The plots in figure 6.8, for the 2D R oberts’ cross on both the half-spaces and ellip
soids, are similar to the gradient responses of figure 6.7. The dynamic range of Q values is slightly less for the 2D Roberts’ cross and the performance in the middle signal-to-noise ratio is slightly more consistent for the 2D Roberts’ cross (i.e. the slope o f the plots is more linear). This also holds for the 3D plots, although the Q response o f the 3D cross appears superior to those o f the 3D gradient for the half-spaces but not for the ellipsoids.
The difference plots of figure 6 . 8 indicate that the 3D operators produce better Q
values than the 2D operators, with the degree o f improvement being small when there is little change over Z and large when there is a significant change over Z. This is similar to the gradient results given above.
The dynamic range of the Roberts’ cross is poorer than the gradient and the results for ellipsoids are fairly disappointing: near-perfect £ 2 values do not occur.
6.2.3 Prewitt
The resultant Q values for Prew itt’s masks on the half-spaces and the ellipsoids (figure 6.9) indicate that the Prewitt operator is more robust in the presense o f noise than the 2x2x2 masks considered previously. In particular the slope o f the merit plot is better for smaller signal-to-noise ratios than those o f the gradient and Roberts’ cross.
The differences between 2D and 3D Prewitt operator merit performances lie for the m ost part in the high noise portion o f the spectrum and tend to increase (with better performance given by the 3D operator) for larger changes in object correlation across slices (changes over Z). For both the half-spaces and ellipsoids the amount o f differ ence is appreciable. The extra ‘dynamic range’ provided by the 3D operator gives the plot a desirable block-like shape (as opposed to the wedge shape o f the gradient and R oberts’ cross), which indicates consistent results for a broad range o f both additive noise and change over Z.
robSd
r o l J ? U
mu
r o l> 3 d
r o b 3 d
Figure 6.8 Resultant merit values for 2D and 3D Roberts’ cross. Top row shows merit values for 2D Roberts’ cross, middle row for 3D Roberts’ cross and bottom row the differences in merit between 3D and 2D.
p**rui?d
GJ
i; • •<>•••
p r e u J d - p r r u c 'd
p r e u 3 d - p r t u f t I
Figure 6.9 Resultant merit values for 2D and 3D Prewitt’s masks. Top row shows merit values for 2D Prewitt’s masks, middle row for 3D Prewitt’s masks and bottom row the differences in merit between 3D and 2D.
6.2.4 Sobel
The results for the Sobel are nearly identical to those o f Prew itt’s masks although the Sobel performed slightly better than the Prewitt. This is indicated in the difference plots in figure 6.10 by the slightly higher differences in m erit than for Prewitt. Every thing that was said about the Prewitt masks holds true for the Sobel masks and thus discussion is not repeated.
6.2.5 L aplacian
In the previous chapter it was mentioned that the Laplacian performs very poorly in the presense o f noise. Since it is typically applied after a smoothing function, the £2 values have been computed only for scenes containing no noise, because this chapter is concerned with surface detection differences between 2D and 3D operators and not with differences between 2D and 3D blurring, which were discussed in chapter 4. The resultant Q values are presented in the 2D plots o f figure 6.11 where the better approxi mations of the Laplacian given in section 5.6 have been used. In all cases the zero- crossing extraction algorithm zc6 (which produces 26-connected surfaces) has been
used. This is appropriate for the 3D Laplacians, given properties I and II of chapter 5, since the 6-neighbour Laplacian will tend to produce a 26-connected surface and both
the 18- and 26-neighbour Laplacians will tend to produce a 6-connected surface (which
will subsequently be thinned to 26-connectivity by zc6). For the sake of consistency in
comparison, zc6 has also been applied to the resultant output o f both the 4- and 8-
neighbour Laplacians rather than using zc4.
The resultant £2 values indicate that the 3D Laplacians performed the same as or better than the 2D Laplacians. The larger neighbourhood 3D Laplacians also tended to perform better than smaller neighbourhood Laplacians. As differences in surface correlation over Z increased, the larger neighbourhood operators performed significantly better. Interestingly enough the inverse o f this was true for the 2D Lapla cians. Overall the performance of the Laplacian on the ellipsoid data set are not very impressive, although the results are in keeping with the general trend of this chapter (i.e. 3D operators perform better on 3D data than do 2D operators).
6.2.6 R esults
The ‘average’ merit values at the S2N marks in the plots for the gradient indicate that they are nearly the same at marks 0 and 1, and for R oberts’ cross at marks 0, 1, and 2, and lastly for both Prewitt and Sobel at marks 0, 1 ,2 , and 3. From this it can be concluded that the Sobel and Prewitt perform better with additive noise than Roberts’ cross, and Roberts’ cross better than the gradient. In a similar fashion, for both
s o t ) 3 d - * o W t l
Figure 6.10 Resultant merit values for 2D and 3D Sobel’s masks. Top row shows merit values for 2D Sobels’s masks, middle row for 3D Sobels’s masks and bottom row the differences in merit between 3D and 2D.
Merit 1.0 0.80 0.60 Merit 0.900 Lap3d6 ~ Lap2U4 Lap3d26 _ _ & Lap2d4 -T Lap2U8 L ap3dl8 — _________ Lap2d8 0.800 0.40 0.700 0 2 3 4 Ellipsoids
additive noise and change over Z (both sharp and gradual), it appears that the 3D operators are more robust and accurate than their 2D counterparts.
Overall, o f the first derivative operators, the 3D Sobel (and Prewitt) masks seem to be the most resistant to noise. They perform significantly better when there are significant changes over Z and marginally better when there is little or no change over Z. The Sobel operator shows the least variability for scenes containing small amounts of noise and little change over Z. Its greatest advantage over its 2D counterpart how ever seems to be when there are significant amounts o f change over Z and/or significant amounts o f noise.
6.3 Summary
A survey o f various parameters over which an analysis o f edge and surface detec tion operators has been carried out. Pratt’s figure o f merit was chosen as a reasonable starting point for such an analysis. Both the 2D and 3D operators were compared, based on their performance using Pratt’s merit on two different data sets. The data sets were chosen to isolate changes over Z (the half-spaces) and exhibit "typical" curved blob like objects (the ellipsoids).
For both robustness to noise and changes over Z it has been shown that the 3D operators perform better than their 2D counterparts. Furthermore it has been shown that the 3D Sobel (and 3D Prewitt) perform better under these conditions than the smaller neighbourhood first derivative operators (gradient and Roberts’ cross).
Results confirming the properties o f the Laplacian given in chapter 5 were also given.