by equi-radius

calibration marks should be equal to each other when the ICP 226 reaches perfect

symmetry. The experimental result shows that the deviation of descriptive

statistics of the equi-radius calibration marks on a selected circle is 0.22 pixel; this

proves that the measuring system can imitate the circular symmetric projection

mechanism and perform satisfactorily in practice.

The optical parameters of the camera are deduced according to the data measured in

the experiment. Taking the first embodiment as an example, referring to FIG. 6 again, if

W_c is the optical origin, the zenithal distance (α), i.e. the angular distance of the sight ray

80 away from the optical axis 224, is formulated as:

α k] = tan 'CR,, Z[p])=(R₂, Z[q]) =(R₃, Z[r])

(4) wherein R_{tl 3]} denotes the radii of the three concentric circles on the target 30, namely the

object heights in absolute space; Z[p] represents the object distance of the target 30 on the Z-axis while k=p, that is to say Z[p]=D here, and Z[q] as well as Z[r] are the denotations

following the same rule. A line segment is determined by W₃₁₃[p], W₃₂₃[q] and W₃₃₃[r] if

W_n[p..r] are given. The extension of the line segment toward the optical axis 224 will

intersect to determine the absolute coordinate of W_c. Similarly in FIG. 9, the positions

(W_c'[p..r]) of the moving camera 22 in the platform coordinate system 29 are observable

and controllable while the target 30 is fixed. The absolute coordinates (W_c[p], W_c[q] and

W_c[r]) of the camera 22 can be accordingly obtained by comparing similar triangles

bounded by both the optical axis 224 and the target 30 which are peφendicular to each

other if the three offsets of the camera 22 are given. These are the two theoretical models

of the invention for solving the FCP 222.

However, considering the limitations of samplings in the experiment, it is hard to

obtain two image heights (or the imaged coordinates mapped by the calibration marks)

exactly coinciding with each other in practice. Moreover, the unavoidable errors caused

by the random quality of image signals while setting the feature coordinates also suggest

that the sight ray 80 should not be directly deduced and, accordingly, the FCP 222 will not

be located thereby, even though exactly coinciding feature coordinates are attained.

In view of the limitations in practice, the present invention proposes an alternative

method to analyze the measured data. There are three groups of data categorized from the

original one, including: the image heights (p_m[0..19] ; m=[1..3]), the object heights ( ^;

m=[1..3]) and the camera's offsets (W_c'[0..19]). The three groups of data are sufficiently

sampled for being over determined, and employed to deduce the FCP 222 (or termed the

viewpoint) and the mapping mechanism of the camera 22.

First, the image heights have an inverse proportion to the object distances (i.e. the distances of the camera 22); FIG 12A shows this phenomenon, from which the image

heights cannot be directly related to the mapping mechanism of the camera 22. However,

on the basis of the postulation about the identical sight ray 80, if the object heights (i.e. the

physical lengths of the radii of the PCP 31) are represented by another kind of aspect, such

as the zenithal distance (α), the mutual meanings of the three profiles in FIG. 12A are

thereby related, namely the overlapping of image heights or/and the overlapping of

angular distances. The fact of one identical sight ray 80 holding one zenithal distance (α)

offers a consistent explanation for all the sampled image heights if the object heights are

replaced with α. Therefore, the FCP 222 (or the viewpoint) ought to be located first in

order to obtain accurate object distances and turn the object heights exactly into the

zenithal distances (α). The overlapping phenomenon of the ranges of different image

heights (p) revealed in the experiment implies that replacing the object heights with the

zenithal distances (α) also presents an overlapping mechanism. Therefore, the object

distance (the distance between the camera 22 and the target 30) is involved as a factor to

deduce the sight ray 80 of the camera 22; then the overlapping phenomenon of the

zenithal distances (α) similar to FIG. 12B would appear as well.

