Image moment

Simple image properties derived via raw moments include:

• Area (for binary images) or sum of grey level (for greytone images):${\displaystyle M_{00}}$
• Centroid: ${\displaystyle \{{\bar {x}},\ {\bar {y}}\}=\left\{{\frac {M_{10}}{M_{00}}},{\frac {M_{01}}{M_{00}}}\right\}}$

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

${\displaystyle \mu '_{20}=\mu _{20}/\mu _{00}=M_{20}/M_{00}-{\bar {x}}^{2}}$

${\displaystyle \mu '_{02}=\mu _{02}/\mu _{00}=M_{02}/M_{00}-{\bar {y}}^{2}}$

${\displaystyle \mu '_{11}=\mu _{11}/\mu _{00}=M_{11}/M_{00}-{\bar {x}}{\bar {y}}}$

The covariance matrix of the image ${\displaystyle I(x,y)}$ is now

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

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 towards the axis closest to this eigenvector. It can be shown that this angle Θ is given by the following formula:

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

The above formula holds as long as:

${\displaystyle \mu '_{20}-\mu '_{02}\neq 0}$

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

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

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

${\displaystyle {\sqrt {1-{\frac {\lambda _{2}}{\lambda _{1}}}}}.}$

The central moments μ_{i j} of any order are, by construction, invariant with respect to translations.

Invariants η_{i j} with respect to both translation and scale can be constructed from central moments by dividing through a properly scaled zero-th central moment:

${\displaystyle \eta _{ij}={\frac {\mu _{ij}}{\mu _{00}^{\left(1+{\frac {i+j}{2}}\right)}}}\,\!}$

where i + j ≥ 2. Note that translational invariance directly follows by only using central moments.

As shown in the work of Hu,[1][2] invariants with respect to translation, scale, and rotation can be constructed:

${\displaystyle I_{1}=\eta _{20}+\eta _{02}}$

${\displaystyle I_{2}=(\eta _{20}-\eta _{02})^{2}+4\eta _{11}^{2}}$

${\displaystyle I_{3}=(\eta _{30}-3\eta _{12})^{2}+(3\eta _{21}-\eta _{03})^{2}}$

${\displaystyle I_{4}=(\eta _{30}+\eta _{12})^{2}+(\eta _{21}+\eta _{03})^{2}}$

${\displaystyle I_{5}=(\eta _{30}-3\eta _{12})(\eta _{30}+\eta _{12})[(\eta _{30}+\eta _{12})^{2}-3(\eta _{21}+\eta _{03})^{2}]+(3\eta _{21}-\eta _{03})(\eta _{21}+\eta _{03})[3(\eta _{30}+\eta _{12})^{2}-(\eta _{21}+\eta _{03})^{2}]}$

${\displaystyle I_{6}=(\eta _{20}-\eta _{02})[(\eta _{30}+\eta _{12})^{2}-(\eta _{21}+\eta _{03})^{2}]+4\eta _{11}(\eta _{30}+\eta _{12})(\eta _{21}+\eta _{03})}$

${\displaystyle I_{7}=(3\eta _{21}-\eta _{03})(\eta _{30}+\eta _{12})[(\eta _{30}+\eta _{12})^{2}-3(\eta _{21}+\eta _{03})^{2}]-(\eta _{30}-3\eta _{12})(\eta _{21}+\eta _{03})[3(\eta _{30}+\eta _{12})^{2}-(\eta _{21}+\eta _{03})^{2}].}$

These are well-known as Hu moment invariants.

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 first six, I₁ ... I₆, are reflection symmetric, i.e. they are unchanged if the image is changed to a mirror image. The last one, I₇, is reflection antisymmetric (changes sign under reflection), which enables it to distinguish mirror images of otherwise identical images.

A general theory on deriving complete and independent sets of rotation moment invariants was proposed by J. Flusser.[3] He showed that the traditional set of Hu moment invariants is neither independent nor complete. I₃ is not very useful as it is dependent on the others (how?). In the original Hu's set there is a missing third order independent moment invariant:

${\displaystyle I_{8}=\eta _{11}[(\eta _{30}+\eta _{12})^{2}-(\eta _{03}+\eta _{21})^{2}]-(\eta _{20}-\eta _{02})(\eta _{30}+\eta _{12})(\eta _{03}+\eta _{21})}$

Like I₇, I₈ is also reflection antisymmetric.

Later, J. Flusser and T. Suk[4] specialized the theory for N-rotationally symmetric shapes case.