Orthographic LODs – Part II

Last time, we got the basic renderer up and running. Now we have more complex issues to deal with. For example, how to light the LOD models.

Lighting

One approach to lighting is to simply light the detailed model, and bake that lighting into the LOD texture. This works, but requires that all lights be static. We can do better.

The standard shading equation has three parts: ambient light, diffuse light, and specular light:
K_a=AT
K_d=DT(N \cdot L)
K_s=S(N \cdot H)^m (N \cdot L > 0)
(For details of how these equations work, see Wikipedia.)

The key thing to notice is that these equations rely only on the texture (which we already have), a unit direction towards the light source, a unit direction towards the camera, and the unit surface normal. By rendering our detailed model down to a texture, we’ve destroyed the normal data. But that’s easy enough to fix: we write it to a texture as well.

(Setup for this texture is more-or-less the same as the color texture, we just bind it to a different output.)

// Detail vertex shader
in vec3 position;
in vec3 normal;

out vec3 Normal;

void main() {
    gl_Position = position;
    Normal = normal;
}

// Detail fragment shader

in vec3 Normal;

out vec4 frag_color;
out vec4 normal_map_color;

void main() {
    vec3 normal = (normalize(Normal) + vec3(1.0)) / 2.0;

    frag_color = vec4(1.0);
    normal_map_color = vec4(normal, 1.0);
}

A unit normal is simply a vector of 3 floats between -1 and 1. We can map this to a RGB color (vec3 clamped to [0, 1]) by adding one and dividing by two. This is, again, nothing new. You may be wondering why we store an alpha channel — we’ll get to that later.

We can now reconstruct the normal from the LOD texture. This is all we need for directional lights:

// Fragment shader
uniform sampler2D color_texture;
uniform sampler2D normal_map_texture;

uniform vec3 sun_direction;
uniform vec3 camera_direction;

in vec2 uv;

out vec4 frag_color;

void main() {
    vec4 normal_sample = texture(normal_map_texture, uv);
    vec3 normal = (normal.xyz * 2.0) - vec3(1.0);
    vec3 view = -1.0 * camera_direction;
    vec3 view_half = (normal + view) / length(normal + view);

    float n_dot_l = dot(normal, sun_direction);
    float n_dot_h = dot(normal, view_half);

    float ambient_intensity = 0.25;
    float diffuse_intensity = 0.5;
    float specular_intensity = 0.0;
    if (n_dot_l > 0) {
        specular_intensity = 0.25 * pow(n_dot_h, 10);
    }

    vec4 texture_sample = texture(color_texture, uv);

    vec3 lit_color = ambient_intensity * texture_sample.rgb +
                     diffuse_intensity * texture_sample.rgb +
                     specular_intensity * vec3(1.0);

    frag_color = vec4(lit_color, texture_sample.a);
}

Result:
lod_test_06

Point / Spot Lighting

So this solves directional lights. However, point and spot lights have attenuation — they fall off over distance. Additionally, they have an actual spatial position, which means the light vector will change.

After rendering the LODs, we no longer have the vertex positions of our detail model. Calculating lighting based on the LOD model’s vertices will, in many cases, be obviously wrong. How can we reconstruct the original points?

By cheating.

Let’s look at what we have to work with:

The camera is orthographic, which means that depth of a pixel is independent of where it is on screen. We have the camera’s forward vector (camera_direction in the above shader).

Finally, and most obviously, we don’t care about shading the points that weren’t rendered to our LOD texture.

This turns out to be the important factor. If we knew how far each pixel was from the camera when rendered, we could reconstruct the original location for lighting.

To put it another way, we need a depth buffer. Remember that normal map alpha value we didn’t use?

