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.
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
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
Non-Premultiplied Alpha
Over operator for non-premultiplied alpha: non-commutative blending of tinted glass
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
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)
Premultiplied alpha is closed under composition
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
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