/* fit.c: turn a bitmap representation of a curve into a list of splines. Some of the ideas, but not the code, comes from the Phoenix thesis. See README for the reference. Copyright (C) 1992 Free Software Foundation, Inc. This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. */ #include <float.h> #include <math.h> #include <assert.h> #include <glib.h> #include "global.h" #include "spline.h" #include "vector.h" #include "curve.h" #include "fit.h" #include "pxl-outline.h" /* If two endpoints are closer than this, they are made to be equal. (-align-threshold) */ real align_threshold = 0.5; /* If the angle defined by a point and its predecessors and successors is smaller than this, it's a corner, even if it's within `corner_surround' pixels of a point with a smaller angle. (-corner-always-threshold) */ real corner_always_threshold = 60.0; /* Number of points to consider when determining if a point is a corner or not. (-corner-surround) */ unsigned corner_surround = 4; /* If a point, its predecessors, and its successors define an angle smaller than this, it's a corner. Should be in range 0..180. (-corner-threshold) */ real corner_threshold = 100.0; /* Amount of error at which a fitted spline is unacceptable. If any pixel is further away than this from the fitted curve, we try again. (-error-threshold) */ /* real error_threshold = .8; ALT */ real error_threshold = .4; /* A second number of adjacent points to consider when filtering. (-filter-alternative-surround) */ unsigned filter_alternative_surround = 1; /* If the angles between the vectors produced by filter_surround and filter_alternative_surround points differ by more than this, use the one from filter_alternative_surround. (-filter-epsilon) */ real filter_epsilon = 10.0; /* Number of times to smooth original data points. Increasing this number dramatically---to 50 or so---can produce vastly better results. But if any points that ``should'' be corners aren't found, the curve goes to hell around that point. (-filter-iterations) */ /* unsigned filter_iteration_count = 4; ALT */ unsigned filter_iteration_count = 4; /* To produce the new point, use the old point plus this times the neighbors. (-filter-percent) */ real filter_percent = .33; /* Number of adjacent points to consider if `filter_surround' points defines a straight line. (-filter-secondary-surround) */ unsigned filter_secondary_surround = 3; /* Number of adjacent points to consider when filtering. (-filter-surround) */ unsigned filter_surround = 2; /* Says whether or not to remove ``knee'' points after finding the outline. (See the comments at `remove_knee_points'.) (-remove-knees). */ boolean keep_knees = false; /* If a spline is closer to a straight line than this, it remains a straight line, even if it would otherwise be changed back to a curve. This is weighted by the square of the curve length, to make shorter curves more likely to be reverted. (-line-reversion-threshold) */ real line_reversion_threshold = .01; /* How many pixels (on the average) a spline can diverge from the line determined by its endpoints before it is changed to a straight line. (-line-threshold) */ /* real line_threshold = 1.0; ALT */ real line_threshold = 0.5; /* If reparameterization doesn't improve the fit by this much percent, stop doing it. (-reparameterize-improve) */ /* real reparameterize_improvement = .10; ALT */ real reparameterize_improvement = .01; /* Amount of error at which it is pointless to reparameterize. This happens, for example, when we are trying to fit the outline of the outside of an `O' with a single spline. The initial fit is not good enough for the Newton-Raphson iteration to improve it. It may be that it would be better to detect the cases where we didn't find any corners. (-reparameterize-threshold) */ /* real reparameterize_threshold = 30.0; ALT */ real reparameterize_threshold = 1.0; /* Percentage of the curve away from the worst point to look for a better place to subdivide. (-subdivide-search) */ real subdivide_search = .1; /* Number of points to consider when deciding whether a given point is a better place to subdivide. (-subdivide-surround) */ unsigned subdivide_surround = 4; /* How many pixels a point can diverge from a straight line and still be considered a better place to subdivide. (-subdivide-threshold) */ real subdivide_threshold = .03; /* Number of points to look at on either side of a point when computing the approximation to the tangent at that point. (-tangent-surround) */ unsigned tangent_surround = 3; /* We need to manipulate lists of array indices. */ typedef struct index_list { unsigned *data; unsigned length; } index_list_type; /* The usual accessor macros. */ #define GET_INDEX(i_l, n) ((i_l).data[n]) #define INDEX_LIST_LENGTH(i_l) ((i_l).length) #define GET_LAST_INDEX(i_l) ((i_l).data[INDEX_LIST_LENGTH (i_l) - 1]) static void append_index (index_list_type *, unsigned); static void free_index_list (index_list_type *); static index_list_type new_index_list (); static void remove_adjacent_corners (index_list_type *, unsigned); static void align (spline_list_type *); static void change_bad_lines (spline_list_type *); static void filter (curve_type); static real filter_angle (vector_type, vector_type); static void find_curve_vectors (unsigned, curve_type, unsigned, vector_type *, vector_type *, unsigned *); static unsigned find_subdivision (curve_type, unsigned initial); static void find_vectors (unsigned, pixel_outline_type, vector_type *, vector_type *); static index_list_type find_corners (pixel_outline_type); static real find_error (curve_type, spline_type, unsigned *); static vector_type find_half_tangent (curve_type, boolean start, unsigned *); static void find_tangent (curve_type, boolean, boolean); static spline_type fit_one_spline (curve_type); static spline_list_type *fit_curve (curve_type); static spline_list_type fit_curve_list (curve_list_type); static spline_list_type *fit_with_least_squares (curve_type); static spline_list_type *fit_with_line (curve_type); static void remove_knee_points (curve_type, boolean); static boolean reparameterize (curve_type, spline_type); static void set_initial_parameter_values (curve_type); static boolean spline_linear_enough (spline_type *, curve_type); static curve_list_array_type split_at_corners (pixel_outline_list_type); static boolean test_subdivision_point (curve_type, unsigned, vector_type *); /* The top-level call that transforms the list of pixels in the outlines of the original character to a list of spline lists fitted to those pixels. */ spline_list_array_type fitted_splines (pixel_outline_list_type pixel_outline_list) { unsigned this_list; unsigned total = 0; spline_list_array_type char_splines = new_spline_list_array (); curve_list_array_type curve_array = split_at_corners (pixel_outline_list); for (this_list = 0; this_list < CURVE_LIST_ARRAY_LENGTH (curve_array); this_list++) { spline_list_type curve_list_splines; curve_list_type curves = CURVE_LIST_ARRAY_ELT (curve_array, this_list); curve_list_splines = fit_curve_list (curves); append_spline_list (&char_splines, curve_list_splines); /* REPORT ("* "); */ } free_curve_list_array (&curve_array); for (this_list = 0; this_list < SPLINE_LIST_ARRAY_LENGTH (char_splines); this_list++) total += SPLINE_LIST_LENGTH (SPLINE_LIST_ARRAY_ELT (char_splines, this_list)); /* REPORT1 ("=%u", total); */ return char_splines; } /* Set up the internal parameters from the external ones */ void fit_set_params(SELVALS *selVals) { align_threshold = selVals->align_threshold; corner_always_threshold = selVals->corner_always_threshold; corner_surround = selVals->corner_surround; corner_threshold = selVals->corner_threshold; error_threshold = selVals->error_threshold; filter_alternative_surround = selVals->filter_alternative_surround; filter_epsilon = selVals->filter_epsilon; filter_iteration_count = selVals->filter_iteration_count; filter_percent = selVals->filter_percent; filter_secondary_surround = selVals->filter_secondary_surround; filter_surround = selVals->filter_surround; keep_knees = selVals->keep_knees; line_reversion_threshold = selVals->line_reversion_threshold; line_threshold = selVals->line_threshold; reparameterize_improvement = selVals->reparameterize_improvement; reparameterize_threshold = selVals->reparameterize_threshold; subdivide_search = selVals->subdivide_search; subdivide_surround = selVals->subdivide_surround; subdivide_threshold = selVals->subdivide_threshold; tangent_surround = selVals->tangent_surround; } void fit_set_default_params(SELVALS *selVals) { selVals->align_threshold = align_threshold; selVals->corner_always_threshold = corner_always_threshold; selVals->corner_surround = corner_surround; selVals->corner_threshold = corner_threshold; selVals->error_threshold = error_threshold; selVals->filter_alternative_surround = filter_alternative_surround; selVals->filter_epsilon = filter_epsilon; selVals->filter_iteration_count = filter_iteration_count; selVals->filter_percent = filter_percent; selVals->filter_secondary_surround = filter_secondary_surround; selVals->filter_surround = filter_surround; selVals->keep_knees = keep_knees; selVals->line_reversion_threshold = line_reversion_threshold; selVals->line_threshold = line_threshold; selVals->reparameterize_improvement = reparameterize_improvement; selVals->reparameterize_threshold = reparameterize_threshold; selVals->subdivide_search = subdivide_search; selVals->subdivide_surround = subdivide_surround; selVals->subdivide_threshold = subdivide_threshold; selVals->tangent_surround = tangent_surround; } /* Fit the list of curves CURVE_LIST to a list of splines, and return it. CURVE_LIST represents a single closed paths, e.g., either the inside or outside outline of an `o'. */ static spline_list_type fit_curve_list (curve_list_type curve_list) { curve_type curve; unsigned this_curve, this_spline; unsigned curve_list_length = CURVE_LIST_LENGTH (curve_list); spline_list_type curve_list_splines = *new_spline_list (); /* Remove the extraneous ``knee'' points before filtering. Since the corners have already been found, we don't need to worry about removing a point that should be a corner. */ if (!keep_knees) { /* LOG ("\nRemoving knees:\n"); */ for (this_curve = 0; this_curve < curve_list_length; this_curve++) { /* LOG1 ("#%u:", this_curve); */ remove_knee_points (CURVE_LIST_ELT (curve_list, this_curve), CURVE_LIST_CLOCKWISE (curve_list)); } } /* We filter all the curves in CURVE_LIST at once; otherwise, we would look at an unfiltered curve when computing tangents. */ /* LOG ("\nFiltering curves:\n"); */ for (this_curve = 0; this_curve < curve_list.length; this_curve++) { /* LOG1 ("#%u: ", this_curve); */ filter (CURVE_LIST_ELT (curve_list, this_curve)); /* REPORT ("f"); */ } /* Make the first point in the first curve also be the last point in the last curve, so the fit to the whole curve list will begin and end at the same point. This may cause slight errors in computing the tangents and t values, but it's worth it for the continuity. Of course we don't want to do this if the two points are already the same, as they are if the curve is cyclic. (We don't append it earlier, in `split_at_corners', because that confuses the filtering.) Finally, we can't append the point if the curve is exactly three points long, because we aren't adding any more data, and three points isn't enough to determine a spline. Therefore, the fitting will fail. */ curve = CURVE_LIST_ELT (curve_list, 0); if (CURVE_CYCLIC (curve) && CURVE_LENGTH (curve) != 3) append_point (curve, CURVE_POINT (curve, 0)); /* Finally, fit each curve in the list to a list of splines. */ for (this_curve = 0; this_curve < curve_list_length; this_curve++) { spline_list_type *curve_splines; curve_type current_curve = CURVE_LIST_ELT (curve_list, this_curve); /* REPORT1 (" %u", this_curve); */ /* LOG1 ("\nFitting curve #%u:\n", this_curve); */ curve_splines = fit_curve (current_curve); if (curve_splines == NULL) printf("Could not fit curve #%u", this_curve); else { /* LOG1 ("Fitted splines for curve #%u:\n", this_curve); */ for (this_spline = 0; this_spline < SPLINE_LIST_LENGTH (*curve_splines); this_spline++) { /* LOG1 (" %u: ", this_spline); */ /* if (logging) */ /* print_spline (log_ file, */ /* SPLINE_LIST_ELT (*curve_splines, this_spline)); */ } /* After fitting, we may need to change some would-be lines back to curves, because they are in a list with other curves. */ change_bad_lines (curve_splines); concat_spline_lists (&curve_list_splines, *curve_splines); /* REPORT1 ("(%u)", SPLINE_LIST_LENGTH (*curve_splines)); */ } } /* We do this for each outline's spline list because when a point is changed, it needs to be changed in both segments in which it appears---and the segments might be in different curves. */ align (&curve_list_splines); return curve_list_splines; } /* Transform a set of locations to a list of splines (the fewer the better). We are guaranteed that CURVE does not contain any corners. We return NULL if we cannot fit the points at all. */ static spline_list_type * fit_curve (curve_type curve) { spline_list_type *fitted_splines; if (CURVE_LENGTH (curve) < 2) { printf ("Tried to fit curve with less than two points"); return NULL; } /* Do we have enough points to fit with a spline? */ fitted_splines = CURVE_LENGTH (curve) < 4 ? fit_with_line (curve) : fit_with_least_squares (curve); return fitted_splines; } /* As mentioned above, the first step is to find the corners in PIXEL_LIST, the list of points. (Presumably we can't fit a single spline around a corner.) The general strategy is to look through all the points, remembering which we want to consider corners. Then go through that list, producing the curve_list. This is dictated by the fact that PIXEL_LIST does not necessarily start on a corner---it just starts at the character's first outline pixel, going left-to-right, top-to-bottom. But we want all our splines to start and end on real corners. For example, consider the top of a capital `C' (this is in cmss20): x *********** ****************** PIXEL_LIST will start at the pixel below the `x'. If we considered this pixel a corner, we would wind up matching a very small segment from there to the end of the line, probably as a straight line, which is certainly not what we want. PIXEL_LIST has one element for each closed outline on the character. To preserve this information, we return an array of curve_lists, one element (which in turn consists of several curves, one between each pair of corners) for each element in PIXEL_LIST. */ static curve_list_array_type split_at_corners (pixel_outline_list_type pixel_list) { unsigned this_pixel_o; curve_list_array_type curve_array = new_curve_list_array (); /* LOG ("\nFinding corners:\n"); */ for (this_pixel_o = 0; this_pixel_o < O_LIST_LENGTH (pixel_list); this_pixel_o++) { curve_type curve, first_curve; index_list_type corner_list; unsigned p, this_corner; curve_list_type curve_list = new_curve_list (); pixel_outline_type pixel_o = O_LIST_OUTLINE (pixel_list, this_pixel_o); CURVE_LIST_CLOCKWISE (curve_list) = O_CLOCKWISE (pixel_o); /* LOG1 ("#%u:", this_pixel_o); */ /* If the outline does not have enough points, we can't do anything. The endpoints of the outlines are automatically corners. We need at least `corner_surround' more pixels on either side of a point before it is conceivable that we might want another corner. */ if (O_LENGTH (pixel_o) > corner_surround * 2 + 2) corner_list = find_corners (pixel_o); else corner_list.length = 0; /* Remember the first curve so we can make it be the `next' of the last one. (And vice versa.) */ first_curve = new_curve (); curve = first_curve; if (corner_list.length == 0) { /* No corners. Use all of the pixel outline as the curve. */ for (p = 0; p < O_LENGTH (pixel_o); p++) append_pixel (curve, O_COORDINATE (pixel_o, p)); /* This curve is cyclic. */ CURVE_CYCLIC (curve) = true; } else { /* Each curve consists of the points between (inclusive) each pair of corners. */ for (this_corner = 0; this_corner < corner_list.length - 1; this_corner++) { curve_type previous_curve = curve; unsigned corner = GET_INDEX (corner_list, this_corner); unsigned next_corner = GET_INDEX (corner_list, this_corner + 1); for (p = corner; p <= next_corner; p++) append_pixel (curve, O_COORDINATE (pixel_o, p)); append_curve (&curve_list, curve); curve = new_curve (); NEXT_CURVE (previous_curve) = curve; PREVIOUS_CURVE (curve) = previous_curve; } /* The last curve is different. It consists of the points (inclusive) between the last corner and the end of the list, and the beginning of the list and the first corner. */ for (p = GET_LAST_INDEX (corner_list); p < O_LENGTH (pixel_o); p++) append_pixel (curve, O_COORDINATE (pixel_o, p)); for (p = 0; p <= GET_INDEX (corner_list, 0); p++) append_pixel (curve, O_COORDINATE (pixel_o, p)); } /* LOG1 (" [%u].