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# Directional light, shadow mapping and matrices..

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14 replies to this topic

### #1KaiserJohan  Members

Posted 20 January 2014 - 04:37 AM

I'm using OpenGL and I've managed to get spotlight shadowmapping to work. I have the position and direction of the spotlight. The view matrix is constructed using glm::lookAt(), and the projection matrix is a perspective matrix.

For directional lights however, all I have is the light direction.

1) What is the view matrix and how to construct it, given only light direction?

2) What is the projection matrix and how is it constructed?

3) Do you still do a shadow map pass with directional lights before the actual lighting pass?

Thanks

### #2lunkhound  Members

Posted 20 January 2014 - 09:31 PM

With a directional light, some things are simpler than a spotlight, like setting up the transforms.  A directional light uses a very simple orthographic projection, as opposed to a perspective projection.

However, most things are more complex because a directional light in theory affects everything out to infinity, but in practice your shadow texture is only a finite size and you don't want to stretch it over too large of an area or the resulting shadows will look very pixely.  Shadowmapping with a directional light is all about optimizing the shadow texture usage to make the projection of shadow map texels onscreen as small as possible.

In addition to the light direction, you'll need to define a frustum for your directional shadow camera.  An orthographic frustum is just a box oriented so that the z-axis is aligned with the projection direction.  The box should include everything that casts or receives a shadow, but at the same time you want the box to be as small as possible because your shadow texture will be stretched over the x and y extents of the box.  Minimizing the z-extents of the box isn't quite as critical, but still important, as this range is mapped into your shadow texture bit depth.

You'll want to calculate the position and size of this box based on your main camera's frustum.  If your main camera can see very far, like say 1000 meters, you won't be able to cover the whole area seen by your main camera with just one shadow texture.  A single shadow texture will only adequately cover a few tens of meters from the camera (depending on shadow texture resolution).  If you want to cover more area then that, you may want to look into cascaded shadow mapping and similar techniques.

Once your shadow camera frustum is defined, the matrices are quite simple to set up.

1. The view matrix is just a transform with only position and rotation, centered in the center of the box and rotated  to align with the box.

2. The projection matrix is just a pure scale matrix that maps the extents of the box into clip space (so each axis is mapped into a -1 to 1 range).

3. Yes you still need to render the shadow texture.  With cascaded shadow mapping, you'll need to render MULTIPLE shadow textures.

### #3KaiserJohan  Members

Posted 21 January 2014 - 03:41 AM

The view matrix - why is it in the center of the box, and not infront of it? Otherwise wont half of the geometry be outside the view? And aligned, as in looking down the -Z axis?

Projection matrix - how would you calculate such a scale matrix? Does glm::ortho, which creates an orthographic matrix, do the same thing?

Edited by KaiserJohan, 21 January 2014 - 03:41 AM.

### #4lunkhound  Members

Posted 21 January 2014 - 01:06 PM

The view matrix, I chose to put the origin of the view transform in the center of the box because it simplifies constructing the projection matrix.  You can put the view transform at one end of the box or the other if you prefer, but then you'll need to introduce a translation component into the projection matrix to compensate.  It's just easier to put it at the center.

And yes, the -Z axis.

The projection matrix would just be:

1/he.x      0          0         0

0         1/he.y      0         0

0          0          1/he.z    0

0          0          0         1

(where "he" is the half-extents of your box).

Also you could use glOrtho( -he.x, he.x, -he.y, he.y, -he.z, he.z ) to define the projection matrix if you like.  (I may have flipped the signs on z there)

It is probably safer to use glOrtho, as I'm not completely sure about the range of clipping coordinates in z.

Edited by lunkhound, 21 January 2014 - 01:28 PM.

### #5KaiserJohan  Members

Posted 26 January 2014 - 12:23 PM

I've got shadow mapping working, using the following matrices:

Mat4 viewMatrix = LookAt(cameraPosition + (directionalLight.mLightDirection * Z_FAR/2.0f), -directionalLight.mLightDirection, Vec3(0.0f, 1.0f, 0.0f));
Mat4 lightVP = CreateOrthographicMatrix(-Z_FAR, Z_FAR, -Z_FAR, Z_FAR, Z_NEAR, Z_FAR) * viewMatrix;


The problem is when moving around the camera, so does the shadows move around, which is unrealistic. What is the proper way to build the view matrix?

Edited by KaiserJohan, 26 January 2014 - 01:02 PM.

### #6lunkhound  Members

Posted 29 January 2014 - 01:02 AM

What you have there doesn't look right at all.  It isn't that easy.

I would do it something like this:

1. Find a vector orthogonal to the light direction.

2. Use the cross-product to find another vector orthogonal to the other two.  Now you have 3 orthogonal unit vectors.

3. Form a rotation matrix from the 3 vectors.  The light direction should be the z-axis, and the other two are x and y.  Make sure the handedness is correct.  The x-axis cross y-axis should equal the z-axis, not the negative z-axis.  This rotation matrix transforms directions from world space into the light space.

4. Generate a list of the 8 corner vertices of the main camera's frustum.  You'll need the main camera's position, z-near, z-far, horizontal field of view, and vertical field of view.

