Analytical study for the identification of the corresponding couples of points on the frames of a plane trajectory. Abstract.

Transcription

1 Analytical stdy for the identification of the corresponding coples of points on the frames of a plane trajectory. Translation of the article: Stdio analitico per lidentificazione di coppie di pnti omologhi slle immagini fotografiche di na traiettoria piana. Riista del Catasto e dei Serizi Tecnici Erariali, Noa Serie Anno X, N., 3-, 955. Abstract. We present a method for the nmerical comptation of the coordinates of corresponding coples on the photographic images of a plane trajectory. The method is based on the analytic projectiity, between the planes of the photographic frames, generated by few known points of the trajectory which may be obtained experimentally. In the srey of space lines, as for instance in the photogrammetric srey of trajectories of bllets or rockets, in order to obtain the trajectory with nmerical method, is of great importance the identification, on the two plates, of the coples of the corresponding points in the images of the trajectory. In terrestrial photogrammetry, in general, is defined normal the position of the plates when are absent conergence, drift of the photographs, obliqe photography and, moreoer, the base is horizontal. In the restittion this positioning allows to determine immediate the coples of the corresponding points in the two images of the trajectory. When the plates are non in the same plane the direct identification is not possible since the ordinates are not the same. The identification of the coples of corresponding points in the two plates is difficlt if one wants to proceed with a graphical method since is reqired the se of the theorem of the nodal points and, conseqently, of their location which, in general, are not in a sefl position to obtain rapidly the graphical identification of the coples of corresponding points. In a stdy Piazzolla Beloch 954 notes that translating the photographic images in the symmetric position, with respect to the optic centre, relatie to that occpied at the time of the taking obtaining ths a perspectie image inersely eqal to the original one, one obtains that the nodal points are near the principal points which gies a sefl setting for the identification of the coples of corresponding points. In the same stdy of is also sggested an apparats for the soltion of the problem in a more general case. I fond therefore of interest to sole the problem of the identification of the corresponding points also analytically.

2 2 We first note that the graphical methods cannot gie an analytical soltion except when we know the eqations of the image cres in the plates in gien coordinate systems, which is difficlt to obtain. The scope of this research is precisely to oercome this obstacle which is present when one wants to gie an analytical treatment to the graphical procedres of restittion. The method sggested here consists in finding the projectiity relation between the planes of the images throgh the plane of the trajectory aoiding ths the need of the eqations of the images of the trajectory in the plates. In order to obtain the eqations of the projectie correspondence in a simple manner it is sefl to discss the following theoretical simpler bt more general problem... We begin considering the problem between two straight lines. Let r and r be non coplanar straight lines as shown in figre, and let t and t linear coordinates in r and r with t = t+/t+ the eqation of a projectiity relating r and r. Let also PX,Y,Z and P X,Y,Z two generic points in space where a Cartesian system Ox,y,z is introdced. Let s now project r and r from P and P respectiely associating the straight lines which project points of r and r corresponding in ; we obtain two sets of projectie straight lines. If s is the straight line intersection of the planes Pr and P r and the projectiity generated by the sets of lines with centres P and P respectiely; the problem of determining the region of the coples P and P for which is the identity, is identical to that of determining the coples P and P for which the sets of straight lines with centres in P and P are perspectie. The problem is also identical to that in which if Q and Q are two corresponding point in r and r respectiely the points P, P, Q, Q are coplanar which sggests a ery simple method for the soltion. In fact assming the straight line r as z axis in the Cartesian system, if x = at + a, y = bt + b, z = ct + c are the eqations of the straight line r, the coplanarity of P, P, Q, Q implies the following identity X Y Z X a t / t a Y b t / t b Z c t / t c t

3 3 The eqation is an identity with respect to t. From the identity we obtain the following 3 eqations X Y-XY +ba+b X-X +a+a Y -Y= ca+c -XY -X Y+b+b X Z-XZ + b+b X-X + 2 +a+a YZ -Y Z+a+a Y -Y= c+c XY -X Y+ b+b X Z-XZ + c+c YZ -Y Z= which are the conditions on the coordinated of P and P for the coplanarity with the generic cople Q and Q, or for to be an identity or, finally, to obtain that the two sets of straight lines with centres in P and P respectiely be perspectie.. We see ths that the eqations 2 define a birational cbic transformation of the space in itself sch that when one of the centres of projection is gien the other is niqely determined. Moreoer it is easily seen that the transformation has the 3 characteristic of an inoltion and that exist coples of points which satisfy the conditions of the problem. 2. For the ftre deelopments it is sefl to consider the inerse problem of the preiosly considered one, that is to determine the projectiity between r and r when are known the positions of r, r, P, and P.. In this case the nknown of the problem are,,, and the eqations 2, which are linear and homogeneos in,,, gie the soltion. In trn,,, determine the projectiity between r and r, that is the cople of corresponding points in the two straight lines. This method may be generalized to the case when r and r are plane cres projection of a gien plane cre, and the following discssion allows to obtain the soltion of this problem when the two plane cres are not gien by means of eqations bt graphically which will also allow to obtain the soltion with simpler algebra. 3. The problem set in the section for two straight lines may be extended to two planes in the following manner. Let p and p be two planes see figre 2 PX,Y,Z and P X,Y,Z two generic points off the two planes whose points are corresponding in a projectiity ; we will determine the cople P and P sch that for any cople Q and Q, in p and p, corresponding in the projectiity, the for points P, P, Q, Q are coplanar. We will proceed as preiosly when discssing the analogos problem for the stright lines preiosly discssed. Let s then assme that the plane p is the xy plane of the Cartesian system, let X = a +a +a = x, Y = b +b +b = y, 3

