WM AU uas
with the terms gg, and gg, expressing the local gradient of
the right image function in the z and y direction respectively
as
E7/1023173) . Ogh(*n. VR)
m e A, IM, e
By differentiating and substituting the affine transformation
parameters, the observation equations become
gi(21,yL) — e(z,y) = 9R(=R,YR) + 9r,da1 + gn, 21da;
ct gn.yrLdas t gn,db, 4- gn, vrdb;
t gn,yrdbs (7)
One observation equation is formed for every pair of pixels
from the left and right image templates, resulting in a total
of n, - ny equations for templates of size n; x n,. Using
matrix notation we have
—e= Az —1 (8)
where the vector of unknowns z is
zT = [da;, da,, das, db,, db», dbs] (9)
and each element of the vector of observations | is of the
form
l — gí(2p, yr) — gn(*«. V) (10)
while each line of the design matrix A is
A — [gn., QR. 1, R.VL; IR,, IR, TL; IR, YL] (11)
The least squares solution is
z — (ATPA) ! AT PI (12)
with P the associated, typically diagonal, weight matrix.
By using the transformation parameters obtained through
the least squares solution to update the coordinates and
resample gray values at integer grid coordinates, a new right
image window gx(zk, yk) centered at
al — (aj -- dai) 4- (a5 4- da;)zr 4- (a3 4- das)yr (13)
and
yl z (57 + dbi) + (55 + dba)xz + (b3 + dbs )yL (14)
is selected as conjugate of the stationary left image template
gr(zr,yr). A new set of observation equations is formed
and solved. In this manner, the true conjugate window
gn(2n,yn) is identified as the window g2(z2,y2) at which
the least squares iterated solution is converging. It is com-
mon practice to use least squares matching as a means for
identifying conjugate points rather than windows. Thus, we
correspond the point (25, y&), center of the right image win-
dow, to the point (zr, yr) of the left image. The maximum
allowable pixel coordinate difference between the initial ap-
proximation and the final solution for which the technique
can still converge is termed pull-in range.
The great advantage of least squares matching is its flexi-
bility and the fact that it is a well-known and documented
technique. The basic model which has been described here
can easily be expanded to accommodate more than two im-
ages or to include various additional constraints. Radiomet-
ric parameters can also be included in an effort to compen-
sate for differences in brightness and contrast between the
803
two images, and are particularly helpful when using digi-
tized images of analog diapositives [Pertl, 1985]. However,
a radiometric adjustment is typically performed prior to the
least squares solution, equalizing the average and the stan-
dard deviation of gray values of the two conjugate windows,
thus accommodating for uneven radiometric properties of
the two images.
3. MULTIPLE IMAGE LEAST SQUARES
MATCHING
3.1 Mathematical Formulation for Multiple Images
Multiple image matching can be performed by simultane-
ously minimizing the gray value differences between all the
possible pairs of conjugate image windows. One image win-
dow has to be kept constant and serves as the matching tem-
plate. For every pair of conjugate image windows (w;, w;),
depicting the same object-space area in the overlapping pho-
tos 1 and j, we form the observation equations
gi(æi, Yi) — 9i(z5,9;) = eij(z,y) (15)
For windows of n; x n; pixels appearing in n overlapping
photographs we have a total of (n — 1) + (n —2) +... + 2+
] pH pairs of conjugate image windows and therefore
no mg observation equations. According to the general
least squares matching approach, each pair of conjugate win-
dows is geometrically related through a six-parameter affine
transformation
2; = af +a + aly; (16)
yj — b7 + bz; + by; (17)
or, conceptually
(25,95) = f(z, 9:) (18)
However, we cannot introduce a set of affine transformation
parameters for every pair of image windows since that leads
to dependency between transformation parameters. Instead,
we can use the set of transformation parameters relating
each window w; to the template window w,
(2; yi) = F (21, yı) for :2 = 2,3...n (19)
which uniquely and sufficiently describes the geometric rela-
tionships between all possible conjugate window pairs [Tsin-
gas, 1991]. Indeed, the transformation between a window w;
in photo j and its conjugate window w; in photo 7 is uniquely
described through the parameters relating each window to
the template window w, as
(25,93) = f (£7)! (zi, y) (20)
with the inverse affine transformation f! — (f!5)-! defined
as
12,18 1: 11 11 Hu
1 — - - - - * - T T 1 T 7 3 - i
alibi 2 alibi: alibi: LA ai by alibi Es ali bi
(21)
alibi -- plight bii —alt
gj, A 201 2 204 2 ji
3 -— * * * * . - 3 - 1 . . * * 1
XN ap a RE ap
(22)
Substituting in equations
1j 1j 1j
z; = Ay -Fayzidctagyy (23)
