Optimisation-Based Image Grid Smoothing for SST Images - Advanced Concepts for Intelligent Vision Systems - page 230

3.2 Second Order Algorithm with Attach (SOWA)

In the SOWA algorithm, the cost function J can be expressed as follows:

Z

Z + θZ t A 2 Z Z t Q Z

J = 1
2

− − (9)

For small size problem, the solution can be found by matrix inversion,
Z opt = I + θ A 2 − 1

Z

(10)

and the solution is unique. For large size problems, the gradient descent method
may be applied. One iteration of the gradient descent method is as follows:

α n ( Z n − Z )+ θ A 2 Z n Z n +1 = Z n − α n ∇ x J = Z n −

(11)

where n is the iteration number and α n is a positive scalar corresponding to the
step in the opposite direction of the gradient. The SOWA method is chosen when
the smoothing needs to conserve the curves of the image. The SOWA algorithm
is applied to Meteosat first generation satellite and the results are depicted in
the results section of the paper. It may be noted that the SOWA algorithm is
only used to filter the gray levels of the image to reduce the level of noise in the
images. It may be shown that this method is more ecient that most of usual
adaptive filtering techniques on our dataset.

4 Optimisation-Based Approach to Grid Smoothing

The goal of the grid smoothing applied to large scale SST images is to enhance
the resolution. The initial uniform grid on which the image is sampled is modified
to fit the content of the image. After modification of the grid using the grid
smoothing approach, the regions with large variance values expose a greater
number of points of the grid while the opposite phenomenon may be seen in
the region with small variance. It may be noted that the total number of points
remains unchanged.

4.1 General Framework

A cost function is introduced to adapt the grid to the information contained in
the image. A cost function J is defined as follows:

J = J X + J Y (12)

where

Z. Z t X. X + θ X

X X t Q X

1
2

J X = − − (13)

and

Z. Z t Y.Y + θ Y

Y Y t Q Y

= 1
2

J Y − − (14)