172	L. Domanski et al.

Fig. 8.5 Illustration of the measurements derived by quantifying linear feature detection results on neuronal cells

8.2.5 Quantifying Branching Structures

Linear features are produced as intermediate representations towards image quantification and image understanding. The conversion of a full image into a set of linear feature entails a considerable compression of the original information. In this section, we describe how to reduce the feature size further while preserving as much information as possible. To this end, the linear features are processed to generate measures of length, branching, and complexity (see Fig. 8.5). This framework is quite general and several applications are described in Sect. 8.5.

A)Feature representation

After preprocessing, the linear features are paths of width equal to only one pixel, often connected in complex ways. The sensitivity of the feature detection process typically leads to false positive detection events. These inaccuracies manifest themselves as small barbs in the skeleton, which can be pruned by a process, where small lateral branches below a chosen length are removed. The skeleton is then divided into unique segments, defined as sections of linear feature between two intersections, or branching points. This division process first requires identifying the branching points as having more than two 4-connected neighbors. Branching points are then removed from the skeleton. In doing so, the skeleton is divided into segments which remain 4-connected and each segment is given a unique label.

8 High-Throughput Detection of Linear Features: Selected Applications...

173

A graph of neighborhood relationships for segments then has to be built. We first morphologically dilate [6] uniquely labeled intersection points with a 3 × 3 structuring element, so that they overlap with the extremities of segments. Segments which overlap a common intersection point are considered neighbors. This information is initially contained in a bivariate histogram of segments versus intersection points. A linked list is then created by scanning across each row of the histogram and locating nonzero entries in the histogram indicating neighborhood relationships among segments.

When the image corresponds to a structure possessing an organizing centre from which branches are protruding (such as the mitotic organizing centre, the trunk of a tree, or the cell body of a neuron), we identify segments that are in contact with that organizing centre as “root” segments. To identify them, we first thicken the labelled centre, so that it overlaps root segments (a thickening is a dilation that preserves an object’s label [6]). Again, we use a bivariate histogram to store the overlap information. Nonzero entries correspond to root segments for a particular cell body.

B)Tree growing using the watershed algorithm

At this stage, we must associate all segments with a particular tree. A tree is a connected network extending from a single root segment. We use the watershed algorithm to derive the association. Typically, the watershed is performed on an image called the segmentation function, which highlights object boundaries. A set of unique seeds are grown on the segmentation function using a priority queue. Seeds are placed in the queue and neighboring pixels are added with priority given to those with the lowest value in the segmentation function. Pixels are repeatedly taken from the top of the queue and added to the object defined by the pixel’s neighboring seed.

We use the watershed methodology to grow all trees from their root segment. The framework for the watershed in our case is different to that which is used for 2D images: we are dealing with graph nodes instead of pixels. The nodes are the individual segments and our seeds are the root segments as found in the previous section. Root segments are initially put in the priority queue and neighboring segments are added with priority given to segments with the highest average brightness. The average brightness is calculated over the pixels that form the segment. Brightness was chosen as priority feature because it is generally found to be preserved along branches. Other criteria for the prioritization could be used such as the relative orientation of the segments. Segments are repeatedly taken from the top of the queue and associated with their neighboring neurite tree until all segments have been removed from the queue.

C)Measures on segments and trees

Various measurements can be accumulated for each branch during the tree growing process. These measurements can in turn be aggregated on a pertree or per-organizing centre basis. It is also common to report measurements on a per-image basis. There are two groups of measurements collected during the watershed process: those relating to length, width or brightness and those relating to complexity. Fig. 8.5 illustrates these measurements.