Mathematics for the treatment and analysis of grey-scale images

The importance of images in the modern world is undeniable. First and foremost, they are intimately integrated into our organic life (visual perception is particularly developed in humans). They are frequently used in our daily lives (video games, magazines, telephones, television...), in our personal lives (medical imaging, biological imaging, photographs...), in our professional lives (office automation, remote surveillance, videoconferencing, industrial vision...), and so on. They are not confined to the various technological sectors, but are a vector for observations and investigations of matter on very small scales (electron microscopes, near-field microscopes, etc.) or of the universe on very large scales (telescopes, space probes, etc.), sometimes leading to major scientific discoveries.

The field of image processing and analysis is broad and multidisciplinary. It encompasses all the theories, methods, techniques, devices, equipment, applications and software relating to images, enabling us to obtain qualitative and/or quantitative information and knowledge for the purposes of investigation, measurement, understanding, interpretation and, ultimately, decision-making. A wide range of scientific and technical disciplines are involved or use images: optics, computer science, physics, electronics, robotics, neurology, medicine, biology, psychology, geology, astronomy... and of course mathematics, with its strengths and limitations.

Mathematics thus plays a decisive role, since images with radiometric values are considered as numerical functions defined spatially on pixels and having as values intensities called gray tones. Applied mathematics, of course (such as numerical or matrix analysis, since grayscale images are often digital and coded in matrix form in digital imaging software), but also, less obviously, so-called fundamental or "pure" mathematics. These include algebra, which provides the notions for defining basic operations (addition and subtraction of two images: what would you do without these two operations?), and topology, the theoretical mathematical discipline par excellence, which is indispensable for defining what a connected region is, and for defining a contour.) Differential calculus (for the study of local variations in an image) and integral calculus (for the study of the average behavior of an image) are two solid pillars of image processing, enabling the implementation of differential operators (gradient, Laplacian for transition detection) and integral operators (Fourier and wavelet transforms for frequency and multiscale analysis). More generally, functional analysis is involved, since the images to be processed and analyzed are represented in function spaces. The calculus of variations is used to formalize certain image restoration...