A Special Case Solution to the Perspective 3-Point Problem William J. Wolfe California State University Channel Islands

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1 A Secial Case Solution to the Persective -Point Problem William J. Wolfe California State University Channel Islands Abstract In this aer we address a secial case of the ersective -oint roblem. We assume that the oints form an equilateral triangle, and that the center of ersective lies on a lane erendicular to the triangle and containing one of its altitudes. We then resent a geometric roof that accounts for all ossible solution sets, from to 4 solutions. We emhasize the shared-oint and shared-side solution airs, as first develoed in reference [5]. Introduction Given the ersective view of non-colinear control oints (i.e.: known ΔABC we wish to determine the ossible camera-triangle configurations. This roblem, called the ersective -oint roblem (PP, has been addressed by several authors. Reference [] shows that there can be at most 4 solutions, and a secific 4-solution case is constructed. Reference [] shows that there are exactly 4 solutions whenever the center of ersective (CP is directly over the orthocenter of the triangle. Reference [] delineates the history of the roblem and discusses the relative effectiveness of the various algebraic solutions that have been roosed in the literature. Reference [4] rovides a comlete algebraic solution along with some geometric techniques to hel visualize configurations with,, or 4 solutions. Reference [5] focuses on the secial case where there is at least one solution with CP on a lane erendicular to the triangle and containing one of the its altitudes, and shows that there are 4 solutions, rovided that CP is within the "danger cylinder". Reference [6] identifies the "danger cylinder" as the region of instability caused by the degeneracy of the Jacobian of the nonlinear equations that define the camera-triangle constraints. Reference [7] suggests that the instability is related to the existence of exactly solutions on the boundary of the danger cylinder. Here we address the same secial case as develoed in [5] but we add an additional assumtion: the known triangle is an equilateral triangle. Then, as should be exected, we can ush the interesting results from [5] even further. In articular, a "Key Theorem" facilitates the derivation of a relatively straightforward geometric solution. To our knowledge, this is the most comlex case for which a comlete geometric characterization of all solutions is unambiguously rovided. The Secial Case We assume a in-hole camera model and that the corresondence between the image oints and the corners of the triangle (A, B, C is known. Figure a shows the notation

2 for the assumed equilateral triangle. As in [5], we assume that there is at least one solution with CP on a lane orthogonal to the triangle and containing one of its altitudes, as shown in figure b. Without any further loss of generality let side AB be the base of that altitude, and label the orthogonal lane π AB. We also know that CPm is a erendicular bisector of AB, and that CPm bisects = ACPB. Figure. The triangle is equilateral with side s and altitude h. O is the orthocenter of the triangle, and m is the midoint of AB. CP is on the lane π AB, which is erendicular to the lane of the triangle and contains the altitudecm. Aroach We start with the triangle in any of the configurations that satisfy our assumtions, such as the configuration shown in figure b. To get multile solutions we maneuver the triangle into other configurations that generate the same ersective measurements. From our assumtions it is clear that the camera's ersective measurements can be used to calculate ( ACPB and φ ( CCPm, as labeled in figure b (and in figure. We think of φ as the "angle of elevation" of the line CPC abovecpm. Because of our assumtions the roblem reduces to rather than measured angles. Generally, the ersective measurements consist of the angles subtended by each of the three sides of the triangle, but in our secial case the roblem reduces to the measurement of, the angle subtended by side AB, and φ, the angle elevation of C. We will use the geometry of the initial configuration to generate all ossible cameratriangle configurations that are consistent with the measured values of and φ. In each

3 case we will rovide geometric constructs and exlicit formulas for the locations of the triangle with resect to the camera. Shared-Side Solution Pair Given the initial configuration (figure b, there is an easy way to generate a second solution. Rotate about AB until C hits the same line of ersective again. This creates a "shared-side solution air" similar to those that are defined in [5] (see figure. This air of solutions shares side AB. We sometimes refer to this air as the "front-back air" because one solution is tyically leaning frontward (toward CP while the other is leaning backward (away from CP. There are a coule of excetions. If the starting solution is such that the triangle's altitude (Cm is erendicular to the line of ersective running through C (CPC, then rotating about AB will not roduce another solution (i.e.: because of tangency. In this case the shared-side "air" reduces to a single solution. This oint of tangency also identifies the largest value of φ that is feasible for the given : φ = arcsin( tan max The other excetion is when the new intersection is behind the camera (i.e.: when the distance from CP to m is less than h. These situations are easy to diagnose and deal with. Figure : The "shared-side" solution air. By rotating about AB the oint C can meet the line of ersective in laces. Both configurations have the same ersective measurements. For our secial case, the ersective measurements reduce to the measurement of angles and φ.

