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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:

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

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

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

How is the filter $G$ derived fromAnd the definition of the bilinear interpolation is defined in wiki as:

$${\displaystyle {\begin{aligned}f(x,y)&\approx {\frac {y_{2}-y}{y_{2}-y_{1}}}f(x,y_{1})+{\frac {y-y_{1}}{y_{2}-y_{1}}}f(x,y_{2})\\&={\frac {y_{2}-y}{y_{2}-y_{1}}}\left({\frac {x_{2}-x}{x_{2}-x_{1}}}f(Q_{11})+{\frac {x-x_{1}}{x_{2}-x_{1}}}f(Q_{21})\right)+{\frac {y-y_{1}}{y_{2}-y_{1}}}\left({\frac {x_{2}-x}{x_{2}-x_{1}}}f(Q_{12})+{\frac {x-x_{1}}{x_{2}-x_{1}}}f(Q_{22})\right)\\&={\frac {1}{(x_{2}-x_{1})(y_{2}-y_{1})}}{\big (}f(Q_{11})(x_{2}-x)(y_{2}-y)+f(Q_{21})(x-x_{1})(y_{2}-y)+f(Q_{12})(x_{2}-x)(y-y_{1})+f(Q_{22})(x-x_{1})(y-y_{1}){\big )}\\&={\frac {1}{(x_{2}-x_{1})(y_{2}-y_{1})}}{\begin{bmatrix}x_{2}-x&x-x_{1}\end{bmatrix}}{\begin{bmatrix}f(Q_{11})&f(Q_{12})\\f(Q_{21})&f(Q_{22})\end{bmatrix}}{\begin{bmatrix}y_{2}-y\\y-y_{1}\end{bmatrix}}.\end{aligned}}}$$$$f(x,y)\approx {\frac {y_{2}-y}{y_{2}-y_{1}}}f(x,y_{1})+{\frac {y-y_{1}}{y_{2}-y_{1}}}f(x,y_{2})\tag{3}$$

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

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:

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

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

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

How is the filter $G$ derived from the definition of the bilinear interpolation:

$${\displaystyle {\begin{aligned}f(x,y)&\approx {\frac {y_{2}-y}{y_{2}-y_{1}}}f(x,y_{1})+{\frac {y-y_{1}}{y_{2}-y_{1}}}f(x,y_{2})\\&={\frac {y_{2}-y}{y_{2}-y_{1}}}\left({\frac {x_{2}-x}{x_{2}-x_{1}}}f(Q_{11})+{\frac {x-x_{1}}{x_{2}-x_{1}}}f(Q_{21})\right)+{\frac {y-y_{1}}{y_{2}-y_{1}}}\left({\frac {x_{2}-x}{x_{2}-x_{1}}}f(Q_{12})+{\frac {x-x_{1}}{x_{2}-x_{1}}}f(Q_{22})\right)\\&={\frac {1}{(x_{2}-x_{1})(y_{2}-y_{1})}}{\big (}f(Q_{11})(x_{2}-x)(y_{2}-y)+f(Q_{21})(x-x_{1})(y_{2}-y)+f(Q_{12})(x_{2}-x)(y-y_{1})+f(Q_{22})(x-x_{1})(y-y_{1}){\big )}\\&={\frac {1}{(x_{2}-x_{1})(y_{2}-y_{1})}}{\begin{bmatrix}x_{2}-x&x-x_{1}\end{bmatrix}}{\begin{bmatrix}f(Q_{11})&f(Q_{12})\\f(Q_{21})&f(Q_{22})\end{bmatrix}}{\begin{bmatrix}y_{2}-y\\y-y_{1}\end{bmatrix}}.\end{aligned}}}$$

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 $(3)$?

2 add meaning of delta p
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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:

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

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

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

How is the filter $G$ derived from the definition of the bilinear interpolation:

$${\displaystyle {\begin{aligned}f(x,y)&\approx {\frac {y_{2}-y}{y_{2}-y_{1}}}f(x,y_{1})+{\frac {y-y_{1}}{y_{2}-y_{1}}}f(x,y_{2})\\&={\frac {y_{2}-y}{y_{2}-y_{1}}}\left({\frac {x_{2}-x}{x_{2}-x_{1}}}f(Q_{11})+{\frac {x-x_{1}}{x_{2}-x_{1}}}f(Q_{21})\right)+{\frac {y-y_{1}}{y_{2}-y_{1}}}\left({\frac {x_{2}-x}{x_{2}-x_{1}}}f(Q_{12})+{\frac {x-x_{1}}{x_{2}-x_{1}}}f(Q_{22})\right)\\&={\frac {1}{(x_{2}-x_{1})(y_{2}-y_{1})}}{\big (}f(Q_{11})(x_{2}-x)(y_{2}-y)+f(Q_{21})(x-x_{1})(y_{2}-y)+f(Q_{12})(x_{2}-x)(y-y_{1})+f(Q_{22})(x-x_{1})(y-y_{1}){\big )}\\&={\frac {1}{(x_{2}-x_{1})(y_{2}-y_{1})}}{\begin{bmatrix}x_{2}-x&x-x_{1}\end{bmatrix}}{\begin{bmatrix}f(Q_{11})&f(Q_{12})\\f(Q_{21})&f(Q_{22})\end{bmatrix}}{\begin{bmatrix}y_{2}-y\\y-y_{1}\end{bmatrix}}.\end{aligned}}}$$

