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 evaluted computationally and used to uniquely describe the information contained in the image.
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:
\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}
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:
\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} \\
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:
\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} \\
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:
\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}}
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
\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}]
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 w_{\mu ij} are defined as
w_{\mu ij} = \sum_{{x}}\sum_{{y}}(x-{\bar {x}})^{i}(y-{\bar {y}})^{j}W(x,y)
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
\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}]
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.