4 Shared-Point Solution Pair Starting with the initial configuration (figure b, while keeing fixed, slide AB toward CP. abel the new location of the triangle A'B'C' (see figure. If A' is closer to CP (as it is in figure then there is a symmetric location with B' closer to CP. This symmetric air will form the "shared-oint solution air" as also described in [5]. We sometimes refer to this air as the "left-right" air. After moving AB to one of these locations we can rotate about A ' B' until C' intersects π AB (see figure 4. Now, we rove the Key Theorem. Figure : While keeing fixed, Δ ABC has been slid to a new location, labeled Δ A'B'C'. Then, the triangle is rotated about A ' B' to move C' into π AB. Figure 4: The triangle can be slid, keeing fixed, to many ossible locations, but in each case we can rotate about A ' B' until C' intersects π AB. On the left side of the figure the triangle has been slid as close to CP as ossible (A', or B', coincident with CP. On the right side of the figure the triangle has been slid as far away from CP as ossible, in which case the left and right handed symmetric air reduces to a single triangle that leans back at the same angle as any of the other configurations shown in the figure. The Key Theorem roves that the height of the intersection (z, as well as the leaning angle, is the same for each of these configurations. 4

5 Key Theorem The height at which C' intersects the orthogonal lane in a shared-oint air, as shown in figures and 4, is given by: tan = h z Proof: Refer to the construction in figure 5. Using a sequence of similar right triangles we have: R B R A Δ R B Δ ( cos( ( sin( sin( ( cos( ( t t + = + = ( ( sin( sin( 5 s x x x B B s s R + = = + Δ Δ Using one more air of similar right triangles, and substituting the results for t and x, we get: tan( sin( 4 s R y t x y B m = = Δ Δ Now, y and z are the legs of a right triangle with hyotenuse h, as shown in figure. Therefore: tan = h z QED. 5

6 Figure 5: ooking down on the lane formed by A, B, and CP. The side AB has been slid, keeing fixed, to an arbitrary osition. For any such osition there is a symmetric osition. The figure labels one of them A B and the other A R BR ( for "left" and R for "right". The Key Theorem shows that the variable y deends only on, indeendent of and. Note that the only roerty of the triangle that is used in the roof of the theorem is that CPm is a erendicular bisector of AB. Therefore the theorem is also true if the triangle is isosceles (with base side AB. 6

7 It is surrising that z deends only on, and not on and. Since z deends only on, we can lot the locus of oints in π AB at height z abovecpm. Each of these oints is the otential location of a share-oint solution air. Consider the line λ, in π AB, at height z above CPm (see figure 4. Whenever the line of ersective (from CP to C' intersects λ we have the otential location of a shared-oint solution air. Figure 6 looks erendicularly at π AB. The circle in the figure has radius h, and it is centered at m, the midoint of AB. The side AB is shown only as a single oint, labeled "m" in the figure. This circle defines the locations of share-side solutions (as was already shown in figure. The shared-side solutions corresond to the intersections of the line of ersective and the circle. The share-oint solutions corresond to the intersections of the line of ersective with λ. Notice that the only intersections on λ that roduce feasible shared-oint candidates are those between c and c. If the intersection is to the left of c then the configuration violates our assumtions (φ > φ max. And, as was reviously exlained, c is the furthest osition that the lean-back solution can reach for our given configuration. Therefore, the only candidates on λ for feasible shared-oint solutions occur between c and c. Figure 6: An orthogonal view of the π AB. ines of ersective emanating from CP that intersect the circle at two laces secify the locations of shared-side solution airs. A line 7

