January 22, 2012
by colorgas

## Why I want to re-understand VSM

In the original VSM paper，William and Andrew derived the MAGIC result from probability theory:

Note:

Note:

- means the shadow map depth or shadow caster depth in shadow map
- means the interpolated( filtered) value in shadow map
- means the interpolated(filtered) value in shadow map
*D* means the shadow receiver’s depth)

The derivation looks very beautiful, but not so easy to understand , especially the idea to treat z as a random variable, so here I’ll try to explain it by middle school math, then we can kill the uncomfortable MAGIC feeling.

## Analysis

Firstly, let’s back to the problem why VSM had been invented : **Soft Shadow**.

In the real word，soft shadow are generated with a physic explanation（eg area light），

but visually, Soft shadow just means a smooth transition between shadowed area and unshadowed area.

so there are many methods to fake it in computer graphics, VSM is just one of them ( note: even with beautiful math, VSM is not a physic correct algorithm at all).

In the above graph, we have 2 neighborhood texels in shadow map corresponding to 2 light rays, their shadow depth value are *Z1* and *Z2*,.

According shadow map algorithm, we can know pixel A is in shadow and pixel B is lit, what we care about is the area between A and B,

If applying regular shadow test, a sudden shadow density changing must happen at somewhere between A and B. Even we can use bilinear filter to fetch shadow, the smooth input value doesn’t generate a smooth output value because shadow testing result is a binary value. That is why we said shadow map is not filterable.

Let’s see what will happen if we apply the MAGIC shadow testing formula from VSM.

First, let’s introduce a interpolating variable t, t will be changing linearly (0-1) from pixel A to pixel B.

Then with linear filtering, between A and B, we can write:

Then

So

here, we define:

Looks still mess, right?

Let’s see 2 special cases :

### Special case 1: Perpendicular Reciever:

when receiver surface perpendicular to light direction (see reciever Rp)

For this scenario , we can easily know ( between t=0 and t=1)

So

Then final result:

So for this case, the shadow result will follow the most simple smooth transition when t change from 0 to 1 : Linear transition.

### Special case 2: Reciever on interp(z):

When Shadow Reciever are on interp(z) ( see the red dash line in above picture).

According

we can know immediately:

Combine the two special cases，and the property of that Pshadow is a Monotonic function decreasing function to D （see Appendix）

For the area between interp(z) (red dash line) and Reciever Rp, we have：

we can say: PShadow is Monotonic changing between t and 1 along D decreasing. it’s a smooth transition too.

For the portion below Rp, the situation is same but the* PShadow < t.*

## Short Summary

- with t increasing to 1, PShadow will increase to 1 smoothly
- with D decreasing to interp(z), PShadow will increasing to 1 smoothly too.
- when ( receiver surfaceperpendicular to light) , it’s a linear transition.

## Conclusion

From the result, we can say ShadowMap filtering start to make sense, since a interpolated shadow depth can generate a shadow value between 0 and 1. that’s why we say VSM is filterable.

we can go one step forward, if we blur the shadow map texture, the same transition effect will expand from 2 neighborhood shadow depth texels to more adjacent texel, then it results a more smooth shadow transition, that’s why we say VSM is pre-filtable.

### Appendix：Pshadow is a Monotonic decreasing funtion to D.

Check the original define:

since if t is fixed,

is a fixed value,

so if D is increasing, then d is increasing, then Pshadow is decreasing,

Then we can know Pshadow is a Monotonic decreasing function to D.

**PS**. the original VSM paper can be downloaded at :

VSM “Variance” http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.104.2569&rep=rep1&type=pdf