2D moments
Idea
Let \(f(x,y)\) be a real valued function at Cartesian
location \((x,y)\). The central moments of \(f(x,y)\) are defined as
\[\mu_{pq}=\int_{a_1}^{a_2} \int_{b_1}^{b_2} (x-\bar{x})^p(y-\bar{y})^q f(x,y) dxdy\]
where \(\bar{x}\) and \(\bar{y}\) are defined as
\[\bar{x} = \frac {M_{10}} {M_{00}}\]
and
\[\bar{y} = \frac {M_{01}} {M_{00}}.\]
The 0-th order moment \(M_{00}\) of function \(f(x,y)\)
\[M_{00} = \int _{a_1}^{a_2} \int _{b_1}^{b_2} f(x,y) dxdy\]
represents the total mass of the
function \(f(x,y)\) and the two 1-st order moments
\[M_{10} = \int _{a_1}^{a_2} \int _{b_1}^{b_2} x f(x,y) dxdy\]
and
\[M_{10} = \int _{a_1}^{a_2} \int _{b_1}^{b_2} y f(x,y) dxdy\]
represent the center of mass of the image \(f(x,y)\). Hu’s Uniqueness Theorem states that if \(f(x,y)\) is piecewise continuous and has nonzero values only in the finite part
of the \((x,y)\) plane, then geometric moments of all orders exist. It can then be shown that the moment set \({\mu_{pq}}\) is
uniquely determined by \(f(x,y)\) and conversely, \(f(x,y)\) is uniquely determined by \({\mu_{pq}}\). Since an image has
finite area, a moment set can be evaluated computationally and used to uniquely describe the information contained in the image.
Raw moments
Considering image pixels \(p(x,y)\) as sampled greyscaled values of \(f(x,y)\) at discrete locations, the moments introduced above can be approximated
by summation, and raw (spatial) moments \(m_{ij}\) are defined as
\[m_{{ij}}=\sum _{x}\sum _{y}x^{i}y^{j}p(x,y)\]
Spatial moment features are calculated as:
\[\begin{split}\text{SPAT_MOMENT_00} &=m_{00} \\
\text{SPAT_MOMENT_01} &=m_{01} \\
\text{SPAT_MOMENT_02} &=m_{02} \\
\text{SPAT_MOMENT_03} &=m_{03} \\
\text{SPAT_MOMENT_10} &=m_{10} \\
\text{SPAT_MOMENT_11} &=m_{11} \\
\text{SPAT_MOMENT_12} &=m_{12} \\
\text{SPAT_MOMENT_20} &=m_{20} \\
\text{SPAT_MOMENT_21} &=m_{21} \\
\text{SPAT_MOMENT_30} &=m_{30}\end{split}\]
Central moments
A central moment \(\mu_{ij}\) is defined as
\[\mu_{{ij}}=\sum_{{x}}\sum _{{y}}(x-{\bar {x}})^{i}(y-{\bar {y}})^{j}p(x,y)\]
Central moment features are calculated as:
\[\begin{split}\text{CENTRAL_MOMENT_02} &=\mu_{02} \\
\text{CENTRAL_MOMENT_03} &=\mu_{03} \\
\text{CENTRAL_MOMENT_11} &=\mu_{11} \\
\text{CENTRAL_MOMENT_12} &=\mu_{12} \\
\text{CENTRAL_MOMENT_20} &=\mu_{20} \\
\text{CENTRAL_MOMENT_21} &=\mu_{21} \\
\text{CENTRAL_MOMENT_30} &=\mu_{20} \\\end{split}\]
Normalized raw moments
Raw (spatial) moments \(m_{ij}\) of a 2-dimensional greyscale image \(p(x,y)\) are calculated by
\[w_{{ij}} = \frac {\mu_{ij}}{\mu_{22}^ {max(i,j)} }\]
Spatial moment features are calculated as:
\[\begin{split}\text{NORM_SPAT_MOMENT_00} =w_{00} \\
\text{NORM_SPAT_MOMENT_01} =w_{01} \\
\text{NORM_SPAT_MOMENT_02} =w_{02} \\
\text{NORM_SPAT_MOMENT_03} =w_{03} \\
\text{NORM_SPAT_MOMENT_10} =w_{10} \\
\text{NORM_SPAT_MOMENT_20} =w_{20} \\
\text{NORM_SPAT_MOMENT_30} =w_{30} \\\end{split}\]
Normalized central moments
A normalized central moment \(\eta_{ij}\) is defined as
\[\eta_{{ij}}={\frac {\mu_{{ij}}}{\mu_{{00}}^{{\left(1+{\frac {i+j}{2}}\right)}}}}\,\]
where \(\mu _{{ij}}\) is central moment.
