As noted above, a purely gray-scale (intensity) based model is inadequate for separating vessel and non-vessel pixels. Even after combined with thresholding as described above, local gray-scale minima are incorrectly identified as vessel pixels. Additional information is needed to perform a more complete separation. Vessel boundaries provide powerful structural information for this purpose. All pixels in a blood vessel are bounded on two sides by a pair of prominent boundaries. Applying this idea, for each gray-scale minima detected, areas on either side of the minima can be searched to find edges by looking for oppositely signed derivatives where the derivative at each pixel is computed using the following equation (for a horizontal profile):
I'(x, y) = I (x + 2,y) + 2I (x + 1, y) — 2I (x — 1, y) — I(x — 2, y). (6.2)
When the pixel is part of the background or inside a vessel (depending on the vessel width), the derivative should be very close to zero. When a pixel is on the left or right vessel boundary, the derivative magnitude is at its maximum and is negative for the left boundary and positive for the right boundary. Only derivatives with the greatest local magnitude (i.e., local minima or local maxima) are considered as boundary points.
The above edge constraints serve to filter out local gray-scale minima that are not associated with boundary edges. This constraint by itself is not sufficient to distinguish vessel and non-vessel pixels. This can be seen by noting that a pair of edges on opposite sides of the image are obviously too far apart to form a vessel that satisfies the above requirements. Therefore, it is necessary to filter the points based on the distance between the edges. If the oppositely signed derivatives are too far apart to be representative of expected vessel widths, they should not be considered matching boundary points of a vessel and the local minima should be considered background.
This type of bound on expected vessel widths is a powerful constraint. The results of using this method to filter image pixels using 50 evenly spaced vertical and 50 evenly space horizontal cross sections (50 x 50 grid) can be seen in Figure 6.10. This method improves upon the results shown in Figure 6.8. This is the method used by Shen et al. in their real-time vessel tracing work .
Even with a bound on vessel width, the relative minima with edges model is often inaccurate in identifying vessel pixels. Specifically, it is still common to incorrectly label background pixels as vessel by encountering multiple oppositely signed derivatives around a minima within a given distance in the background regions. One method to be more selective is to check if the magnitudes of the oppositely signed derivatives are roughly equal. While this is effective in eliminating some of the false positives, there is still one more step that can be applied to eliminate even more false positives. This involves using a thresholding technique similar to that described in Equation 6.1 above, but applied to the edges and based on the edge strength.
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