Therefore, on the basis of the conception described in FIG. 9, the method of trial-

and-error is utilized in searching for the FCP 222 along the optical axis 224 (note: at this

time the optical axis 224 has been positioned). In other words, taking the successive

target points one by one on the optical axis 224, each point is supposed to be the FCP 222,

so that the initial distance (D[p]) between W_c[p] and the target 30 is accordingly

determined. The offsets between W_c[p], W_c[q] and W_c[r] are already given so D[q] and

D[r] can be derived from D[p]. Based on the three given coordinates (W_c[p], W_c[q] and W_c[r]) and referring to a equi-length image height (I₃ι₃[p]=I₃₂₃[q]⁼I₃₃₃[r]), only when D[p]

is accurate can the three corresponding zenithal distances (i.e. α₃₁₃, α₃₂₃ and α₃₃₃),

transformed from the object heights by the tangent function, be equal to each other.

In the experiment, the object-to-image conjugate coordinate array

(W_c'(x',y',z')[0..19], I_n(u,v)[0..19]) is extracted at 20 positions from k=0 to k=19.

Namely, aiming at every object height (P , extract the image-height profile (p_m[0..19]) at

twenty given positions of the camera 22 (W_c[0..19]). Twenty object distances (D[0:19])

would be obtained as well in the course of the experiment, while the distance of W_c[0] is

assumed to be D[0]. The α-profiles (α_m[0:20] ' m=[1..3]) are accordingly deduced by

referring to both D[0:19] and the object heights, or the radii of the concentric circles. The

task of posing the camera 22 is realized by the overlapping degree of the traces disturbed

by the zenithal distances (α_m[0:20] > m=[l ..3]), referring to the image heights (p_m[0..19] '

m=[1..3]); it is tenned the first overlapping index in the invention. The overlapping

phenomenon will appear only if the FCP 222, i.e. the value of D[0], is accurately fixed.

This is the first method for posing the camera 22 disclosed in the invention. FIG. 13

illustrates the traces of the data points of α_m[0..19] to p_m[0..19] when D[0] is accurately

acquired; it reveals an extremely good overlapping phenomenon on the profiles

corresponding to the three concentric circles. The functional relationship between the

image height (p) and the zenithal distance (α) is exactly the projection function of the

camera 22, and hence the curve shown in FIG. 13 is the so-called projection curve or

projection function in optics. To date, the measurement for the projection function of the

camera 22 with a nonlinear perspective projection model is still unattainable in the related

art (note: not for the lens); however, the present invention can achieve the function only with the assistance of simple equipment. On the other hand, if D[0] is shifted from the

accurate value, taking 50 mm as an example, an obvious divergent phenomenon of the

α-traces will appear, as shown in FIG. 14.

In conclusion, the object-to-image conjugate-coordinate array is found to be capable

of deducing the projection function and posing the camera 22 (i.e. positioning the FCP

222). Although the experimental result shows that the projection function is close to the

EDP (equidistant projection), this is just a special case and will not cause any limitation in

the projection model in the invention. The method disclosed in the invention can be

widely applied in various sorts of projection curves.

The projection curve is able to describe the mapping mechanism of the camera 22,

but unable to quantify the distortion degree of the camera system. From the experimental

result, it is clear that the lens used in the embodiment is similar to the one with the EDP so

that its projective curve is supposedly a straight line, and the result is quite close. In the

aspect of a rectilinear projection model, there is a nonlinear relationship (minus the

direct-proportions) between the distortion degrees and the image heights. For the

convenience of further explaining the distortion mechanism of the camera system, the

invention defines an optical parameter, termed the zenithal focal length (simplified as the

zFL hereinafter). Referring to FIG. 6, the zFL (also termed the focal length constant in the

Gaussian optics model) is the distance between BNP 223' and the principal point 227,

which can express the specific mapping mechanism of the sight ray 80 of a particular

zenithal distance (α), and formulized as follows:

ZFL 0..19] = p_m[0..19]*cot(α_m[0..19])

(5) In the aspect of the one-to-one corresponding relationship between the imaged

coordinate I(u,v) and the identical sight ray 80, the zFL can be regarded as the focal length

constant in conformity with the rectilinear perspective projection model while only one

specific imaged coordinate is considered. The zFL will vary along with different image

heights (p); the bigger the margins of different zFLs, the severer the radial distortion of

the camera system. Therefore, an image height can be explained as a zenithal distance (α)

in the object space; however, in the matter of the mapping mechanism, the image height is

also dependent on the zFL, so the function of zFL(p) can directly reveal the distortion

degrees of the camera system, subsequently presented in their totality as "the zFL-curve"

or "the zFL-function".