8 of ersective that intersects λ, anywhere between c and c, is the otential location of a shared-oint solution air. Combining Shared-Side and Shared-Point Solutions Position in figure 6 shows a side view of the triangle as it leans back at the angle: arctan tan This corresonds to the far right art of figure 4, where the triangle is in the osition with AB erendicular to CPm and leaning back at the same angle as each of the sharedoint configurations. That is, this is where the "left" and "right" members of the sharedoint air become a single solution. In articular, lines of ersective that intersect λ beyond Position do not corresond to alternate solutions since the triangle cannot be that far away from CP and still have AB erendicular to CPm and subtending the angle. Position corresonds to the smallest value of φ for which shared-oint solutions are ossible. ines of ersective emanating from CP and intersecting the circle below Position corresond to configurations with exactly solutions, coming from shared-side airs. These are somewhat extreme front-back airings, with the extreme case being φ = 0 (CP in the lan of the triangle. As the line of ersective rises (i.e.: φ increases from Position to Position we have both shared-side and shared-oint solution airs, as indicated by the intersection twice with the circle and once with λ. This results in configurations with exactly 4 solutions. Position is where CP is directly over the orthocenter of the triangle. Notice its symmetric ositioning with resect to Position (a consequence of the equilateral triangle assumtion. At Position, each of the shared-oint solutions share a common side with one of the shared-side solutions (the "lean-back" solution, which is consistent with the solutions derived in []. Between Position and Position we still have 4 solutions coming from distinct sharedside and shared-oint airs, but the shared-side solutions get closer and closer as we aroach Position. At Position the line of ersective is tangent to the circle (φ = φ max, in which case the shared-side configuration becomes a single solution, but the shared-oint solutions are still resent as is indicated by the intersection of the line of ersective with λ at c. Therefore, there are solutions at Position. Given the angles φ and, the exact ositions of each triangle in solutions described above can be derived in a straightforward manner from the equations in the Key Theorem and the labels in figures 5 and 6. 8

9 Varying That accounts for all the feasible solutions when CP is at least a distance h from m, as it is in figure 6. If CP gets closer to the triangle, the same tye of sketch shows that there are situations with exactly solution because the "front" member of the shared-side solution is now behind the camera. However, for large enough value of φ, it is ossible to have shared-oint solutions, which in combination with the single shared-side solution roduces exactly solutions. For a systematic look at such ossibilities, we let vary. et d = CPm be the distance from CP to m. From figure 5 we see that d is inversely related to : d = h tan et φ, φ and φ be the elevation angles of Position, Position and Position resectively, in figure 6. Given any value of d, we can comute the values of φ, φ and φ (from figure 6 as functions of : T T φ = arctan( + T T T φ = arctan( T φ = arctan( T = tan T T From these equations we can lot φ, φ and φ as a function of the distance d, as shown in figure 7. If φ, the angle of elevation of the line of ersective, is less than or equal to φ then we will have exactly solutions, the shared-side air. If φ < φ < φ there are exactly 4 solutions, the combined shared-oint and shared-side airs. When φ = φ, CP is directly over the orthocenter. φ cannot be greater than φ (= φ max. As d increases, decreases gradually to 0 while Positions, and in figure 6 move toward the to of the circle (converging to a single oint at the to of the circle when d =, = 0. When d = h, the oint of tangency (Position is coincident with CP and we have φ = π/ and = π /. 9

10 Figures 7 and 8 show the ossible solution sets for d h. As d decreases, Positions,, and move down the circle. When d = h, CP is coincident with the oint of tangency (Position and φ = π/, = π /. When d = h, CP can no longer be directly over the orthocenter because φ = π/. At d = h/ the shared-oint solutions are no longer geometrically ossible (φ = 0, z = 0, = π/. See Chart for a summary of combinations of d, and φ. Figure 8 shows the same grah a figure 7, but this time with labels indicating the exact number of solutions associated with each configuration. Figure 8 omits φ since it simly tells us when CP is directly over the orthocenter, and does not cause a change in the number of solutions. Figure 7: When CP is at a distance h from m, Position (from figure 6 is coincident with CP. At this distance we lose one of the shared-side solutions because the leanforward solution is behind the camera. At d = h the CP can no longer be over the orthocenter (φ = π/. At d = h/ the shared-oint solutions are lying flat (φ = 0, z = 0, = π/, and if d < h/ ( > π/ the shared-oint solutions are no longer ossible. d φ φ φ h π tan ( tan ( π π tan ( π * π 0 * * 0 π * * * h h 0