Normalized central moment features are calculated as:
\[\begin{split}\text{NORM_CENTRAL_MOMENT_02} &=\eta_{{02}} \\
\text{NORM_CENTRAL_MOMENT_03} &=\eta_{{03}} \\
\text{NORM_CENTRAL_MOMENT_11} &=\eta_{{11}} \\
\text{NORM_CENTRAL_MOMENT_12} &=\eta_{{12}} \\
\text{NORM_CENTRAL_MOMENT_20} &=\eta_{{20}} \\
\text{NORM_CENTRAL_MOMENT_21} &=\eta_{{21}} \\
\text{NORM_CENTRAL_MOMENT_30} &=\eta_{{30}}\end{split}\]
Hu moments
Using nonlinear combinations of geometric moments, M.K. Hu derived a set of invariant moments which has the desirable properties of
being invariant under image translation, scaling, and rotation. Hu moments HU_M1 through HU_M7 are calculated as
\[\begin{split}\text{HU_M1} =& \eta_{{20}}+\eta _{{02}} \\
\text{HU_M2} =& (\eta_{{20}}-\eta_{{02}})^{2}+4\eta_{{11}}^{2} \\
\text{HU_M3} =& (\eta_{{30}}-3\eta_{{12}})^{2}+(3\eta_{{21}}-\eta _{{03}})^{2} \\
\text{HU_M4} =& (\eta_{{30}}+\eta_{{12}})^{2}+(\eta_{{21}}+\eta _{{03}})^{2} \\
\text{HU_M5} =& (\eta_{{30}}-3\eta_{{12}})(\eta_{{30}}+\eta_{{12}})[(\eta_{{30}}+\eta_{{12}})^{2}-3(\eta_{{21}}+\eta_{{03}})^{2}]+ \\
&(3\eta_{{21}}-\eta_{{03}})(\eta_{{21}}+\eta_{{03}})[3(\eta_{{30}}+\eta_{{12}})^{2}-(\eta_{{21}}+\eta _{{03}})^{2}] \\
\text{HU_M6} =& (\eta_{{20}}-\eta_{{02}})[(\eta_{{30}}+\eta_{{12}})^{2}-(\eta_{{21}}+\eta_{{03}})^{2}]+4\eta_{{11}}(\eta_{{30}}+\eta_{{12}})(\eta_{{21}}+\eta_{{03}}) \\
\text{HU_M7} =& (3\eta_{{21}}-\eta_{{03}})(\eta_{{30}}+\eta_{{12}})[(\eta_{{30}}+\eta_{{12}})^{2}-3(\eta_{{21}}+\eta_{{03}})^{2}]- \\
&(\eta_{{30}}-3\eta_{{12}})(\eta_{{21}}+\eta_{{03}})[3(\eta_{{30}}+\eta_{{12}})^{2}-(\eta_{{21}}+\eta _{{03}})^{2}]\end{split}\]
Weighted raw moments
Let \(W(x,y)\) be a 2-dimensional weighted greyscale image such that each pixel of \(I\) is weighted with respect to its distance to the nearest contour pixel:
\[W(x,y) = \frac {p(x,y)} {\min_i d^2(x,y,C_i)}\]
where C - set of 2-dimensional ROI contour pixels, \(d^2(.)\) - Euclidean distance norm. Weighted raw moments \(w_{Mij}\) are defined as
\[w_{Mij}=\sum_{x}\sum _{y}x^{i}y^{j}W(x,y)\]
Weighted central moments
Weighted central moments \(w_{\mu ij}\) are defined as
\[w_{\mu ij} = \sum_{{x}}\sum_{{y}}(x-{\bar {x}})^{i}(y-{\bar {y}})^{j}W(x,y)\]
Weighted Hu moments
A normalized weighted central moment \(w_{\eta ij}\) is defined as
\[w_{{\eta ij}}={\frac {w_{{\mu ij}}}{w_{{\mu 00}}^{{\left(1+{\frac {i+j}{2}}\right)}}}}\,\]
where \(w _{{\mu ij}}\) is weighted central moment. Weighted Hu moments are defined as
\[\begin{split}\text{WEIGHTED_HU_M1} =& w_{\eta 20}+w_{\eta 02} \\
\text{WEIGHTED_HU_M2} =& (w_{\eta 20}-w_{\eta 02})^{2}+4w_{\eta 11}^{2} \\
\text{WEIGHTED_HU_M3} =& (w_{\eta 30}-3w_{\eta 12})^{2}+(3w_{\eta 21}-w _{\eta 03})^{2} \\
\text{WEIGHTED_HU_M4} =& (w_{\eta 30}+w_{\eta 12})^{2}+(w_{\eta 21}+w _{\eta 03})^{2} \\
\text{ WEIGHTED_HU_M5} =& (w_{\eta 30}-3w_{\eta 12})(w_{\eta 30}+w_{\eta 12})[(w_{\eta 30}+w_{\eta 12})^{2}-3(w_{\eta 21}+ w_{\eta 03})^{2}]+ \\
&(3w_{\eta 21}-w_{\eta 03})(w_{\eta 21}+w_{\eta 03})[3(w_{\eta 30}+w_{\eta 12})^{2}-(w_{\eta 21}+w _{\eta 03})^{2}] \\
\text{WEIGHTED_HU_M6} =& (w_{\eta 20}-w_{\eta 02})[(w_{\eta 30}+w_{\eta 12})^{2}-(w_{\eta 21}+w_{\eta 03})^{2}]+ \\
&4w_{\eta 11}(w_{\eta 30}+w_{\eta 12})(w_{\eta 21}+w_{\eta 03})\\
\text{WEIGHTED_HU_M7} =& (3w_{\eta 21}-w_{\eta 03})(w_{\eta 30}+w_{\eta 12})[(w_{\eta 30}+w_{\eta 12})^{2}-3(w_{\eta 21}+w_{\eta 03})^{2}]- \\
&(w_{\eta 30}-3w_{\eta 12})(w_{\eta 21}+w_{\eta 03})[3(w_{\eta 30}+w_{\eta 12})^{2}-(w_{\eta 21}+w _{\eta 03})^{2}]\end{split}\]
References
M.K. Hu. Pattern recognition by moment invariants, proc. IRE 49, 1961, 1428.
M.K. Hu. Visual problem recognition by moment invariant. IRE Trans. Inform. Theory, Vol. IT-8, pp. 179-187, Feb. 1962.
T.H. Reiss. The Revised Fundamental Theorem of Moment Invariants. IEEE Trans. Pattern Anal. Machine Intell., Vol. PAMI-13. No. 8, August 1991. pp. 830-834.