Chapter 4
Multiscale Segmentation of
Microscopic Images
Dimiter Prodanov
Abstract
The chapter introduces multiscale methods for image analysis and their applications to segmentation of microscopic images. Specifically, it presents mathematical
morphology and linear scale-space theories as overarching signal processing frameworks without excessive mathematical formalization. The chapter introduces several differential invariants, which are computed from parametrized Gaussian
kernels and their derivatives. The main application of this approach is to build a
multidimensional multiscale feature space, which can be subsequently used to learn
characteristic fingerprints of the objects of interests. More specialized applications,
such as anisotropic diffusion and detection of blob-like and fiber-like structures, are
introduced for two-dimensional images, and extensions to three-dimensional
images are discussed. Presented approaches are generic and thus have broad applicability to time-varying signals and to two- and three-dimensional signals, such as
microscopic images. The chapter is intended for biologists and computer scientists
with a keen interest in the theoretical background of the employed techniques and
is in part conceived as a tutorial.
Keywords: Laplacian of Gaussian, scale spaces, mathematical morphology, Fourier
domain
1. Introduction
Neurophysiological data are very much variable, and while certain patterns are
prominent and reproducible (e.g., action potentials, tissue textures, and cells) they
by no means can be easily defined precisely in a quantitative way. Data are enriched
with unwanted patterns having complicated temporal and spatial structure, which
are misleadingly called “noise.” Unlike the noise, natural objects have features on a
limited number of spatial or temporal scales. This observation is the starting point
of all available multiscale methods of analysis. The main focus of the chapter are
digital images; however the presented approaches can be applied in the more simple
setting of time-varying one-dimensional signals, such as voltage electrophysiological recordings. In the subsequent presentation, the images will always be considered
as two-dimensional signals sampled on a rectangular spatial grid. The reason is that
all common microscopic approaches acquire images on a plane of illumination;
thus three-, four-, and five-dimensional images are essentially sets of correlated
planar signals. The third dimension can represent depth, time, or an acquisition
channel. Obviously, in the case of four and five dimensions, the number of combinations increases. Therefore, one cannot assume isotropic resolution of the transfer
function for more than two dimensions. This situation introduces anisotropy in
47
Multiscale Segmentation of
Microscopic Images
Dimiter Prodanov
Abstract
The chapter introduces multiscale methods for image analysis and their applications to segmentation of microscopic images. Specifically, it presents mathematical
morphology and linear scale-space theories as overarching signal processing frameworks without excessive mathematical formalization. The chapter introduces several differential invariants, which are computed from parametrized Gaussian
kernels and their derivatives. The main application of this approach is to build a
multidimensional multiscale feature space, which can be subsequently used to learn
characteristic fingerprints of the objects of interests. More specialized applications,
such as anisotropic diffusion and detection of blob-like and fiber-like structures, are
introduced for two-dimensional images, and extensions to three-dimensional
images are discussed. Presented approaches are generic and thus have broad applicability to time-varying signals and to two- and three-dimensional signals, such as
microscopic images. The chapter is intended for biologists and computer scientists
with a keen interest in the theoretical background of the employed techniques and
is in part conceived as a tutorial.
Keywords: Laplacian of Gaussian, scale spaces, mathematical morphology, Fourier
domain
1. Introduction
Neurophysiological data are very much variable, and while certain patterns are
prominent and reproducible (e.g., action potentials, tissue textures, and cells) they
by no means can be easily defined precisely in a quantitative way. Data are enriched
with unwanted patterns having complicated temporal and spatial structure, which
are misleadingly called “noise.” Unlike the noise, natural objects have features on a
limited number of spatial or temporal scales. This observation is the starting point
of all available multiscale methods of analysis. The main focus of the chapter are
digital images; however the presented approaches can be applied in the more simple
setting of time-varying one-dimensional signals, such as voltage electrophysiological recordings. In the subsequent presentation, the images will always be considered
as two-dimensional signals sampled on a rectangular spatial grid. The reason is that
all common microscopic approaches acquire images on a plane of illumination;
thus three-, four-, and five-dimensional images are essentially sets of correlated
planar signals. The third dimension can represent depth, time, or an acquisition
channel. Obviously, in the case of four and five dimensions, the number of combinations increases. Therefore, one cannot assume isotropic resolution of the transfer
function for more than two dimensions. This situation introduces anisotropy in
47