3.3.1
The Euler equation (3.25) can be treated as a boundary
value problem. It can also be
solved by minimizing the corresponding energy (3.19) because
a minimum of the energy is (sufficiently) a solution of the equation
(With 
, the form of the energy needs not to be
known in order to minimize it -- see below). The energy minimization
approach is chosen here.
Because both 
and 
are bounded below, so
is the energy. This means that a minimal solution, and hence a solution
to the Euler equation, exists. If 
is chosen with C=0, the
corresponding energy 
is non-convex with respect to f, and the
minimization is subject local minima. See [Blake and Zisserman 1987] for an
analysis of convexity in string and membrane models. The following
presents a discrete method for finding a local minimum.
Sample the integral interval 
into m uniformly spaced points:
. For clarity and without loss of generality,
let us assume that the bounds a and b are re-scaled in such a way
that the point spacing is one unit (More advanced numerical methods using varying spacing, such as [Weiss 1990] may be advantageous in obtaining more accurate solutions) i.e.
. Approximate the first derivative 
by the first
order backward difference 
. Then the energy (3.19) is approximated by
where 
consists of the neighbors of i. Using the
gradient-descent method, we can obtain the following updating equation:
,
The solution 
takes prescribed values at the boundary points at
i=1 and i=m to meet the boundary condition (3.22)
and the values may be estimated from data 
near the boundaries.
With initial 
, the solution is in the limit 
.
Equation (3.51) helps us see more of how
the DA works. The smoothing at i is due to 
where 
. The contributions to
smoothing are from the two neighboring points. That from site 
is
proportional to the product of the two factors, 
and
. This is why we relate 
to the strength of smoothing performed by regularizers. On
the other hand, 
acts as an adaptive weighting function to
control the smoothing due to the difference 
. Therefore
we regard it as the interaction. Two more remarks are made: First, the
contributions from the two sides, i-1 and i+1 are treated separately
or non-symmetrically. Second, the sum of the contributions to smoothing
is zero if the three points are aligned when 
and 
is even since in this situation 
.
The updating rule for the adaptive membrane on 2D can be easily derived. In the 2D case (3.42) , there is an additional term in the y dimension. This leads to the following energy
Letting 
leads to the following updating rule
where 
is the set
of the four neighboring points (With the 4-neighbour system, the model considers derivatives in the horizontal and vertical directions. With the 8-neighbour system, the regularizer also includes the diagonal derivatives weighted by 1/******) of 
.
The updating on 2D grid can be performed on the white and black sites of
a checkerboard alternately to accelerate convergence.
There are three parameters in (3.51) which
shall be determined for the DA model: 
, 
and 
.
Parameter 
is related to the convergence of the relaxation
algorithm. There is an upper bound for 
for the system to be
stable. There also exists an optimal value for 
[Blake and Zisserman 1987]. Optimal choices of 
for quadratic
regularization may be made using cross-validation [Wahba 1980 ; Shahraray and Anderson 1989]. In [Nadabar and Jain 1992], a
least squares method [Derina and Elliott 1987] is presented for estimating MRF
clique potentials for a LP model. Automated selection of the 
and 
parameters for the DA model is an unsolved problem. They
are currently chosen in an ad hoc
way.
Figure 3.3: 3D plots of an image (left) and its reconstruction using DA
(right). The plots are drawn after sampling the images at every 4
pixels in both directions into the size of 
.
From (Li 1995b) with permission; © 1995 IEEE.
The DA model with C=0 leads to non-convex dynamic systems and direct minimization using gradient descent only guarantees to find a local minimum. A GNC-like algorithm can be constructed for approximating the global solution; see Section 9.3.
