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Let $W_{\text e}$ denote the camera sensor responsivity (i.e. importance) and $h_j$ and $W_{\text e}^{(j)}$ denote the image reconstruction filter and sensor response associated with the $j$th pixel, respectively, i.e. $$W_{\text e}^{(j)}=h_jW_{\text e}.\tag1$$ The measurement of the $j$th pixel is given by \begin{equation}\begin{split}\Phi_j&:=\sigma_M({\rm d}x_0)\int\sigma_M({\rm d}x_1)g(x_0\leftrightarrow x_1)W_{\text e}^{(j)}(x_1\to x_0)L_{\text o}(x_1\to x_0)\\&=\int\lambda({\rm d}x)h_j(x_1\to x_0)f(x),\end{split}\tag2\end{equation} where $g$ denotes the geometry term, $\lambda$ is the infinite-product area measure on the path space and $f$ is the measurement contribution function, i.e. $$f(x)=g(x_0\leftrightarrow x_1)W_{\text e}(x_1\to x_0)\prod_{i=2}^kg(x_{i-1}\leftrightarrow x_i)f_{\text s}(x_i\to x_{i-1}\to x_{i-2})L_{\text e}(x_k\to x_{k-1}),\tag3$$ if $x$ is a path of length $k\in\mathbb N$.

Usually, in the context of Metropolis Light Transport, the scalar contribution function $p$ is defined to be the luminance of $f$. So, by definition, $p(x)=0$ whenever $(x_0,x_1)$ doesn't belong to $$B:=\left\{(x_0,x_1):W_{\text e}(x_1\to x_0)\ne 0\right\}.$$ Now let $$B_j:=\left\{(x_0,x_1):W_{\text e}^{(j)}(x_1\to x_0)\ne 0\right\}.$$ We clearly have $$B=\bigcup_jB_j.\tag4$$ However, in light of the way samples are splat to image (see http://www.pbr-book.org/3ed-2018/Light_Transport_III_Bidirectional_Methods/Metropolis_Light_Transport.html#fragment-Splatbothcurrentandproposedsamplestomonofilm-0 or https://pdfs.semanticscholar.org/a386/55316ee7f438ba115b38e8d0b1410c691a26.pdf and my other question), it seems like it is assumed that the union in $(4)$ is disjoint. Is this really guaranteed to be the case?

Taking a look in Section 2.6.4, on page 51, of this PhD thesis, this seems at least to be the case when the camera is a pinhole camera. The authors write: $W_{\text e}^{(j)}$ is non-zero only for the set of directions within the pyramid formed by the pinhole and the four corners of the corresponding pixel $j$.

This seems to indicate that (at least for the pinhole camera) the "support" of $W_{\text e}^{(j)}$ is limited to the image rectangle region occupied by the $j$th pixel. But is this really true for any filter $h_j$ or only for a special one?

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    $\begingroup$ The $B_j$ are not necessarily disjoint - consider having an aperture with some area. Furthermore, a reconstruction filter $h$ in theory (I am considering what PBRT does here), can have an effect over the whole screen if desired. Consider a Laplace or Poisson reconstruction problem from sparse samples (the Dirichlet data). The latter is even the case if you consider gradient domain path tracing. $\endgroup$ – lightxbulb Feb 21 at 12:25
  • $\begingroup$ @lightxbulb You're right. Meanwhile, I've figured the source of my understandings out: computergraphics.stackexchange.com/a/9598/9254. $\endgroup$ – 0xbadf00d Feb 21 at 14:28
  • $\begingroup$ On second thought - the $B_j$ are disjoint. I was thinking in terms of directions: (x_1 - x_0), in which case they would not have been. But as a tuple, they are, since one of the points (depending on your convention) is on the film, and those sets (of the points corresponding to a specific pixel) are disjoint. Basically pixels do not overlap. Now I am curious what the result will be of having overlapping pixels. $\endgroup$ – lightxbulb Feb 22 at 8:34
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    $\begingroup$ Depends on your convention/implementation - pixels can have any size you would like, as long as you account for this properly (I am not sure whether pbrt uses size 1). You don't have to think too hard - just check PBRT's film functions. As for Halton - it's deterministic - then taking the expectation of the error obviously does nothing and you get that it's biased, sure. However, low-discrepancy samplers have a different property that guarantees bounds on the error and the equivalent of consistency also. Look into Koksma-Hlawka's inequality. $\endgroup$ – lightxbulb Feb 22 at 14:38
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    $\begingroup$ @lightxbulb I'm sorry to bother you again (it's the last time on this, I promise), but the whole scaling thing is continuing to confuse me. Can you tell me what they mean by the "image radiance function" $L$ here pbr-book.org/3ed-2018/Monte_Carlo_Integration/Bias.html#? $\endgroup$ – 0xbadf00d Feb 23 at 16:01

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