## Axioms and fundamental equations of image processing.(English)Zbl 0788.68153

The (nearly 60 pages) paper under review gives a full axiomatic base for the multiscale analysis of a picture in image analysis. The authors define a multiscale analysis as a family of transforms $$(T_ t)_{t \geq 0}$$ defined on a class of functions on $$\mathbb{R}^ n$$, which gives rise to a family of pictures $$(T_ tf)_{t \geq 0}$$ for a given original picture $$f$$; the parameter $$t$$ defines the size of the neighbourhood; the operators $$T_ t$$ are not necessarily linear. Besides the so-called comparison principle $$(T_ t(f) \leq T_ t(g)$$ for all $$t \geq 0$$ and for $$f \leq g)$$, two classes of axioms are distinguished. It is shown that, under these axioms, there exists a continuous function $$F$$ such that, when for a given picture $$f$$ we put $$u(t,x)=(T_ tf)(x)$$, $$u(x,t)$$ satisfies a partial differential equation of the form $${\partial u \over \partial t}=F(D^ 2u,Du)$$. The already existing classical models of multiscale analysis (such as the Marr and Hildreth theory) are shown to be special cases of the developed general theory. The same methodology is applied to the analysis of movies.
Reviewer: G.Crombez (Gent)

### MSC:

 68U10 Computing methodologies for image processing
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