Calculate magnitude of the gradient using higher order statistics

Question

I am making a model for detecting blurred part of an image. I'm using features described in the paper Blurred Image Region Detection And Segmentation by Hyukzae Lee and Changick Kim, and I have a problem with their equation for computing magnitude of the gradient.

In the paper they talk about

using HOS enhance strong edges while suppressing weak edges and noise components

This is equation for calculating MG:

$$ MG(i) = log \left[ \frac{1}{N_{p_{i}}} \sum_{j \in P_i }\biggl\{\sqrt{\frac{I_x^2(j)+I_y^2(j)}{2}}\biggr\}^t\right]$$

where $I_x$ and $I_y$ stand for the gradient of intensity value in horizontal and vertical directions, respectively. $P_i$ represents the patch centered at the ith pixel position and $N_{P_i}$ denotes the number of pixels in $P_i$.

For the value $k$, we select it as 4 in terms of trade-off between performance and complexity of MG feature.

My main question is:

How do I calculate the $\{\}^t$ part? I was thinking this was just normal power but it isn't. I believe it's some statistical moment but how can I calculate it? It follows form notation that we calculate $\{\}^t$ from every number in the sum and then add them up. Or maybe I have to first add them up, and then calculate $\{\}^t$?

Answer 1

First, on the copy I see of the paper I can read:

Namely there is a match between $ k $ in the text and the power in the equation.

If you read about Higher Order Statistics (HOS) then you see the idea is to use high power of the sample.

If you treat the magnitude of the pixel as it is defined as a sample, then what they do is just raising it by $ k $.
They also specify that they chose $ k = 4 $.