5. Loop over the 8 corner vertices, and apply the rotation matrix to each in turn and record the minimum and maximum on the x, y, and z axes.  This gives the extents of the box I mentioned in an earlier post.

6. Find the coordinates of the center of the box in light space i.e. ((xmin + xmax)*0.5, (ymin + ymax)*0.5, (zmin + zmax)*0.5)

Now it is straightforward to assemble the shadowmap view matrix.  The 3x3 rotation matrix goes into the rotation part of the 4x4 matrix.  For the translation part of the matrix, you want to the box-center coordinates NEGATED.  And the last row of the matrix is just (0 0 0 1).

### #7KaiserJohan  Members

Posted 31 January 2014 - 02:47 PM

Thanks for the reply. I've tried it, but I do not seem to get any shadows. Here is an image:

I tried it with the following code, as per your suggestions:

        for (const RenderableLighting::DirectionalLight& directionalLight : lighting.mDirectionalLights)
{
Vec3 lightDir = directionalLight.mLightDirection;   // = Vec3(0.0f, 1.0f, 1.0f)
Vec3 perpVec1(1.0f, 0.0f, 0.0f);
Vec3 perpVec2(0.0f, 1.0f, -1.0f);
Mat3 rotationMatrix;                // using glm, so matrices are column-major
rotationMatrix[0] = perpVec1;
rotationMatrix[1] = perpVec2;
rotationMatrix[2] = lightDir;

assert(glm::cross(perpVec1, perpVec2) == lightDir);

Vec3 corners[8] = {Vec3(-25.0f, -25.0f, 25.0f), Vec3(-25.0f, -25.0f, -25.0f), Vec3(-25.0f, 25.0f, 25.0f), Vec3(-25.0f, 25.0f, -25.0f),
Vec3(25.0f, -25.0f, 25.0f), Vec3(25.0f, -25.0f, -25.0f), Vec3(25.0f, 25.0f, 25.0f), Vec3(25.0f, 25.0f, -25.0f) };

for (uint32_t i = 0; i < 8; i++)
corners[i] = corners[i] * rotationMatrix;

float minX = corners[0].x, minY = corners[0].y, minZ = corners[0].z, maxX = corners[0].x, maxY = corners[0].y, maxZ = corners[0].z;
for (uint32_t i = 0; i < 8; i++)
{
if (corners[i].x < minX)
minX = corners[i].x;
if (corners[i].x > maxX)
maxX = corners[i].x;
if (corners[i].y < minY)
minY = corners[i].y;
if (corners[i].y > maxY)
maxY = corners[i].y;
if (corners[i].z < minZ)
minZ = corners[i].z;
if (corners[i].z > maxZ)
maxZ = corners[i].z;
}

Mat4 viewMatrix(rotationMatrix);
viewMatrix[3][0] = -(minX + maxX) * 0.5f;
viewMatrix[3][1] = -(minY + maxY) * 0.5f;
viewMatrix[3][2] = -(minZ + maxZ) * 0.5f;
viewMatrix[0][3] = 0.0f;
viewMatrix[1][3] = 0.0f;
viewMatrix[2][3] = 0.0f;
viewMatrix[3][3] = 1.0f;

Mat4 lightVP = glm::ortho(minX, minX, minY, maxY, minZ, maxZ) * viewMatrix;

GLCALL(glDrawBuffer(GL_NONE));

GLCALL(glBindFramebuffer(GL_DRAW_FRAMEBUFFER, mGBuffer.mFramebuffer));
GLCALL(glDrawBuffer(GBuffer::GBUFFER_COLOR_ATTACHMENT_FINAL));
DirLightLightingPass(directionalLight, gNDCBiasMatrix * lightVP, lighting.mGamma, lighting.mScreenSize);
}


Do you spot anything that stands out? Besides the view/projection calculations, the rest should be fine

Edited by KaiserJohan, 31 January 2014 - 03:59 PM.

### #8lunkhound  Members

Posted 31 January 2014 - 04:02 PM

Well, the rotation matrix has some problems for starters.  The rows of the rotation matrix all need to be unit magnitude (some of yours aren't), and they need to be perpendicular to each other (also not true for your matrix).  So what you've got there is a matrix that scales, skews, and rotates.

Another problem that jumps out is that you aren't calculating the corners of the main camera's frustum.  Instead you've got the corners of a 50 x 50 x 50 box centered on the origin.  Your main camera has a perspective projection matrix, so the shape of it isn't a box, it is more of a truncated pyramid.

If you take your main camera's view-projection matrix, invert it (note, NOT a simple transpose), and then run the corners of a 2 x 2 x 2 box centered on the origin through it, that should give the corners of your camera's frustum.

Also your orthographic matrix is also wrong.  You should calculate the half-extents of the box from your minX, maxX, etc, and express the ortho in terms of the half-extents.

### #9KaiserJohan  Members

Posted 31 January 2014 - 04:51 PM

I'm using the cross product to find out the perpendicular vectors; I forgot to normalize lightDirection - the following should give me the proper rotation matrix no?

