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 BÃ©ziers. Meanwhile it has been extended to render third-order BÃ©zier 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) BÃ©zier 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) BÃ©zier 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 BÃ©zier arcs. The parametric form of a second-order BÃ©zier 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 BÃ©zier 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 BÃ©zier 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 BÃ©zier 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 BÃ©zier 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 BÃ©zier 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. BÃ©zier arcs have the special property of being very easily decomposed into two sub-arcs, which are themselves BÃ©zier arcs. Moreover, it is easy to prove that there is at most one vertical extremum on each BÃ©zier 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 BÃ©zier 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 BÃ©zier 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 BÃ©zier arcs, the job is a little more complicated. We assume that all BÃ©ziers 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 2003, 2007 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: