In classical mathematics, various kinds of transforms (Fourier, Laplace, intergral, wawelet) are used as powerful methods for construction of approximation models and their further utilization. The main idea of these transforms consists in transforming the original model into a special space where the computation is simpler. The transform back to the original space produces an approximate model or an approximate solution. There is a bridge put between these well known classical methods and methods for construction of fuzzy approximation models in this research. We have developed the general method called fuzzy transform (or shortly F-Transform) that encompasses both classical transforms as well as approximation methods based on elaboration of fuzzy IF-THEN rules studiet in fuzzy modeling.
This transform we used, i.a., in image processing, especially in image compression and image fusion.
The other application we deal with in research is image fusion. The main idea is: there is a set of source images (photos are expected) with differ with focus; and therefore is fuzzy in different parts. The target is to analyse the source images, find the best parts of each image and create a result image as a composition best part of each image.
Again, the discrete fuzzy transform is used in this process. The general idea is build on the weak point of the fuzzy transform - sharp edges. Simply said, if very fuzzy image is used as a source, the result after forward and inverse fuzzy transform looks like the origin image. If very sharp image is used (with lot of sharp edges), the result after forward and inverse fuzzy transform is fuzzy and therefore differs a lot from the source image. The main idea is to apply a forward and inverse fuzzy transform over the image and find parts, where the image was deformed. Those parts are sharp parts of the image and are taken to the construction of the final image.
The complete algorithm is fully explained and described in . All the source images are taken and with increasing number of resulting components are processed iteratively until the reconstructed images are enough similar to source (the meaing of "enough" is set as an input parameter of the algorithm). Then, from calculated componets those with the highest differences are taken and used to reconstruct final image.
The main advantage of this method is that the process itself can (via the fuzzy transform) find the best part form the image, so the user does not have to select it manually.
List of examples:
We use simple F-transform based image fusion algorithm (SA) and complete F-transform based algorithm (CA), which are described in . In the case of the SA and CA, an important role is played by the initial settings of the values of the algorithm parameters: the number of basic functions m, n in the case of SA and CA and the values of the increment step and the number of iterations kmax in the case of CA.
Multi-focus input images
In this section we demonstrate a solution of multi-focused image fusion problem using F-transform technique.
Multi-sensor input images
This section presents example of multi-sensor images and their fusion using F-transform technique.
 Research report 58. Irina Perfilieva: Fuzzy Transforms, 2004. Download Report 58.
 Research report 122. Irina Perfilieva, Martina Daňková: Image Fusion on the Basis of Fuzzy Transforms, 2004. Download Report 122.
Chapters in monographs
 I. Perfilieva, Fuzzy Transforms and Their Applications to Image Compression, in: Springer, Heidelberg, 2006, pp. 19-31.
 I. Perfilieva, M. Daňková, P. Hoďáková, M. Vajgl: F-Transform Based Image Fusion, in: INTECH ... in treatment.