5.4 Characterisation of data variability
6.1.4 Analysis of between-class variances
Analysis of the behaviour of the between-class variances σB2 as a function of the thresholds confirms the above. First we note that, as illustrated in figures 6.6 and 6.7 respectively for the intensity and colour-based segmentations of image 45, at the first level of recursion when the lower threshold T1 is being determined in order to eliminate spurious, small objects from the segmentations, σB2 has a broad plateau in the lower half of the range of image object areas A.
The threshold T1 determined by the Otsu algorithm is located within this plateau but to see this as a well-defined maximum of the between-class variance it is necessary to plot σ2B as a function of the rank of the object areas A. This mapping is monotonic and does not create new extrema but distorts the ordinate so that it is clear where the maximum lies.
The behaviour of this between-class variance is similar for the other two images. Using the Otsu algorithm to determine T1thus works well and small, spurious objects may be successfully eliminated from the segmentations.
In section 6.1.2 the Otsu algorithm was then used again on remaining objects with areas A > T1to determine the second threshold T2and the means µ(1) and µ(2) of putative singlet and (mostly) doublet RBCs. In this case, the between-class variances for image 45 show mul-tiple extrema which are rather more pronounced when the colour-based segmentation is used (figure 6.9) than when the intensity-based segmentation is used (figures 6.8) . For this image, the upper thresholds T2= 3469 and 3430 (Table 6.1) determined by the Otsu algorithms corre-spond to the local maxima of the between-class variances at low values of the thresholds – best (but even then, only just) discernible in figure 6.8 (b) and figure 6.9 (b) between histogram bin ranks 70 − 71 and 77 − 78 respectively.
It is clear that in this case the behaviour of the between-class variance is more complicated than it was at the first level when T1was being determined. Inspection of the numerical values
(a)
(b)
Figure 6.6: Between-class variances of the object areas at the first level for image 45 where the lower threshold T1 is being determined using the intensity-based segmentation of the image.
σB2 is shown as a function of the threshold, top (a); and with the threshold mapped onto the rank of the image object area bins in the histograms, bottom (b).
(a)
(b)
Figure 6.7: Between-class variances of the object areas at the first level for image 45 where the lower threshold T1 is being determined using the colour-based segmentation of the image. σB2 is shown as a function of the threshold, top (a); and with the threshold mapped onto the rank of the image object area bins in the histograms, bottom (b).
(a)
(b)
Figure 6.8: Between-class variances of the object areas at the second level for the intensity-based segmentation of image 45 after removal of spurious, small objects. σB2 is shown, top (a), as a function of the threshold for T2 > T1and, bottom (b), with the threshold mapped onto the rank of the image object areas for A > T1.
(a)
(b)
Figure 6.9: Between-class variances of the object areas at the second level for image 45 after removal of spurious, small objects. The second, upper threshold T2 is being determined for objects segmented using the colour-based segmentation of the image. σ2Bis shown, top (a), as a function of the threshold for T2 > T1and, bottom (b), with the threshold mapped onto the rank of the image object areas A > T1.
reveals that in both the intensity and colour-based interpretations of image 45 at this second level of recursion σ2Bhas at least five extrema as summarised in table 6.3. In both cases these extrema are alternately maxima and minima with, in figure 6.9 (b) the final and highest maximum at the end of the range. The three intermediate extrema are weak and, indeed, there may be other pairs of very weak extrema which we ignore in regions where σB2 is flattish – hence we said
“at least five extrema” above. In both cases the Otsu algorithm produces a threshold (table 6.1) corresponding to (in fact probably owing to the sparseness of the data just below) the first maximum at the lowest value of A in table 6.3. However, it can be seen from table 6.3 that the highest maximum of the between-class variances occurs at the extrema at the largest values of the object area A given. Detailed inspection of the object area histograms reveals that, again because of the sparseness of the data, these correspond to the areas of almost the largest image objects found in the segmentations (see table 6.1 for comparison). In contrast, for image 1 where both the intensity and colour-based interpretations appear to be successful, the between-class variances at the second level are much simpler with only a single maximum in each case near the largest values of the object areas A (figure 6.10).
Intensity-based Colour-based
Table 6.3: Extrema of the between-class variances of image object areas at the second level of recursion for the intensity and colour-based interpretations of image 45.
As the tests summarised in table 6.2 suggest, the colour-based interpretation of image 4 generates a between-class covariance which has multiple extrema similar to those found for image 45. Placing the boundary between the RBC singlet and multiplet clusters near the largest maximum of the between-class variances for the colour-based interpretation of image 4 and both interpretations of image 45 leads to the results summarised in table 6.4 with the first three test parameters, t1, t2, t3 small and cell counts and number estimates consistent to within ap-proximately 1%. Admittedly the values of t2and t3for the colour-based segmentation of image 4 are larger than the largest of those in the first three columns of table 6.2 by approximately a factor of two, but it can be seen by comparison with table 6.2 that the cell counts and number
(a)
(b)
Figure 6.10: Similar results to those shown in figures 6.8 and 6.9 for image 1 when the be-haviour of the between-class variances is simpler with only a single peak when either: (a) the intensity-based segmentation of RBCs is used, or (b) the colour-based segmentation is used.
estimates are the same as those obtained from the intensity-based interpretation.
Image #4 Image #45 Image attributes used Colour Intensity Colour Segmentation algorithm Otsu 3D 1D Otsu Otsu 3D Image object characteristics T2 from max(σ2B)
T2 5005.2 4829.6 4965.8
#pixels(T1 < N < T2) 425696.0 437183.5 440113.5
#(image objects|T1 < N < T2) 114 131 133
#pixels(N > T2) 37731 18965.5 13245.0
#(image objects|N > T2) 6 3 2
# inconsistent object labels 1 0 0
tL 0.008 0.000 0.000
# cells inferred from #pixels
#pixels(N > T1)/µ(1) 123.1 136.7 137.0
tN 0.007 0.002 0.000
Table 6.4: Interpretation of images 4 and 45 at the second level when the between-class vari-ances have multiple extrema and the thresholds T2 are set to correspond to the largest maxima of σ2B.