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:

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

Where $q$ is the point from the source image, and $p$ is the points on the target image. $G$ is defined as

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

How is the filter $G$ derived from the definition of the bilinear interpolation:

$${\displaystyle {\begin{aligned}f(x,y)&\approx {\frac {y_{2}-y}{y_{2}-y_{1}}}f(x,y_{1})+{\frac {y-y_{1}}{y_{2}-y_{1}}}f(x,y_{2})\\&={\frac {y_{2}-y}{y_{2}-y_{1}}}\left({\frac {x_{2}-x}{x_{2}-x_{1}}}f(Q_{11})+{\frac {x-x_{1}}{x_{2}-x_{1}}}f(Q_{21})\right)+{\frac {y-y_{1}}{y_{2}-y_{1}}}\left({\frac {x_{2}-x}{x_{2}-x_{1}}}f(Q_{12})+{\frac {x-x_{1}}{x_{2}-x_{1}}}f(Q_{22})\right)\\&={\frac {1}{(x_{2}-x_{1})(y_{2}-y_{1})}}{\big (}f(Q_{11})(x_{2}-x)(y_{2}-y)+f(Q_{21})(x-x_{1})(y_{2}-y)+f(Q_{12})(x_{2}-x)(y-y_{1})+f(Q_{22})(x-x_{1})(y-y_{1}){\big )}\\&={\frac {1}{(x_{2}-x_{1})(y_{2}-y_{1})}}{\begin{bmatrix}x_{2}-x&x-x_{1}\end{bmatrix}}{\begin{bmatrix}f(Q_{11})&f(Q_{12})\\f(Q_{21})&f(Q_{22})\end{bmatrix}}{\begin{bmatrix}y_{2}-y\\y-y_{1}\end{bmatrix}}.\end{aligned}}}$$

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:

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

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

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

How is the filter $G$ derived from the definition of the bilinear interpolation:

$${\displaystyle {\begin{aligned}f(x,y)&\approx {\frac {y_{2}-y}{y_{2}-y_{1}}}f(x,y_{1})+{\frac {y-y_{1}}{y_{2}-y_{1}}}f(x,y_{2})\\&={\frac {y_{2}-y}{y_{2}-y_{1}}}\left({\frac {x_{2}-x}{x_{2}-x_{1}}}f(Q_{11})+{\frac {x-x_{1}}{x_{2}-x_{1}}}f(Q_{21})\right)+{\frac {y-y_{1}}{y_{2}-y_{1}}}\left({\frac {x_{2}-x}{x_{2}-x_{1}}}f(Q_{12})+{\frac {x-x_{1}}{x_{2}-x_{1}}}f(Q_{22})\right)\\&={\frac {1}{(x_{2}-x_{1})(y_{2}-y_{1})}}{\big (}f(Q_{11})(x_{2}-x)(y_{2}-y)+f(Q_{21})(x-x_{1})(y_{2}-y)+f(Q_{12})(x_{2}-x)(y-y_{1})+f(Q_{22})(x-x_{1})(y-y_{1}){\big )}\\&={\frac {1}{(x_{2}-x_{1})(y_{2}-y_{1})}}{\begin{bmatrix}x_{2}-x&x-x_{1}\end{bmatrix}}{\begin{bmatrix}f(Q_{11})&f(Q_{12})\\f(Q_{21})&f(Q_{22})\end{bmatrix}}{\begin{bmatrix}y_{2}-y\\y-y_{1}\end{bmatrix}}.\end{aligned}}}$$

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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:

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

Where $q$ is the point from the source image, and $p$ is the points on the target image. $G$ is defined as

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

How is the filter $G$ derived from the definition of the bilinear interpolation:

$${\displaystyle {\begin{aligned}f(x,y)&\approx {\frac {y_{2}-y}{y_{2}-y_{1}}}f(x,y_{1})+{\frac {y-y_{1}}{y_{2}-y_{1}}}f(x,y_{2})\\&={\frac {y_{2}-y}{y_{2}-y_{1}}}\left({\frac {x_{2}-x}{x_{2}-x_{1}}}f(Q_{11})+{\frac {x-x_{1}}{x_{2}-x_{1}}}f(Q_{21})\right)+{\frac {y-y_{1}}{y_{2}-y_{1}}}\left({\frac {x_{2}-x}{x_{2}-x_{1}}}f(Q_{12})+{\frac {x-x_{1}}{x_{2}-x_{1}}}f(Q_{22})\right)\\&={\frac {1}{(x_{2}-x_{1})(y_{2}-y_{1})}}{\big (}f(Q_{11})(x_{2}-x)(y_{2}-y)+f(Q_{21})(x-x_{1})(y_{2}-y)+f(Q_{12})(x_{2}-x)(y-y_{1})+f(Q_{22})(x-x_{1})(y-y_{1}){\big )}\\&={\frac {1}{(x_{2}-x_{1})(y_{2}-y_{1})}}{\begin{bmatrix}x_{2}-x&x-x_{1}\end{bmatrix}}{\begin{bmatrix}f(Q_{11})&f(Q_{12})\\f(Q_{21})&f(Q_{22})\end{bmatrix}}{\begin{bmatrix}y_{2}-y\\y-y_{1}\end{bmatrix}}.\end{aligned}}}$$