[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
[freetype2] anuj-distance-field 363f1e8 29/93: [sdf] Added Newton's meth
|
From: |
Anuj Verma |
|
Subject: |
[freetype2] anuj-distance-field 363f1e8 29/93: [sdf] Added Newton's method for conic curve. |
|
Date: |
Sun, 2 Aug 2020 07:04:15 -0400 (EDT) |
branch: anuj-distance-field
commit 363f1e8de1297cc4f042faf54af78e9e6a9ff927
Author: Anuj Verma <anujv@iitbhilai.ac.in>
Commit: anujverma <anujv@iitbhilai.ac.in>
[sdf] Added Newton's method for conic curve.
---
[GSoC]ChangeLog | 14 ++++
src/sdf/ftsdf.c | 245 +++++++++++++++++++++++++++++++++++++++++++++++++++++++-
2 files changed, 258 insertions(+), 1 deletion(-)
diff --git a/[GSoC]ChangeLog b/[GSoC]ChangeLog
index 723a648..fe90c78 100644
--- a/[GSoC]ChangeLog
+++ b/[GSoC]ChangeLog
@@ -1,3 +1,17 @@
+2020-07-02 Anuj Verma <anujv@iitbhilai.ac.in>
+
+ [sdf] Added Newton's method for shortest distance
+ from a point to a conic.
+
+ * src/sdf/ftsdf.c (get_min_distance_conic): Created
+ a new function with same name which uses Newon't
+ iteration for finding shortest distance fom a point
+ to a conic curve. This dosen't causes underfow.
+
+ * src/sdf/ftsdf.c (USE_NEWTON_FOR_CONIC): This macro
+ can be used to toggle between Newton or analytical
+ cubic solving method.
+
2020-07-01 Anuj Verma <anujv@iitbhilai.ac.in>
* src/sdf/ftsdf.c (get_min_distance_conic): Add more
diff --git a/src/sdf/ftsdf.c b/src/sdf/ftsdf.c
index 6fb8ec5..8ceb7dd 100644
--- a/src/sdf/ftsdf.c
+++ b/src/sdf/ftsdf.c
@@ -18,9 +18,22 @@
/* a chance of overflow and artifacts. You can safely use it upto a */
/* pixel size of 128. */
#ifndef USE_SQUARED_DISTANCES
- # define USE_SQUARED_DISTANCES 0
+ # define USE_SQUARED_DISTANCES 1
#endif
+ /* If it is defined to 1 then the rasterizer will use Newton-Raphson's */
+ /* method for finding shortest distance from a point to a conic curve. */
+ /* The other method is an analytical method which find the roots of a */
+ /* cubic polynomial to find the shortest distance. But the analytical */
+ /* method has underflow as of now. So, use the Newton's method if there */
+ /* is any visible artifacts. */
+ #ifndef USE_NEWTON_FOR_CONIC
+ # define USE_NEWTON_FOR_CONIC 1
+ #endif
+
+ #define MAX_NEWTON_ITERATION 4
+ #define MAX_NEWTON_STEPS 4
+
/**************************************************************************
*
* macros
@@ -1136,6 +1149,8 @@
return error;
}
+#if !USE_NEWTON_FOR_CONIC
+
/**************************************************************************
*
* @Function:
@@ -1145,6 +1160,10 @@
* This function find the shortest distance from the `conic' bezier
* curve to a given `point' and assigns it to `out'. Only use it for
* conic/quadratic curves.
+ * [Note]: The function uses analytical method to find shortest distance
+ * which is faster than the Newton-Raphson's method, but has
+ * underflows at the moment. Use Newton's method if you can
+ * see artifacts in the SDF.
*
* @Input:
* [TODO]
@@ -1358,6 +1377,230 @@
return error;
}
+#else
+
+ /**************************************************************************
+ *
+ * @Function:
+ * get_min_distance_conic
+ *
+ * @Description:
+ * This function find the shortest distance from the `conic' bezier
+ * curve to a given `point' and assigns it to `out'. Only use it for
+ * conic/quadratic curves.
+ * [Note]: The function uses Newton's approximation to find the shortest
+ * distance, which is a bit slower than the analytical method
+ * doesn't cause underflow. Use is upto your needs.
