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This file is an attempt at explaining the internals of the FreeType
rasterizer. This component is quite general purpose and could
easily be integrated into other programs (but still under the
current license).
--------------------------------------------------------------------
The HOWs and WHYs of the FreeType rasterizer
by David Turner
I. Introduction
II. Rendering Technology
III. Implementation Details
IV. Gray-Level Support
I. Introduction
===============
A rasterizer is a library in charge of converting a vectorial
representation of a shape into a bitmap. The FreeType rasterizer
has been developed to render the glyphs found in TrueType files,
made up of segments and second-order Beziers. This document is an
explanation of its design and implementation.
Though these explanations start from the basics, a knowledge of
common rasterization techniques is assumed.
--------------------------------------------------------------------
II. Rendering Technology
========================
1. Requirements
---------------
We will assume that all scaling/rotating/hinting/whatever has been
already done. The glyph is thus described, as in the TrueType
specification, by a list of points. Each point has an x and y
coordinate, as well as a flag that indicates whether the point is
_on_ or _off_ the curve.
More precisely:
- All point coordinates are in the 26.6 fixed float format as
defined by the specification. The orientation used is:
^ y
| reference orientation
|
*----> x
0
This means that the `distance' between two neighbouring pixels
is 64 `units' (1 unit = 1/64th of a pixel).
Note that, for the rasterizer, pixel centers are located at
integer coordinates, i.e., (0.0, 0.0) is the coordinate of the
origin's center (unlike what happens within the TrueType
bytecode interpreter where this point's center lies at (0.5,
0.5)).
A pixel line in the target bitmap is called a `scanline'.
- A glyph is usually made of several contours, also called
outlines. A contour is simply a closed curve that delimits an
outer or inner region of the glyph. It is described by a series
of successive points of the points table.
Each point of the glyph has an associated flag that indicates
whether it is `on' or `off' the curve. Two successive `on'
points indicate a line segment joining the two points.
One `off' point amidst two `on' points indicates a second degree
Bezier parametric arc, defined by these three points (the `off'
point being the control point, and the `on' ones the start and
end points).
Finally, two successive `off' points forces the rasterizer to
create, during rendering, an `on' point amidst them, at their
exact middle. This greatly facilitates the definition of
successive Bezier arcs.
* # on curve
* off curve
__---__
#-__ _-- -_
--__ _- -
--__ # \
--__ #
-#
Two `on' points
Two `on' points and one `off' point
between them
*
# __ Two `on' points with two `off'
\ - - points between them. The point
\ / \ marked `0' is the middle of the
- 0 \ `off' points, and is a `virtual
-_ _- # on' point where the curve passes.
-- It does not appear in the point
* list.
The FreeType rasterizer, as intended to render TrueType glyphs,
does not support third order Beziers, usually found in Type 1
fonts. Type 1 support may lead to further development of the
engine (it is already part of FreeType 2.0).
The parametric form of a second-order Bezier is:
P(t) = (1-t)^2*P1 + 2*t*(1-t)*P2 + t^2*P3
with t a real number in the range [0..1]
P1 and P3 are the endpoints, P2 the control point.
Note that the rasterizer does not use this formula. It exhibits,
however, one very useful property of Bezier arcs: Each point of
the curve is a weighted average of the control points.
As all weights are positive and always sum up to 1, whatever the
value of t, each arc point lies within the triangle defined by the
arc's three control points.
2. Profiles and Spans
---------------------
The following is a basic explanation of the _kind_ of computations
made by the rasterizer to build a bitmap from a vector
representation. Note that the actual implementation is slightly
different, due to performance tuning and other factors.
However, the following ideas remain in the same category, and are
more convenient to understand.
a. Sweeping the shape
The best way to fill a shape is to decompose it into a number of
simple horizontal segments, then turn them on in the target
bitmap. These segments are called `spans'.
__---__
_-- -_
_- -
- \
/ \
/ \
| \
__---__ Example: filling a shape
_----------_ with spans.
_--------------
----------------\
/-----------------\ This is typically done from the top
/ \ to the bottom of the shape, in a
| | \ movement called a `sweep".
V
__---__
_----------_
_--------------
----------------\
/-----------------\
/-------------------\
|---------------------\
In order to draw a span, the rasterizer must compute its
coordinates, which are simply the shape's contours'
x-coordinates taken on the y-scanlines.
/---/ |---| Note that there are usually
/---/ |---| several spans per scanline.
| /---/ |---|
| /---/_______|---| When rendering this shape to the
V /----------------| current scanline y, we must
/-----------------| compute the x values of the
a /----| |---| points a, b, c, and d.
- - - * * - - - - * * - - y -
/ / b c| |d
/---/ |---|
/---/ |---| And then turn on the spans a-b
/---/ |---| and c-d.
