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Attempt at Fourier analysis:

Underlying continous image: $I_{cont}(x,y)$.
Introduce comb function: $$C(x,y)=\sum_{k=-\infty}^{k=\infty} \sum_{l=-\infty}^{l=\infty}\delta(x-k\Delta x) \delta(y-l\Delta y) $$

Original image:
$I = I_{cont}*C(x,y)$

$\mathcal{F}(I)=\mathcal{F}(I_{cont})\mathcal{F}(C)$

Binned image: $I_b = 0.25I_{cont}*C+0.25I_{cont}*C_{10}+0.25I_{cont}*C_{01}+0.25I_{cont}*C_{11}$

where $C_{10}=C(x-\Delta x,y)$, $C_{01}=C(x,y-\Delta y)$ and $C_{11}=C(x-\Delta x,y-\Delta y)$

$\mathcal{F}(I_b)=0.25\mathcal{F}(I_{cont})[\mathcal{F}(C)+e^{-i2\pi u\Delta x}\mathcal{F}(C)+e^{-i2\pi v\Delta y}\mathcal{F}(C)+e^{-i2\pi(u\Delta x + v\Delta y)}\mathcal{F}(C)]$

$\mathcal{F}(I_b)=0.25\mathcal{F}(I_{cont})\mathcal{F}(C)[1+e^{-i2\pi u\Delta x}+e^{-i2\pi v\Delta y}+e^{-i2\pi(u\Delta x + v\Delta y)}]$

Attempt at Fourier analysis:

Underlying continous image: $I_{cont}(x,y)$.
Introduce comb function: $$C(x,y)=\sum_{k=-\infty}^{k=\infty} \sum_{l=-\infty}^{l=\infty}\delta(x-k\Delta x) \delta(y-l\Delta y) $$

Original image:
$I = I_{cont}*C(x,y)$

$\mathcal{F}(I)=\mathcal{F}(I_{cont})\mathcal{F}(C)$

Binned image: $I_b = 0.25I_{cont}*C+0.25I_{cont}*C_{10}+0.25I_{cont}*C_{01}+0.25I_{cont}*C_{11}$

where $C_{10}=C(x-\Delta x,y)$, $C_{01}=C(x,y-\Delta y)$ and $C_{11}=C(x-\Delta x,y-\Delta y)$

$\mathcal{F}(I_b)=0.25\mathcal{F}(I_{cont})[\mathcal{F}(C)+e^{-i2\pi u\Delta x}\mathcal{F}(C)+e^{-i2\pi v\Delta y}\mathcal{F}(C)+e^{-i2\pi(u\Delta x + v\Delta y)}\mathcal{F}(C)]$

$\mathcal{F}(I_b)=0.25\mathcal{F}(I_{cont})\mathcal{F}(C)[1+e^{-i2\pi u\Delta x}+e^{-i2\pi v\Delta y}+e^{-i2\pi(u\Delta x + v\Delta y)}]$

$H(u,v)=0.25*[1+e^{-i2\pi u\Delta x}+e^{-i2\pi v\Delta y}+e^{-i2\pi(u\Delta x + v\Delta y)}]$

Attempt at Fourier analysis:

Underlying continous image: $I_{cont}(x,y)$.
Introduce comb function: $$C(x,y)=\sum_{k=-\infty}^{k=\infty} \sum_{l=-\infty}^{l=\infty}\delta(x-k\Delta x) \delta(y-l\Delta y) $$

Original image:
$I = I_{cont}*C(x,y)$

Binned image: $I_b = 0.25I_{cont}*C_{00}+0.25I_{cont}*C_{01}+0.25I_{cont}*C_{01}+0.25I_{cont}*C_{11}$

$\mathcal{F}(I_b)=0.25\mathcal{F}(I_{cont})[\mathcal{F}(C)+e^{-i2\pi u\Delta x}\mathcal{F}(C)+e^{-i2\pi v\Delta y}\mathcal{F}(C)+e^{-i2\pi(u\Delta x + v\Delta y)}\mathcal{F}(C)]$

Attempt at Fourier analysis:

Underlying continous image: $I_{cont}(x,y)$.
Introduce comb function: $$C(x,y)=\sum_{k=-\infty}^{k=\infty} \sum_{l=-\infty}^{l=\infty}\delta(x-k\Delta x) \delta(y-l\Delta y) $$

Original image:
$I = I_{cont}*C(x,y)$

$\mathcal{F}(I)=\mathcal{F}(I_{cont})\mathcal{F}(C)$

Binned image: $I_b = 0.25I_{cont}*C+0.25I_{cont}*C_{10}+0.25I_{cont}*C_{01}+0.25I_{cont}*C_{11}$

where $C_{10}=C(x-\Delta x,y)$, $C_{01}=C(x,y-\Delta y)$ and $C_{11}=C(x-\Delta x,y-\Delta y)$

$\mathcal{F}(I_b)=0.25\mathcal{F}(I_{cont})[\mathcal{F}(C)+e^{-i2\pi u\Delta x}\mathcal{F}(C)+e^{-i2\pi v\Delta y}\mathcal{F}(C)+e^{-i2\pi(u\Delta x + v\Delta y)}\mathcal{F}(C)]$

$\mathcal{F}(I_b)=0.25\mathcal{F}(I_{cont})\mathcal{F}(C)[1+e^{-i2\pi u\Delta x}+e^{-i2\pi v\Delta y}+e^{-i2\pi(u\Delta x + v\Delta y)}]$

Attempt at Fourier analysis:

Underlying continous image: $I_{cont}(x,y)$.
Introduce comb function: $$C(x,i,y,j)=\sum_{k=-\infty}^{k=\infty} \sum_{l=-\infty}^{l=\infty}\delta(x-k\Delta x) \delta(y-l\Delta y) $$

Original image:
$I = I_{cont}*C(x,i,y,j)$

Binned image: $I_b = 0.25I_{cont}*C(x,2i,y,2j)+0.25I_{cont}*C(x,2i+1,y,2j)+0.25I_{cont}*C(x,2i,y,2j+1)+0.25I_{cont}*C(x,2i+1,y,2j+1)$

$\mathcal{F}(I_b)=0.25\mathcal{F}(I_{cont})[\mathcal{F}(C)+e^{-i2\pi u\Delta x}\mathcal{F}(C)+e^{-i2\pi v\Delta y}\mathcal{F}(C)+e^{-i2\pi(u\Delta x + v\Delta y)}\mathcal{F}(C)]$

Attempt at Fourier analysis:

Underlying continous image: $I_{cont}(x,y)$.
Introduce comb function: $$C(x,y)=\sum_{k=-\infty}^{k=\infty} \sum_{l=-\infty}^{l=\infty}\delta(x-k\Delta x) \delta(y-l\Delta y) $$

Original image:
$I = I_{cont}*C(x,y)$

Binned image: $I_b = 0.25I_{cont}*C_{00}+0.25I_{cont}*C_{01}+0.25I_{cont}*C_{01}+0.25I_{cont}*C_{11}$

$\mathcal{F}(I_b)=0.25\mathcal{F}(I_{cont})[\mathcal{F}(C)+e^{-i2\pi u\Delta x}\mathcal{F}(C)+e^{-i2\pi v\Delta y}\mathcal{F}(C)+e^{-i2\pi(u\Delta x + v\Delta y)}\mathcal{F}(C)]$