+ *
+ * @Input:
+ * [TODO]
+ *
+ * @Return:
+ * [TODO]
+ */
+ static FT_Error
+ get_min_distance_conic( SDF_Edge* conic,
+ FT_26D6_Vec point,
+ SDF_Signed_Distance* out )
+ {
+ /* This method uses Newton-Raphson's approximation to find the */
+ /* shortest distance from a point to conic curve which does */
+ /* not involve solving any cubic equation, that is why there */
+ /* is no risk of underflow. The method is as follows: */
+ /* */
+ /* p0 = first endpoint */
+ /* p1 = control point */
+ /* p3 = second endpoint */
+ /* p = point from which shortest distance is to be calculated */
+ /* ----------------------------------------------------------- */
+ /* => the equation of a quadratic bezier curve can be written */
+ /* B( t ) = ( ( 1 - t )^2 )p0 + 2( 1 - t )tp1 + t^2p2 */
+ /* here t is the factor with range [0.0f, 1.0f] */
+ /* the above equation can be rewritten as */
+ /* B( t ) = t^2( p0 - 2p1 + p2 ) + 2t( p1 - p0 ) + p0 */
+ /* */
+ /* now let A = ( p0 - 2p1 + p2), B = 2( p1 - p0 ) */
+ /* B( t ) = t^2( A ) + t( B ) + p0 */
+ /* */
+ /* => the derivative of the above equation is written as */
+ /* B`( t ) = 2t( A ) + B */
+ /* */
+ /* => further derivative of the above equation is written as */
+ /* B``( t ) = 2A */
+ /* */
+ /* => the equation of distance from point `p' to the curve */
+ /* P( t ) can be written as */
+ /* P( t ) = t^2( A ) + t^2( B ) + p0 - p */
+ /* Now let C = ( p0 - p ) */
+ /* P( t ) = t^2( A ) + t( B ) + C */
+ /* */
+ /* => finally the equation of angle between curve B( t ) and */
+ /* point to curve distance P( t ) can be written as */
+ /* Q( t ) = P( t ).B`( t ) */
+ /* */
+ /* => now our task is to find a value of t such that the above */
+ /* equation Q( t ) becomes zero. in other words the point */
+ /* to curve vector makes 90 degree with curve. this is done */
+ /* by Newton-Raphson's method. */
+ /* */
+ /* => we first assume a arbitary value of the factor `t' and */
+ /* then we improve it using Newton's equation such as */
+ /* */
+ /* t -= Q( t ) / Q`( t ) */
+ /* putting value of Q( t ) from the above equation gives */
+ /* */
+ /* t -= P( t ).B`( t ) / derivative( P( t ).B`( t ) ) */
+ /* t -= P( t ).B`( t ) / */
+ /* ( P`( t )B`( t ) + P( t ).B``( t ) ) */
+ /* */
+ /* P`( t ) is noting but B`( t ) because the constant are */
+ /* gone due to derivative */
+ /* */
+ /* => finally we get the equation to improve the factor as */
+ /* t -= P( t ).B`( t ) / */
+ /* ( B`( t ).B`( t ) + P( t ).B``( t ) ) */
+ /* */
+ /* [note]: B and B( t ) are different in the above equations */
+
+ FT_Error error = FT_Err_Ok;
+
+ FT_26D6_Vec aA, bB, cC; /* A, B, C in the above comment */
+ FT_26D6_Vec nearest_point; /* point on curve nearest to `point' */
+ FT_26D6_Vec direction; /* direction of curve at `nearest_point' */
+
+ FT_26D6_Vec p0, p1, p2; /* control points of a conic curve */
+ FT_26D6_Vec p; /* `point' to which shortest distance */
+
+ FT_16D16 min_factor; /* factor at `nearest_point' */
+ FT_16D16 cross; /* to determine the sign */
+ FT_16D16 min = FT_INT_MAX; /* shortest squared distance */
+
+ FT_UShort iterations;
+ FT_UShort steps;
+
+ if ( !conic || !out )
+ {
+ error = FT_THROW( Invalid_Argument );
