Edited by author 10-09-2001 01:38 PM
Dave:
The two papers by Meila and Shi that I'm going to be presenting on Thursday give some insight into how and why the normalized cut method works by studying the problem as a random walk on a graph. These papers extend the study by Weiss and provide some interesting connections between spectral segmentation and concepts from the theory of Markov chains such as aggregation, conductance and mixing time.

Andrew:
The reason for _not_ using the first eigenvector is that it is always equal to 1 (unless the smallest eigenvalue = 0 is degenerate). This can be seen both in the original NCut formulation and especially through the random walk interpretation. The reason for using the other k-1 first eigenvectors is a bit more complicated so that I'd better leave it for my presentation.

Junwen:
The generalized eigenvector used for segmentation will have one component for each pixel of the image. Figures 6-9 all show particular eigenvectors plotted as the value of the particular component at the corresponding position of the pixel in the image. The cross-section is just a particular subset of components of the eigenvector obtained e.g. by slicing through the image in the x direction at a particular value of y (the value of y is not mentioned anywhere). The image is like a map where lighter shade indicates high elevation and darker shade low elevation and the cross-section is just an elevation profile along a particular line in the map.