Mathematics for the treatment and analysis of binary images

A first article in the Techniques de l'Ingénieur collection [E 6 610] gave a synthetic overview of the main concepts, notions and mathematical frameworks involved in the processing and analysis of grayscale images. Grayscale images are mathematically considered as numerical functions defined spatially on pixels and having as values intensities called grayscale tones. Binary images are usually the result of prior processing of grayscale images. They consist of functions defined spatially on pixels and taking only two values: 0 and 1. The value 1 represents informative pixels and the value 0 the other pixels. As with grayscale images, mathematics plays an important role, since binary images are considered to be composed of spatial objects. This is the case for geometry, a discipline all too often forgotten in today's higher education, which plays a central role in binary imaging.

From a technological point of view, this importance is boosted by the performance of imaging investigation systems and the computing power of computers, which developed considerably in the second half of the 20th century. Binary imaging has thus enabled a remarkable return to the "hit parade" of many "old" results (19th century: Cauchy's and Crofton's theorems for measuring the perimeter of an object), and even medieval ones (16th century: Cavalieri's principle on measuring volume by "slicing" a solid object). It is based on two pillars: differential geometry (19th century: study of local variations in objects) and integral geometry (19th and 20th centuries: measurement of the content and contour of an object). In the second half of the 20th century, it led to the emergence of specific branches of mathematics such as stereology (the study of the transition from one- or two-dimensional spatial measurements to the third dimension) and stochastic geometry (the study of the spatial distribution of objects from a probabilistic point of view). Set theory and convex geometry (Steiner's formula from the 18th century, Minkowski's addition from the early 20th century) have also found a new lease of life, serving as a basis for mathematical morphology (second half of the 20th century: erosion and dilation of shapes, granulometric spectra...). Topology, algebra and measurement theory, combined with geometry, gave rise in the 20th century to new branches of mathematics (topological geometry, algebraic geometry, geometric measurement theory) of great interest to binary imaging, particularly for characterizing objects...