Optimisation-Based Image Grid Smoothing for SST Images - Advanced Concepts for Intelligent Vision Systems - page 232

α n u i g i t CX + X

X

X n +1 = X n − α n ∇ x J X = X n −

− (22)

where n is the iteration number and α n is a positive scalar corresponding to the
step in the opposite direction of the gradient.

4.3 Grid Smoothing with Linear Equality Constraints

The adaptation of the grid presented in the previous section does not ensure
that the initial size of the image remains the same. It broader terms it can be
useful in various cases to fix the coordinates of some points of the image.
The linear constraints can be expressed as

X ) t B = 0

( X − (23)

where B is a vector which size is the total number of points in the image and
the values of B verify the following properties
B = 1 if the points belongs to Φ
0otherwise

(24)

where, Φ is the set of points which coordinates will remained unchanged.
Using the properties of the vector B , the grid smoothing with fixed points
can be formalize as an nonlinear optimisation problem with linear constraints
as follows:

minimise J ( X, Y )
X

X t B =0

(25)

−

The optimisation problem can be solved using the Lagrangian parameters. Fixing
certain point in the image can be convenient if, for example, the size of the image
needs to be unchanged. In this case, B would be equal to the concatenated rows
of the following matrix Φ :

⎛ ⎞

···

11 11

⎝ 10 ··· 01 ⎠

. . . .

Φ = (26)

10 ··· 01

11 ··· 11

4.4 Grid Smoothing with Inequality Constraints

As described in the sections above, the points of the grid can move according to
the change in temperature. However, the connection between the points remains
the same. A constraint about the region of the plane where the points can move
have to be introduced to make sure that the optimisation will not end up with
a graph containing intersecting connections. The constraint can be expressed as