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APPENDIX

B

**Estimation of the Moment Axis of a Cell and the Equivalent Ellipse [29] **

The orientation of any object in an image is simply reduced to defining the slope

*m*

of the moment axis through the center of area (

*x*

*0*

*, y*

*0*

). The perpendicular distance of any point (

*x i*

*, y i*

) within the perimeter from this imagined moment axis is:

*p*

=

** Equation **

1

+

*m*

2

0

**B**

**. 1 **

The sum of the inertial contribution from each point, or the rotational moment of inertia of the areal projection of the object is therefore:

*I*

=

*p*

2

=

[(

*y i*

−

*y*

0

)

−

1

+

*m*

(

*x i m*

2

−

*x*

0

)] 2

** Equation **

**B**

**. 2 **

The moment axis is the line that would result in the least moment of inertia of the object area. Minimizing Equation 3 we get:

∂

*I*

∂

*m*

=

∂

∂

*m*

[(

*y i*

−

*y*

0

)

−

1

+

*m*

(

*x i m*

2

−

*x*

0

)] 2

=

0

** Equation **

**B**

**. 3 **

*m*

2

(

*y i*

−

*y*

0

)(

*x i*

−

*x*

0

)

−

*m*

[(

*y i*

−

*y*

0

) 2

−

(

*x i*

−

*x*

0

) 2 ]

−

(

*y i*

−

*y*

0

)(

*x i*

−

*x*

0

)

=

0

*m*

2 (

∑

*y i x i*

−

∑

*y i*

*N*

∑

*x i*

)

−

*m*

[

∑

*y i*

2

−

( )

2

*N*

]

−

[

∑

*x i*

2

−

( )

2

*N*

]

−

(

∑

*y i x i*

−

∑

*y i*

*N*

∑

*x i*

)

=

0

** Equation **

**B**

**. 4 **

The summations may be easily estimated from the image matrix and expressed in terms of the individual moments:

µ

*yx*

=

(

∑

*y i x i*

−

∑

*y i*

*N*

∑

*x i*

);

µ

*yy*

=

[

∑

*y*

2

*i*

−

( )

2

*N*

];

µ

*xx*

=

[

∑

*x*

2

*i*

−

( )

2

*N*

] .

The quadratic equation may be solved to yield the orientation from the x-axis:

*m*

= tan(

θ

)

=

(

µ

*yy*

−

µ

*xx*

)

±

(

µ

*yy*

2

µ

*yx*

−

µ

*xx*

)

2

+

4

µ

2

*yx*

** Equation **

**B**

**. 5 **

Once the orientation axis has been determined, the co-ordinates (

*x i*

*, y i*

) in the mask image of each particle (cell) may be clockwise rotated to new co-ordinates (

*X i*

*, Y i*

) by the transformation:

*X*

*Y i i*

=

− cos

θ sin

θ sin

θ cos

θ

*x y i i*

** Equation **

**B**

**. 6 **

The major axis is now parallel to the x-axis and the modified moment of rotation about the major axis is estimated to be:

*I a*

=

*yy*

=

∑

*Y*

2 −

(

*N*

)

2

µ

'

** Equation **

**B**

**. 7 **

*i*

∑

*Y i*

The moment of inertia of an equivalent elliptical slab about the major axis (major axis

*a*

, minor axis

*b*

, and thickness

*h,*

and mass

*M*

) may be equated:

*I a*

=

1

6

*M*

( 3

*b*

2

+

4

*h*

)

=

*N*

2

*b*

2

**B**

**. 8 **

assuming

*h *

= 0, and the mass to be equivalent to the area of the ellipse

*M= A = N = *

π

* ab*

.

The above treatment allows rigorous estimation of the major and minor axis of the equivalent ellipse to the particle and its orientation to the x-axis.