Estimation of the Moment Axis of a Cell and the... The orientation of any object in an image is simply... ). The perpendicular distance of

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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.

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