We can get the depth value like so:

// Map depth from [near, far] to [-1, 1]
float depth = (2.0 * gl_FragCoord.z - gl_DepthRange.near - gl_DepthRange.far) / (gl_DepthRange.far - gl_DepthRange.near);

// Remap normal and depth from [-1, 1] to [0, 1]
normal_depth_color = (vec4(normalize(normal_vec), depth) + vec4(1.0)) / 2.0;

However, we now run into a problem. This depth value is in the LOD render’s window space. We want the position in the final render’s camera space — this way we can transform the light position to the same space, which allows us to do the attenuation calculations.

There are a few ways to handle this. One is to record the depth range for our LOD model after rendering it, and then use that to scale the depth properly. This is complex (asymmetrical models can have different depth sizes for each rotation) and quite likely inconsistent, due to precision errors.

A simpler solution, then, is to pick an arbitrary depth range, and use it consistently for both the main render and LODs. This is not ideal — one getting out of sync with the other may lead to subtle errors. In a production design, it may be a good idea to record the camera matrix used in the model data, so that it can be confirmed on load.

We need two further pieces of data to proceed: the viewport coordinates (the size of the window) and the inverse projection matrix (to “unproject” the position back into camera space). Both of these are fairly trivial to provide. To reconstruct the proper position, then:

vec3 window_to_camera_space(vec3 window_space) {
    // viewport = vec4(x, y, width, height)

    // Because projection is orthographic, NDC == clip space
    vec2 clip_xy = ((2.0 * window_space.xy) - (2.0 * viewport.xy)) / (viewport.zw) - 1.0;
    // Already mapped to [-1, 1] by the same transform that extracts our normal.
    float clip_z = window_space.z

    vec4 clip_space = vec4(clip_xy, clip_z, 1.0);

    vec4 camera_space = camera_from_clip * clip_space;
    return camera_space.xyz;
}

(Note that this is rather inefficient. We *know* what a projection matrix looks like, so that last multiply can be made a lot faster.)

Calculating the light color is the same as for directional lights. For point lights, we then divide by an attenuation factor:

float attenuation = point_light_attenuation.x +
                    point_light_attenuation.y * light_distance +
                    point_light_attenuation.z * light_distance * light_distance;

point_light_color = point_light_color / attenuation;

This uses 3 factors: constant, linear, and quadratic. The constant factor determines the base intensity of the light (a fractional constant brightens the light, a constant greater than 1.0 darkens it.) The linear and quadratic factors control the fall-off curve.

A spot light has the same attenuation factors, but also has another term:

float spot_intensity = pow(max(dot(-1.0 * spot_direction, light_direction), 0.0), spot_exponent);

The dot product here limits the light to a cone around the spotlight’s facing direction. The exponential factor controls the “tightness” of the light cone — as the exponent increases, the fall-off becomes much sharper (remember that the dot product of unit vectors produces a value in [0, 1]).

So that covers lighting. Next time: exploring options for shadows.

lod_test_09

NOTE:

I made some errors with terminology in my last post. After the projection transform is applied, vertices are in *clip* space, not eye space. The series of transforms is as follows:

Model -> World -> Camera -> Clip -> NDC
(With an orthographic projection, Clip == NDC.)

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Orthographic LODs – Part I

NOTE: I’m going to try something new here: rather than write about something I have already solved, this series will be my notes as I work through this problem. As such, this is *not* a guide. The code is a pile of hacks and I will make silly mistakes.

The Problem:

I am working on a city-building game. I want to be able to use detailed models with large numbers of polygons and textures, but without the runtime rendering cost that entails. To do so, I want to pre-render detailed models as textures for lower level-of-detail (LOD) models. This is not a particularly original solution (it is the system SimCity 4 uses). There are a few restrictions implied by this choice:

  • Camera projection must be orthographic.
    • A perspective projection will appear to skew the model the further it gets from the center of the view. Orthographic projection will not.
  • Camera angle must be fixed (or a finite set of fixed positions).
    • While camera *position* is irrelevant (because an orthographic projection doesn’t affect depth), its rotation is not. Each camera angle needs its own LOD image. Therefore, we need a small number of them.