4 4 Z = c +c +c = z, be the parametric eqations of the plane p and let =+ + /+ + =+ + /+ + 4 be the eqations of the projectie correspondence between the plane p and p where,, and are Cartesian coordinates in the planes p and p with, sch that the x and y axes coincide with the and axes respectiely. The conditions that the points P, P, Q, Q be coplanar that is the sets of straight lines throgh P and P, obtained projecting the corresponding points of p and p, be a perspectiity may be written X Y Z X x, Y y, Z z, 5 t where the coordinates of Q and Q are as Q,,, Q x,,y,,z,. We exclde the case when p and p are parallel and assme the reference systems sch that the x axis and the axis be parallel to the line of intersection of the planes p and p as in figre 2; then in eqations 3 x = and y and z depend only on as follows x = y= b +b 3 x=c +c which imply that the identity 5 may be written as X Y Z X Y Z / b / b c / c 5 where now the identity is meant for the ariables and. Sbtracting the last row from the first three and expanding with the elements of the last colmn, the matrix 5 is redced to the following one X X Y Y b b 5 Z Z c c

5 5 In order to satisfy the identity 5 for all coples,, mst be nil all coefficients of,,, 2, 2 and the constant term, which gie the following eqations c +c Y-Y -b +b Z-Z + ZY -Z Y = c +c X-X + Z-Z + Z Y-ZY = c +c X -X-c +c Y-Y + Z-Z + Z X-ZX + Z -Zb +b + ZY -Z Y = c +c Y-Y -b +b Z -Z ++ - Z Y-ZY + +c +c XY -X Y+b +b ZX -Z X = c +c X -X + Z-Z + Z Y-ZY + c +c XY -X Y+ 6 b +b - ZX -Z X= Z Y-ZY +c +c XY -X Y+b +b ZX -Z X= We note that the 6 eqations of system 6 are not independent and are eqialent to fie eqations as we show in the following. To this prpose let s consider the inerse problem; that is gien p, p, P and P find the eqations of the projectiity between p and p sch that a generic cople of points Q and Q, corresponding in, are coplanar with P and P, Q, Q for all coples Q,Q. Projectiities of sch type may be obtained projecting oer p and p from P and P respectiely the 3 planes of the space, then the system in which we consider,,,,,,,, as nknown has at least 3 soltions. We may then conclde that since the system of 6 homogeneos eqations 6 in the 9 nknowns,,,,,,,, has at least 3 soltions, at most 5 eqations are independent. We note that the space of the intersections of the cople of straight lines obtained projecting the corresponding points of p and p from P and P respectiely is a deelopable srface where each straight line intersects all the others; this srface is therefore a plane. It is ths clear that the projectiities of the type are only those which one obtains projecting oer p and p from P and P respectiely the 3 planes of the space, which in trn implies that the 6 eqations of system 6 are actally 5 independent eqations. We hae ths established that the projectiities between the planes p and p obtained projecting from P and P respectiely the 3 planes of the space hae eqations 4 whose coefficients satisfy eqations 6. Gien in an appropriate Cartesian system the planes p and p and the points P and P, the coefficients of the

