<p>The specific chapter about this is here - <a href="http://www.pbr-book.org/3ed-2018/Sampling_and_Reconstruction/Sampling_Theory.html" rel="nofollow noreferrer">Sampling Theory</a><br>
Unlike what I read anywhere else, it defines Shah as: <span class="math-container">$$Ш_T(x)=T\sum\nolimits_i{\delta{(x-Ti)}}$$</span>
And the T is still present in the reconstructed function: <span class="math-container">$$f\tilde(x)=T\sum\limits_{i=-\infty}^\infty{f(iT)r(x-iT)}$$</span>
where r(x) is a reconstruction filter.<br>
Everywhere else I can find gives: <span class="math-container">$$s_T{(x)}=\sum\nolimits_i{\delta{(x-Ti)}}$$</span>
Can someone work out the maths for me? Why is the difference?</p>
<p>I haven't been doing maths for years, and even back in my school days, I don't think I was a good student, so please be gentle.</p>

CC BY-SA 4.0
2020-10-26T17:10:42.447Z
`Why is the difference?` - they're just using a different definition with a scaling factor, it shouldn't really matter.
lightxbulb
CC BY-SA 4.0
2020-10-27T01:42:09.120Z
@lightxbulb Thanks for your reply! I thought so at first, but the triangle filter r(x) = max(0, 1 - |x|) that they give as an example in 7.1.2, doesn't seem to unscale it (while the box function given later does). Maybe it's simply too trivial that they overlooked? Still what good does it do to have the T factor and then 1/T it?
Eugene
CC BY-SA 4.0
2020-10-27T07:17:58.377Z
It doesn't change anything if you have $T$ and then unscale it. It could be that they wanted a specific relationship to the Fourier transform. Similar to how one would choose different scaling factors for the DFT, e.g. to make it unitary.
lightxbulb
<pbrt><pbrt-v3>
Why is there a T factor in the definition of Shah given in Matt Pharr's Physically Based Rendering?
Eugene
# Why is there a T factor in the definition of Shah given in Matt Pharr's Physically Based Rendering?

(Asked by: username_0 on 2020-10-26)

The specific chapter about this is here - [Sampling Theory](http://www.pbr-book.org/3ed-2018/Sampling_and_Reconstruction/Sampling_Theory.html)  
Unlike what I read anywhere else, it defines Shah as: $$Ш\_T(x)=T\sum\nolimits\_i{\delta{(x-Ti)}}$$
And the T is still present in the reconstructed function: $$f\tilde(x)=T\sum\limits\_{i=-\infty}^\infty{f(iT)r(x-iT)}$$
where r(x) is a reconstruction filter.  
Everywhere else I can find gives: $$s\_T{(x)}=\sum\nolimits\_i{\delta{(x-Ti)}}$$
Can someone work out the maths for me? Why is the difference?

I haven't been doing maths for years, and even back in my school days, I don't think I was a good student, so please be gentle.

**Tags:** pbrt, pbrt-v3

> **Comments:**
> **username_1**: `Why is the difference?` - they're just using a different definition with a scaling factor, it shouldn't really matter.  
> **username_0**: @username_1 Thanks for your reply! I thought so at first, but the triangle filter r(x) = max(0, 1 - |x|) that they give as an example in 7.1.2, doesn't seem to unscale it (while the box function given later does). Maybe it's simply too trivial that they overlooked? Still what good does it do to have the T factor and then 1/T it?  
> **username_1**: It doesn't change anything if you have $T$ and then unscale it. It could be that they wanted a specific relationship to the Fourier transform. Similar to how one would choose different scaling factors for the DFT, e.g. to make it unitary.  

---

## 0 answers

