# Downsampling an image using sinc interpolation?

I have a discrete image of size $$2^N \times 2^N$$ which I would like to iteratively downsample to produce a pyramid with image sizes $$2^k\times 2^k, \, k=0,\ldots,N$$. That is, each subsequent image will have half the resolution along each axis. So I would like to better understand how this is modeled mathematically. My current understanding of the ideal downsampling of a continuous image $$f$$ is that one wants to compute the discrete image $$\boldsymbol{g} \in \mathbb{R}^{2^{N-1}\times 2^{N-1}}$$ such that:

$$g_{ij}=\int_{x_{2i}}^{x_{2(i+1)}}\int_{y_{2j}}^{y_{2(j+1)}}f(x,y)\,dy\,dx,$$

where pixel $$(i,j)$$ on the coarser level corresponds to some rectangle $$[x_{2i},x_{2(i+1)}]\times[y_{2j},y_{2(j+1)}]$$. The main trouble is that I am not given the continuous function, but instead some discrete data $$\boldsymbol{h} \in \mathbb{R}^{2^N \times 2^N}$$. To reconstruct a continuous function using a linear model I would have something of the form:

$$f(\boldsymbol{z}) = \sum_{i=1}^{2^N}\sum_{j=1}^{2^N}c_{ij}\phi_{ij}(\boldsymbol{z}),$$

where $$c_{ij}$$ are some coefficients, and $$\phi_{ij}$$ are some predetermined basis functions. A standard constraint used to derive $$c_{ij}$$ is the interpolation requirement: $$h_{ij} = f(\boldsymbol{z}_{ij})$$, where for example the samples are taken in the middle of each pixel: $$\boldsymbol{z}_{ij} = \frac{1}{2}(x_i+x_{i+1}, y_j+y_{j+1})$$. This is a linear system, and provided that the basis functions are linearly independent, then there is a unique set of coefficient $$\boldsymbol{c} \in \mathbb{R}^{2^N\times 2^N}$$ corresponding to $$\boldsymbol{h}\in \mathbb{R}^{2^N\times 2^N}$$.

Correct me if I am wrong, but I believe that if $$\phi_{ij}$$ are box functions, then computing the integral for $$g$$ is equivalent to using bilinear basis functions interpolation and then sampling at the midpoint. I believe a similar thing holds for higher degrees, where if $$f$$ is assumed to be formed by polynomials of degree $$n$$ then evaluating the integral would be equivalent to sampling a $$n+1$$ degree interpolated version of $$\boldsymbol{h}$$. The box filter version seems to be special in that it requires the basis functions to be Dirac deltas with a scaling factor of $$\frac{1}{4}$$ and $$\boldsymbol{c} = \boldsymbol{h}$$.

And this is the point where I am not certain about things anymore. I have seen the $$sinc$$ function described as an ideal interpolator in the context of the Nyquist-Shannon sampling theorem, and I have also seen it discussed in the context of downsampling. How does such a $$sinc$$ downsampling relate to the above mathematical modelling of the problem? I am not quite sure how this downsampling is achieved in the first place or what it means when a discrete set of samples are considered. I understand that $$\mathcal{F}[f * sinc] = \mathcal{F}[f] \cdot box$$ can be used to cut off frequencies above the Nyquist frequency when $$f$$ is a continuous function. On the other hand I have seen people just put zeroes as DFT coefficients (notably this is the setting of a discrete signal). Is the idea that they consider $$f$$ a sum of Dirac deltas weighted by $$h_{ij}$$, and then consider the convolution with an appropriate $$sinc$$ to cut-off frequencies that are too high? This would essentially result in considering $$sinc$$ as basis functions. Where is the integration step that I have at the very beginning however? Am I misunderstanding how $$sinc$$ downsampling is supposed to work? Also there seem to be some concerns with ringing artifacts arising from the above? What is the mathematical reason for this?

• My understanding is that the sampling theorem covers continuous functions, and infinitely many samples of those. Instead in the above we start with a discrete function $\boldsymbol{h}$. How does that fit into the $sinc$ downscaling? Should I mirror or toroidally extend the samples of $\boldsymbol{h}$ and assume that those were the product of sampling $f$? Thus the assumption being that: $f(z) = \sum_{k=-\infty}^{\infty} h_k sinc\left(\frac{z-kT}{T}\right)$, where $h_k = f(kT)$? Where does the integration over each pixel come into play? I am looking for the mathematical model's motivation. Commented Jan 26, 2022 at 22:51