/---/_______|---|
/----------------|
/-----------------|
a /----| |---|
- - - ####### - - - - ##### - - y -
/ / b c| |d
b. Decomposing outlines into profiles
For each scanline during the sweep, we need the following
information:
o The number of spans on the current scanline, given by the
number of shape points intersecting the scanline (these are
the points a, b, c, and d in the above example).
o The x coordinates of these points.
These are computed before the sweep, in a phase called
`decomposition' which converts the glyph into *profiles*.
Put it simply, a `profile' is a contour's portion that can only
be either ascending or descending, i.e., it is monotonic in the
vertical direction (we will also say y-monotonic). There is no
such thing as a horizontal profile, as we shall see.
Here are a few examples:
this square
1 2
---->---- is made of two
| | | |
| | profiles | |
^ v ^ + v
| | | |
| | | |
----<----
up down
this triangle
P2 1 2
|\ is made of two | \
^ | \ \ | \
| | \ \ profiles | \ |
| | \ v ^ | \ |
| \ | | + \ v
| \ | | \
P1 ---___ \ ---___ \
---_\ ---_ \
<--__ P3 up down
A more general contour can be made of more than two profiles:
__ ^
/ | / ___ / |
/ | / | / | / |
| | / / => | v / /
| | | | | | ^ |
^ | |___| | | ^ + | + | + v
| | | v | |
| | | up |
|___________| | down |
<-- up down
Successive profiles are always joined by horizontal segments
that are not part of the profiles themselves.
Note that for the rasterizer, a profile is simply an *array*
that associates one horizontal *pixel* coordinate to each bitmap
*scanline* crossed by the contour's section containing the
profile. Note also that profiles are *oriented* up or down
along the glyph's original flow orientation.
In other graphics libraries, profiles are also called `edges' or
`edgelists'.
c. The Render Pool
FreeType has been designed to be able to run well on _very_
light systems, including embedded systems with very few memory.
A render pool will be allocated once; the rasterizer uses this
pool for all its needs by managing this memory directly in it.
The algorithms that are used for profile computation make it
possible to use the pool as a simple growing heap. This means
that this memory management is actually easy, and faster than
any kind of malloc()/free() combination.
Moreover, we'll see later that the rasterizer is able, when
dealing with profiles too large and numerous to lie all at once
in the render pool, to immediately decompose recursively the
rendering process into independent sub-tasks, each taking less
memory to be performed (see `sub-banding' below).
The render pool doesn't need to be large. A 4kByte pool is
enough for nearly all renditions, though nearly 100% slower than
a more confortable 16 or 32kByte pool (that was tested with
complex glyphs at sizes over 500 pixels).
d. Computing Profiles Extents
Remember that a profile is an array, associating a _scanline_ to
the x pixel coordinate of its intersection with a contour.
Though it's not exactly how the FreeType rasterizer works, it is
convenient to think that we need a profile's height before
allocating it in the pool and computing its coordinates.
The profile's height is the number of scanlines crossed by the
y-monotonic section of a contour. We thus need to compute these
sections from the vectorial description. In order to do that,
we are obliged to compute all (local and global) y-extrema of
the glyph (minima and maxima).
P2 For instance, this triangle has only
two y-extrema, which are simply
|\
| \ P2.y as an y-maximum
| \ P3.y as an y-minimum
| \
| \ P1.y is not an y-extremum (though it is
| \ a x-minimum, which we don't need).
P1 ---___ \
---_\
P3
Note that the extrema are expressed in pixel units, not in
scanlines. The triangle's height is certainly (P3.y-P2.y+1)
pixel units, but its profiles' heights are computed in
scanlines. The exact conversion is simply:
- min scanline = FLOOR ( min y )
- max scanline = CEILING( max y )
A problem arises with Bezier Arcs. While a segment is always
necessarily y-monotonic (i.e., flat, ascending, or descending),
which makes extrema computations easy, the ascent of an arc can
vary between its control points.
P2
*
# on curve
* off curve
__-x--_
_-- -_
P1 _- - A non y-monotonic Bezier arc.
# \
- The arc goes from P1 to P3.
\
\ P3
#
We first need to be able to easily detect non-monotonic arcs,
according to their control points. I will state here, without
proof, that the monotony condition can be expressed as:
P1.y <= P2.y <= P3.y for an ever-ascending arc
P1.y >= P2.y >= P3.y for an ever-descending arc
with the special case of
P1.y = P2.y = P3.y where the arc is said to be `flat'.
As you can see, these conditions can be very easily tested.
They are, however, extremely important, as any arc that does not
satisfy them necessarily contains an extremum.
Note also that a monotonic arc can contain an extremum too,
which is then one of its `on' points:
P1 P2
#---__ * P1P2P3 is ever-descending, but P1
-_ is an y-extremum.
-
---_ \
-> \
\ P3
#
Let's go back to our previous example:
P2
*
# on curve
* off curve
__-x--_
_-- -_
P1 _- - A non-y-monotonic Bezier arc.
