nential image model with white noise the effective bandwidth is /2/a and v3/a for one dimensional
and two dimensional isotropic signals. P can be interpreted as the standard deviation of the
frequency having the standardized SNE?(u) as probability density function.
The parameter » can be estimated from the empirical autocovariance function. The variance of
the noise can directly be measured if one uses flat imagery with no structure. The variation of
g then reflects the (film grain) noise (cf. Helava 1976, Ryan 1980). But P (0) and cz can rigo-
rously estimated using variance component estimation techniques (cf. Koch 1980) which for digi-
tal terrain profiles has given realistic results (cf. Lindlohr).
3.2 Precision of object location
We will now derive formulas for determining the variance of the estimated shifts after object lo-
cation. We start with the discrete two-dimensional model
g;0= y; = g(z 2, y 79) + Yeo 22 0, 0a (9)
This equation holds for all » pixels which need not form a grid.
The idea is to linearize eq. (9) and solve the linearized problem using least squares tech-
nique (cf. Limb and Murphy 1975, Burckhardt and Moll 1978, Wild 1979, Meyers and Frank 1980,
Thurgood and Mikhail 1982). Hence since nearly 10 years this method has been applied for image
sequence analysis (Cafforio and Rocca 1976, Schalkoff and McVey 1978, Fennema and Thompson 1979,
Dinse et. al. 1981, Huang 1981) and recently tor target location and point transfer in photo-
grammetry and remote sensing (Ackermann and Pert] 1982, Fórstner 1982, Mikhail 1982).
Starting from approximate values =, and 9. and setting Ag,-bg(z.,y )7g. (zy og (ze »U
Lu 7 Qu eo
and VEE sy JERE YL) the unknown differnces £ and 7 (assuming = _=y =0 for the moment) can
17
z
L Quo
be estimated from the n equations
Ty Jj
ic. 0.79. .2*5...1 Tz 1, .…. ?
La, +9, 5 9, 2 ET 9,42 Jo L038, (10)
where g_ = 3g/3x and gy = dg/3y are taken at the approximate values. The overdeterminec equation
system (10) leads to the normal equations
(195 — te, e) f£ (Te. Ag \
lee wm Ag ls es
Mg: 6, Ig; J 2] Le, Mj
if the disturbing noise is white, c = 1. It can be rewritten if the sums are replaced by the cor-
responding variances or covariances, e.g. Ig.9, 7 ^ Og g
fci 9g e. (8) [Cg ag) |
n| “x dd = nl =¥ior NA z = h (115)
Uo. e 02 iz les ve
\ 2274 9; 7] «€ Ve, Ag )
Eq. (11) yields optimal estimators for the unknown shifts. Their covariancematrix for general
is
^5
me
-
n
ie
; |
a TN. mg a 2)
Vm V) A %,
1. The precision of the location is determined by three parameters
- the image noise variance
- the number of pixels and
- the variance and the covariance of the gradient image. This is a specification of the edge
business in an image which proves to be decisive for the precision of object location.
image is isotropic, i.e. the covariance of the gradients is zero we obtain the variances
Fn
na
