Automated Axon Tracking of 3D Confocal Laser Scanning ...

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distribution and used to train the model. The mean and the covariance matrices for each axon arecalculated using these features. The similarity of the shape and size of an axon cross-sectionbetween two slices is dependent on the distance between them. In other words, the similaritybetween the cross-sections of an axon lying close together is more than those that are far-spaced.Thus, the feature vectors used for training are weighted to imply the similarity between theobjects in the current slice and the training feature. These weights are set to:( N −n+1)wn= e ,where n is the distance between the current slice and the slice where the features were extracted,and N is the number of slices used to train the model. As it can be seen, the influence of thefeature vectors decreases exponentially as we move away from the current slice. The weights arethen normalized as:wwˆ nn= , where W =W ∑ wNn=1n.where,The weighted mean for the feature vectors are then calculated as:matrix is then computed as:rμw=N∑rwˆ. fn=1rfnis the feature vector at n slices away from the current slice. The weighted covariancer N r r r rR wˆw= ∑ n.(fn− μw)*(fn− μw)n=1Once the model is built using Equations (4) and (5), it acts as the central force driving thennT(4)(5)region growing. The likelihood of occurrence of a region with the feature vector, f r , in terms ofthe probabilistic model can be written as:Guided region growingr r r rT −1r[(f −μ) * R *( f −μ)]11 − w w w2pfr = r e1/ 2(6)2πRwOur algorithm starts with the seed regions in the image as defined in Figure 9(b). As pixels are20
analyzed, they are marked with a label that indicates the region to which they belong. The wateris filled in the image based on a threshold that is increased in subsequent iterations. In order tomake the algorithm less computationally demanding, only the pixels lying inside a region ofinterest are considered for analysis. This is computed by binarizing the image. If Otsu’s method(Otsu, 1979), a popular algorithm for determining the threshold for binarization, were to be used,the axons with low intensity are not properly binarized.In order to correct this, we adopt fuzzy c-means clustering (Dunn, 1973) to compute thethreshold. The first step is to cluster the image pixels into three groups based on their intensity.The threshold is then calculated by averaging the highest intensity pixel of the lowest intensitygroup and the lowest intensity pixel of the medium intensity group, which can be written as:( I low+ Imid)T =(7)2where, I low is the highest intensity pixel of the lowest intensity group and I mid is the lowestintensity pixel of the medium intensity pixel group. This makes sure that the threshold is lowenough to properly binarize the fainter axon cross-sections. Figure 8 compares the binarizedimages obtained using the thresholds determined by Otsu’s method and the fuzzy c-meansclustering method. It can be easily seen, by comparing Figure 8(b) and (c), that thethresholdcomputed using Equation (7) is better in binarizing fainter axons in the image.(a)(b)(c)Figure 8 – Determining the threshold for binarization of cross-sectional slices: (a) Original Image, (b) Binarizationusing Otsu’s method, and (c) Binarization using Fuzzy c-means clustering. The scale bar in (a) corresponds to 2μm.21
Page 2 and 3: AbstractThis paper presents a new a
Page 4 and 5: the 3D stack. Several automatic alg
Page 6 and 7: Track the centerlines in the MIP im
Page 8 and 9: 2.1 Tracking the Axons in the MIP I
Page 10 and 11: algorithm, and therefore must be in
Page 12 and 13: towards the center. Similarly, all
Page 14 and 15: algorithm. The blue line in Figure
Page 16 and 17: To avoid this problem, we introduce
Page 18 and 19: (a)(b)(c)Figure 7 - Segmentation of
Page 22 and 23: After the region of interest is ide
Page 24 and 25: The number of cross-sections to be
Page 26 and 27: (e)Figure 10 - Tracking results in
Page 28 and 29: Figure 11 - Tracking results in dat
Page 30 and 31: In comparison with the manual track
Page 32 and 33: algorithm worked for five consecuti
Page 34 and 35: dataset in a direction normal to th
Page 36 and 37: ReferencesAbramoff M.D., Magelhaes,
Page 38: Imaging. 24(12), 1529- 1539.Wang M.

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