            Vec3 lightDir = glm::normalize(directionalLight.mLightDirection);   // = Vec3(0.0f, 1.0f, 1.0f)
Vec3 perpVec1 = glm::cross(lightDir, Vec3(0.0f, 0.0f, 1.0f));
Vec3 perpVec2 = glm::normalize(glm::cross(lightDir, perpVec1));
Mat3 rotationMatrix;                // using glm, so matrices are column-major
rotationMatrix[0] = perpVec1;
rotationMatrix[1] = perpVec2;
rotationMatrix[2] = lightDir;

assert(glm::dot(lightDir, perpVec1) == 0);
assert(glm::dot(lightDir, perpVec2) == 0);


As for the corners of the camera frustrum, I'm thinking one step at a time, the simplest would be to just use static boundaries like the box I posted, it would work just as well for the moment, right?

When you say half-extents, you mean the min/max divided by two, like this?

Mat4 lightVP = glm::ortho(minX/2.0f, minX/2.0f, minY/2.0f, maxY/2.0f, minZ/2.0f, maxZ/2.0f) * viewMatrix;


### #10lunkhound  Members

Posted 31 January 2014 - 06:21 PM

You also need to normalize the first cross product there (perpVec1).

If the 50x50x50 box encloses your scene, then yes that should work fine as a temporary hack.

The extents would be:

Vec3 extents( maxX - minX, maxY - minY, maxZ - minZ );

The half extents would just be 0.5 * extents:

Vec3 he = extents * 0.5;

Then you want something like ortho( -he.x, he.x, -he.y, he.y, -he.z, he.z );

Otherwise, you'll have unwanted translation in the projection matrix.  Translation that already exists in the view matrix so you'd be applying it twice.

### #11KaiserJohan  Members

Posted 01 February 2014 - 01:14 PM

Thanks, I've got shadows working with the static bounding box, I'll be trying what you wrote in an earlier post about inverting the camera view/projection matrix. One thing though, what do you mean by

run the corners of a 2 x 2 x 2 box centered on the origin through it

?

### #12lunkhound  Members

Posted 01 February 2014 - 10:22 PM

The corners of a 2x2x2 box at the origin are 8 vertices where each component is either a 1 or a -1.  Vec3(1,1,1), Vec3(-1,1,1), .. and so on.  This corresponds to the view volume in clip-space.  The view-projection matrix maps from world space to clip space (it is the concatenation of the view matrix and the projection matrix).  So if you invert the view-projection matrix, it will map from clip space to world space.  If you apply the inverse-view-projection matrix to each of the 8 corners, you should end up with the 8 corners of the view volume in world-space.  The inverse-view-projection matrix will be a 4x4 matrix, so each of your vertices will need to be "promoted" to a Vec4 with a w-component of 1, and the result will be in homogeneous coordinates so you will need to divide by w after the transform.

### #13KaiserJohan  Members

Posted 02 February 2014 - 08:01 AM

If I understod correctly;

        Mat4 rotationMatrix(Vec4(perpVec1, 0.0f), Vec4(perpVec2, 0.0f), Vec4(lightDir, 0.0f), Vec4(0.0f, 0.0f, 0.0f, 1.0f));  // again, column major

Vec4 corners[8] = { Vec4(-1.0f, -1.0f, 1.0f, 1.0f), Vec4(-1.0f, -1.0f, -1.0f, 1.0f), Vec4(-1.0f, 1.0f, 1.0f, 1.0f), Vec4(-1.0f, 1.0f, -1.0f, 1.0f),
Vec4(1.0f, -1.0f, 1.0f, 1.0f), Vec4(1.0f, -1.0f, -1.0f, 1.0f), Vec4(1.0f, 1.0f, 1.0f, 1.0f), Vec4(1.0f, 1.0f, -1.0f, 1.0f) };
const Mat4 inverseVPMatrix = glm::inverse(viewProjectionMatrix);    // the cameras view-projection matrix
for (uint32_t i = 0; i < 8; i++)
{
corners[i] = corners[i] * inverseVPMatrix;
corners[i] /= corners[i].w;
}

for (uint32_t i = 0; i < 8; i++)
corners[i] = corners[i] * rotationMatrix;


Looking at the corners after multiplying with the inverse VPMatrix and dividing by w, the values all seem to be in the range of [-2.0, 2.0]. which can't be right?

Edited by KaiserJohan, 02 February 2014 - 08:08 AM.

### #14lunkhound  Members

Posted 02 February 2014 - 11:09 AM

That doesn't sound right.  I don't see anything wrong with the code there.  There must be something wrong with the viewProjection matrix, make sure view and projection are concatenated in the correct order.  For column major matrices, it should be projection * view, not the other way around.

### #15KaiserJohan  Members

Posted 03 February 2014 - 01:13 PM

Turned out the issue was with wrong order of multiplication of the corners and inverse VP matrix. This is the correct one:

corners[i] = inverseVPMatrix * corners[i];

Perhaps due to glm being column major?

I consider this topic solved; big thanks for all the help! I'm gonna investigate cascaded shadow mapping next,

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