# \
- Here we have
\ P2.y >= P1.y &&
\ P3 P2.y >= P3.y (!)
#
We need to compute the y-maximum of this arc to be able to
compute a profile's height (the point marked by an `x'). The
arc's equation indicates that a direct computation is possible,
but we'll rely on a different technique, which use will become
apparent a bit later.
Bezier arcs have the special property of being very easily
decomposed into two other sub-arcs, which are themselves Beziers
arcs. Moreover, it is easy to prove that there is at most one
y-extremum on each Bezier arc (for second degree ones).
For instance, the following arc P1P2P3 can be decomposed into
two sub-arcs Q1Q2Q3 and R1R2R3 that look like:
P2
*
# on curve
* off curve
Original Bezier Arc P1P2P3.
__---__
_-- --_
_- -_
- -
/ \
/ \
# #
P1 P3
P2
*
Q3 Decomposed into two subarcs
Q2 R2 Q1Q2Q3 and R1R2R3
* __-#-__ *
_-- --_
_- R1 -_ Q1 = P1 R3 = P3
- - Q2 = (P1+P2)/2 R2 = (P2+P3)/2
/ \
/ \ Q3 = R1 = (Q2+R2)/2
# #
Q1 R3 Note that Q2, R2, and Q3=R1
are on a single line which is
tangent to the curve.
We have then decomposed a non-y-monotonic bezier into two
smaller sub-arcs. Note that in the above drawing, both sub-arcs
are monotonic, and that the extremum is then Q3=R1. However, in
a more general case, only one sub-arc is guaranteed to be
monotonic. Getting back to our former example:
Q2
*
__-x--_ R1
_-- #_
Q1 _- Q3 - R2
# \ *
-
\
\ R3
#
Here, we see that, though Q1Q2Q3 is still non-monotonic, R1R2R3
is ever descending: we thus know that it doesn't contain the
extremum. We can then re-subdivide Q1Q2Q3 into two sub-arcs and
go on recursively, stopping when we encounter two monotonic
subarcs, or when the subarcs become simply too small.
We will finally find the y-extremum. Note that the iterative
process of finding an extremum is called `flattening'.
e. Computing Profiles coordinates
Once we have the height of each profile, we are able to allocate
it in the render pool. We now have to compute its coordinate
for each scanline.
In the case of segments, the computation is straightforward, and
uses good old Euclide (also known as Bresenham ;-). However,
for Bezier arcs, things get a little more complicated.
We assume that all Beziers that are part of a profile are the
result of `flattening' the curve, which means that they are all
y-monotonic (ascending or descending, and never flat). We now
have to compute the arcs' intersections with the profile's
scanlines. One way is to use a similar scheme to `flattening',
called `stepping'.
Consider this arc, going from P1 to
--------------------- P3. Suppose that we need to
compute its intersections with the
drawn scanlines. Again, this is
--------------------- feasible directly, if we dare
to compute one square root per
* P2 _---# P3 scanline (how great!).
------------- _-- --
_-
_/ Rather, it is still possible to use
---------/----------- the decomposition property in the
/ same recursive way, i.e. subdivide
| the arc into subarcs until these
------|-------------- get too small to cross more than
| one scanline!
|
-----|--------------- This is very easily done using a
| rasterizer-managed stack of
| subarcs.
# P1
f. Sweeping and Sorting the spans
Once all our profiles have been computed, we begin the sweep to
build (and fill) the spans.
As the TrueType specification uses the winding fill rule, we
place on each scanline the profiles present in two separate
lists.
One list, called the `left' one, only contains ascending
profiles, while the other `right' list contains the descending
profiles.
As each glyph is made of closed curves, a simple geometric
property is that the two lists necessarily contain the same
number of elements.
Creating spans is there straightforward:
1. We sort each list in increasing x order.
2. We pair each value of the left list, with its corresponding
value in the right one.
/ / | | For example, we have here
/ / | | four profiles. Two of
>/ / | | | them are ascending (1 &
1// / ^ | | | 2 3), while the two others
// // 3| | | v are descending (2 & 4).
/ //4 | | | On the given scanline,
a / /< | | the left list is (1,3),
- - - *-----* - - - - *---* - - y - and the right one is
/ / b c| |d (4,2) (sorted).
There are then two spans, joining
1 to 4 (i.e. a-b) and 3 to 2
(i.e. c-d)!
Sorting doesn't necessarily take much time, as in 99 cases out
of 100, the lists' order is kept from one scanline to the next.
We can thus implement it with two simple singly-linked lists,
sorted by a classic bubble-sort, which takes a minimum amount of
time when the lists are already sorted.
A previous version of the rasterizer used more elaborate
structures, like arrays to perform `faster' sorting. It turned
out that this old scheme is not faster than the one described
above.
Once the spans have been `created', we can simply draw them in
the target bitmap.
g. Drop-out control
To be continued.
--- end of raster.txt ---