8.3 Applications
8.3.3 Detection of Circular Objects
As discussed in Section 8.2.2, the convolution with the 2D mexican hat yields strong responses when it is applied to circularly shaped objects. Therefore the problem of vesicle detection in the Lamina of Drosophila melanogaster described in Section 7.4.4 (Figure 7.9.a) is reconsidered here. First a method, basing on the multiscale Laplace transform will be presented. However, the final detection relies here, similarly to the procedure from Section 7.4.4 on a rather tedious threshold selection. Therefore, the two methods are used for reciprocal validation before thresholding, which requires less parameters, is easier to apply and is more reliable (see also Diagram 8.2).
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a)
b)
Figure 8.3: Points of Minimal Negative Third Scale Curvature for the Neuron Dataset. a)Points of minimal curvature (shown in green) computed on the original gray value image of the honey bee neuron (shown in Figure 6.4) and overlaid with it. Strong negative responses are found also in regions of high gray value variations (red arrows) inside the branches (i.e. at varicosities). At branch points results are therefore difficult to interpret (circles) and some sharp bend-ing points are not detected (rectangles), due to their relatively weaker responses;
b) Points of minimal curvature computed on the binary segmented neuron of Figure 7.4 overlaid with it. Green dots indicate voxels for which the curvature is smaller than 0.2 times its minimal value. Two or more minimal negative curvature regions are detected at branching points (circles) and only one negative curvature region is detected at high bending points (rectangles). The curvature values are computed in both images at the third wavelet scale of the underlying analyzed data. Both images are additive superpositions of (x, y)-slices forz ∈[91,110].
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0 10 20 30 40 50 60 70
150 160 170 180 190 200 210 220 230
a) b)
Figure 8.4: Zoom-In on the Lamina of Drosophila melanogaster: a) of the boxed region from Figure 7.9.a;b) the gray values along the red line drawn in a);
Circular Object Detection, basing on the Multiscale Laplace Transform Figure 8.4.a shows again the considered zoom in of the Lamina of Drosophila melanogaster. The red line indicates the cut through the middle of a vesicle along which the gray values are shown (Figure 8.4.b). The gray values have the shape of a mexican hat with a higher brim than its central bump. This creates problems for the direct detection of these objects by a simple Laplace transform (see Step 2 below), such that a more elaborated algorithm (left branch of Diagram 8.2) has to be applied for this purpose, as follows:
Algorithm 8.2 (Vesicle Detection)
1. Compute the Laplace transform on several scales2. Figure 8.5 exemplifies the Laplace transform on scales 1 to 4 on the vesicle indicated by the red line in Figure 8.4.a. Negative values correspond to bright circular areas, positive values to dark circular areas, i.e. the Laplace transformed vesicles posess negative centers and positive surround.
2. Select a scale for which the size of the mexican hat function matches the size of the analyzed shapes (for the presented dataset it is scale 4). The negative minima of the centers are of interest. Due to the low contrast between the vesicles’ centers and their surround (cf. Figure 8.4.b), the operator’s response at the vesicle center is not an extremum (Figure 8.6.c). Therefore further processing is needed as described below.
2Here the scale-space approach (i.e. low-pass filter with a Gaussian function) was applied, because the variance of the Gaussian function - which should match as well as possible with the vesicle size - can be chosen more freely than for the dyadic WT. Thus the Gaussian function with variance 4 was better suited for the current analysis than any of scales 2 or 3 of the wavelet transform (see also Appendix B.3 for a comparison of the Spline and Gauss filters).
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Figure 8.5: The Laplace Transform on 3 Scales of a Vesicle: Top: the original zoom-in and its Gaussian low-pass filtered versions at scales of σ = 1,2, and 4. Bottom: the corresponding Laplace transforms. Note the very noisy Laplace transform on the original.
3. Flatten the image into negative (at -1) and positive (at +1) values to elimi-nate background fluctuations and brightness differences between centers and surround (Figures 8.6.d and 8.7.a).
4. Convolve the flattened image with a ring mask (shown in Figure 8.7.c) cor-responding to the mean shape of the vesicles (i.e. having a center with value -1 and a surround with value 1 and inner and outer radiuses matching those of the vesicles) to highlight the structures which have a negative center and a positive surround (Figure 8.7.b).