\n", corner_list.length); */ /* Add `curve' to the end of the list, updating the pointers in the chain. */ append_curve (&curve_list, curve); NEXT_CURVE (curve) = first_curve; PREVIOUS_CURVE (first_curve) = curve; /* And now add the just-completed curve list to the array. */ append_curve_list (&curve_array, curve_list); } /* End of considering each pixel outline. */ return curve_array; } /* We consider a point to be a corner if (1) the angle defined by the `corner_surround' points coming into it and going out from it is less than `corner_threshold' degrees, and no point within `corner_surround' points has a smaller angle; or (2) the angle is less than `corner_always_threshold' degrees. Because of the different cases, it is convenient to have the following macro to append a corner on to the list we return. The character argument C is simply so that the different cases can be distinguished in the log file. */ #define APPEND_CORNER(index, angle, c) \ do \ { \ append_index (&corner_list, index); \ /*LOG4 (" (%d,%d)%c%.3f", */ \ /* O_COORDINATE (pixel_outline, index).x,*/ \ /* O_COORDINATE (pixel_outline, index).y,*/ \ /* c, angle);*/ \ } \ while (0) static index_list_type find_corners (pixel_outline_type pixel_outline) { unsigned p; index_list_type corner_list = new_index_list (); /* Consider each pixel on the outline in turn. */ for (p = 0; p < O_LENGTH (pixel_outline); p++) { real corner_angle; vector_type in_vector, out_vector; /* Check if the angle is small enough. */ find_vectors (p, pixel_outline, &in_vector, &out_vector); corner_angle = Vangle (in_vector, out_vector); if (fabs (corner_angle) <= corner_threshold) { /* We want to keep looking, instead of just appending the first pixel we find with a small enough angle, since there might be another corner within `corner_surround' pixels, with a smaller angle. If that is the case, we want that one. */ real best_corner_angle = corner_angle; unsigned best_corner_index = p; index_list_type equally_good_list = new_index_list (); /* As we come into the loop, `p' is the index of the point that has an angle less than `corner_angle'. We use `i' to move through the pixels next to that, and `q' for moving through the adjacent pixels to each `p'. */ unsigned q = p; unsigned i = p + 1; while (true) { /* Perhaps the angle is sufficiently small that we want to consider this a corner, even if it's not the best (unless we've already wrapped around in the search, i.e., `q<i', in which case we have already added the corner, and we don't want to add it again). We want to do this check on the first candidate we find, as well as the others in the loop, hence this comes before the stopping condition. */ if (corner_angle <= corner_always_threshold && q >= p) APPEND_CORNER (q, corner_angle, '\\'); /* Exit the loop if we've looked at `corner_surround' pixels past the best one we found, or if we've looked at all the pixels. */ if (i >= best_corner_index + corner_surround || i >= O_LENGTH (pixel_outline)) break; /* Check the angle. */ q = i % O_LENGTH (pixel_outline); find_vectors (q, pixel_outline, &in_vector, &out_vector); corner_angle = Vangle (in_vector, out_vector); /* If we come across a corner that is just as good as the best one, we should make it a corner, too. This happens, for example, at the points on the `W' in some typefaces, where the ``points'' are flat. */ if (epsilon_equal (corner_angle, best_corner_angle)) append_index (&equally_good_list, q); else if (corner_angle < best_corner_angle) { best_corner_angle = corner_angle; /* We want to check `corner_surround' pixels beyond the new best corner. */ i = best_corner_index = q; free_index_list (&equally_good_list); equally_good_list = new_index_list (); } i++; } /* After we exit the loop, `q' is the index of the last point we checked. We have already added the corner if `best_corner_angle' is less than `corner_always_threshold'. Again, if we've already wrapped around, we don't want to add