# Four different types of third order image moments (skewness), what are the differences?

I am attempting to analyze images using the third order image moment (aka skewness), but I am having trouble figuring out how to do so. As it turns out, there are four different types of third order image moments:

$$\begin{array}{rcl} \mu_{21} &=& M_{21} + 2\bar{x}M_{11} - \bar{y}M_{20} + 2\bar{x}^2M_{01}\\ \mu_{12} &=& M_{12} + 2\bar{y}M_{11} - \bar{x}M_{02} + 2\bar{y}^2M_{10}\\ \mu_{30} &=& M_{30} - 3\bar{x}M_{20} + 2\bar{x}^2M_{10}\\ \mu_{03} &=& M_{03} - 3\bar{y}M_{02} + 2\bar{y}^2M_{01} \end{array}$$

$$\begin{array}{rcl} M_{p,q} &=& \mbox{Raw moment of order}\ (p+q)\\ \mu_{p,q} &=& \mbox{Central moment of order}\ (p+q)\\ \end{array}$$

Formulas pulled from: https://en.wikipedia.org/wiki/Image_moment

It's been very difficult to find out what the differences are between these 4 (if there are any) on the internet. If there's anyone here that could explain in plain English how they differ, I would greatly appreciate it.

This question is most likely asking something that's more fundamental to the calculation of moments themselves than specifically with regard to image moments. However, I have seen several questions about them here, so I figured it would be a good place to start.

• It would be good to get the definitions of $M_{kl}$ in your equations! – Peter K. Aug 13 '15 at 12:17
• @PeterK. Appropriate edits made. – cadams Aug 13 '15 at 18:53

It it based on the definition of Central Moments on 2D grid given in the Wikipedia Page.

$${\mu}_{pq} = \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} (x - \bar{x})^p(y - \bar{y})^q f(x,y) \, dx \, dy$$

Since we're dealing with 2 random variables, a multiplication which the powers sum to 3 can be achieved in 4 different methods.

It's up to you to decide how to calculate the third moment.

Usually I'd use the Rotation Invariant Moments.
You can find the definition on the same page.
Enforcing the Rotation (And Scale) Invariance yields single definition.

• Earlier today, I became familiarized with the rotation invariant moments, realizing that they were what I was needing! Would you happen to know which of the 7 would be able to provide the most effective separation between images containing a small line of high pixel intensity and images with a small curve of high pixel intensity? – cadams Aug 14 '15 at 13:21