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.