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Bilinear interpolation implemented by convolution

I read the paper Deep Feature Flow for Video Recognition https://arxiv.org/abs/1611.07715.

In Sec.3, the author implements bilinear interpolation like this:

$$f_i^c(p)=\sum\limits_{q}G(q,p+\delta p)f_k^c(q) \tag{1}$$

Where $q$ is the point from the source image, and $p$ is the points on the target image. $\delta p$ is the distance the point moved each point $p$ (not $\delta \bullet p$). $G$ is defined as

$$G(q,p+\delta p)=g(q_x,p_x+\delta p_x)g(q_y,p_y+\delta p_y)\tag{2}$$

And the bilinear interpolation is defined in wiki as:

$$f(x,y)\approx {\frac {y_{2}-y}{y_{2}-y_{1}}}f(x,y_{1})+{\frac {y-y_{1}}{y_{2}-y_{1}}}f(x,y_{2})\tag{3}$$

I think the operation $(1)$ and $(3)$ is equivalent. How can I derive the filter $(1)$ from $(3)$?