While turning the image heights (p_m[0..19]) in FIG. 12A into the zFLs (zFL_m[0..19])₅

it is necessary to refer to the object distances, and the overlapping phenomenon of the

zFL-profiles also has the capability of locating the FCP 222 of the camera 22. This is the

first method for posing the camera 22 disclosed in the invention. The image heights

(ρ_m[0..19j) shown in FIG. 12A can be replaced by the zFLs (zFL JO..19]) with a

consistent explanation. Therefore, the method of trial-and-error is employed for searching

the FCP 222 along the optical axis 224; namely, by taking the successive target points on

the optical axis 224 one by one, each of which point is supposed to be the FCP 222 so that

the initial distance (D[0]) is accordingly determined. Further, the image heights

(P_m[0--ⁱ9],nι=[1..3]) are accordingly turned into the corresponding zFLs (zFL

_m[0..19],m=[1..3]), and the task of posing the camera 22 is realized by the overlapping

degree of the zFL-pro files; this is termed the second overlapping index in the invention.

FIG. 15 illustrates the profiles of the zFL-function showing a pretty good overlapping phenomenon. It also signifies that the FCP 222 on the optical axis 224 can be truly

positioned with the aid of the practical results of the experiment. On the other hand, while

D[0] is shifted from the accurate value, taking 5 mm as an example, an obvious divergent

phenomenon appears among the three zFL-pro files, as shown in FIG. 16. Besides, FIG.

15 also directly exposes the distortion degrees of the image or the distortion mechanism

of the camera system.

It is worth noting that, comparing FIG. 16 with FIG. 14, the divergent phenomenon

in FIG. 16, with only a 5-mm shift of D-value, is much more apparent than the one in FIG.

14 with a 50-mm shift of D-value. This proves the sensitivity of zFL(p) is much higher

than the one of α(p) to the position of the FCP 222 of the camera 22. It also reveals that, in

practice, utilizing the overlapping degree of the zFL-curve to fix the FCP 222 of the

camera 22 is a superior method. Moreover, when the image height (p_m[0..19]) approaches

zero, the zFL will be the focal length of the lens mounted in the camera 22; generally ideal

lenses take this value as their focal length constant.

To enable the method for positioning the viewpoint in the invention to be more

multi-functional and suitable to any kind of projection function, the invention further

discloses a method for verifying the overlapping degree of the profiles. Owing to the high

sensitivity of the zFL-function, taking it as an example, a "divergent length" (also called

the "feature length") on the overlapping portion of the profiles is calculated after

rearranging the three groups of data in FIGs. 15 and 16 to evaluate the overlapping degree

of the curve of zFL(p), as shown in FIG. 17. The method connects all adjacent points

representing the relationships of the zFLs to the image heights (p), and then the total

length (i.e. the divergent length) of the overlapping portion is calculated. If the divergent length is the minimum, like the curve notated as zFL in FIG. 17, the overlapping degree of

the tracks of zFL(p) is supposed to be optimum, and accordingly, the tested target point

on the optical axis 224 is the FCP 222 (or the viewpoint) of the camera 22. Otherwise, the

tracks of zFL(p) reflect a longer divergent length, like the one notated as zFL_shift.