11 Chart : Values of, φ, φ and φ corresonding to samled values of d between h and 0. The *'s indicate infeasible configurations. For examle, when d = h, φ = π/, but when d < h Position goes behind the camera, so φ is ignored. Figure 8: Same as figure 7, but with φ omitted, and labeled with the number of solutions associated with each region of values for φ and d. When d is between 0 to h/ we get a single solution: the lean-back member of the shared-side air. No other solution is geometrically ossible. When the distance is between h/ and there are two ossibilities. If φ φ, then we again have only one solution, the lean-back member of the shared-side air. If φ > φ then the shared-oint solutions are ossible and together with the lean-back solution make distinct solutions. For d > h, we have two ossibilities. If φ φ we get exactly solutions, the shared side air. If φ > φ we get exactly 4 solutions u until φ = φ, which is where the tangency occurs, so we lose one of the shared-side solutions, resulting in exactly solutions. Fixed Triangle Figure 9 shows the same configurations as figure 6 but this time with the triangle fixed s and CP moving along a circle in the orthogonal lane. This circle has radius r =, tan and it is centered at m. Included in the icture is the location of the "danger cylinder", the right cylinder whose base is the circle that circumscribes the triangle. Positions and corresond to being on the boundary of the danger cylinder, and Position corresonds to being directly over the triangle's orthocenter.

12 Figure 9: The same configurations as figure 6, but with a fixed triangle, and including the location of the danger cylinder. Positions and are on the boundary of the cylinder and Position is directly over the orthocenter. Position is clearly the lace where the triangle's altitude is tangent to the line of ersective. Shared-oint solutions are ossible when CP is between Position and Position, as are the shared-side solutions, for a total of 4 solutions. It is easy to see the symmetry between Positions and since they are located directly above oints at distance h/ from m, which is how we know that Position is on the boundary of the cylinder. If CP is on the same circular trajectory but outside the cylinder, then there are exactly solutions, the front-back air. Conclusions For the secial case of an equilateral triangle, and the center of ersective in one of the orthogonal lanes, we have rovided a thorough geometric solution. Some of the analysis can be directly extended to the case of an isosceles triangle, but then the results would be restricted to the secific orthogonal lane that is orthogonal to the base of the triangle, whereas the results resented here aly to any of the three orthogonal lanes. Another extension of these results would be to consider a small erturbation off the orthogonal lane. If CP were close to, but not on, the orthogonal lane then the sharedside solutions would be eliminated. That is, the shared-side solutions deend heavily on the symmetry created by the orthogonal lane. Once that symmetry is gone, it aears that the only ossible solutions are those that are small adjustments to the share-oint solutions, which would no longer share a oint but would be close to their configurations as in the orthogonal case. This leads us to the conjecture that -solution case is the norm

13 and that the 4-solution case results from geometrically rare (set of measure 0 symmetries. References [] Martin A. Fischler, Robert C. Bolles, "Random samle consensus: a aradigm for model fitting with alications to image analysis and automated cartograhy", Communications of the ACM, Volume 4, Issue 6 (June 98, Pages: 8-95 [] William J. Wolfe, Donald Mathis, Cheryl Weber Sklair, Michael Magee, "The Persective View of Three Points", IEEE Transactions on Pattern Analysis and Machine Intelligence, Volume, Issue (January 99, Pages: [] Robert M. Haralick, Chung-Nan ee, Karsten Ottenberg, Michael Nölle, "Review and analysis of solutions of the three oint ersective ose estimation roblem", International Journal of Comuter Vision, Volume, Issue (December 994, Pages: [4] Xiao-Shan Gao, Xiao-Rong Hou, Jianliang Tang, Hang-Fei Cheng, "Comlete Solution Classification for the Persective-Three-Point Problem", IEEE Transactions on Pattern Analysis and Machine Intelligence, Volume 5, Issue 8 (August 00, Pages: [5] Cai-Xia Zhang, Zhan-Yi Hu, "A general sufficient condition of four ositive solutions of the PP roblem", Journal of Comuter Science and Technology, Volume 0, Issue 6 (November 005, Pages: [6] E. H. Thomson "SPACE RESECTION : FAIURE CASES", The Photogrammetric Record 5 (7, 0 07, 966. [7] ZHANG Cai-Xia, HU Zhan-Yi, "Why is the Danger Cylinder Dangerous in the PP Problem?", Acta Automatica Sinica, 006, (4: