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C00003 00002 5.8 Camera Locus Solving - the iron triangle algorithm.
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5.8 Camera Locus Solving - the iron triangle algorithm.
IRON TRIANGLE CAMERA LOCUS ALGORITHM.
A mobile robot with only visual perception must determine where
it is going by what it sees. Specifically, the position of the
robot is found relative to the position of the lens center of its
camera. The algorithm explained below if for computing the locus
of a camera's lens center from three landmark points.
Consider four non-coplanar points A, B, C and L. Let L be the
unknown camera's lens center, here after to be called the camera
locus, also let A, B and C be the landmark points whoes loci
either are given on a map or are found by stereo from two already
known viewing positions.
Assuming for a moment an ideal camera which can see all 4π
steradians at once, the camera can measure the angles formed by
the rays from the camera locus to the landmark points. Let these
angles be called α, β and g where α is <BLC, β is <ALC and g is
<ALB. The camera also measures whether the landmarks appear to
be in clockwise or counter clockwise order as seen from L. If the
landmarks are counterclockwise then B is swaped with C and β with
g.
A mechanical analog of the problem would be to position a rigid
triangular piece of sheet metal between the legs of a tripod so
that its corners touch each leg. The metal triangle is the same
size as the triangle ABC and the legs of the tripod are rigidly
held forming the angles α, β and g.
The algorithm was developed by thinking in terms of this analogy.
First, the triangle edge BC is placed
between the tripod legs of angle α. Let
a be the length of BC, likewise b and c
are the lengths of AC and AB.
Restricting attention to the plane of B,
L & C; Consider the points L' arrived at
by sliding the tripod and maintaining
contacts at B and C.
Recalling that in a circle, a chord
subtends equal angles at any two points
on the same one of the two arcs
determined by the chord; it can be seen
that the set of possible L' points lie
on a circular arc. Let this arc be
called L's arc, which is part of L's
circle.
Also in a circle the angle at the center
is double the angle at the
circumference, when the rays forming the
angles meet the circumference in the
same two points.
And the perpendicular bisector of a
chord passes thru the center of the
chord's circle bisecting the central
angle. Let S be the distance between the
center of the circle and the chord BC.
So by trigonometric definitions
R = a / 2sin(α)
S = R cos(α)
The position of L on its arc in the plane BLC can be expressed in
terms of one parametric variable w, where w is the counter
clockwise angular displacement of L from the perpendicular
bisector such that for w=π-α L is at B and for w=α-π L is at C.
Next, spin the metal triangle about BC sweeping the vertex A thru
a circle. Let this circle be called A's circle.
Let H be the radius of A's circle and let D be the directed
distance between te center of A's circle and the midpoint of BC.
By Trigonometric relations on triangle ABC:
cos(C) = (a↑2 + b↑2 - c↑2)/2ab
sin(C) = sqrt(1 - cos(C)↑2)
H = b sin(C)
D = b cos(C) - a/2
Now consider the third leg of the tripod which forms the angles β
and g. The third leg intersects the BLC plane at point L and
extends into the appropriate halfspace so that the landmark
points appear to be in clockwise order as seen from L. Let A' be
the third leg's point of intersection with the plane containing
A's circle.
Let the distance between the point A' and the center of A's
circle less the radius H of A's circle be called "The Gap". The
Gap's value is negative if A' falls within A's circle.
By constructing an expression for the value of the Gap as a
function of the parametric variable w, a root solving routine can
find the w for which the Gap is zero thus determining the
orientation of the triangle with respect to the Tripod and in
turn the locus of the point L in space.
Lapsing now into vector geometry place an origin at the midpoint
of BC. Establish the unit y-vector j pointing towards the vertex
B. Let the plane BCL be the xy plane and orient the unit
x-vector i pointing into L's halfplane. For right handedness
sake set the unit z-vector k to i cross j.