Furthermore, the invention utilizes the nature of the image projected from the

innovative PCP 31 to estimate the quality of the arrangement of the measuring system, to

modify the arrangement accordingly and predict whether the camera system is capable of

being examined or not. The mapping mechanisms of some cameras with defects are

apparently below expectations because the distortion model of the camera 22 is

unpredictable. For example, if the optical axis 224 of a lens-set is not peφendicular to the

image plane 225 in the camera 22, it is impossible to get an utterly symmetric image no

matter how much effort is expended. However, the method disclosed in the invention can

eliminate these cameras, with all sorts of defects, from calibration beforehand.

In conclusion, the method disclosed in the invention attains the goal of evaluating the

specifications and obtaining the optical parameters of the camera 22 either by the

projection function or the zFL-function of the said camera 22.

Therefore, the method and the measuring system disclosed in the invention perform

most satisfactorily indeed in analyzing the mapping mechanism of the camera 22. Further,

the present invention can guide or modify the arrangement of the measuring system as

well as determine the reliability of the measured parameters by the distribution of the

measured data, and is finally applied in calibrating cameras or employed to develop

image-processing / image-transfonned technologies. Overall, the invention has the following advantages:

1. The capabilities of accurately locating the optical axis 224, posing the camera 22

(namely, fixing the absolute coordinate of the FCP 222) and evaluating the

projection function and the focal length constant of the camera 22.

2. The capabilities of quantifying the distortion of the imaged points through the

zFL-function.

3. The capability of verifying the reliability of the measuring system through the

measured data.

4. The capability of verifying the quality of the target camera 22 through the

measured data.

5. The capability of directly turning the imaged points into the corresponding

projection angles (i.e. the zenithal distances) in space.

6. The capability of being applied in stereoscopic applications.

7. The merits of simplicity and low cost make the method suitable for any kind of

nonlinear mapping mechanism of the camera 22.

The invention being thus described, it will be obvious that the same may be varied in

many ways. Such variations are not to be regarded as a departure from the spirit and scope

of the invention, and all such modifications as would be obvious to one skilled in the art

are intended to be included within the scope of the following claims.

Claims

CLAIMSWhat is claimed is:

1. A method for obtaining the optical parameters of a camera, which utilizes the

specific characteristic that one single sight ray in space corresponds to one single imaged

point on an image plane to obtain the optical parameters of a camera, the method

comprises:

placing a target with a physical central-symmetric pattern (PCP) thereon in

the field of view (FOV) of the camera, in which the PCP is composed of a

central mark, located at the geometric center thereof, and at least two

calibration marks, individually termed the first calibration mark and the

second calibration mark, located on a straight radial line centered at the

central mark;

collimating the target and the camera to enable an optical axis of the camera

to peφendicularly pass through the central mark;

recording the pixel coordinate of an imaged point imaged by the first

calibration mark;

moving the target along the optical axis by locking the central mark thereon

in order to enable the second calibration mark to image in an overlapping

manner at the same pixel coordinate of the imaged point;

extracting both the spatial absolute coordinates of the first and second

calibration marks and deducing a sight ray defined by the two spatial

absolute coordinates; and

regarding the point of intersection of the sight ray and the optical axis as a viewpoint of the camera.

2. The method according to claim 1, wherein the step of collimating the target and

the camera is fulfilled by locating a principal point on the image plane, hence a spatial

sight ray peφendicularly passing through both the principal point and the central mark

representing the optical axis.

3. The method according to claim 2, wherein the method of locating the principal

point comprises:

further providing a plurality of center-symmetric geometric figures to the

PCP;

placing the target in the FOV of the camera to allow the PCP to image on the

image plane;

adjusting the relative position between the target and the camera until the

image of the PCP turns into an imaged central-symmetric pattern (ICP); and

examining the symmetry of the ICP with at least one symmetric index to

ensure that the imaged traces of the plurality of geometric figures are

symmetrical, the feature coordinate of the imaged point mapped by the

central mark locating the principal point.

4. The method according to claim 3, wherein the plurality of geometric figures is

selected from the group comprising concentric circles, concentric rectangles, concentric

triangles and concentric polygons.

