# TubeTK/RadiusEstimationFunction

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# Background

- In the original 1996 publication, we used an oriented Laplacian of a Guassian (Lpp) to estimate local radius. The scale that returned the local maximum response was chosen as the radius for the local tube.
- There were numerous problems with this approach:
- Interference from adjacent vessels: the Lpp function was evaluated out to 2.5 to 3.5 standard deviations. This meant that structures within a radius of the vessel would influence the estimations, and vessel estimates would be erroneous at points of high curvature.
- Interference from intensity variations within a vessel: The most strongly weighted intensities were along the centerline, yet in time-of-flight MRA, the intensity within a vessel may be dimmer, perhaps as dim as the background. Furthermore, vessel interior intensity would vary as a function of vessel radius, creating a complex coupling between medialness response and actual radius.
- Limited parameterizations: the Lpp function is defined by a limited number of variables which eliminates the option of shaping the kernel based on, for example, the local straightness or curvature of the vessel.

- There were numerous problems with this approach:
- The long-running MIDAS vessel extraction system used an set of consentric spheres that bounded and extended along the expected vessel wall. Inner spheres has positive kernel weights and outer spheres had negative kernel weights. The weighted sum of intensities for all spheres defined the medialness value for the corresponding scale. Care was taken to ensure balance in size and coverage between and across scales. The sequence of spheres could be aligned to the curvature of the tube, and intensities near the center of the tube were not used in the computations.
- There was two remaining problems with this approach:
- It integrated over only a small portion of the tube, and the discrete boundaries of each sphere created multiple local minima when applied to the sparse grid of an image (i.e., small changes in scale could result in drastic changes in medialness due to cross boundaries between adjacent voxels).
- The method could be slow to apply, it was applied at distant intervals along a tube to reduce total computational costs.

- There was two remaining problems with this approach:
- The final approach uses an ordered listing of adjacent intensities that allows for fast computation at multiple scales while maximizing available data to minimize the effect of image discretization.