The paper considers the problem of image representation in the form of global and detailing structural elements. It is proposed to use topological decomposition to extract structural components of connectivity in an image. The application of matrix algebra to image components derived from topological decomposition is investigated. The components correspond to individual objects of interest in the image, the information about which is contained in the decomposition matrices. Algebraic addition of the decomposition matrices and multiplication of the matrix by a number are shown. The algebraic addition allows us to separate objects of interest into separate groups based on similar topological features. The number of such groups depends on a particular practical problem. The most commonly used division of an image into two groups in the form of low-frequency and high-frequency components. The possibilities of topological decomposition to obtain an arbitrary number of groups are shown. In order to underestimate or increase the visibility of individual components, matrix multiplication by a number is used, which allows us to process only individual objects of interest without affecting the whole image as it happens in spectral wavelet and Fourier transforms.