5. The method according to claim 3, wherein the plurality of geometric figures is a

combination of any number of concentric-and-symmetric circles, rectangles, triangles

and/or polygons.

6. The method according to claim 3, wherein the at least one symmetric index

comprises imaged-distortion indexes, a horizontal deviation index and a vertical

deviation index.

7. A method for obtaining the optical parameters of a camera, which utilizes the

specific characteristic that one single sight ray in space coπesponds to one single imaged

point on an image plane to obtain the optical parameters of a camera, the method

comprises:

placing a target with a physical central-symmetric pattern (PCP) thereon in

the field of view (FOV) of the camera, in which the PCP is composed of a

central mark, located at the geometric center thereof, and a plurality of

calibration marks defined by a plurality of geometric figures;

collimating the target and the camera to enable an optical axis of the camera

to peφendicularly pass through the central mark;

varying the relative position between the target and the camera along the

optical axis and recording a plurality of object-to-image conjugate-

coordinate pairs corresponding to the plurality of calibration marks

separately in different the relative positions to form an object-to-image

conjugate-coordinate array; and

searching a target point along the optical axis to enable an overlapping

index to approach optimal trace-overlap by means of analyzing the object-

to-image conjugate-coordinate aπay on the basis of the target point, in

which the target point is a viewpoint of the camera.

8. The method according to claim 7, wherein the step of collimating the target and the camera is fulfilled by locating a principal point on the image plane, hence a spatial

sight ray peφendicularly passing through both the principal point and the central mark

representing the optical axis.

9. The method according to claim 8, wherein the method for locating the principal

point further comprises:

placing the target in the FOV of the camera to allow the PCP to image on the

image plane;

adjusting the relative position between the target and the camera until the

image of the PCP turns into an imaged central-symmetric pattern (ICP); and

examining the symmetry of the ICP with at least one symmetric index to

ensure that the imaged traces of the plurality of geometric figures achieve a

symmetry request, the feature coordinate of the imaged point mapped by the

central mark locating the principal point.

10. The method according to claim 9, wherein the at least one symmetric index

comprises imaged-distortion indexes, a horizontal deviation index and a vertical

deviation index.

11. The method according to claim 7, wherein the plurality of geometric figures is

selected from the group comprising concentric circles, concentric rectangles, concentric

triangles and concentric polygons.

12. The method according to claim 7, wherein the plurality of geometric figures is a

combination of any number of concentric-and-symmetric circles, rectangles, triangles

and/or polygons.

13. The method according to claim 7, wherein the plurality of object-to-image conjugate-coordinate pairs is composed of the absolute coordinates of the plurality of

calibration marks, or the coordinates of the camera, and the pixel coordinates

coπesponding to the plurality of calibration marks, in which three parameters, including

the image height, an object height and an object distance, can be deduced by means of

analyzing the plurality of object-to-image conjugate-coordinate pairs.

14. The method according to claim 7, wherein the overlapping index is a divergent

length, which is deduced by the steps comprising:

analyzing the object-to-image conjugate-coordinate aπay to obtain a

plurality of data points; and

adjacently connecting the plurality of data points to form the divergent

length which is minimized to make the overlapping index optimal.

15. The method according to claim 14, wherein the plurality of data points

expresses the relationship between two variables of the zenithal distance (α) and the

image height (p), representing a projection curve of the camera as a whole and being

obtained by means of analyzing the object-to-image conjugate-coordinate aπay and the

postulated locations of the target point.

16. The method according to claim 14, wherein the plurality of data points

expresses the relationship between two variables of the zenithal focal length (zFL) and the

image height (p), representing the level of distortion of the camera as a whole and being

obtained by means of analyzing the object-to-image conjugate-coordinate aπay and the

postulated locations of the target point.

17. The method according to claim 16, wherein the zenithal focal length (zFL) is

determined by the mathematic equation as follows: zFL = p*cot(α)

wherein:

p is the image height, which is the distance between an imaged point and a

principal point on the image plane; and

α is the zenithal distance, which is the angular distance of a sight ray away

from the optical axis.