5. Select a threshold value to keep only the maxima corresponding to vesicle centers (Figure 8.8.a).
6. Perform a Seed-Fill (as described in Section 7.3) on the thresholded and bi-narized maxima, to merge multiple peaks in single vesicles, which can appear due to the sometimes elongated shapes of the vesicles. The base image for the Seed-Fill is the flattened Laplace transformed image (shown in Figure 8.7.a).
Thereafter the center of each segmented region is computed (Figure 8.8.b).
The convolution result from Figure 8.7.b (the ring mask had inner radius of 5 and outer radius of 10 pixels) was thresholded in Figure 8.8.a at 54% of the maximum positive value (the threshold was chosen visually and corresponds to the 0.0195 positive percentile of the normalized distribution - Figure 8.7.d). The Seed-Fill computed on the binarized convolution maxima and with the flattened image
8.3 Applications 107
a) b) c) d)
Figure 8.6: The Fourth Scale of the Laplace Transform: a) all values; b) only positive values; c) reversed negative values - the absolute values inside the vesicle centers are lower than those of the surround; d)binarized positive (at +1) and negative (at -1) values;
as base (Ni = 10, Nr = 50, Vs = 7, cf. Section 7.3) enlarges the detected zones.
Less false positives (12, representing 19% of the total nr. of detected regions) arise than in the Gradient Tracing result (29), but also less vesicles are detected correctly (51 out of 61 manually labeled - representing 83.6%, as compared to 56 detected by the Gradient Tracing).
The same parameters used on the whole image result in 132 detected regions from which only 100 (i.e. 61% of the 164 manually labeled vesicles) are correctly detected vesicles and 32 (i.e. 24% of the total number of detected regions) are false positives. This proves that the threshold which was well suited for the small patch is too high for the whole image (first two rows in Table 8.1). A lower threshold at (41% of the maximum convolution value) combined with the same Seed-Fill parameters gives 140 correctly detected vesicles (i.e. 85%), 56 false positives (i.e.
28.6%) and 24 not detected vesicles (i.e. 15%) (third row in Table 8.1).
To reduce the problem of threshold selection and parameter choice for the Seed-Fill procedure a cross-validation of the responses given by the convolution operation (Step 4) and the Gradient Tracing (Section 7.4.4) is considered in the next section.
Cross-Validation of Circular Objects Detected by Gradient Tracing and by the Multiscale Laplace Transform.
It was shown in this and in the previous chapter, how the same features can be detected using the Gradient Tracing (Section 7.4.4) and using the multiscale Laplace transform. These are two independent methods and therefore their output can be cross-validated as follows:
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a)
b)
c) d)
Figure 8.7: Convolution of the Flattened Image with a Ring Mask: a) The flattened Laplace transform at scale 4 (Step 3 in Section 8.3.3);b)The result of the convolution of the image in a) with the mask shown in c); c) The zoom-in of the convolution mask, with zoom-inner radius 5 and outer radius 10 pixels; d) The histogram of the values in image b) (the red line marks the visually chosen threshold). White stars in a) and b) are the manual vesicle labels.
8.3 Applications 109
a)
b)
Figure 8.8: The Detected Vesicles by the Laplace Transform: The original gray value image with manually labeled vesicles (white stars) is overlaid with:
a) the thresholded maxima (at 54%) of the image shown in Figure 8.7.b; b) the enlarged regions from a) (green spots) by the Seed-Fill post-processing (Ni = 10, Nr = 50, Vs = 7). The correctly detected vesicles are marked by yellow balls (51 vesicles out of 61, i.e. 83.6%). Isolated green spots denote false positives (these are 12, i.e. 19% of 63 detected vesicles), whereas isolated white stars denote not detected vesicles (the 6 vesicles located on the border are lost due to the implicit border artifacts of the transform).
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Algorithm 8.3 (Vesicle Detection by Cross-Validation)
I. Compute the dot product (shown in Figure 8.9.a) between the Gradient Tracing result (shown in Figure 7.10.b) and the convolution of the ring mask with the flattened Laplace transform of the image3 (Step 4 in the previous paragraph, Figure 8.7.b)
II. Threshold the dot product result to eliminate spurious points located outside the vesicles (Figure 8.9.b).
III. Select connected regions of a certain minimal size. Their centers represent the detected vesicles of the Lamina dataset (Figure 8.9.b).
