I am currently reading this paper which discusses several image interpolation methods, such as nearest neighbor and linear interpolation, using convolution filters. I first want to do this in 1D with discrete-time signals (e.g. [0 0 1 0 0]), however, I have some trouble understanding on how to derive the convolution filters from the mathematical definitions:

$$h_{NN}(x)=\cases{ 1, &if$\quad-\frac{1}{2} \leqslant x < \frac{1}{2},$\\ 0, &otherwise, }$$ $$h_{Lin}(x)=\cases{ 1-|x|, &if$\quad 0 \leqslant x < 1,$\\ 0, &if$\quad 1 \leqslant x.$ }$$

My main question is, how do I calculate the filter weights as well as the filter size? Furthermore, I am aware of the fact that I need to convolve the signal multiple times to upsample it, which the following equation in the paper also shows.

$$I(\mathbf{x}) = (\dots((I_s(\mathbf{p})*h(x_1))*h(x_2))*\dots)*h(x_N),$$

What are the several x in there? In case of images, they are the positions of the original image (before it is reconstructed from $I_s(p)$), but how do I apply this concept to one-dimensional discrete-time signals?

I would be thankful for any help!

I am currently reading this paper which discusses several image interpolation methods, such as nearest neighbor and linear interpolation, using convolution filters. I first want to do this in 1D with discrete-time signals (e.g. [0 0 1 0 0]), however, I have some trouble understanding on how to derive the convolution filters from the mathematical definitions:

$$h_{NN}(x)=\cases{ 1, &if$\quad-\frac{1}{2} \leqslant x < \frac{1}{2},$\\ 0, &otherwise, }$$ $$h_{Lin}(x)=\cases{ 1-|x|, &if$\quad 0 \leqslant |x| < 1,$\\ 0, &if$\quad 1 \leqslant |x|.$ }$$

My main question is, how do I calculate the filter weights as well as the filter size? Furthermore, I am aware of the fact that I need to convolve the signal multiple times to upsample it, which the following equation in the paper also shows.

$$I(\mathbf{x}) = (\dots((I_s(\mathbf{p})*h(x_1))*h(x_2))*\dots)*h(x_N),$$

What are the several x in there? In case of images, they are the positions of the original image (before it is reconstructed from $I_s(p)$), but how do I apply this concept to one-dimensional discrete-time signals?

I would be thankful for any help!

I am currently reading this paper which discusses several image interpolation methods, such as nearest neighbor and linear interpolation, using convolution filters. I first want to do this in 1D with discrete-time signals (e.g. [0 0 1 0 0]), however, I have some trouble understanding on how to derive the convolution filters from the mathematical definitions:

My main question is, how do I calculate the filter weights as well as the filter size? Furthermore, I am aware of the fact that I need to convolve the signal multiple times to upsample it, which the following equation in the paper also shows.

What are the several x in there? In case of images, they are the positions of the original image (before it is reconstructed from $I_s(p)$), but how do I apply this concept to one-dimensional discrete-time signals?

I would be thankful for any help!

I am currently reading this paper which discusses several image interpolation methods, such as nearest neighbor and linear interpolation, using convolution filters. I first want to do this in 1D with discrete-time signals (e.g. [0 0 1 0 0]), however, I have some trouble understanding on how to derive the convolution filters from the mathematical definitions:

$$h_{NN}(x)=\cases{ 1, &if$\quad-\frac{1}{2} \leqslant x < \frac{1}{2},$\\ 0, &otherwise, }$$ $$h_{Lin}(x)=\cases{ 1-|x|, &if$\quad 0 \leqslant x < 1,$\\ 0, &if$\quad 1 \leqslant x.$ }$$

My main question is, how do I calculate the filter weights as well as the filter size? Furthermore, I am aware of the fact that I need to convolve the signal multiple times to upsample it, which the following equation in the paper also shows.

$$I(\mathbf{x}) = (\dots((I_s(\mathbf{p})*h(x_1))*h(x_2))*\dots)*h(x_N),$$

What are the several x in there? In case of images, they are the positions of the original image (before it is reconstructed from $I_s(p)$), but how do I apply this concept to one-dimensional discrete-time signals?

I would be thankful for any help!