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So I've been reading Physicall Based Rendering which in section 16.4.3 defines the "image contribution function" to be the following

$$I_j = \int_\Omega h_j(X)L(X)d\Omega$$

where the variables are defined in the following way as I understand them

  • The domain $\Omega$ is the space of paths in the scene
  • $h_j$ is the "image reconstruction filter" for the $j$th pixel on a given path. My understanding is that this really only depends on where the path is on image plane.
  • $I_j$ is the value of the $j$th pixel
  • $L$ is the radiance delivered on a given path

They also give the following equation which is just an application of Monte Carlo importance sampling when drawing from a sample distribution $p$

$$\int_\Omega h_j(X)L(X)d\Omega \approx \frac{1}{N}\sum_i^N \frac{h_j(X_i)L(X_i)}{p(X_i)}$$

Earlier in section 7.8 the define equation 7.12 to be the following

$$I(x, y) = \frac{\sum_i f(x - x_i, y - y_i)w(x_i, y_i)L(x_i, y_i)}{\sum_i f(x - x_i, y - y_i)}$$

Where the variables are defined in the following way as I understand them

  • $(x_i, y_i)$ is the location on the image plane of the $i$th sample

  • $L(x_i, y_i)$ is the radiance delivered to $(x_i, y_i)$ by the $i$th sample

  • $I(x, y)$ is the value of the pixel at $(x, y)$

  • $f$ is a filter function like a tent or sinc filter

  • $w$ is a sample contribution weight. For simplicity of this question. I'll assume this to just be the constantly $1$ function. In the later chapters where the "importance function" appears to subsume this or be the same thing

So for a given pixel $j$ at location $(x, y)$ on the image I interpret $I(x, y) = I_j$ and for a given sample $L(X_i) = L(x_i, y_i)$ and given that I'm ignoring $w$ the following equation should hold. Taking $f(X_i) = f(x_i, y_i)$ as well

$$\frac{\sum_i^N f(X_i)L(X_i)}{\sum_i^N f(X_i)} = \frac{1}{N}\sum_i^N \frac{h_j(X_i)L(X_i)}{p(X_i)}$$

While its clear how to implement $f$ it isn't clear to me how to implement $h_j$. I considered that $L(x_i, y_i)$ might actually be $\frac{L(X_i)}{p(X_i)}$ in the importance sampling case which gets rid of the discrepancy caused by the probability distribution but that implies that $h_j$ must be defined to be the following

$$h_j(X_i) = \frac{N \cdot f(X_i)}{\sum_i^N f(X_i)}$$

But it doesn't seem to me that $h_j$ should be dependent on the number of samples $N$ which means I must have made a mistake in some assumption that I've made. I made a lot of assumptions to attempt to connect these to equations but I'm still at a loss. What mistakes have I made so far? How does one implement $h_j$ in the first equation from section 16.4.3?

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