263
(21)
(17)
a jj = a ÿ( 1 + ^)
a jk =QL jk (j * k)
called the “chi-square”, where y ; are the measured
values, y(x ; ,a) are the values derived by model and
vector parameters a=[ai,a 2 ,....,a M ], while o, denote the
stdev of errors on y 5 values (if unknown they can be all
set to 1). In our case this chi square quantity is replaced
by (14), in which (ri,cO are the measured values while
(r’j,c’i) are the image coordinates computed by camera
model with lens distortions.
The gradient of % 2 with respect to the parameters a has
following components
dT_ = _ 2 y [y, -y(x,;a)]dy(x,;a)
da k tT a,. 2 da k (18)
while additional partial derivative components are
(neglecting second derivative):
dy = 2 y J_ ^yfc;a) dy{xj ; a)
da k da, ~fa ; 2 da k da,
k = 1,2, ,M
(19)
Removing the factors of 2 from (18) and (19) by
defining
P*
isy
2 da k
1 ay
2 da k da,
(20)
the equations of Steepest Descent and Hessian minimi
zation procedures, derived in terms of these factors,
can be combined in a unique expression (22) if we de
fine a new matrix a’ of partial derivative of X [4]:
M
Yj a u da ' = Pi (22)
1 = 1
where 5aj are the increments that added to the current
approximation, give us the next one. When X is very
large, the matrix of becomes diagonally dominant, so
(22) goes over to be identical to Steepest Descent mini
mization formula, while as X approaches zero, the equ
ation (22) goes over the Hessian matrix formula. Given
an initial estimate for the set of parameters a, the steps
of Marquardt algorithm are as follows:
a) Compute x 2 (a);
b) Choose a modest value for X, say A.=0.001;
c) Solve (22) for 5a and evaluate x 2 (a + 5a);
d) If x 2 (a + 5a)> x 2 (a), increase X, i.e. by a factor
of 10, and go back to c);
e) If x 2 (a + 5a)< x 2 ( a ), decrease X, i.e. by a
factor of 10, update the current solution a <— a
+ 5a, and go back to c).
As regards the linear estimate of distortion parameters,
this step is accomplished by least square solution of
linear system Ax = b resulting from (8), (9) and (12).
Given the coordinates of N control points, both in the
world reference system E w and in the image plane refe
rence system Z u v , A becomes the 2n x 6 matrix, con
taining the coefficients of distortion parameters, and b
the 2n-dimensional vector of known terms, as reported
below at the bottom of the page:
A =
y i +vj ) u\(u\ +vj ) :
v,>; 2 +v; 2 ) v>;-+v;-) 2
(3w’"+vj 2 ) 2u\v\ (m, 2 +v, ) 0
2 u\v\ (3 u\ 2 +v\ 2 ) 0 (u\ 2 +v\~)
(23)
X =
k x
1
k 2
Pi
Pi
II
O
s i
_ S 2
r o+S u *
c n + 5.. *
r r u X wi + r n Y+ r r z w +T X
\ r 3\ X w j
21**
r,,X^+r n Y wi +r„Z wi +T :J
f + r 22 Y wj + r 23 Z wi +T y ^
- r.
V 31 wi
x.., + r 32 Y wi + r„z w) . + T,
- J
(24)
