# Image contribution function and reconstruction using filters

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?