realistically, with supersample, your z-buffer is per sample not pixel.
Model transparency: with some probability we don't see the object at all, with \alpha probability, we see the object.
object edge: can be thought of a layer of transparency object
combine transparency: say you have a \alpha_1 in front and a \alpha_2 in the back, the combined probability is \alpha_1 + (1 - \alpha_1)\alpha_2 (the probability of hitting the first layer plus the probability of miss the first layer and hit the second)
But this isn't good, we need to assume two events are independent (well, in some case it is a bad assumption, but let's go with it)
One example of non-independent is: when two triangles both occupy unoccupied space in one pixel. In this case, we will underestimate coverage.
With assumption of independence (both color value and opaque value), we can use the following equation for calculate color: c_{12} = \frac{\alpha_1 c_1 + (1 - \alpha_1)\alpha_2 c_2}{\alpha_1 + (1 - \alpha_1)\alpha_2}
We can transform the above equation be defining pre-multiplied alpha C_1 = \alpha_1 \cdot c_1, C_2 = \alpha_2 \cdot c_2 (but with pre-multiplied alpha, we lose feudality because we are storing (r, g, b, a) into (r, g, b))
depth order: if we rasterize front to back, then we can "early out"
So in practice:
render opaque objects only with z-buffer
render transparency back to front with painter's algorithm (sorting transparent triangles first) with no update to z-buffer but with depth test.
For better version of painter's algorithm: search Order-independent transparency.
Given depth of d_i we can easily figure out pixel depth by linear interpolation
Depth Buffer Example
Depth Buffer: we keep track the depth of the closest triangle seen so far
0. initialize depth for each pixel (super sample) to infinity
randomly select a not drawn triangle from buffer
for each pixel (super sample), draw that triangle if its depth value is smaller than stored value, don't draw if its depth value is larger than stored value
update new depth buffer if pixel (super sample) get drawn
repeat until all triangles are drawn
Depth Buffer Code
The above technique will work fine with supersample. You obtain multiple copies of frame buffer and then merge them into one.
Space: constant space per sample for depth buffer, don't depend on overlapping primitives
Time: constant time per covered sample
Transparency and Alpha
Why do we need alpha for non-transparent: to represent things thinner than one pixel
Non-Premultiplied Alpha
Over operator for non-premultiplied alpha: non-commutative blending of tinted glass
Over operator for non-premultiplied alpha
Given A = (A_r, A_g, A_b) with alpha \alpha_A and B = (A_r, A_g, A_b) with alpha \alpha_B, to compute B over A, we get: C = \alpha_BB+ (1 - \alpha_B)\alpha_AA, \alpha_C = \alpha_B + (1 - \alpha_B)\alpha_A
Fringing: dark halo in non-premultiplied alpha
Color got Darken with Non-Premultiplied Alpha
Premultiplied Alpha
Premultiplied Alpha: compute B over A
A' = (\alpha_A A_r, \alpha_A A_g, \alpha_A A_b, \alpha_A)
B' = (\alpha_B B_r, \alpha_B B_g, \alpha_B B_b, \alpha_B)
This is exactly how we compose RGB value (we are expressing color in homogeneous coordinates)
Alpha Blending with Premultiplied Alpha Works Well with Upsampling
Alpha Blending with Premultiplied Alpha Works Well with MIP Map
Premultiplied alpha is closed under composition
Semi-tranparent Object is a Pain in Non-Ray Tracing Because of Strage in Depth Buffer
Render mixture of opaque and transparent triangles
render all opaque primitives in any order
disable write to depth buffer, render semi-transparent triangles in back-to-front order. If depth test passed, triangle is over color buffer, otherwise don't draw. (we need to sort semi-transparent triangles and hope they don't intersect)
Full Rasterization Pipeline
Input
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
Step 7
Step 8
Steps:
Transform triangle vertices into camera-centered world space (inverse of camera transform)
Apply perspective projection into normalized coordinate space
Clipping: discard triangles lie outside (culling) and clip triangles to box (possibly generate new triangles)
Transform normalized coordinates into screen coordinates and to image coordinates
Pre-compute data in Barycentric Coordinates
Sample Coverage
Filtering, MIP Map, Sample Texture, Interpolation
Depth Test and Update Depth Value
GPUs
Modern Rasterization Pipeline: OpenGL / Direct3D
Integrated GPU
GPU has very specialized parts that only deals with rendering triangles (fixed functions)
GPU are becoming more programmable, control staging