The problems of de-blurring, de-noising, compression and segmenta-

tion are fundamental problems in image processing. Each of these prob-

lems can be formulated as a problem to find some approximation of an

initial image. To find this approximation one needs to specify the ap-

proximation space and in what space the error between the image and its

approximation should be calculated.

Using the space of Bounded Variation, BV, became very popular in

the last decade. However it was later proved that for a rich variety of nat-

ural images it is more effective to use spaces of smooth functions that are

called Besov spaces instead of BV. In the previous papers two methods

for classifying the smoothness of images were suggested. The DeVore’s

method based on the wavelet transform and Carasso’s method based on

singular integrals are reviewed.

The classical definition of Besov spaces is based on the modulus of

continuity. In this master thesis a new method is suggested for classifying

the smoothness of images based on this definition. The developed method

was applied to some images to classify the smoothness of them.