Compute eye space from window space
This page will explain how to recompute eye-space vertex positions given window-space vertex positions. This will be shown for several cases.
Contents
Definitions
Before we begin, we need to define some symbols:
Symbol | Meaning |
---|---|
M | The projection matrix |
P | The eye-space position, 4D vector |
C | The clip-space position, 4D vector |
N | The normalized device coordinate space position, 3D vector |
W | The window-space position, 3D vector |
V_{x, y} | The X and Y values passed to glViewport |
V_{w, h} | The width and height values passed to glViewport |
D_{n, f} | The near and far values passed to glDepthRange |
From gl_FragCoord
gl_FragCoord.xyz is the window-space position W, a 3D vector quantity. gl_FragCoord.w contains the inverse of the clip-space W: .
Given these values, we have a fairly simple system of equations:
In a GLSL fragment shader, the code would be as follows:
vec4 ndcPos;
ndcPos.xy = ((2.0 * gl_FragCoord.xy) - (2.0 * viewport.xy)) / (viewport.zw) - 1;
ndcPos.z = (2.0 * gl_FragCoord.z - gl_DepthRange.near - gl_DepthRange.far) /
(gl_DepthRange.far - gl_DepthRange.near);
ndcPos.w = 1.0;
vec4 clipPos = ndcPos / gl_FragCoord.w;
vec4 eyePos = invPersMatrix * clipPos;
This assumes the presence of a uniform called viewport, which is a vec4, matching the parameters to glViewport, in the order passed to that function. Also, this assumes that invPersMatrix is the inverse of the perspective projection matrix (it is a really bad idea to compute this in the fragment shader). Note that gl_DepthRange is a built-in variable available to the fragment shader.
From XYZ of gl_FragCoord
This case is mostly useful for deferred rendering techniques, but the last method is also very usefu. In deferred rendering, we render the material parameters of our objects to images. Then, we make several passes over these images, loading those material parameters and performing lighting computations on them.
In the light pass, we need to reconstruct the eye-space vertex position in order to do lighting. However, we do not actually have gl_FragCoord; not for the fragment that produced the material parameters. Instead, we have the window-space X and Y position, from gl_FragCoord.xy, and we have the window-space depth, sampled by accessing the depth buffer, which was also saved from the deferred pass.
What we are missing is the original window-space W coordinate.
Therefore, we must find a way to compute it from the window-space XYZ coordinate and the perspective projection matrix. This discussion will assume your perspective projection matrix is of the following form:
[ xx xx xx xx ] [ xx xx xx xx ] [ 0 0 T1 T2 ] [ 0 0 E1 0 ]
The xx mean "anything;" they can be any value you use in your projection. The 0's must be zeros in your projection matrix. T1, T2, and E1 can be any arbitrary terms, depending on how your projection matrix works.
If your projection matrix does not fit this form, then the following code will get a lot more complicated.
From window to ndc
We have the XYZ of window space:
Computing the NDC space from window space is the same as the above:
Just remember: the viewport and depth range parameters are, in this case, the parameters that were used to render the original scene. The viewport should not have changed of course, but the depth range certainly could (assuming you even have a depth range in the lighting pass of a deferred renderer).
From NDC to clip
For the sake of simplicity, here are the equations for going from NDC space to clip space:
Derivation
Deriving those two equiations is very non-trivial; it's a pretty big stumbling block. Let's start with what we know.
We can convert from clip space to NDC space, so we can go back:
The problem is that we don't have C_{w}. We were able to use gl_FragCoord.w to compute it before, but that's not available when we're doing this after the fact in a deferred lighting pass.
