Introduction to Image Restoration Methods – Part 2 Iterative Algorithms

Introduction to Image Restoration Methods – Part 2 Iterative Algorithms

This is the second part of a small series of articles on various image restoration methods used in digital image processing applications, in which we try to present the bird’s-eye perspective of some concepts of different restoration techniques without diving too deep into the math and theoretical intricacies. We assume that the reader has some understanding of discrete mathematics and signal processing basics.

Click here to read the first part.

Iterative algorithms

The simplest iterative algorithm for image restoration was first described by Van Cittert in 1930. The equation of this algorithm can be written as

f n+1 = f n + (g — h* f n),

where f n+1 is the new estimate of the image matrix derived from the previous one — f n,В f 0 = g is the blurred image, n is the number of the current step of the iteration, f is the blur kernel (point spread function, or PSF), and * sign denotes a convolution operator.

The Van Cittert algorithm (see more in German wiki) has many advantages, such as rapid deblurring, contains only a few variables and rather simple mathematical operations, and does not require smoothness restrictions or additional information. On the other hand, it has some major limitations. For example, the algorithm is sensitive to the presence of noise, and it increases the noise amounts in the deblurred image tremendously. Also, it becomes unstable if the number of iterations exceeds a certain limit, and the resultant image starts to look shaky.

The Landweber iteration, or Landweber algorithm, is an algorithm most frequently used to solve ill-posed linear inverse problems, although it has been extended to solve non-linear problems that involve constraints as well. The method was first proposed in the 1950s. It is now considered to be a base for many other iterative image restoration methods.

The equation of the Landweber algorithm is (using the same notation as for the previous formula):