18. A method for obtaining the optical parameters of a camera, which utilizes the

specific characteristic that one single sight ray in space coπesponds to one single imaged

point on an image plane to obtain the optical parameters of a camera, the method

comprises:

looking for at least two different absolute coordinates in space, all of which

project at the same imaged point, in order to define the sight ray;

deducing a zenithal distance (α) on behalf of the sight ray, which is the

angular distance of the sight ray away from an optical axis of the camera;

further deducing a plurality of zenithal distances (α) on behalf of a plurality

of sight rays separately coπesponding to a plurality of imaged points; and

obtaining a projection function describing the projecting behavior of the

camera from the relationship between the plurality of imaged points and the

plurality of zenithal distances (α).

19. The method according to claim 18, wherein the method for defining the sight

ray further comprises:

placing a target with a physical central-symmetric pattern (PCP) thereon in

the field of view (FOV) of the camera, in which the PCP is composed of a central mark, located at the geometrical center thereof, and at least two

calibration marks, individually termed the first calibration mark and the

second calibration mark, located on a straight radial line centered at the

central mark;

collimating the target and the camera to enable the optical axis of the camera

to peφendicularly pass through the central mark;

recording the pixel coordinate of the imaged point imaged by the first

calibration mark;

moving the target along the optical axis by locking the central mark thereon

in order to enable the second calibration mark to image in an overlapping

manner at the same pixel coordinate of the imaged point; and

extracting both the spatial absolute coordinates of the first and second

calibration marks, the two spatial absolute coordinates defining the sight ray.

20. The method according to claim 19, wherein the point of intersection of the sight

ray and the optical axis is a viewpoint of the camera.

21. The method according to claim 19, wherein the step of collimating the target

and the camera is fulfilled by locating a principal point on the image plane, hence a spatial

sight ray peφendicularly passing through both the principal point and the central mark

representing the optical axis, the method further comprising:

further providing a plurality of center-symmetric geometric figures to the

PCP;

placing the target in the FOV of the camera to allow the PCP to image on the

image plane; adjusting the relative position between the target and the camera until the

image of the PCP turns into an imaged central-symmetric pattern (ICP);

examining the symmetry of the ICP with at least one symmetric index to

ensure that the imaged traces of the plurality of geometric figures are

symmetrical, the feature coordinate of the imaged point mapped by the

central mark locating the principal point; and

according to the given position of the target, picking the spatial sight ray

peφendicularly passing through both the principal point and the central mark

as the optical axis.

22. The method according to claim 21, wherein the at least one symmetric index

comprises imaged-distortion indexes, a horizontal deviation index and a vertical

deviation index.

23. The method according to claim 21, wherein the plurality of geometric figures is

selected from the group comprising concentric circles, concentric rectangles, concentric

triangles and concentric polygons.

24. The method according to claim 21, wherein the plurality of geometric figures is

a combination of any number of concentric-and-symmetric circles, rectangles, triangles

and/or polygons.

25. The method according to claim 18, wherein through analyzing the relationship

between the plurality of imaged points and the plurality of zenithal distances (α) a

viewpoint of the camera is obtained.

26. The method according to claim 25, wherein the viewpoint is obtained by the

steps comprising: placing a target with a physical central-symmetric pattern (PCP) thereon in

the field of view (FOV) of the camera, in which the PCP is composed of a

central mark, located at the geometric center thereof, and a plurality of

calibration marks defined by a plurality of geometric figures;

collimating the target and the camera to enable the optical axis of the camera

to peφendicularly pass through the central mark;

varying the relative position between the target and the camera along the

optical axis and recording a plurality of object-to-image conjugate-

coordinate pairs coπesponding to the plurality of calibration marks

separately in different the relative positions to fonn an object-to-image

conjugate-coordinate aπay; and

searching a target point along the optical axis to enable an overlapping index

to approach optimal trace-overlap by means of analyzing the object-to-image

conjugate-coordinate array on the basis of the target point, in which the target

point is the viewpoint of the camera.

