moments calculates all of the moments up to the third
order of a polygon or rasterized shape.
Arguments
- x
Either a Nx2 matrix of the X-Y coordinates of a polygon (e.g., a contour produced by
findContours), or a single-channelImageobject.- binary
If set to TRUE (default: FALSE), all non-zero image pixels are treated as 1's. The parameter is used for images only.
Value
A data frame with 2 columns:
- "moment":
the name of the moment. See Note below.
- "value":
the value of the moment.
Note
The spatial moments \(m_{ji}\) are computed as: $$m_{ji}= \sum _{x,y} \left ( \texttt{contour} (x,y) \cdot x^j \cdot y^i \right )$$
The central moments \(\mu_{ji}\) are computed as: $${\mu_{ji}}= \sum _{x,y} \left ( \texttt{contour} (x,y) \cdot (x - \bar{x} )^j \cdot (y - \bar{y} )^i \right )$$ where \((\bar{x}, \bar{y})\) is the mass center: $$\bar{x} = \frac{m_{10}}{m_{00}} , \; \bar{y} = \frac{m_{01}}{m_{00}}$$
The normalized central moments \(\eta_{ji}\) are computed as: $$\eta_{ji}= \frac{\mu_{ji}}{m_{00}^{(i+j)/2+1}} .$$
Author
Simon Garnier, garnier@njit.edu
Examples
dots <- image(system.file("sample_img/dots.jpg", package = "Rvision"))
dots_gray <- changeColorSpace(dots, "GRAY")
dots_bin <- dots_gray < 200
contours <- findContours(dots_bin)
contour0 <- contours$contours[contours$contours[, 1] == 0, 2:3]
moments(contour0)
#> moment value
#> 1 m00 8.958500e+03
#> 2 m10 5.043943e+06
#> 3 m01 4.582793e+05
#> 4 m20 2.846367e+09
#> 5 m11 2.575802e+08
#> 6 m02 2.981324e+07
#> 7 m30 1.609871e+12
#> 8 m21 1.450999e+11
#> 9 m12 1.674067e+10
#> 10 m03 2.181108e+09
#> 11 mu20 6.454539e+06
#> 12 mu11 -4.467378e+05
#> 13 mu02 6.369592e+06
#> 14 mu30 3.439845e+05
#> 15 mu21 -5.259778e+06
#> 16 mu12 5.018900e+05
#> 17 mu03 4.303903e+06
#> 18 nu20 8.042566e-02
#> 19 nu11 -5.566499e-03
#> 20 nu02 7.936719e-02
#> 21 nu30 4.528461e-05
#> 22 nu21 -6.924352e-04
#> 23 nu12 6.607243e-05
#> 24 nu03 5.665970e-04