$w$ and $h$ are the width and height of the image respectively. At each randomly selected pixel position $(x,y)\in \mathcal{R}$, we consider a $N\times N$ local patch $P_{x,y}^N$ around the pixel. A motion vector field $V_{x,y} = (u, v)$ is then generated in $P_{x,y}^N$. For each $p\in P_{x,y}^N$, the vector $(u(p), v(p))$ is sampled from a normal distribution, smoothened by a Gaussian kernel and entry-wisely multiplied by a strength value of distortion. Mathematically, the vector field $V_{x,y}$ can be written as: $$V_{x,y} = S\ (G_{\sigma}*\mathcal{N}_1,G_{\sigma}*\mathcal{N}_2),$$ where $G_{\sigma}$ is the Gaussian kernel with standard deviation $\sigma$, $S$ is the strength value, $\mathcal{N}_1$ and $\mathcal{N}_2$ are randomly selected from a normal distribution. $V_{x,y}$ is then extended to the whole image domain by setting zero outside $P_{x,y}^N$. It is then employed to wrap the original image $I$ to get a transformed image. We repeat this process by $M$ iterations.

Essentially, the overall motion vector field $V=(u,v)$ after $M$ iterations is defined by fusing the vector patches together wherever overlapping. Mathematically, $$V = \sum_{(x,y)\in \mathcal{R}} V_{x,y},$$ where $\mathcal{R}$ is the collection of $Q$ randomly selected pixel positions.

We denote the transformed image by $I^t$. The transformed image $I^t$ is further blurred by a Gaussian 