So how do we compute it? Well, we know that the clip space position was originally computed like this:
Therefore, we know that C_{w} was computed by the dot-product of P with the fourth row of M. And given our above definition of the fourth row of M, we can conclude:
Of course, this just trades one unknown for another. But we can use this. It turns out that N_{z} has something in common with this:
It's interesting to look at where C_{z} comes from. As before, we know that it was computed by the dot-product of P with the third row of M. And again, given our above definition for M, we can conclude:
We still have two unknown values here, P_{z} and P_{w}. However, we can assume that P_{w} is 1.0, as this is usually the case for eye space positions. Given that assumption, we only have one unknown, P_{z}, which we can solve for:
Now armed with P_{z}, we can compute C_{w}:
And thus, we can compute the rest of C from this:
From clip to eye
With the full 4D vector C computed, we can compute P just as before:
GLSL example
Here is some GLSL sample code for what this would look like:
uniform mat4 persMatrix;
uniform mat4 invPersMatrix;
uniform vec4 viewport;
uniform vec2 depthrange;
vec4 CalcEyeFromWindow(vec3 windowSpace)
{
vec3 ndcPos;
ndcPos.xy = ((2.0 * windowSpace.xy) - (2.0 * viewport.xy)) / (viewport.zw) - 1;
ndcPos.z = (2.0 * windowSpace.z - depthrange.x - depthrange.y) /
(depthrange.y - depthrange.x);
vec4 clipPos;
clipPos.w = persMatrix[3][2] / (ndcPos.z - (persMatrix[2][2] / persMatrix[2][3]));
clipPos.xyz = ndcPos * clipPos.w;
vec4 eyePos = invPersMatrix * clipPos;
}
viewport is a vector containing the viewport parameters. depthrange is a 2D vector containing the glDepthRange parameters.
Optimized method from XYZ of gl_FragCoord
The previous method is certainly useful, but it's a bit slow. We can significantly aid the computation of the eye-space position by using the vertex shader to provide assistance. This allows us to avoid the use of the inverse perspective matrix entirely.
This method is a two step process. We first compute P_{z}, the eye-space Z coordinate. Then use that to compute the full eye-space position.
The first part is actually quite easy. Most of the computations we used above were necessary because we needed C_{w}, which we had to do since we needed a full clip-space position. This optimized method only needs to get P_{z}, which we can compute directly from W_{z}, the depth range, and the three components of the projection matrix:
Note that this also means that we don't need the viewport settings. We only need the depth range.
The trick to this method is what follows: how we go from P_{z} to the full eye-space position. To understand how this works, here's a quick bit of geometry:
The E in the diagram represents the eye position, which is the origin. P is the position we want, and P_{z} is what we have. So, what do we need to get P from P_{z}? What we need is a vector direction that points towards P, but has a z component of 1.0. With that, we just multiply that vector by P_{z}; the result will necessarily be P.
So how do we get this vector?
That's where the vertex shader comes in. In deferred rendering, the Vertex Shader is often a simple pass-through shader, performing no actual computation and passing no user-defined outputs. So we are free to use it for something.
In the vertex shader, we simply construct a vector from the origin towards each vertex coordinate. In the deferred pass, vertex positions are typically provided in two dimensions and in NDC space (since the clip-space W is 1.0, clip-space and NDC space are identical). The extents of the X and Y will be [-1, 1]. But we set the Z-distance of the vector to be -1.0. This constructs a vector that points into the scene in front of the camera.
Linear interpolation of this value will make sure that every vector computed for a fragment will have a Z-value of -1.0. And linear interpolation will also guarantee that it points directly towards the fragment generated.
We could have computed this in the fragment shader, but why bother? That would require providing the viewport transform to the fragment shader(so that we can transform W_{xy} to NDC space). And it's not like the VS is doing anything...
Once we have the value, we simply multiply it by P_{z} to get our eye-space position P.
Here is some shader code.
//Vertex shader
in vec2 clipPos;
out vec3 eyeDirection;
void main()
{
gl_Position = vec4(clipPos, 0, 1);
eyeDirection = vec3(clipPos.xy, -1);
}
//Fragment shader
uniform mat4 persMatrix;
uniform vec2 depthrange;
vec4 CalcEyeFromWindow(vec3 windowSpace, vec3 eyeDirection)
{
float ndcZ = (2.0 * windowSpace.z - depthrange.x - depthrange.y) /
(depthrange.y - depthrange.x);
float eyeZ = persMatrix[3][2] / ((persMatrix[2][3] * ndcZ) - persMatrix[2][2]);
return vec4(eyeDirection * eyeZ, 1);
}