of the image. The spatial integration of explicit information 
is due to the process memory which is derived from the 
model state equation. The main interest of this approach 
comes from its ability to remove the underdetermination 
bound to motion estimation. 
In order to use this model, three main hypotheses must 
be satisfied: 
1. the illumination is supposed to be constant between 
two consecutive images of the sequence; 
2. the displacement between two consecutive images can 
be locally assimilated to a translation; 
3. the image noise is supposed to be white and of known 
variance. 
2.1 Model description 
The model equation of the filter provides the relation be- 
tween the displacement vectors associated to two consecu- 
tive pixels. It can be expressed as a function of a transition 
matrix ® and of a white noise (w;): 
D(i) = $D(i — 1) + wi-1 
The model makes the assumption of a parametric tran- 
sition matrix whose type is: 
® = X 
with 0 < A < 1 and I; being the identity matrix. The 
choice of A determines the estimator’s memory, therefore 
it strongly influences the trade-off between the integration 
capability and the adaptability when being faced to a sudden 
change in the estimated process. 
In the following, Ey will denote the estimator parame- 
terized by A. 
In every pixel, the measure is equal to the component of 
the searched displacement along the spatial intensity gradi- 
ent. The fact that this component is the only information to 
be intrinsically available provides an intuitive justification. 
2.2 Estimator running 
Between two consecutive pixels, the estimation is updated 
both in the direction of the spatial intensity gradient and 
in the directly orthogonal direction, under the premise that 
there exists a significant variation of the gradient along the 
current image line. When the preceding condition is satis- 
fied, and according to the operator memory, the estimation 
process progressively removes the underdetermination due 
to the aperture phenomenon. The estimation error remains 
minimum along the spatial intensity gradient direction and 
is maximum along the orthogonal direction. Associated to 
each pixel, the value of this second component of the error 
is a measure of the residual underdetermination. 
The parametric Kalman model represents a continuum 
between two extreme estimators (resp. À = 0 and À = 1) 
(refer to figure 1). These two extreme estimators are radi- 
cally opposed from the regularization point of view. Finally, 
each instance of the model is a trade-off between a good inte- 
gration capability and a good adaptability to the underlying 
estimation process. 
304 
3 Using two cooperating Kalman 
filters 
3.1 Extreme models 
Closely looking at the two extreme cases generated by the 
extrema of A in the [0, 1] range is extremely valuable. The 
Eo and E, estimators are well suited to the motion estima- 
tion of a rigid object (F1) and to the motion estimation of 
a totally random motion (white motion process) ( Eo). 
Eg estimator: As the process memory is non-existent, 
no spatial integration is performed. In this case, it is easily 
shown that the gain vector is oriented in the direction of the 
intensity spatial gradient The model converges toward the 
estimate of the local orthogonal displacement vector field. 
E, estimator: As the process memory is total the spatial 
integration is maximum. À significant change in the gra- 
dient direction between two consecutive pixels leads to an 
optimal update of the gain, in the direction perpendicular to 
the gradient one. In this case the model converges towards 
the estimate of the true displacement vector field. 
From the above, one may infer that: 
e The F, estimator allows an optimal removal of the un- 
derdetermination bound to motion estimation. But, as 
a consequence, the estimator’s memory is detrimental 
to its adaptability. 
e The E, estimator does not allow any underdetermi- 
nation removal at all. But its adaptability, that is its 
ability to quickly adapt itself to a sudden change of 
the estimated signal is excellent. 
The principle of our method is based on a cooperative 
usage of the E, and Fo models to provide an accurate motion 
estimation. It is justified by the simultaneous exploitation 
of the good integration characteristics of E; and of the good 
adaptability of Eo. 
3.2 Cooperation principle 
We have highlighted the E, estimator capability to, accord- 
ing to the parametric family, optimally remove the under- 
determination bound to the motion estimation. The major 
problem related to the usage of E; is its lack of ability to 
adapt itself to a qualitative change in the estimated system. 
The basic principle of the cooperation relies on the par- 
allel activation of the E, and Ej estimators in order to use 
the adaptability of the second to the benefit of the first. 
The useful information of Eg resides in the innovation 
term. In this specific case, the innovation - the difference 
between a measure and the predicted value of this measure - 
is easily expressed as a function of the spatial gradient G(:), 
the searched displacement D(?) and a white noise v; accord- 
ing to the equation: 
zi = G(i)! D(3) - vi 
As a consequence, every qualitative modification in the 
“intensity spatial gradient” vector field G;, as well as in the 
“true displacement” vector field D(z) induces a change of 
the Eg estimator innovation. 
  
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