| |
| How FreeType's rasterizer work |
| |
| by David Turner |
| |
| Revised 2007-Feb-01 |
| |
| |
| This file is an attempt to explain the internals of the FreeType |
| rasterizer. The rasterizer is of quite general purpose and could |
| easily be integrated into other programs. |
| |
| |
| I. Introduction |
| |
| II. Rendering Technology |
| 1. Requirements |
| 2. Profiles and Spans |
| a. Sweeping the Shape |
| b. Decomposing Outlines into Profiles |
| c. The Render Pool |
| d. Computing Profiles Extents |
| e. Computing Profiles Coordinates |
| f. Sweeping and Sorting the Spans |
| |
| |
| 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 originally developed to render the glyphs found in |
| TrueType files, made up of segments and second-order Béziers. |
| Meanwhile it has been extended to render third-order Bézier curves |
| also. This document is an explanation of its design and |
| implementation. |
| |
| While these explanations start from the basics, a knowledge of |
| common rasterization techniques is assumed. |
| |
| |
| II. Rendering Technology |
| ======================== |
| |
| 1. Requirements |
| --------------- |
| |
| We assume that all scaling, rotating, hinting, etc., has been |
| already done. The glyph is thus described by a list of points in |
| the device space. |
| |
| - All point coordinates are in the 26.6 fixed float format. The |
| used orientation is: |
| |
| |
| ^ y |
| | reference orientation |
| | |
| *----> x |
| 0 |
| |
| |
| `26.6' means that 26 bits are used for the integer part of a |
| value and 6 bits are used for the fractional part. |
| Consequently, 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. The TrueType bytecode interpreter, |
| however, assumes that the lower left edge of a pixel (which is |
| taken to be a square with a length of 1 unit) has integer |
| coordinates. |
| |
| |
| ^ y ^ y |
| | | |
| | (1,1) | (0.5,0.5) |
| +-----------+ +-----+-----+ |
| | | | | | |
| | | | | | |
| | | | o-----+-----> x |
| | | | (0,0) | |
| | | | | |
| o-----------+-----> x +-----------+ |
| (0,0) (-0.5,-0.5) |
| |
| TrueType bytecode interpreter FreeType rasterizer |
| |
| |
| 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 |
| (conic) Bézier parametric arc, defined by these three points |
| (the `off' point being the control point, and the `on' ones the |
| start and end points). Similarly, a third-degree (cubic) Bézier |
| curve is described by four points (two `off' control points |
| between two `on' points). |
| |
| Finally, for second-order curves only, 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 Bézier arcs. |
| |
| The parametric form of a second-order Bézier curve is: |
| |
| P(t) = (1-t)^2*P1 + 2*t*(1-t)*P2 + t^2*P3 |
| |
| (P1 and P3 are the end points, P2 the control point.) |
| |
| The parametric form of a third-order Bézier curve is: |
| |
| P(t) = (1-t)^3*P1 + 3*t*(1-t)^2*P2 + 3*t^2*(1-t)*P3 + t^3*P4 |
| |
| (P1 and P4 are the end points, P2 and P3 the control points.) |
| |
| For both formulae, t is a real number in the range [0..1]. |
| |
| Note that the rasterizer does not use these formulae directly. |
| They exhibit, however, one very useful property of Bézier 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 (polygon) |
| defined by the arc's three (four) control points. |
| |
| In the following, only second-order curves are discussed since |
| rasterization of third-order curves is completely identical. |
| |
| Here some samples for second-order curves. |
| |
| |
| * # 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. |
| |
| |
| 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 x coordinates of the shape's |
| contours, 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. |
| |
| x coordinates 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 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. |
| |
| 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 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 quite 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 comfortable 16KByte 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 a vertical maximum |
| | \ P3.y as a vertical minimum |
| | \ |
| | \ P1.y is not a vertical extremum (though |
| | \ it is a horizontal minimum, which we |
| P1 ---___ \ don't need). |
| ---_\ |
| 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 simple: |
| |
| - min scanline = FLOOR ( min y ) |
| - max scanline = CEILING( max y ) |
| |
| A problem arises with Bézier 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 Bézier 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 Bézier arc. |
| # \ |
| - Here we have |
| \ P2.y >= P1.y && |
| \ P3 P2.y >= P3.y (!) |
| # |
| |
| |
| We need to compute the vertical 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 rely on a different technique, which use will become |
| apparent soon. |
| |
| Bézier arcs have the special property of being very easily |
| decomposed into two sub-arcs, which are themselves Bézier arcs. |
| Moreover, it is easy to prove that there is at most one vertical |
| extremum on each Bézier arc (for second-degree curves; similar |
| conditions can be found for third-order arcs). |
| |
| For instance, the following arc P1P2P3 can be decomposed into |
| two sub-arcs Q1Q2Q3 and R1R2R3: |
| |
| |
| P2 |
| * |
| # on curve |
| * off curve |
| |
| |
| original Bézier 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 Bézier curve 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 vertical 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. The next task is to compute coordinates |
| for each scanline. |
| |
| In the case of segments, the computation is straightforward, |
| using the Euclidean algorithm (also known as Bresenham). |
| However, for Bézier arcs, the job is a little more complicated. |
| |
| We assume that all Béziers 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 intersections of arcs 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. As already |
| --------------------- mentioned this can be done |
| directly, but the involved |
| * P2 _---# P3 algorithm is far too slow. |
| ------------- _-- -- |
| _- |
| _/ Instead, 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 both the TrueType and Type 1 specifications use the winding |
| fill rule (but with opposite directions), we place, on each |
| scanline, the present profiles 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 ensures that the two lists contain the same number of |
| elements. |
| |
| Creating spans is thus straightforward: |
| |
| 1. We sort each list in increasing horizontal order. |
| |
| 2. We pair each value of the left list with its corresponding |
| value in the right list. |
| |
| |
| / / | | 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. |
| |
| ------------------------------------------------------------------------ |
| |
| Copyright (C) 2003-2022 by |
| David Turner, Robert Wilhelm, and Werner Lemberg. |
| |
| This file is part of the FreeType project, and may only be used, |
| modified, and distributed under the terms of the FreeType project |
| license, LICENSE.TXT. By continuing to use, modify, or distribute this |
| file you indicate that you have read the license and understand and |
| accept it fully. |
| |
| |
| --- end of raster.txt --- |
| |
| Local Variables: |
| coding: utf-8 |
| End: |
