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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 ---