27. The method according to claim 26, wherein the plurality of object-to-image

conjugate-coordinate pairs is composed of the absolute coordinates of the plurality of

calibration marks, or the coordinates of the camera, and the pixel coordinates

coπesponding to the plurality of calibration marks, in which tliree parameters, including

the image height, object height and object distance, can be deduced by means of

analyzing the plurality of object-to-image conjugate-coordinate pairs.

28. The method according to claim 26, wherein the overlapping index is a divergent

length, which is deduced by the steps comprising: analyzing the object-to-image conjugate-coordinate array to obtain a

plurality of data points; and

adjacently connecting the plurality of data points to form the divergent length

which is minimized to make the overlapping index optimal.

29. The method according to claim 28, wherein the plurality of data points

expresses the relationship between two variables of the zenithal distance (α) and the

image height (p), representing a projection curve of the camera as a whole and being

obtained by means of analyzing the object-to-image conjugate-coordinate array and the

postulated locations of the target point.

30. The method according to claim 28, wherein the plurality of data points

expresses the relationship between two variables of the zenithal focal length (zFL) and the

image height (p), representing the level of distortion of the camera as a whole and being

obtained by means of analyzing the object-to-image conjugate-coordinate aπay and the

postulated locations of the target point.

31. The method according to claim 30, wherein the zenithal focal length (zFL) is

determined by the mathematic equation as follows:

zFL = p*cot(α)

wherein:

p is the image height, which is the distance between an imaged point and a

principal point on the image plane; and

α is the zenithal distance, which is the angular distance of a sight ray

proceeding away from the optical axis.

32. A system for obtaining the optical parameters of a camera, which is employed to analyze the relationship between a plurality of sight rays in object space and a plurality of

imaged points on an image plane, the system comprises:

a target possessed of a physical central-symmetric pattern (PCP) which is

composed of a central mark and a plurality of center-symmetric geometric

figures defining a plurality of calibration marks;

a camera equipped with a non-linear perspective projection lens used to

capture the rays from the PCP and fonn a coπesponding image on the image

plane;

an adjusting platform possessed of three rigid axes which are peφendicular

to one another in order to define a coordinate system used to adjust the

relative position between the target and the camera;

a platform controller connected with the adjusting platform and used to

provide power to and limit the moving range of the adjusting platform; and

a processing unit connected with the camera and the platform controller,

which is used to command the platform controller to adjust the positions of

the three rigid axes and grab the absolute coordinates of the plurality of

calibration marks and their coπesponding imaged pixel coordinates in order

to form an object-to-image conjugate-coordinate aπay as the basis of

calculation for obtaining the camera's projection function representing the

relationship between the object space and the image plane as the mapping

mechanism of the camera.

33. The system according to claim 32, wherein the system further comprises an

illuminant used to light up the target.

34. The system according to claim 32, wherein the plurality of calibration marks is

individually constructed by an active lighting element.

35. The system according to claim 34, wherein the active lighting element is a LED

(Light Emitting Diode).

36. The system according to claim 32, wherein the processing unit further

comprises:

a frame grabber connected with the camera, which is employed to turn the

analogical signals captured by the camera into digital ones;

a digital image processor connected with the frame grabber, which is

employed to process the digital signals in order to extract the imaged pixel

coordinates; and

a CPU in charge of controlling the frame grabber and the digital image

processor.

37. The system according to claim 32, wherein the processing unit is a personal

computer (PC).

38. The system according to claim 32, wherein the camera is selected from the

group comprising a CCD camera, a CMOS camera and the one mounting an image

sensor.

39. The system according to claim 32, wherein the plurality of geometric figures is

selected from the group comprising concentric circles, concentric rectangles, concentric

triangles and concentric polygons.

40. The system according to claim 32, wherein the plurality of geometric figures is a

combination of any number of concentric-and-symmetric circles, rectangles, triangles and/or polygons.