World Library  
Flag as Inappropriate
Email this Article

Image moment

Article Id: WHEBN0002132859
Reproduction Date:

Title: Image moment  
Author: World Heritage Encyclopedia
Language: English
Subject: Moment (mathematics)
Publisher: World Heritage Encyclopedia

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_{pq}=\int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} x^py^qf(x,y) \,dx\, dy

for p,q = 0,1,2,... Adapting this to scalar (greyscale) image with pixel intensities I(x,y), raw image moments Mij are calculated by

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

\sum_x \sum_y I(x,y) \,\!

A uniqueness theorem (Hu [1962]) 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.


Simple image properties derived via raw moments include:

  • Area (for binary images) or sum of grey level (for greytone images): M00
  • Centroid: { x, y } = {M10/M00, M01/M00 }

Central moments

Central moments are defined as

\mu_{pq} = \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} (x - \bar{x})^p(y - \bar{y})^q f(x,y) \, dx \, dy

where \bar{x}=\frac{M_{10}}{M_{00}} and \bar{y}=\frac{M_{01}}{M_{00}} are the components of the centroid.

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

\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:

\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}.

It can be shown that:

\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.


Information about image orientation can be derived by first using the second order central moments to construct a covariance matrix.

\mu'_{20} = \mu_{20} / \mu_{00} = M_{20}/M_{00} - \bar{x}^2
\mu'_{02} = \mu_{02} / \mu_{00} = M_{02}/M_{00} - \bar{y}^2
\mu'_{11} = \mu_{11} / \mu_{00} = M_{11}/M_{00} - \bar{x}\bar{y}

The covariance matrix of the image I(x,y) is now

\operatorname{cov}[I(x,y)] = \begin{bmatrix} \mu'_{20} & \mu'_{11} \\ \mu'_{11} & \mu'_{02} \end{bmatrix}.

The eigenvectors of this matrix correspond to the major and minor axes of the image intensity, so the orientation can thus be extracted from the angle of the eigenvector associated with the largest eigenvalue. It can be shown that this angle Θ is given by the following formula:

\Theta = \frac{1}{2} \arctan \left( \frac{2\mu'_{11}}{\mu'_{20} - \mu'_{02}} \right)

The above formula holds as long as:

\mu'_{20} - \mu'_{02} \ne 0

The eigenvalues of the covariance matrix can easily be shown to be

\lambda_i = \frac{\mu'_{20} + \mu'_{02}}{2} \pm \frac{\sqrt{4{\mu'}_{11}^2 + ({\mu'}_{20}-{\mu'}_{02})^2 }}{2},

and are proportional to the squared length of the eigenvector axes. The relative difference in magnitude of the eigenvalues are thus an indication of the eccentricity of the image, or how elongated it is. The eccentricity is

\sqrt{1 - \frac{\lambda_2}{\lambda_1}}.

Scale invariant moments

Moments ηi j where i + j ≥ 2 can be constructed to be invariant to both translation and changes in scale by dividing the corresponding central moment by the properly scaled (00)th moment, using the following formula.

\eta_{ij} = \frac{\mu_{ij}} {\mu_{00}^{\left(1 + \frac{i+j}{2}\right)}}\,\!

Rotation invariant moments

It is possible to calculate moments which are invariant under translation, changes in scale, and also rotation. Most frequently used are the Hu set of invariant moments:[1][2]

I_1 = \eta_{20} + \eta_{02}

I_2 = (\eta_{20} - \eta_{02})^2 + 4\eta_{11}^2

I_3 = (\eta_{30} - 3\eta_{12})^2 + (3\eta_{21} - \eta_{03})^2

I_4 = (\eta_{30} + \eta_{12})^2 + (\eta_{21} + \eta_{03})^2

I_5 = (\eta_{30} - 3\eta_{12}) (\eta_{30} + \eta_{12})[ (\eta_{30} + \eta_{12})^2 - 3 (\eta_{21} + \eta_{03})^2] + (3 \eta_{21} - \eta_{03}) (\eta_{21} + \eta_{03})[ 3(\eta_{30} + \eta_{12})^2 - (\eta_{21} + \eta_{03})^2]

I_6 = (\eta_{20} - \eta_{02})[(\eta_{30} + \eta_{12})^2 - (\eta_{21} + \eta_{03})^2] + 4\eta_{11}(\eta_{30} + \eta_{12})(\eta_{21} + \eta_{03})

I_7 = (3 \eta_{21} - \eta_{03})(\eta_{30} + \eta_{12})[(\eta_{30} + \eta_{12})^2 - 3(\eta_{21} + \eta_{03})^2] - (\eta_{30} - 3\eta_{12})(\eta_{21} + \eta_{03})[3(\eta_{30} + \eta_{12})^2 - (\eta_{21} + \eta_{03})^2].

The first one, I1, is analogous to the moment of inertia around the image's centroid, where the pixels' intensities are analogous to physical density. The last one, I7, is skew invariant, which enables it to distinguish mirror images of otherwise identical images.

A general theory on deriving complete and independent sets of rotation invariant moments was proposed by J. Flusser[3] and T. Suk.[4] They showed that the traditional Hu's invariant set is not independent nor complete. I3 is not very useful as it is dependent on the others. In the original Hu's set there is a missing third order independent moment invariant:

I_8 = \eta_{11}[ ( \eta_{30} + \eta_{12})^2 - (\eta_{03} + \eta_{21})^2 ] - (\eta_{20}-\eta_{02}) (\eta_{30}+\eta_{12}) (\eta_{03}+\eta_{21})

External links

  • Analysis of Binary Images, University of Edinburgh
  • Statistical Moments, University of Edinburgh
  • Variant moments, Machine Perception and Computer Vision page (Matlab and Python source code)
  • Hu Moments introductory video on YouTube


  1. ^ M. K. Hu, "Visual Pattern Recognition by Moment Invariants", IRE Trans. Info. Theory, vol. IT-8, pp.179–187, 1962
  2. ^ Hu Moments' OpenCV method
  3. ^ J. Flusser: "On the Independence of Rotation Moment Invariants", Pattern Recognition, vol. 33, pp. 1405–1410, 2000.
  4. ^ J. Flusser and T. Suk, "Rotation Moment Invariants for Recognition of Symmetric Objects", IEEE Trans. Image Proc., vol. 15, pp. 3784–3790, 2006.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Hawaii eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.