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Table 3 Formal definition of computed features

From: Visual4DTracker: a tool to interact with 3D + t image stacks

Preliminary symbols definitions
\(\mathbf{F}\) is the set including the index of frames which contain one marker of the granule under study
\(\mathbf{x}\), \(\mathbf{y}\) and \(\mathbf{z}\) are vectors containing the coordinates of the marker of the granule under study at each frame \(f \in \mathbf{F}\)
\({\mathbf{p}} _f = ({\mathbf{x}} _{f}, {\mathbf{y}} _{f}, {\mathbf{z}} _{f})\), where \(f \in [0;|{\mathbf{F}} |-1]\)
Set of displacements: \({\mathbf{D}} = \{ ||{\mathbf{p}} _{f + 1} - {\mathbf{p}} _{f}||, f \in [0;|{\mathbf{F}} | - 1]\}\)
\(pdf(\cdot )\): probability density function of \(\cdot\)
Set of angles: \({\mathbf{R}} = \left\{ \arccos \left( ({\mathbf{p}} _{f + 1} - {\mathbf{p}} _{f})\cdot ({\mathbf{p}} _{f + 2} - {\mathbf{p}} _{f + 1})\over {||{\mathbf{p}} _{f + 1} - {\mathbf{p}} _{f}||\cdot ||{\mathbf{p}} _{f + 2} - {\mathbf{p}} _{f + 1}||} \right) , f \in [0;|{\mathbf{F}} | - 2] \right\}\)
Set of velocity: \({\mathbf{V}} = \left\{ {\frac{{{\mathbf{P}}_{f} }}{{{\mathbf{F}}(f + {\mathbf{1}}) - {\mathbf{F}}(f)}},f \in [0;|{\mathbf{F}}| - 1]} \right\}\)
Set of acceleration: \({\mathbf{A}} = \left\{ {\frac{{{\mathbf{V}}_{{{\mathbf{F}}_{f} }} }}{{{\mathbf{F}}(f + {\mathbf{1}}) - {\mathbf{F}}(f)}},f \in [0;|{\mathbf{F}}| - 2]} \right\}\)
Features  
Route-based  Hourly law-based
Total displacement \((\Delta ) = \sum _{i = 1}^{|{\mathbf{D}} |} {\mathbf{D}} _i\) Average velocity \(\left( \bar{{\mathbf{V }}} \right) = {{1}\over {|{\mathbf{V}} |}} \sum _{i = 1}^{|{\mathbf{V}} |} {\mathbf{V}} _i\)
Average angles \(\left( \bar{{\mathbf{R }}} \right) = {{1}\over {|{\mathbf{R}} |}} \sum _{i = 1}^{|{\mathbf{R}} |} {\mathbf{R}} _i\) Velocity standard deviation \(= \sqrt{{{\sum _{i = 1}^{|{\mathbf{V}} |} ({\mathbf{V}} _i - \bar{{\mathbf{V} }})^2}\over {|{\mathbf{V}} |} - 1}}\)
Angles standard deviation \(= \sqrt{{{\sum _{i = 1}^{|{\mathbf{R}} |} ({\mathbf{R}} _i - \bar{{\mathbf{R} }})^2} \over {|{\mathbf{R}} |} - 1}}\) Velocity skewness \(= {{{{1}\over {|{\mathbf{V}} |}}\sum _{i = 1}^{|{\mathbf{V}} |}({\mathbf{V}} _i - \bar{{\mathbf{V }}})^3}\over {\left( \sqrt{{{1}\over {|{\mathbf{V}} |}} \sum _{v = 1}^{|{\mathbf{V}} |}({\mathbf{V}} _i - \bar{{\mathbf{V} }})^2}\right) ^3}}\)
Tortuosity \(= {{\Delta }\over {|{\mathbf{p}} _{|{\mathbf{F}} |-1} - {\mathbf{p}} _1|}}\) Velocity kurtosis \(= {{{{1}\over {|{\mathbf{V}} |}}\sum _{i = 1}^{|{\mathbf{V}} |}({\mathbf{V}} _i - \bar{{\mathbf{V} }})^4}\over {\left( {{1}\over {|{\mathbf{V}} |}}\sum _{i = 1}^{|{\mathbf{V}} |}({\mathbf{V}} _i - \bar{{\mathbf{V} }})^2\right) ^2}}\)
Energy \(= pdf({\mathbf{x}} )^2 + pdf({\mathbf{y}} )^2 + pdf({\mathbf{z}} )^2\) Average acceleration \(\left( \bar{{\mathbf{A} }} \right) = {{1}\over {|{\mathbf{A}} |}} \sum _{i = 1}^{|{\mathbf{A}} |} {\mathbf{A}} _i\)
Entropy \(= -(pdf({\mathbf{x}} )\log _2{(pdf({\mathbf{x}} ))}~+\) Acceleration standard deviation \(= \sqrt{{{\sum _{i = 1}^{|{\mathbf{A}} |} ({\mathbf{A}} _i - \bar{{\mathbf{A} }})^2}\over {|{\mathbf{A}} |} - 1}}\)
\(+~pdf({\mathbf{y}} )\log _2{(pdf({\mathbf{y}} ))}~+\) Acceleration skewness \(= {{{{1}\over {|{\mathbf{A}} |}}\sum _{i = 1}^{|{\mathbf{A}} |}({\mathbf{A}} _i - \bar{{\mathbf{A} }})^3}\over {\left( \sqrt{{{1}\over {|{\mathbf{A}} |}}\sum _{i = 1}^{|{\mathbf{A}} |}({\mathbf{A}} _i - \bar{{\mathbf{A} }})^2}\right) ^3}}\)
\(+~pdf({\mathbf{z}} )\log _2{(pdf({\mathbf{z}} ))})\) Acceleration kurtosis \(= {{{{1}\over {|{\mathbf{A}} |}}\sum _{i = 1}^{|{\mathbf{A}} |}({\mathbf{A}} _i - \bar{{\mathbf{A} }})^4}\over {\left( {{1}\over {|{\mathbf{A}} |}}\sum _{i = 1}^{|{\mathbf{A}} |}({\mathbf{A}} _i - \bar{{\mathbf{A} }})^2\right) ^2}}\)
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