galaxy-classification

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Calculation

To find the major and minor axis for disk sources we can use the raw and central image moments within the effective radius of the source.

More on image moments can be found here and a great walkthrough for application can be found here.

The raw image moments are defined as:

$$M_{ij}=\sum_x\sum_yx^iy^jI(x,y)$$

Where $I(x,y)$ is the value at pixel $x,y$.

The central image moments we use are:

To get the major and minor axis of the source we need to construct a covariance matrix for the pixels using the second order central moments.

$$\text{cov}[I(x,y)]=\left[\begin{array}{ c c }\mu'_{20} & \mu'_{11} \\ \mu'_{11} & \mu'_{02} \end{array} \right]$$

Where :

$$\bar{x} = M_{10}/M_{00}$$ $$\bar{y} = M_{01}/M_{00}$$ $$\mu'_{20} = \mu_{20}/\mu_{00} = M_{20}/M_{00} - (\bar{x})^2$$ $$\mu'_{02} = \mu_{02}/\mu_{00} = M_{02}/M_{00} - (\bar{y})^2$$ $$\mu'_{11} = \mu_{11}/\mu_{00} = M_{11}/M_{00} - \bar{x}\bar{y}$$

The eigenvectors of this matrix correspond to the major and minor axis of the source.

Example

The source we’ll use is Deep 2 10064:

With the segmap

Here are the major(red) and minor(blue) axis as measured from the h band plotted over the segmap with the effective radius highlighted in yellow.