In the newly defined coordinates points B, C, & L
are reached by vectors:
B = (-s, +a/2, 0);
C = (-s, -a/2, 0)
L = (Rcos(w), Rsin(w), 0)
Introducing two unknowns xx and zz the locus
of point A' as a vector is:
A' = (xx, D, zz)
The vectors corresponding to the legs of the tripod are:
LB = B - L = (-s-Rcos(w), +a/2-Rsin(w), 0)
LC = C - L = (-s-Rcos(w), -a/2-Rsin(w), 0)
LA = A'- L = (xx-Rcos(w), D-Rsin(w), zz)
Since the third leg forms the angles β and g:
LA . LC = |LA| |LC| cos(β)
LA . LB = |LA| |LB| cos(g)
Solving each equation for |LA| yields:
|LA| = (LA . LC)/|LC|cos(β) = (LA . LB)/|LB|cos(g)
Multiplying by |LB| |LC| cos(β) cos(g) gives:
(LA . LC)|LB| cos(g) = (LA . LB)|LC| cos(β)
Expressing the vector quantites in terms of their components:
|LB| = sqrt((-S-Rcos(w))↑2 + (+a/2-Rsin(w))↑2)
|LC| = sqrt((-S-Rcos(w))↑2 + (-a/2-Rsin(w))↑2)
LA . LC = (xx-Rcos(w))(-s-Rocs(w)) + (D-Rsin(w))(-a/2-Rsin(w))
LA . LC = (xx-Rcos(w))(-s-Rocs(w)) + (D-Rsin(w))(+a/2-Rsin(w))
Substituting:
((xx-Rcos(w))(-s-Rcos(w)) + (D-Rsin(w))(-a/2-Rsin(w))) |LB|cos(g)
= ((xx-Rcos(w))(-s-Rcos(w)) + (D-Rsin(w))(+a/2-Rsin(w))) |LC|cos(β)
The previous equation is linear in xx, so solving for xx:
xx = P/Q + Rcos(w)
where
P = (-s-Rcos(w))(|LB|cos(g) - |LC|cos(β))
Q = (D-Rsin(w))((+a/2-Rsin(w))|LC|cos(β)
- (-a/2-Rsin(w))|LB|cos(g))
The unknown zz can be found from the definition of |LA|
|LA| = sqrt( (xx-Rcos(w))↑2 + (D-Rsin(w))↑2 + zz↑2)
so zz = sqrt( |LA|↑2 - (P/Q)↑2 - (D-Rsin(w))↑2)
and since:
|LA| = (LA . LC) / |LC|cos(β)
The negative values of zz are precluded by the clockwise ordering
of the landmark points. Thus the expression for the Gap can be
formed:
GAP = sqrt( (XX+S)↑2 + zz↑2 ) - H
As mentioned above, when the gap is zero the problem is solved
since the locus of A' then must be on A's circle, so the triangle
touches the third leg. The gap function looks like a cubic on its
interval [α-π,π-α], and it crosses zero just once if A≠b and b≠c
and c≠a. If the triangle ABC is isosceles or equilateral then
there are two or three zero crossings respectively.
Having found the locus of L in the specially defined coordinate
system all that remains to do is to solve for the components of L
in the coordinate system that A, B and C were given. This is
done by considering three vector expressions which are not
dependent on the frame of reference and do not have second order
L terms, namely:
CA . CL
CB . CL
(CA x CB) . CL
Let the locus of L in the given frame of reference be (X,Y,Z) and
let the components of the points A, B & C be (XA,YA,ZA),
(XB,YB,ZB) & (XC,YC,ZC) respectively. Listing all four points in
both frames of reference:
A = (xx, D, zz) = (XA, YA, ZA)
B = (-s, +a/2, 0) = (XB, YA, ZA)
C = (-s, -a/2, 0) = (XC, YC, ZC)
L = (Rcos(w),Rsin(w),0) = ( X, Y, Z)
Evaluating the vector expressions which are invariant:
CA = A - C = (XA-XC. YA-YC, ZA-ZC)
CB = B - C = (0, a, 0) = (XB-XC, YB-YC, ZB-ZC)
CL = L - C = (Rcos(w)+s,Rsin(w)+a/2,0) = ( X-XC, Y-YC, Z-ZC)
CA . CL = (xx+S)(Rcos(w)+s)+(D+a/2)(Rsin(w)+A/2)
= (XA-XC)(X-XC) + (YA-YC)(Y-YC) + (ZA-ZC)(Z-ZC)
CB . CL = a(Rsin(w) + a/2)
= (XB-XC)(X-XC) + (YB-YC)(Y-YC) + (ZB-ZC)(Z-ZC)
(CA x CB) . CL = -a zz(Rcos(w) + s)
= ((YA-YC)(ZB-ZC) - (ZA-ZC)(YB-YC)) (X-XC)
- ((XA-XC)(ZB-ZC) - (ZA-ZC)(XB-XC)) (Y-YC)
+ ((XA-XC)(YB-YC) - (YA-YC)(XB-XC)) (Z-ZC)
The last three equations are linear equations in the three
unknowns X, Y & Z which are readily isolated by Cramer's Rule.