Image moment

In image processing, computer vision and related fields, an image moment is a certain particular weighted average (moment) of the image pixels' intensities, or a function of such moments, usually chosen to have some attractive property or interpretation.

Image moments are useful to describe objects after segmentation. Simple properties of the image which are found via image moments include area (or total intensity), its centroid, and information about its orientation.

Raw moments

For a 2D continuous function f(x,y) the moment (sometimes called "raw moment") of order (p + q) is defined as

M

p

q

=

∫

−

∞

∞

∫

−

∞

∞

x

p

y

q

f

(

x

,

y

)

d

x

d

y

{\displaystyle M_{pq}=\int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty }x^{p}y^{q}f(x,y)\,dx\,dy}

for p,q = 0,1,2,...

Adapting this to scalar (grayscale) image with pixel intensities I(x,y), raw image moments Mij are calculated by

M

i

j

=

∑

x

∑

y

x

i

y

j

I

(

x

,

y

)

{\displaystyle M_{ij}=\sum _{x}\sum _{y}x^{i}y^{j}I(x,y)\,\!}

In some cases, this may be calculated by considering the image as a probability density function, i.e., by dividing the above by

∑

x

∑

y

I

(

x

,

y

)

{\displaystyle \sum _{x}\sum _{y}I(x,y)\,\!}

A uniqueness theorem states that if f(x,y)

is piecewise continuous and has nonzero values only in a finite part of the xy

plane, moments of all orders exist, and the moment sequence (Mpq) is uniquely determined by f(x,y). Conversely, (Mpq) uniquely determines f(x,y). In practice, the image is summarized with functions of a few lower order moments.

Examples

Simple image properties derived via raw moments include:

Area (for binary images) or sum of grey level (for greytone images):

M

00

{\displaystyle M_{00}}

Centroid:

{

x

¯

,

y

¯

}

=

{

M

10

M

00

,

M

01

M

00

}

{\displaystyle \{{\bar {x}},\ {\bar {y}}\}=\left\{{\frac {M_{10}}{M_{00}}},{\frac {M_{01}}{M_{00}}}\right\}}

Central moments

Central moments are defined as

μ

p

q

=

∫

−

∞

∞

∫

−

∞

∞

(

x

−

x

¯

)

p

(

y

−

y

¯

)

q

f

(

x

,

y

)

d

x

d

y

{\displaystyle \mu _{pq}=\int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty }(x-{\bar {x}})^{p}(y-{\bar {y}})^{q}f(x,y)\,dx\,dy}

where

x

¯

=

M

10

M

00

{\displaystyle {\bar {x}}={\frac {M_{10}}{M_{00}}}}

and

y

¯

=

M

01

M

00

{\displaystyle {\bar {y}}={\frac {M_{01}}{M_{00}}}}

are the components of the centroid.

If ƒ(x, y) is a digital image, then the previous equation becomes

μ

p

q

=

∑

x

∑

y

(

x

−

x

¯

)

p

(

y

−

y

¯

)

q

f

(

x

,

y

)

{\displaystyle \mu _{pq}=\sum _{x}\sum _{y}(x-{\bar {x}})^{p}(y-{\bar {y}})^{q}f(x,y)}

The central moments of order up to 3 are:

μ

00

=

M

00

,

μ

01

=

0

,

μ

10

=

0

,

μ

11

=

M

11

−

x

¯

M

01

=

M

11

−

y

¯

M

10

,

μ

20

=

M

20

−

x

¯

M

10

,

μ

02

=

M

02

−

y

¯

M

01

,

μ

21

=

M

21

−

2

x

¯

M

11

−

y

¯

M

20

+

2

x

¯

2

M

01

,

μ

12

=

M

12

−

2

y

¯

M

11

−

x

¯

M

02

+

2

y

¯

2

M

10

,

μ

30

=

M

30

−

3

x

¯

M

20

+

2

x

¯

2

M

10

,

μ

03

=

M

03

−

3

y

¯

M

02

+

2

y

¯

2

M

01

.

{\displaystyle {\begin{aligned}\mu _{00}&=M_{00},&\mu _{01}&=0,\\\mu _{10}&=0,&\mu _{11}&=M_{11}-{\bar {x}}M_{01}=M_{11}-{\bar {y}}M_{10},\\\mu _{20}&=M_{20}-{\bar {x}}M_{10},&\mu _{02}&=M_{02}-{\bar {y}}M_{01},\\\mu _{21}&=M_{21}-2{\bar {x}}M_{11}-{\bar {y}}M_{20}+2{\bar {x}}^{2}M_{01},&\mu _{12}&=M_{12}-2{\bar {y}}M_{11}-{\bar {x}}M_{02}+2{\bar {y}}^{2}M_{10},\\\mu _{30}&=M_{30}-3{\bar {x}}M_{20}+2{\bar {x}}^{2}M_{10},&\mu _{03}&=M_{03}-3{\bar {y}}M_{02}+2{\bar {y}}^{2}M_{01}.\end{aligned}}}

It can be shown that:

μ

p

q

=

∑

m

p

∑

n

q

(

p

m

)

(

q

n

)

(

−

x

¯

)

(

p

−

m

)

(

−

y

¯

)

(

q

−

n

)

M

m

n

{\displaystyle \mu _{pq}=\sum _{m}^{p}\sum _{n}^{q}{p \choose m}{q \choose n}(-{\bar {x}})^{(p-m)}(-{\bar {y}})^{(q-n)}M_{mn}}

Central moments are translational invariant.