6 6 eqations of these projectiities are related by 5 independent homogeneos eqations and is possible to express the 9 coefficients by means of 3 of them for instance,,. One obtains ths the eqations 4 expressed in sch form that the coefficients appearing in them depend linearly from,,. 4. Before proceeding to the comptation which will lead to the determination of the correspondence between the planes p and p few considerations are sefl. It is clear that three coples of points on p and p respectiely which are projection from P and P of three points located on a generic plane s, bt not on a straight line, of the space determine niqely a correspondence of the type. Moreoer if we want to proceed to the nmerical comptation of the coefficients of sch projectiity sing these elements, it is sfficient to sbstitte in eqations 4 here all the coefficients are expressed as fnctions of,,, the coordinates,,, of these three coples of corresponding points; we find ths 6 linear eqations in the 3 nknowns,,. It is clear that only 3 of these 6 eqations are independent; in fact, with reference with figre 3 we see that the cople Q and Q cannot be gien arbitrarily since, when we chose the cople Q, Q we mst find the straight line Q C2 de to the nodal points theorem; ths fixing Q the point Q depends on only one parameter. That is each point of a corresponding cople determines only one of the parameters,, and not 2 as it may seem when sbstitting the coordinates,,, in the eqations 4 where the coefficients are all expressed as fnctions of,, finding two eqations relating,,. It is ths clear that 3 coples of corresponding points on p and p respectiely, which are not on the same straight line determine one and only one projectiity. 5. Going to the application assme that the planes p and p are photographic plates and P and P the objecties of the cameras where are set the plates, It is clear that if we take a pictre of a plane trajectory and we hae in the plates 3 coples of points reslting from 3 points on the trajectory we may assme to hae a pictre of the plane of the trajectory and with the 3 coples of the corresponding points single ot the projectiity between the two plates. The by means of these 3 coples of corresponding points we may obtain all the coples of corresponding points which reslt from the same points of the trajectory considering them as corresponding point in the projectiity established between the plates. As an application of the reslts preiosly obtained let s consider the case when the cameras hae the objectie at the same eleation and, indicating with p and p the

7 7 plates and O and O2 the centres of projections, that is the centres of the objecties, assme the reference systems Ox, y, z;,,, sch that O coincides with the first principal point of the first plate C, x =, y = and the origin of the system, coinciding with the principal point of the second plate C2 as shown in the figre 4. Moreoer let s indicate with d = O C = O2 C2 the principal distances, with ω the angle p y, with g the angle p z and b, b2 the angles, redced to the horizon of the planes p and p. It is seen that g = 9 - ω = b + b2 = -9 + b + b2 7 and the coefficients of the eqations 3 are b = -cosb + b2 b = y2 + d sin b + b2 c = sin b + b2 c = z2 d cos b + b2 8 Indicating with, y2, z2 the coordinates of O and with,, d those of O2 the eqation 5 becomes 2 2 c c b b z y d 9 Sbtracting the first row form the others and expanding with the elements of the last colmn we find d d c c d z b b y where the identity is with reference to the ariables and. Expanding the determinant into an eqation and setting to zero the coefficients of all coefficients of,,, 2, 2 and the constant term, we obtain the following system d-z2b β+b γ + y2 c β+c γ y2 dγ = d-z2α = d-z2 b β +b γ- α+ y2 c β +c γ - y2 dγ = y2 dα + d-z2 b β +b γ + y2 c β +c γ - y2 dγ = d-z2α - y2 dα = y2 dα = To simplify the form of the soltion we set y 2 d = B

8 8 d z 2 = A d-z2b - y2 c = M d-z2b + y2 c d = N and find B+ M ++N = -A+M +N = M M+N = = = and setting = we find = - N + B/M =-N +A/M 2 = -N/M = = The reslt is that the projectiity between the planes p and p which one obtains with takes of a generic plane is of the type: = / + + = /M N + A - N B- N/ In eqations 3 the coefficients A, B, M, N, obtained from eqation are determined by means of d, y2, z2, b, b2 appearing in eqation 8 which shold be measred before the shooting. In order to determine,, one shold hae 3 coples of corresponding points on p and p which reslt from 3 not aligned points of the plane of the trajectory. If the cameras are opened at the same time and closed at the same time the initial and end points of the trajectory gie 2 coples of corresponding points. In order to obtain the third cople one may close the cameras at the same time for a ery short interal which gies the third cople of points. Haing obtained the 3 coples of corresponding points, sbstitting their coordinates In eqations 3 which gie the projectiity between p and p, we find 6 eqations, in the 3 nknowns,,, of which only 3 are independent. Considering that the measres are affected by errors one may se all six eqations with the least sqare

9 9 method. Ths the coefficients,, and the eqations of the projectiity 3 are determined and one may proceed to the determination of the coples of corresponding points in the frames. References Piazzolla Beloch M., Identificazione strmentale dei coppie di pnti omologhi slle immagini di traiettorie di proiettili e missili, Riista del Catasto e dei Serizi Tecnici Erariali, Noa Serie, IX, 3, 954. Fig.. Q and Q are corresponding points in the projectiity relating r and r. When Q and Q and the projecting points P and P are coplanar the projectiity inferred on s, the intersection of the planes Pr and P r, is a perspectiity. Fig. 2. Q, and Q, are corresponding points in the projectiity relating p and p.

10 Fig. 3. P and P are the projecting points in the frames p and p. Becase of the theorem of the nodal points the cople of corresponding points in p and p respectiely may not be gien arbitrarily since the lines QC and Q C2 mst intersect. Fig. 4. The frames are in the planes x,y and x,y and O and O2 are the centres of the objecties of the cameras. The origins of the Cartesian systems C and C2 mst be in the nodal points of the respectie cameras. The principal distances O C and O2 C2 mst be eqal. is the angle p y, g is the angle p z and b and b2 are the angles of p and p respectiely redced to the horizon.