Fundamental Linear Transforms

Abstract

In this chapter, the fundamental linear transformations that have immediate application in the field of image processing, in particular, to extract the essential characteristics contained in the images are described. These characteristics, which effectively summarize the global informational features of the image, are then used for other image processing tasks such as classification, compression, description, etc. Linear transforms are also used, as global operators, to improve the visual qualities of an image (enhancement), to attenuate noise (restoration), or to reduce the dimensionality of the data (data reduction). Typically, a linear transform, geometrically, can be seen as a mathematical operator that projects (transforms) the input data into a new output space, which in many cases better highlights the content of the input data. The one of greatest interest is the unitary transform, that is, a linear operator with the characteristic of being invertible, with a kernel (the transformation matrix) that satisfies the orthogonality conditions. It follows that the inverse transform is also realized as an inner product between the coefficients and the rows of the inverse matrix of the transformation. Each transformation is characterized by a transformation matrix. The desired effects on an image can be made by operating directly in the spectral domain of the transforms and then reconstructing the image by the inverse transform, thus observing the results on the spatial domain. With the unitary transformations, the concept of digital filtering is generalized by operating in the spectral domain of a generic transformation, such as the transforms: Cosine, Haar, Walsh, etc. For some of these transformations (DCT, Hadamard, Walsh, Haar) the transformation matrix is not characterized by input data. With those transformation matrices that are independent of the input data, it is possible to implement fast image compression algorithms, especially useful for transmission. Finally, the wavelet transform is described which is characterized for its ability to be accurate in detecting frequency content together with spatial or temporal localization information, thus overcoming the limits of the Fourier transform with which spatial localization of spectral content is lost.