The term moment stems from mathematics, where it describes quantitative measures of the shape of a set of points. In physics, the moment can refer to many different concepts, but all have in common that some physical quantity is multiplied by a length to give a moment. A moment used in image processing and related fields is some weighted average of pixel intensities. Image moments are useful to describe objects. Some simple features of images which are found via image moments include the area, the accumulated intensity, the centroid of center of gravity and other information about orientation.

Moments

of order (p + q) are defined as:

M_pq = ∑x∑yxpyqIxy

where x and y are the horizontal and vertical pixel coordinates and I_xy is the pixel intensity at location (x, y).

If pixel intensities are not important, so called binary moments can also be calculated using the following formula:

M_pq = ∑_x∑_yx^py^q

Moments are not invariant with respect to translation, scaling and rotation.

The region class in nGI has a moments property that calculates moments up to order 3 and returns them in the moments class. You can use the properties m00, m10, m01, m20, m11, m02, m30, m21, m12 and m03 to read the respective moments.

Simple image properties derived via moments are the area for binary images (m00) or the sum of grey levels for greyscale images (m00).

Normalized moments

of order (p + q) are defined as:

$${N}_{{pq}}={\frac{{M}_{{pq}}}{{M}_{{00}}}}$$

Normalized moments are invariant with respect to scaling, but are not invariant with respect to translation and rotation.

The normalized_moments class in nGI calculates normalized moments up to order 3 when given a moments instance. You can use the properties n10, n01, n20, n11, n02, n30, n21, n12 and n03 to read the respective normalized moments. There are also static methods calculate_n10, calculate_n01, calculate_n20, calculate_n11, calculate_n02, calculate_n30, calculate_n21, calculate_n12 and calculate_n03 that you can use to calculate the respective moment only.

A simple image property derived via normalized moments is the centroid:

$$\left( \begin{matrix} x \\ y \end{matrix} \right) =\left( \begin{matrix} \frac { { N }_{ 10 } }{ { N }_{ 00 } } \\ \frac { { N }_{ 01 } }{ { N }_{ 00 } } \end{matrix} \right)$$

Central moments

of order (p + q) are defined as:

$${ \mu }_{ pq }=\frac { 1 }{ { M }_{ 00 } } \sum _{ x }^{ }{ \sum _{ y }^{ }{ { (x-\overline { x } ) }^{ p }{ (y-\overline { y } ) }^{ q }{ I }_{ xy } } }$$

If pixel intensities are not important, so called binary central moments can also be calculated using the following formula:

$${ \mu }_{ pq }=\frac { 1 }{ { M }_{ 00 } } \sum _{ x }^{ }{ \sum _{ y }^{ }{ { (x-\overline { x } ) }^{ p }{ (y-\overline { y } ) }^{ q } } }$$

Central moments are invariant with respect to translation and scaling, but not with rotation.

The central_moments class in nGI calculates central moments up to order 3 when given a moments instance. You can use the properties mu20, mu11, mu02, mu30, mu21, mu12 and mu03 to read the respective central moments. There are also static methods calculate_mu20, calculate_mu11, calculate_mu02, calculate_mu30, calculate_mu21, calculate_mu12 and calculate_mu03 that you can use to calculate the respective moment only.

Simple image properties derived via central moments are major and minor axes as well as the rotation angle of the equivalent ellipse.

From the equivalent ellipses major and minor axes, the eccentricity can be calculated. The eccentricity can have values ranging from 0 to 1. It is zero for a perfectly round object and 1 for a line shaped object.

Scale invariant moments

of order (p + q) are defined as:

$${\eta}_{{pq}}={\frac{{\mu}_{{pq}}}{{\mu}_{{00}}^{\frac{p+q}{2}+1}}}$$

Scale invariant moments are invariant with respect to translation and scaling, but are not invariant with respect to rotation.

The scale_invariant_moments class in nGI calculates scale invariant moments up to order 3 when given a central_moments instance. You can use the properties eta20, eta11, eta02, eta30, eta21, eta12 and eta03 to read the respective scale invariant moments. There are also static methods calculate_eta20, calculate_eta11, calculate_eta02, calculate_eta30, calculate_eta21, calculate_eta12 and calculate_eta03 that you can use to calculate the respective moment only.

Hu’s moments

are defined as:

$$\begin{aligned} {\phi}_{{1}} & = & {{\eta}_{{20}}+{\eta}_{{02}}} \\ {\phi}_{{2}} & = & {({\eta}_{{20}}-{\eta}_{{02}})^2+4{\eta}_{{11}}^2} \\ {\phi}_{{3}} & = & {({\eta}_{{30}}-3{\eta}_{{12}})^2+(3{\eta}_{{21}}-{\eta}_{{03}})^2} \\ {\phi}_{{4}} & = & {({\eta}_{{30}}+{\eta}_{{12}})^2+({\eta}_{{21}}+{\eta}_{{03}})^2} \\ {\phi}_{{5}} & = & {({\eta}_{{30}}-3{\eta}_{{12}})({\eta}_{{30}}+{\eta}_{{12}})\left[({\eta}_{{30}}+{\eta}_{{12}})^2-3({\eta}_{{21}}+{\eta}_{{03}})^2\right]} \nonumber \\ & & {}{+(3{\eta}_{{21}}-{\eta}_{{03}})({\eta}_{{21}}+{\eta}_{{03}})\left[3({\eta}_{{30}}+{\eta}_{{12}})^2-({\eta}_{{21}}+{\eta}_{{03}})^2\right]} \\ {\phi}_{{6}} & = & {({\eta}_{{20}}-{\eta}_{{02}})\left[({\eta}_{{30}}+{\eta}_{{12}})^2-({\eta}_{{21}}+{\eta}_{{03}})^2\right]} \nonumber \\ & & {}{+4{\eta}_{{11}}({\eta}_{{30}}+{\eta}_{{12}})({\eta}_{{21}}+{\eta}_{{03}})} \\ {\phi}_{{7}} & = & {(3{\eta}_{{21}}-3{\eta}_{{03}})({\eta}_{{30}}+{\eta}_{{12}})\left[({\eta}_{{30}}+{\eta}_{{12}})^2-3({\eta}_{{21}}+{\eta}_{{03}})^2\right]} \nonumber \\ & & {}{-({\eta}_{{30}}-3{\eta}_{{12}})({\eta}_{{21}}+{\eta}_{{03}})\left[3({\eta}_{{30}}+{\eta}_{{12}})^2-({\eta}_{{21}}+{\eta}_{{03}})^2\right]}\end{aligned}$$

Hu’s moments are invariant with respect to translation, scaling and rotation.

The hu_moments class in nGI calculates Hu’s moments when given a scale_invariant_moments instance. You can use the properties phi1, phi2, phi3, phi4, phi5, phi6 and phi7 to read the respective Hu moment. There are also static methods calculate_phi1, calculate_phi2, calculate_phi3, calculate_phi4, calculate_phi5, calculate_phi6 and calculate_phi7 that you can use to calculate the respective moment only.

Flusser’s moments

are defined as:

$$\begin{aligned} {I}_{{1}} & = & {\frac{{\mu}_{{20}}{\mu}_{{02}}-{\mu}_{{11}}^2}{{\mu}_{{00}}^4}} \\ {I}_{{2}} & = & {\frac{{\mu}_{{30}}^2{\mu}_{{03}}^2-6{\mu}_{{30}}{\mu}_{{21}}{\mu}_{{12}}{\mu}_{{03}}+4{\mu}_{{30}}{\mu}_{{12}}^3+4{\mu}_{{21}}^3{\mu}_{{03}}-3{\mu}_{{21}}^2{\mu}_{{12}}^2}{{\mu}_{{00}}^10}} \\ {I}_{{3}} & = & {\frac{{\mu}_{{20}}({\mu}_{{21}}{\mu}_{{03}}-{\mu}_{{12}}^2)-{\mu}_{{11}}({\mu}_{{30}}{\mu}_{{03}}-{\mu}_{{21}}{\mu}_{{12}}))}{{\mu}_{{00}}^7}} \nonumber \\ & & {+\frac{{\mu}_{{02}}({\mu}_{{30}}{\mu}_{{12}}-{\mu}_{{21}}^2)}{{\mu}_{{00}}^7}} \\ {I}_{{4}} & = & {\frac{{\mu}_{{20}}^3{\mu}_{{03}}^2-6{\mu}_{{20}}^2{\mu}_{{11}}{\mu}_{{12}}({\mu}_{{03}}-6{\mu}_{{20}}^2{\mu}_{{02}}{\mu}_{{21}}){\mu}_{{03}}}{{\mu}_{{00}}^{11}}} \nonumber \\ & & {+\frac{9{\mu}_{{20}}^2{\mu}_{{02}}{\mu}_{{12}}^2+12{\mu}_{{20}}{\mu}_{{11}}^2{\mu}_{{21}}{\mu}_{{03}}+6{\mu}_{{20}}{\mu}_{{11}}{\mu}_{{02}}{\mu}_{{30}}{\mu}_{{03}}}{{\mu}_{{00}}^{11}}} \nonumber \\ & & {-\frac{18{\mu}_{{20}}{\mu}_{{11}}{\mu}_{{02}}{\mu}_{{21}}{\mu}_{{12}}+8{\mu}_{{11}}^3{\mu}_{{30}}{\mu}_{{03}}+6{\mu}_{{20}}{\mu}_{{02}}^2{\mu}_{{30}}{\mu}_{{12}}}{{\mu}_{{00}}^{11}}} \nonumber \\ & & {+\frac{9{\mu}_{{20}}{\mu}_{{02}}^2{\mu}_{{21}}^2+12{\mu}_{{11}}^2{\mu}_{{02}}{\mu}_{{30}}{\mu}_{{12}}-6{\mu}_{{11}}{\mu}_{{02}}^2{\mu}_{{30}}{\mu}_{{21}}}{{\mu}_{{00}}^{11}}} \nonumber \\ & & {+\frac{{\mu}_{{02}}^3{\mu}_{{30}}^2}{{\mu}_{{00}}^{11}}}\end{aligned}$$

Flusser moments are invariant with respect to affine transformations.

The flusser_moments class in nGI calculates flusser moments when given a central_moments instance. You can use the properties i1, i2, i3, and i4 to read the respective flusser moments. There are also static methods calculate_i1, calculate_i2, calculate_i3 and calculate_i4 that you can use to calculate the respective moment only.