Transformed image processing methods are methods that operate in image transformation domains, such as Discrete Fourier, Discrete Cosine, Wavelet, and the like. They proved to be very efficient in image compression, image restoration, image resampling and geometric transformations, and can be traced back to the early 1970s. It will not be an exaggeration to claim that digital image processing came into force with the introduction, in 1965 by Cooley and Tukey, of the Fast Fourier Transform (FFT) algorithm to calculate the Discrete Fourier Transform (DFT). This publication immediately led to a rapid growth of activity in all branches of digital signal and image processing and applications.
The second wave in this process was inspired by the introduction in communication engineering and digital image processing in the 1970s of the transformation of Walsh-Hadamard and the transformation of Haar and the development of a large family of rapid transformations with algorithms of type FFT. While the transformations of Walsh-Hadamard and Haar have already been known in mathematics, other transformations, for example, very popular at the time of Slant Transform, were invented "from scratch." the processing utility of the transformed domain for restoration and image enhancement was also recognized very soon. This period ended with the introduction of the Discrete Cosine Transformation (DCT), which was soon widely recognized as the best choice among all available at that time and which translated into JPEG and MPEG standards for image, audio and video compression .
The third wave of transformational activities for signal and image processing was caused by the introduction, in the 1980s, of a family of transformations that was coined as the "wavelet transform". The main motivation was to achieve a better local representation of signals and images in contrast to the "global" representation that is characteristic of discrete Fourier, DCT, Walsh-Hadamard and other fast transformations available at that time. During the 1980s and 1990s a variety of discrete wave transformations were suggested to solve various tasks in signal and image processing.
Today, fast transforms with FFT fast algorithms and wavelet transform are the basic instrumentation tool for digital image processing. The main distinguishing feature of the transformations that make them so efficient in the processing of digital images is their capacity of energy compaction. In regular representations of images, in the form of sets of ordered pixels, some pixels, for example those belonging to the edges of objects, are more important than the others and there are always some pixels in each particular image that do not have such importance and can be removed from the image representation and restored from the remaining "important" pixels. But the problem is that one never knows in advance which pixels in the image are "important" and which are not.
The situation is totally different in the representation of the image in the domain of transformation. For orthogonal transformations with a good energy compaction capacity, a lion's share of the total energy of the image (sum of transformation coefficients) is concentrated in a small fraction of transformation coefficients, which are, by rule general, type of images or can be easily detected. This characteristic of the transformations is called its capacity of compaction of energy. It allows to replace images with their "limited band", in terms of a specific transformation, approximations, that is, approximations defined by a sufficiently small fraction of the image transformation coefficients.