The process is simple enough: render the detail model for each view. Projecting the vertices of the LOD model with the same view produces the proper texture coordinates for the LOD. Then, pack all of the rendered textures into an atlas.

Orthographic Rendering

Rendering an orthographic projection is, conceptually, very straightforward. After transforming the world to camera space (moving the camera to the origin, and aligning the z-axis with the camera’s forward view), we select a box of space and map it onto a unit cube (that is, a cube with a minimum point of (-1, -1, -1) and a maximum point of (1, 1, 1).) The z-component of each point is then discarded.

I will not get into the math here. Wikipedia has a nicely concise explanation.

What this *means* is that we can fit this box tightly to the model we want to render. With clever application of glViewport, we could even render all of our views in one go. (At the moment, I have not yet implemented this.) This makes building the texture atlas much simpler.

Calculating Camera-space LOD

To get this tight fit, we need to know the bounds of the rendered model in camera space. Since we are projecting onto the LOD model, *its* bounds are what we’re concerned with. (This implies, by the way, that the LOD model must completely enclose the detail model.)

struct AABB {
    vec3 min;
    vec3 max;
};

AABB
GetCameraSpaceBounds(Model * model, mat4x4 camera_from_model) {
    AABB result;

    vec4 * model_verts = model->vertices;
    vec4 camera_vert = camera_from_model * model_verts[0];

    // We don't initialize to 0 vectors, because we don't
    // guarentee the model is centered on the origin.
    result.min = vec3(camera_vert);
    result.max = vec3(camera_vert);

    for (u32 i = 1; i < model->vertex_count; ++i) {
        camera_vert = camera_from_model * model_verts[i];

        result.min.x = min(result.min.x, camera_vert.x);
        result.min.y = min(result.min.y, camera_vert.y);
        result.min.z = min(result.min.z, camera_vert.z);

        result.max.x = max(result.max.x, camera_vert.x);
        result.max.y = max(result.max.y, camera_vert.y);
        result.max.z = max(result.max.z, camera_vert.z);
    }

    return result;
}

This bounding box gives us the clip volume to render.

AABB lod_bounds = GetCameraSpaceBounds(lod_model, camera_from_model);

mat4x4 eye_from_camera = OrthoProjectionMatrix(
    lod_bounds.max.x, lod_bounds.min.x,  // right, left
    lod_bounds.max.y, lod_bounds.min.y,  // top, bottom
    lod_bounds.min.z, lod_bounds.max.z); // near, far

mat4x4 eye_from_model = eye_from_camera * camera_from_model;

Rendering LOD Texture

Rendering to a texture requires a framebuffer. (In theory, we could also simply render to the screen and extract the result with glGetPixels. However, that limits us to a single a single color output and makes it more difficult to do any GPU postprocessing.)

After binding a new framebuffer, we need to create the texture to render onto:

glGenTextures(1, &result->texture);
glActiveTexture(GL_TEXTURE0);
glBindTexture(GL_TEXTURE_2D, result->texture);

// render_width and render_height are the width and height of the 
// camera space lod bounding box.
glTexImage2D(
    GL_TEXTURE_2D, 0, GL_RGBA, render_width, render_height, 0, GL_RGBA, GL_UNSIGNED_BYTE, 0
);

glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_MAX_LEVEL, 0);
glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_WRAP_S, GL_CLAMP_TO_EDGE);
glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_WRAP_T, GL_CLAMP_TO_EDGE);

glFramebufferTexture(GL_FRAMEBUFFER, GL_COLOR_ATTACHMENT0, result->texture, 0);

glViewport(0, 0, render_width, render_height);

This gives us a texture large enough to fit our rendered image, attaches it to the framebuffer, and sets the viewport to the entire framebuffer. To render multiple views to one texture, we’d allocate a larger texture, and use the viewport call to selected the target region.

Now we simply render the detail model with the eye_from_model transform calculated earlier. The result:

lod_test_02
(The detail shader used simply maps position to color.)

That’s all for now. Next time: how do we light this model?