An interpretation of depth value

Posted 2021-11-29 dydx

An Interpretation of Depth Value

 

Recently when I am working on Screen Space Reflection, I noticed there are some subtleties in the computation of depth value.

 

In this article I will explain the deeper meaning in the computation of depth value.

 

Perspective Projection Matrix

 

First, it is well known that the DirectX perspective projection matrix is:

In this matrix:

 is width of the screen

 is height of the screen

 is near clipping plane

 is far clipping plane

All these values are measured in camera space.

 

Since DirectX uses row vector, the z and w component are:

After perspective division, the actual depth value is:

But what does it mean? Is there any deeper meaning in it? Why not use a simpler one, like linear interpolation?

 

The answer to the second question is yes.

To dive into the deeper meaning, first we express  as an expression of , we get:

 

Something interesting happens! The depth value is linear interpolation weight for the reciprocal of near and far value!

 

Sounds great, but who cares? Yes, in most cases, you don’t need to care about that. However, in some specific case, it will bring you a lot of convenience. Screen space reflection is an example.

 

Ray Marching in Screen Space

 

In screen space reflection, one need to perform ray marching and find the intersection of a ray and depth buffer. Of course, ray marching can be performed in view space, but it’s difficult to choose a proper step size because the same step size in view space appear smaller and smaller when the ray is moving away from the camera. Alternatively, performing ray marching in screen space can avoid this problem because you can choose the step size based on the pixel size. However, as the depth value is a non-linear function of z value in view space, the step size for depth value is not constant. If you use the projection matrix to calculate the depth value based on the x and y values in every step, that will cause a lot of computation. Is there a fast way to calculate the depth value? Yes! And the linear interpolation nature of the reciprocal of depth value is the heart of this method.

 

To explain this, I would like to formulate the problem first. Suppose you know the view space coordinates of the both ends of a line segment AB, denoted as  and . The projection matrix is also known, so you can calculate their screen space coordinates  and . You want to perform linear interpolation on image plane with  the interpolation weight. The xy coordinate of point C is easy:

 

But how can I get  conveniently, given  ? I can calculate  given , it’s

So, the next step is to know the relationship between s and t.

Now let’s look at the picture below, it’s the equivalent version of the last picture.

Let

,

.

Then we have:

Also we know

and

Put them together, we have

 

Put it into  , we have:

 

Finally, we get

That is to say, instead of using  to linear interpolate  ,we can use  to linear interpolate ! We’ve already use  to linear interpolate xy coordinate. Furthermore, we can consider  as a special coordinate axis. So, in the (x’, y’, 1/z) coordinate space, every axis can be linear interpolated!

 

Why is the depth value like that?

From the first section, we know the depth value is the linear interpolation weight for the reciprocal of near and far value.

From the second section, we know (x’, y’, 1/z) can be linear interpolated.

Then we have

That means  linearly interpolates depth value, and in the (x’, y’, depth) coordinate space, every axis can be linear interpolated.

To show the benefit of this, let’s see the picture above, suppose there is a line segment in view space, if we choose a na?ve depth (for example, linear interpolating near and far value), the line segment will become a curve in (x’, y’, depth) space. However, by using DirectX depth (OpenGL depth is similar), the line segment will be still a line segment!

 

Things that appear straight/planar in view space, will also appear straight/planar in clip space. That is the idea behind the depth computation formula.

 

 

来自为知笔记(Wiz)

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