SUMMARY Up to now, image processing and image analysis techniques have borrowed their basic tools from functional analysis: Fourier filtering, differential and integral calculus, and so on. These tools, however, only realize their efficiency when they are put into a well‐defined algebraic frame, most of the time of a vectorial nature. Unfortunately, the class of functions modelling ‘images’, commonly referred to as ‘grey tone functions’ does not necessarily present this very type of structure. We present here an operation for the ‘addition’ of two images, with a physical justification in the context of transmitted light. Such an addition permits the construction of the family of ‘positive homothetics' of the grey tone function at hand. The vectorial context sought is well defined: The class of images associated with the class of their grey tone functions naturally becomes the positive cone of an ordered real vector space. Furthermore, the proposed model holds for logarithmic imaging and is compatible with what is known about the human visual process. This model has been called ‘LIP’ (logarithmic image processing model).

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SUMMARYLogarithmic images, such as images obtained by transmitted light or those produced by the human visual system, differ from linear images. Their processing and analysis require consequently specific laws and structures. The latter have been developed in the concept of a logarithmic image processing (LIP) model (Jourlin & Pinoli, 1987, 1988; Pinoli, 1987a). This model permits the introduction of a well‐justified contrast definition: from a physical point of view, it is closely linked with logarithmic images and from a mathematical point of view, it is set up in an algebraic structure.The applications presented at the end of this paper concern image preprocessing and segmentation. In particular, in the case of microscopic images, the proposed method of segmentation gives good results with transmitted light (thin foils in biology or transmitted electronic microscopy). However, images obtained by reflected light microscopy are not within the scope of this model.

The so-called General Adaptive Neighborhood Image Processing (GANIP) approach is presented in a two parts paper dealing respectively with its theoretical and practical aspects. The Adaptive Neighborhood (AN) paradigm allows the building of new image processing transformations using context-dependent analysis. Such operators are no longer spatially invariant, but vary over the whole image with ANs as adaptive operational windows, taking intrinsically into account the local image features. This AN concept is here largely extended, using well-defined mathematical concepts, to that General Adaptive Neighborhood (GAN) in two main ways. Firstly, an analyzing criterion is added within the definition of the ANs in order to consider the radiometric, morphological or geometrical characteristics of the image, allowing a more significant spatial analysis to be addressed. Secondly, general linear image processing frameworks are introduced in the GAN approach, using concepts of abstract linear algebra, so as to develop operators that are consistent with the physical and/or physiological settings of the image to be processed. In this paper, the GANIP approach is more particularly studied in the context of Mathematical Morphology (MM). The structuring elements, required for MM, are substituted by GAN-based structuring elements, fitting to the local contextual details of the studied image. The resulting transforms perform a relevant spatially-adaptive image processing, in an intrinsic manner, that is to say without a priori knowledge needed about the image structures. Moreover, in several important and practical cases, the adaptive morphological operators are connected, which is an overwhelming advantage compared to the usual ones that fail to this property.

The human corneal endothelium can be observed with two types of microscopes: classical optical microscope for ex-vivo imaging, and specular optical microscope for in-vivo imaging. The quality of the cornea is correlated to the endothelial cell density and morphometry. Automatic methods to analyze the human corneal endothelium images are still not totally efficient. Image analysis methods that focus only on cell contours do not give good results in presence of noise and of bad conditions of acquisition. More elaborated methods introduce regional informations in order to performthe cell contours completion, thus implementing the duality contour-region. Their good performance can be explained by their connections with several basic principles of human visual perception (Gestalt Theory and Marr's computational theory)

This paper deals with spatially adaptive morphological filtering, extending the theory of mathematical morphology to the paradigm of adaptive neighborhood. The basic idea in this approach is to substitute the extrinsically-defined, fixed-shape, fixed-size structuring elements generally used by morphological operators, by intrinsically-defined, variable-shape, variable-size structuring elements. These last so-called intrinsic structuring elements fit to the local features of the image, with respect to a selected analyzing criterion such as luminance, contrast, thickness, curvature or orientation. The resulting spatially-variant morphological operators perform efficient image processing, without any a priori knowledge of the studied image and some of which satisfy multiscale properties. Moreover, in a lot of practical cases, the elementary adaptive morphological operators are connected, which is topologically relevant. The proposed approach is practically illustrated in several application examples, such as morphological multiscale decomposition, morphological hierarchical segmentation and boundary detection.