microscopic images, compared to other imaging modalities, such as magnetic
resonance imaging or computer tomography. In microscopy, in such way, the
default case is a planar image.
All physical signals are bounded and of finite duration. Such signals are acquired
as discrete samples from an underlying physical process which, as an idealization,
can be considered as continuous. The physical signals are naturally related to the
properties of the measurement apparatus. As another idealization, these properties
are described by a linear transfer function so that the measurement process
becomes a convolution (denoted further by ⋆ ). On the other hand, the measurement is always contaminated by an unwanted signal, which is denoted broadly as
“noise.” The noise process can be identified with the nonlinearities of the measurement process. In many occasions because of its apparent irregularity in time, it can
be treated as a purely random process. Since the physical measurement is a repeated
process, the Gaussian noise comes as a very common and useful model by virtue of
the central limit theorem of probability theory. The theorem roughly states that the
weighted sum of uncorrelated random variables having finite variance approaches a
normally distributed random variable. Hence, if the noise is not spatially and temporally correlated, methods suitable for treating Gaussian or Poisson noise are
applicable. Evidently, very fast sampling or sampling from processes having long
memories, such as viscoelastic interactions, can violate these requirements. In such
settings, other noise models can become more suitable. The readers are directed to
[1] for a useful noise classification.
The chapter is organized as follows. Section 2 discusses the segmentation problem in general. Section 3 gives an overview of the mathematical morphology theory
and provides examples. Section 4 gives an overview of the geometrical image
features from the perspective of differential geometry. Section 5 introduces several
types of scale spaces and their application in segmentation. Section 6 discussed
implementation details. The chapter is intended for biologists and data scientists
with keen interest in theoretical background of the employed techniques and is in
part conceived as a tutorial. The references cited in the chapter are suitable for
introductions on the mentioned topics.
2. Brief overview of image segmentation approaches
Extraction of an object’s boundaries from a digital image is called segmentation.
Image segmentation is related also to object classification, which does not require
precise delineation of the object boundary. Therefore, segmentation can be also
viewed as classification on a pixel level.
The image segmentation is a nontrivial problem. For a successful image segmentation, it is important to have prior knowledge of the image composition, that
is, the texture properties of the background and the objects of interest. Segmentation generates a mask consisting of a binary image delimiting the objects of interest
present in the raw image. The challenge is to define an accurate segmentation
methodology or at least an approach that enables segmentation of biologically
relevant features. There are several classes of methods, which can be applied in
different circumstances. These can be classified broadly into two classes: (i) intensity based, where the hypothesis is that only differences in the image intensity
histogram can be sufficient for segmentation and (ii) geometry based, where the
image is transformed so that the geometrical features of interest become enhanced.
Historically, the first and simplest segmentation methods are based on global
thresholding of the histogram. Classical threshold-based methods consist of identifying a given pixel intensity level that allows for separating the object of interest
48
Advances in Neural Signal Processing
resonance imaging or computer tomography. In microscopy, in such way, the
default case is a planar image.
All physical signals are bounded and of finite duration. Such signals are acquired
as discrete samples from an underlying physical process which, as an idealization,
can be considered as continuous. The physical signals are naturally related to the
properties of the measurement apparatus. As another idealization, these properties
are described by a linear transfer function so that the measurement process
becomes a convolution (denoted further by ⋆ ). On the other hand, the measurement is always contaminated by an unwanted signal, which is denoted broadly as
“noise.” The noise process can be identified with the nonlinearities of the measurement process. In many occasions because of its apparent irregularity in time, it can
be treated as a purely random process. Since the physical measurement is a repeated
process, the Gaussian noise comes as a very common and useful model by virtue of
the central limit theorem of probability theory. The theorem roughly states that the
weighted sum of uncorrelated random variables having finite variance approaches a
normally distributed random variable. Hence, if the noise is not spatially and temporally correlated, methods suitable for treating Gaussian or Poisson noise are
applicable. Evidently, very fast sampling or sampling from processes having long
memories, such as viscoelastic interactions, can violate these requirements. In such
settings, other noise models can become more suitable. The readers are directed to
[1] for a useful noise classification.
The chapter is organized as follows. Section 2 discusses the segmentation problem in general. Section 3 gives an overview of the mathematical morphology theory
and provides examples. Section 4 gives an overview of the geometrical image
features from the perspective of differential geometry. Section 5 introduces several
types of scale spaces and their application in segmentation. Section 6 discussed
implementation details. The chapter is intended for biologists and data scientists
with keen interest in theoretical background of the employed techniques and is in
part conceived as a tutorial. The references cited in the chapter are suitable for
introductions on the mentioned topics.
2. Brief overview of image segmentation approaches
Extraction of an object’s boundaries from a digital image is called segmentation.
Image segmentation is related also to object classification, which does not require
precise delineation of the object boundary. Therefore, segmentation can be also
viewed as classification on a pixel level.
The image segmentation is a nontrivial problem. For a successful image segmentation, it is important to have prior knowledge of the image composition, that
is, the texture properties of the background and the objects of interest. Segmentation generates a mask consisting of a binary image delimiting the objects of interest
present in the raw image. The challenge is to define an accurate segmentation
methodology or at least an approach that enables segmentation of biologically
relevant features. There are several classes of methods, which can be applied in
different circumstances. These can be classified broadly into two classes: (i) intensity based, where the hypothesis is that only differences in the image intensity
histogram can be sufficient for segmentation and (ii) geometry based, where the
image is transformed so that the geometrical features of interest become enhanced.
Historically, the first and simplest segmentation methods are based on global
thresholding of the histogram. Classical threshold-based methods consist of identifying a given pixel intensity level that allows for separating the object of interest
48
Advances in Neural Signal Processing