The dot product of the two transforms (computed on the zoom-in) and its thresh-olded and binarized version are shown overlaid with the labels of the original image in Figure 8.9. A threshold of 6% of the positive maximum value is chosen for the selection of significant maxima. In general, this threshold needs only to away the spurious isolated rays between spots, making thus the choice significantly easier and less critical than for the threshold needed for the simple Laplace method.
The cross-validation procedure has a comparable result with the simple Laplace method, since it detects 50 vesicles and has 10 false positive responses.
Instead, the same threshold application detects on the whole image 190 regions, from which 145 (i.e. 88%) are correctly detected vesicles, 45 (i.e. 23% of the total number of detected regions) are false positives and 19 (12%) vesicles are not detected (row 4 of Table 8.1).
Therefore, the method has only slightly less correctly detected vesicles and significantly less false positives than the Gradient Tracing, and a slightly higher number of correctly detected vesicles along with a lower number of false positives than the Laplace method (see Table 8.1 for a summary of vesicle detection results presented here and in Section 7.4.4). This result makes it a good compromise between the two methods.
The same procedure is applied with the same relative parameters on another lamina scan (figure not shown). Here only 136 vesicles are manually labeled. 190 total vesicles are detected by the cross-validation method, from which 123 (90% of the manually labeled 136) vesicles are correctly detected, 67 (35% of the total nr., 190, of detected regions) are false positives, and 13 (10%) vesicles are not found.
This result shows that the method is stable in the relative amount of correctly detected vesicles4, and that the amount of false positives has a mean of 30%.
3Due to the performed low-pass filtering, the Laplace transform introduces a slight shift into the data, which has to be corrected before computing the dot product.
4Testing on more datasets is necessary for a more reliable quantification of the algorithm’s performance. However, the current presentation has the purpose to show that the multiscale Laplacian operator - similarly to the Gaussian curvature operator - cannot provide by itself satisfactory results, but needs a validation with “gradient traced” data.
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a)
b)
Figure 8.9: Cross-Validation of Gradient Tracing and Laplace Transform:
a) the dot product between the segmentation by Gradient Tracing (Fig. 7.10.b) and the convolution result from Fig. 8.7.b; colors: green is zero, blue is negative, red is positive; b) the positive regions in a) thresholded at 6% of the maximum image value and binarized. Only regions containing at least 30 pixels (green) are considered. They detect correctly 50 (i.e. 82% of the total number of 61 manually labeled) vesicles (yellow balls). The isolated green spots are false positives (10 of 60, i.e. 16%), whereas the isolated white stars are not detected vesicles (11 of 61, i.e. 18%).
112 Feature Detection
Detected regions
Correctly detected (% of 164 manually labeled vesicles)
False positives5 (% of:
detected regions | labeled vesicles )
Not detected (% of 164 manually labeled vesicles) 1. Gradient Tracing
threshold: 40% of max 321 149 (91) 172 (53 | 105)
15 (9) 2. Laplace Transform
threshold: 54% of max 132 100 (61) 32
(24 |20)
64 (39) 2. Laplace Transform
threshold: 41% of max 196 140 (85) 56
(29 |34)
24 (15) 3. Cross-Validation
of un-thresholded 1 & 2 190 145 (88) 45 (23 |27)
19 (12) 3. Cross-Validation
of second test set
190 (of 136 man.
labeled)
123 (90% of 136)
67 (35% of 190 49% of 136)
13 (10% of 136)
Table 8.1: Summary of Vesicle Detection Results:
for the Gradient Tracing method (row 1), the Laplace method with two different thresholds (rows 2 and 3) and the cross-validation of the former two unthresholded partial results (row 3) computed on the large ROI in the Lamina of Drosophila melanogaster (Figure 7.9.b); row 4 gives the cross-validation result on a second test set (figure not shown).
The advantage of this method compared to the Gradient Tracing and the simple Laplace method is that the product of the two results enhances regions, where both methods have overlapping high responses. Therefore the vesicle centers are represented by relatively large spots which can be selected directly by setting a minimal allowed spot size. In contrast, for the simple methods, the detected spots have first to be enlarged by a Seed-Fill step, for which three additional parameters are required. Since these methods demand more expertise from the user, they are more exposed to faults.