An original approach to quantization is presented. It is based on operator-valued measures, under the generic name of integral quantization. The Wigner-Weyl and Berezin-Klauder-Toeplitz quantization, and more generally coherent states quantization, are particular (and mostly manageable) cases of this approach. We particularly insist on the probabilistic aspects appearing at each stage of this quantization procedure and its subsequent “dequantization” through semi-classical expressions (lower symbols). The link with quantum measurement is sketched.

The method is illustrated by the example of the sea star or starfish. Motivated by its five-fold symmetry, we develop in a comprehensive way the quantization of functions on a set with five elements, which yields in particular quantum and semi-classical angles from a starfish point of view.