Image Upload

Image Upload.md

For image uploading only

S(k_1, k_2) = \sum_{n_1=0}^{N_1-1}\sum_{n_2=0}^{N_2-1} s(n_1, n_2)g(n_1, n_2, k_1, k_2)

s(n_1, n_2) = \sum_{k_1=0}^{N_1-1}\sum_{k_2=0}^{N_2-1} S(k_1, k_2)h(n_1, n_2, k_1, k_2)

g(n_1, n_2, k_1, k_2) &= exp(-j\frac{2\pi(n_1k_1+n_2k_2)}{N^2}) \
&=exp(-j\frac{2\pi n_1k_1}{N^2})exp(-j\frac{2\pi n_2k_2}{N^2})

S = \sum_{k_1=0}^{N-1}\sum_{k_2=0}^{N-1}S(k_1, k_2)H_{k_1, k_2}

H_{k_1, k_2} =
\begin{bmatrix}
h(0, 0, k_1, k_2) & h(0, 1, k_1, k_2) & \dots & h(0, N-1, k_1, k_2) \
h(1, 0, k_1, k_2) & & & \vdots \
\vdots & & & \vdots \
h(N-1, 0, k_1, k_2) & \dots & \dots & h(N-1, N-1, k_1, k_2)
\end{bmatrix}

S(k_1, k_2) &= \sqrt{\frac{4}{N^2}} C(k_1)C(k_2) \sum_{n_1=0}^{N-1} \sum_{n_2=0}^{N-1}s(n_1, n_2) \
& \times{cos(\frac{\pi (2n_1+1)k_1}{2N})} \times{cos(\frac{\pi (2n_2+1)k_2}{2N})}

C(k)=
\begin{cases}
\frac{1}{\sqrt{2}},& \text{for } k=0\
1, & \text{otherwise}
\end{cases}