+ goto Exit;
+ }
+
+ if ( conic->edge_type != SDF_EDGE_CONIC )
+ {
+ error = FT_THROW( Invalid_Argument );
+ goto Exit;
+ }
+
+ /* assign the values after checking pointer */
+ p0 = conic->start_pos;
+ p1 = conic->control_a;
+ p2 = conic->end_pos;
+ p = point;
+
+ /* compute substitution coefficients */
+ aA.x = p0.x - 2 * p1.x + p2.x;
+ aA.y = p0.y - 2 * p1.y + p2.y;
+
+ bB.x = 2 * ( p1.x - p0.x );
+ bB.y = 2 * ( p1.y - p0.y );
+
+ cC.x = p0.x;
+ cC.y = p0.y;
+
+ /* do newton's iterations */
+ for ( iterations = 0; iterations <= MAX_NEWTON_ITERATION; iterations++ )
+ {
+ FT_16D16 factor = FT_INT_16D16( iterations ) / MAX_NEWTON_ITERATION;
+ FT_16D16 factor2;
+ FT_16D16 length;
+
+ FT_16D16_Vec curve_point; /* point on the curve */
+ FT_16D16_Vec dist_vector; /* `curve_point' - `p' */
+
+ FT_26D6_Vec d1; /* first derivative */
+ FT_26D6_Vec d2; /* second derivative */
+
+ FT_16D16 temp1;
+ FT_16D16 temp2;
+
+ for ( steps = 0; steps < MAX_NEWTON_STEPS; steps++ )
+ {
+ factor2 = FT_MulFix( factor, factor );
+
+ /* B( t ) = t^2( A ) + t( B ) + p0 */
+ curve_point.x = FT_MulFix( aA.x, factor2 ) +
+ FT_MulFix( bB.x, factor ) + cC.x;
+ curve_point.y = FT_MulFix( aA.y, factor2 ) +
+ FT_MulFix( bB.y, factor ) + cC.y;
+
+ /* convert to 16.16 */
+ curve_point.x = FT_26D6_16D16( curve_point.x );
+ curve_point.y = FT_26D6_16D16( curve_point.y );
+
+ /* B( t ) = t^2( A ) + t( B ) + p0 - p. P( t ) in the comment */
+ dist_vector.x = curve_point.x - FT_26D6_16D16( p.x );
+ dist_vector.y = curve_point.y - FT_26D6_16D16( p.y );
+
+ length = VECTOR_LENGTH_16D16( dist_vector );
+
+ if ( length < min )
+ {
+ min = length;
+ min_factor = factor;
+ nearest_point = curve_point;
+ }
+
+ /* This the actual Newton's approximation. */
+ /* t -= P( t ).B`( t ) / */
+ /* ( B`( t ).B`( t ) + P( t ).B``( t ) ) */
+
+ /* B`( t ) = 2tA + B */
+ d1.x = FT_MulFix( aA.x, 2 * factor ) + bB.x;
+ d1.y = FT_MulFix( aA.y, 2 * factor ) + bB.y;
+
+ /* B``( t ) = 2A */
+ d2.x = 2 * aA.x;
+ d2.y = 2 * aA.y;
+
+ dist_vector.x /= 1024;
+ dist_vector.y /= 1024;
+
+ /* temp1 = P( t ).B`( t ) */
+ temp1 = VEC_26D6_DOT( dist_vector, d1 );
+
+ /* temp2 = ( B`( t ).B`( t ) + P( t ).B``( t ) ) */
+ temp2 = VEC_26D6_DOT( d1, d1 ) +
+ VEC_26D6_DOT( dist_vector, d2 );
+
+ factor -= FT_DivFix( temp1, temp2 );
+
+ if ( factor < 0 || factor > FT_INT_16D16( 1 ) )
+ break;
+ }
+ }
+
+ /* B`( t ) = 2tA + B */
+ direction.x = 2 * FT_MulFix( aA.x, min_factor ) + bB.x;
+ direction.y = 2 * FT_MulFix( aA.y, min_factor ) + bB.y;
+
+ /* determine the sign */
+ cross = FT_MulFix( nearest_point.x - FT_26D6_16D16( p.x ), direction.y ) -
+ FT_MulFix( nearest_point.y - FT_26D6_16D16( p.y ), direction.x );
+
+ /* assign the values */
+ out->distance = min;
+ out->nearest_point = nearest_point;
+ out->sign = cross < 0 ? 1 : -1;
+
+ FT_Vector_NormLen( &direction );
+
+ out->direction = direction;
+
+ Exit:
+ return error;
+ }
+
+#endif
+
/**************************************************************************
*
* @Function:
| [Prev in Thread] |
Current Thread |
[Next in Thread] |
- [freetype2] anuj-distance-field 363f1e8 29/93: [sdf] Added Newton's method for conic curve.,
Anuj Verma <=