The operation (filtering an ideal, mathematically perfect image) can be described in two equivalent ways:
- You take a square a single pixel spacing wide by its center and attach it to a sampling point (“center of a pixel”). The value of that pixel is then your mathematically perfect image (of a polygon) integrated over that square (and normalized). This is perhaps the more intuitive definition.
- You take a box kernel (the indicator function of that square, centered, normalized), take the convolution[1] of it with the original perfect image, then sample the result at the final points (“pixel centers”). This is the standard definition, which yields exactly the same result as long as your kernel is symmetric (which the box kernel is).
The connection with the pixel-image filtering case is that you take the perfect image to be composed of delta functions at the original pixel centers and multiplied by the original pixel values. That is, in the first definition above, “integrate” means to sum the original pixel values multiplied by the filter’s value at the original pixel centers (for a box filter, zero if outside the box—i.e. throw away the addend—and a normalization constant if inside it). Alternatively, in the second definition above, “take the convolution” means to attach a copy of the filter (still sized according to the new pixel spacing) multiplied by the original pixel value to the original pixel center and sum up any overlaps. Try proving both of these give the answer you’re already accustomed to.
This is the most honest signal-processing answer, and it might be a bit challenging to work through but my hope is that it’ll be ultimately doable. I’m sure there’ll be neighboring answers in more elementary terms, but this is ultimately a (two-dimensional) signal processing task and there’s value in knowing exactly what those signal processing people are talking about.
[1] (f∗g)(x) = (g∗f)(x) = ∫f(y)g(x-y)dy is the definition you’re most likely to encounter. Equivalently, (f∗g)(x) is f(y)g(z) integrated over the line (plane, etc.) x=y+z, which sounds a bit more vague but exposes the underlying symmetry more directly. Convolving an image with a box filter gives you, at each point, the average of the original over the box centered around that point.
- You take a square a single pixel spacing wide by its center and attach it to a sampling point (“center of a pixel”). The value of that pixel is then your mathematically perfect image (of a polygon) integrated over that square (and normalized). This is perhaps the more intuitive definition.
- You take a box kernel (the indicator function of that square, centered, normalized), take the convolution[1] of it with the original perfect image, then sample the result at the final points (“pixel centers”). This is the standard definition, which yields exactly the same result as long as your kernel is symmetric (which the box kernel is).
The connection with the pixel-image filtering case is that you take the perfect image to be composed of delta functions at the original pixel centers and multiplied by the original pixel values. That is, in the first definition above, “integrate” means to sum the original pixel values multiplied by the filter’s value at the original pixel centers (for a box filter, zero if outside the box—i.e. throw away the addend—and a normalization constant if inside it). Alternatively, in the second definition above, “take the convolution” means to attach a copy of the filter (still sized according to the new pixel spacing) multiplied by the original pixel value to the original pixel center and sum up any overlaps. Try proving both of these give the answer you’re already accustomed to.
This is the most honest signal-processing answer, and it might be a bit challenging to work through but my hope is that it’ll be ultimately doable. I’m sure there’ll be neighboring answers in more elementary terms, but this is ultimately a (two-dimensional) signal processing task and there’s value in knowing exactly what those signal processing people are talking about.
[1] (f∗g)(x) = (g∗f)(x) = ∫f(y)g(x-y)dy is the definition you’re most likely to encounter. Equivalently, (f∗g)(x) is f(y)g(z) integrated over the line (plane, etc.) x=y+z, which sounds a bit more vague but exposes the underlying symmetry more directly. Convolving an image with a box filter gives you, at each point, the average of the original over the box centered around that point.