Fourier Transform Methods Applied To The Analysis Of Microphotometric Images Of Histologic Sections

Implementation of one - dimensional Fourier transform techniques for the analysis of histologic sections is discussed, as is the motivation for their use. Features of the frequency domain representation derived by such a transform are shown to be related to several important diagnostic clues. The interpretation of the Fourier magnitude spectrum in histologically relevant terms is examined by means of Fourier transforms of idealized tissue simulations. Some of the perturbations of these ideal spectra produced by biologic reality are discussed. Three classic types of cervical epithelial tissue are modeled, and their representation as Fourier magnitude spectra interpreted in the light of the previous results: characteristic frequency domain signatures are obtained for each. It is concluded that these techniques may provide diagnostically important objective measures, and may be applied to otherwise intractable histologic specimens with crowded and overlapping nuclei. ABSTRACT Implementation of one-dimensional Fourier transform techniques for the analysis of histologic sections is discussed, as is the motivation for their use. Features of the frequency domain representation derived by such a transform are shown to be related to several important diagnostic clues. The interpretation of the Fourier magnitude spectrum in histologically relevant terms is examined by means of Fourier transforms of idealized tissue simulations. Some of the perturbations of these ideal spectra produced by biologic reality are discussed. Three classic types of cervical epithelial tissue are modeled, and their representation as Fourier magnitude spectra interpreted in the light of the previous results: characteristic frequency domain signatures are obtained for each. It is concluded that these techniques may provide diagnostically important objective measures, and may be applied to otherwise intractable histologic specimens with crowded and overlapping nuclei. The quantitative analysis of images derived from microscopic observation of tissue sections for the purpose of objective pathologic diagnosis has, largely for historical reasons, followed two divergent paths. On the one hand, stereologic methods have continued to be applied to relatively large scale histologic structures, while high resolution image cytometry has rapidly developed as a discipline dealing with the detailed photometric analysis of individual nuclei and cells. A rather conspicuous gap in the continuum of the biologic hierarchy exists between these two levels: one which contains important diagnostic information regarding the relationships of nuclei to each other and to the larger structures of the tissue. As part of the development of an expert diagnostic system for the use of pathologists this gap became evident, and was the motivation behind the development of a system for the quantitative contextual analysis of histologic specimens at an intermediate scale.1 There exists, however, a rather substantial class of specimens which may be analyzed by these techniques with difficulty, if at all, due to problems of scene segmentation. These include cases with crowded and overlapping nuclei often the very cases (i.e. the more advanced grades of neoplasia) from which we would particularly like to obtain quantitative data in order to obtain prognostic clues by examining relationships with follow-up data. It was these considerations which led to the investigations of the application and interpretation of Fourier transform techniques which will be discussed in this paper. 2. FUNDAMENTALS OF THE APPLICATION OF THE ONE-DIMENSIONAL FOURIER TRANSFORM TO THE ANALYSIS OF HISTOLOGIC SECTIONS We begin by defining, in the spatial domain, a function whose value is the local optical density of a digitized histologic image as measured in terms of arc length along some curve transecting the image field. This curve may simply be a straight line of given length (as will be used in these examples), but need not, and probably ought not, be. In the case of squamous or stratified epithelia such as that derived from cervical sections, a curve locally parallel to the stratum germinativum would be appropriate, while in the case of glandular epithelia such as colonic sections, one through the locus of the nuclear centers of the acinar formation would be the proper choice. Typically this function will appear as a pulse train of optical densities corresponding to the transections of nuclei cut by the curve, with some regular or irregular width and spacing, and amplitudes determined by the local optical density of each nucleus and their overlap. A close-up of this schema is shown in Fig. 1.