Morphological Operators

I have taken a different approach to the pulmonary vein ostium search problem. 'Primitive' machine vision techniques such as opening and closing can be applied to certain binary images in order to decompose them into their meaningful parts and separate them from their extraneous parts. These happen to be very useful operators. It is derived from mathematical morphology which is a theoretical model for digital images built using lattice theory and topology. So it is back to the basics of machine vision - and I am a little more than glad that I am doing hands-on morphology. I take it as one of the 'must-know' concepts in machine vision.

I will be looking to separate the atrium from its pulmonary veins using these operators. Although surely the operators, on its own, will not be sufficient to do such a task. At this end, I am also looking to possibly implement knowledge-based operators. These operators will incorporate knowledge about the subdivisions (such as their radii, etc.)

Although reading about the fundamentals of mathematical morphology was not great fun, however, I found this book by Haralick to be really helpful. I also found these lecture notes from the U. of Edinburgh informatics site easy to read and understand.

This Friday I picked up more atrium data from my clinician. We are looking at the possibility of exploring a different technique to the atrium segmentation problem.

Removal of Partial Volume Effected Voxels

As I had pointed out earlier in previous posts, perhaps the one and only reason the segmentation algorithm leaks into surrounding structures especially the aorta and the pulmonary artery is because of partial volume effected voxels. We have observed that the segmentation at times will leak into the pulmonary artery through a pulmonary vein. Although, it is physically possible, atleast in theory, for the pulmonary vein to touch the pulmonary artery. However, our images are Angiographic scans where the blood was Gadolinium-enhanced and the only thing that we see after subtracting the pre-Angio from the post-Angio is the blood pool. So, we can rightfully state that the blood pools of the pulmonary artery and the pulmonary vein cannot physically touch and this is definitely caused by partial volume effected voxels. Partial volume effected voxels are voxels that image two or more tissue types, thus giving an intensity level that is a weighted average of the intensity levels of each tissue type.

So here arises a dire requirement to identify partial volume effected voxels and remove them as much as possible. I have been looking at some papers recently such as Ballester et. al., and my task for this week will mainly be literature review of partial volume effect removal techniques.

About subdivisions

A lot of the atrium segmentation computations depend on how the MRI image is subdivided into subdivisions (or basic components) and also partly on the location and spatial distribution of the local maximum and saddle points. Subdivisions are the collection of voxels each of which lead to a local maximum by following a path of increasing EDT values. Now, although intuitively these subdivisions could be thought of as one group of neighboring voxels, however, as I just found out this is not the case. The subdivisions can actually be a disjoint group of voxels as shown below in the figure. Notice that this is one single subdivision that is centered by the local maximum voxel marked in green (by the red arrow). The trail of points indicate the path that leads to the local maximum for a point in the smaller disjoint subdivision.

Here is another image of the same subdivision with the opacity changed to show the opaqued surface.

This is the main reason behind why the boundaries between certain subdivisions dont have a saddle point. For example if we look at the smaller disjoint group in the subdivision shown above in the figure, this smaller group may not have a saddle point with its neighboring subdivisions, since the other larger disjoint subdivision have satisfied the saddle point requirement by having a saddle point with a neighboring subdivision. Ideally we wouldn't have wanted subdivisions to be disjointed in this manner. As explained, this causes some subdivison boundaries to not have saddle points. This has implications in the work we are currently doing where we are trying to locate ostium centers automatically.