Image Segmentation

This document describes an algorithm implemented using the Insight Toolkit ITK www.itk.org. This paper is accompanied with the source code, input data, parameters and output data that the authors used for validating the algorithm described in this paper. This adheres to the fundamental principle that scientific publications must facilitate reproducibility of the reported results.


Detection of Discontinuities
• This is usually accomplished by applying a suitable mask to the image. • The mask output or response at each pixel is computed by centering centering the mask on the pixel location.
• When the mask is centered at a point on the image boundary, the mask response or output is computed using suitable boundary condition. Usually, the mask is truncated.

Point Detection
• This is used to detect isolated spots in an image.
• The graylevel of an isolated point will be very different from its neighbors.
• It can be accomplished using the following 3 3× mask: • The output of the mask operation is usually thresholded.
• We say that an isolated point has been detected if

Detection of lines
• This is used to detect lines in an image.
• It can be done using the following four masks:

Edge Detection
• Isolated points and thin lines do not occur frequently in most practical applications.
• For image segmentation, we are mostly interested in detecting the boundary between two regions with relatively distinct gray-level properties.
• We assume that the regions in question are sufficiently homogeneous so that the transition between two regions can be determined on the basis of gray-level discontinuities alone.
• An edge in an image may be defined as a discontinuity or abrupt change in gray level. • These are ideal situations that do not frequently occur in practice.
Also, in two dimensions edges may occur at any orientation.
• Edges may not be represented by perfect discontinuities. Therefore, the task of edge detection is much more difficult than what it looks like.
• A useful mathematical tool for developing edge detectors is the first and second derivative operators.
• From the example above it is clear that the magnitude of the first derivative can be used to detect the presence of an edge in an image.
• The sign of the second derivative can be used to determine whether an edge pixel lies on the dark or light side of an edge.
• The zero crossings of the second derivative provide a powerful way of locating edges in an image.
• We would like to have small-sized masks in order to detect fine variation in graylevel distribution (i.e., micro-edges).
• On the other hand, we would like to employ large-sized masks in order to detect coarse variation in graylevel distribution (i.e., macro-edges) and filter-out noise and other irregularities.
• We therefore need to find a mask size, which is a compromise between these two opposing requirements, or determine edge content by using different mask sizes.
• Most common differentiation operator is the gradient. • Other discrete approximations to the gradient (more precisely, the appropriate partial derivatives) have been proposed (Roberts, Prewitt).
• Because derivatives enhance noise, the previous operators may not give good results if the input image is very noisy.
• One way to combat the effect of noise is by applying a smoothing mask. The Sobel edge detector combines this smoothing operation along with the derivative operation give the following masks: • Since the gradient edge detection methodology depends only on the relative magnitudes within an image, scalar multiplication by ( n m f ∠∇ factors such as 1/2 or 1/8 play no essential role. The same is true for the signs of the mask entries. Therefore, masks like • However when the exact magnitude is important, the proper scalar multiplication factor should be used.
• All masks considered so far have entries that add up to zero. This is typical of any derivative mask.

The Laplacian Edge Detector
• In many applications, it is of particular interest to construct derivative operators, which are isotropic (rotation invariant). This means that rotating the image f and applying the operator gives the same result as applying the operator on f and then rotating the result.
• We want the operator to be isotropic because we want to equally sharpen edges which run in any direction.
• This can be accomplished by another detector, which is based on the Laplacian of a two-dimensional function.
• The Laplacian of a two-dimensional function ) , ( y x f is given by • The Laplacian is rotation invariant! • A discrete approximation can be obtained as: • The Laplacian operator suffers from the following drawbacks: ¦ It is a second derivative operator and is therefore extremely sensitive to noise. ¦ Laplacian produces double edges and cannot detect edge direction.
• Laplacian normally plays a secondary role in edge detection.
• It is particularly useful when the graylevel transition at the edge is not abrupt but gradual.
Laplacian of image Thresholded output Bacteria image