What Do Skewness and Kurtosis Represent?

I understand that the question could mean a lot of things but I am thinking specifically to image processing. For example, I know that the mean can be a basic texture feature that represents the average pixel value of the image. That is, it is useful for determining an image background. Then I know that the standard deviation can be an indicator of how spread is the histogram and is useful for indicating what other pixel values also belong to the background.

In similar terms to this, what do Kurtosis and Skewness represent? I have read about them in various posts but I can't find anything related to what they represent in terms of what I have explained above.

I am not sure if I have came to the right place but if I haven't feel free to point me in the right direction.

This paper has a nice write up on those and many other measures.

For skewness:

In terms of digital image processing, Darker and glossier surfaces tend to be more positively skewed than lighter and matte surfaces. Hence we can use skewness in making judgements about image surfaces.

This is because skewness measures how "lopsided" the distribution of pixel values are.

For kurtosis:

In digital image processing kurtosis values are interpreted in combination with noise and resolution measurement. High kurtosis values should go hand in hand with low noise and low resolution.

I'm not sure I agree with this. Images with moderate amounts of salt and pepper noise are likely to have a high kurtosis value.

Mean, standard deviation, skewness and kurtosis are based on geometrical moments of patches of images. Being homogeneous ratios, and generally centered, skewness and kurtosis have the advantage of being invariant to affine luminance changes in images. Based on degree $3$ and $4$ moments, they are sometimes termed Higher-order-statistics.

Regarding skewness, it was used to detect edges in dark objects on white background, having a sign change at luminance edges, and could replace, with some Gaussian prefiltering, a Laplacian, see Performance of the Skewness-of-Gaussian (SoG) edge extractor, Seventh European Signal Processing Conf. 1994.

Kurtosis somehow detects if a distribution is flat or peaky, and later was associated to perceptual aspects of sparse coding. It is often considered as a measure a sparsity, and used in early deconvolution methods.

Both have also been used on transformations of images: co-occurence matrices, wavelets, etc., to assess or provide parameters for selection, detection, clustering.

You can find additional literature with "Moment Functions" in "Image Analysis". Caveat: if pixel distributions are quite multimodal, such simple estimators do no say much.

• So would it be safe to say that skewness and kurtosis tell us something about the shape of an image? It was my understanding it was texture! Doh!! @laurent May 3, 2016 at 19:22
• I would not call it safe. Shape in images is a complicated topic, which cannot (so far) be summarized by simple local measures like skeness and kurtosis. Additionnaly, those can be used for textural characterization, for instance on second order image statistics (co-occurence matrix). May 4, 2016 at 6:41

this ain't image processing, but think of it regarding flipping a coin. say the value of heads is +1 and of tails is -1.

if the coin ain't honest, the skewness will increase in magnitude (and so will the mean, but if you were to subtract the mean, the skewness would still increase.

the kurtosis of the coin flipping is zero whether the coin is honest or not. but if somehow the results of the coin flip could be something deviating from +1 or -1, like sometimes the coin comes out 95% heads and 5% tails and the value of the flip comes out as 0.9 or -1.04 or something like that. then the kurtosis would be greater than one. like the other even-numbered moments, i don't think kurtosis can be negative.