the corner again. */ if (best_corner_angle > corner_always_threshold && best_corner_index >= p) { unsigned i; APPEND_CORNER (best_corner_index, best_corner_angle, '/'); for (i = 0; i < INDEX_LIST_LENGTH (equally_good_list); i++) APPEND_CORNER (GET_INDEX (equally_good_list, i), best_corner_angle, '@'); free_index_list (&equally_good_list); } /* If we wrapped around in our search, we're done; otherwise, we don't want the outer loop to look at the pixels that we already looked at in searching for the best corner. */ p = (q < p) ? O_LENGTH (pixel_outline) : q; } /* End of searching for the best corner. */ } /* End of considering each pixel. */ if (INDEX_LIST_LENGTH (corner_list) > 0) /* We never want two corners next to each other, since the only way to fit such a ``curve'' would be with a straight line, which usually interrupts the continuity dreadfully. */ remove_adjacent_corners (&corner_list, O_LENGTH (pixel_outline) - 1); return corner_list; } /* Return the difference vectors coming in and going out of the outline OUTLINE at the point whose index is TEST_INDEX. In Phoenix, Schneider looks at a single point on either side of the point we're considering. That works for him because his points are not touching. But our points *are* touching, and so we have to look at `corner_surround' points on either side, to get a better picture of the outline's shape. */ static void find_vectors (unsigned test_index, pixel_outline_type outline, vector_type *in, vector_type *out) { int i; unsigned n_done; coordinate_type candidate = O_COORDINATE (outline, test_index); in->dx = 0.0; in->dy = 0.0; out->dx = 0.0; out->dy = 0.0; /* Add up the differences from p of the `corner_surround' points before p. */ for (i = O_PREV (outline, test_index), n_done = 0; n_done < corner_surround; i = O_PREV (outline, i), n_done++) *in = Vadd (*in, IPsubtract (O_COORDINATE (outline, i), candidate)); #if 0 /* We don't need this code any more, because now we create the pixel outline from the corners of the pixels, rather than the edges. */ /* To see why we need this test, consider the following case: four pixels stacked vertically with no other adjacent pixels, i.e., * *x * * *** (etc.) We are considering the pixel marked `x' for cornerhood. The out vector at this point is going to be the zero vector (if `corner_surround' is 3), because the first pixel on the outline is the one above the x, the second pixel x itself, and the third the one below x. (Remember that we go around the edges of the pixels to find the outlines, not the pixels themselves.) */ if (magnitude (*in) == 0.0) { WARNING ("Zero magnitude in"); return corner_threshold + 1.0; } #endif /* 0 */ /* And the points after p. */ for (i = O_NEXT (outline, test_index), n_done = 0; n_done < corner_surround; i = O_NEXT (outline, i), n_done++) *out = Vadd (*out, IPsubtract (O_COORDINATE (outline, i), candidate)); #if 0 /* As with the test for the in vector, we don't need this any more. */ if (magnitude (*out) == 0.0) { WARNING ("Zero magnitude out"); return corner_threshold + 1.0; } #endif /* 0 */ } /* Remove adjacent points from the index list LIST. We do this by first sorting the list and then running through it. Since these lists are quite short, a straight selection sort (e.g., p.139 of the Art of Computer Programming, vol.3) is good enough. LAST_INDEX is the index of the last pixel on the outline, i.e., the next one is the first pixel. We need this for checking the adjacency of the last corner. We need to do this because the adjacent corners turn into two-pixel-long curves, which can only be fit by straight lines. */ static void remove_adjacent_corners (index_list_type *list, unsigned last_index) { unsigned j; unsigned last; index_list_type new = new_index_list (); for (j = INDEX_LIST_LENGTH (*list) - 1; j > 0; j--) { unsigned search; unsigned temp; /* Find maximal element below `j'. */ unsigned max_index = j; for (search = 0; search < j; search++) if (GET_INDEX (*list, search) > GET_INDEX (*list, max_index)) max_index = search; if (max_index != j) { temp = GET_INDEX (*list, j); GET_INDEX (*list, j) = GET_INDEX (*list, max_index); GET_INDEX (*list, max_index) = temp; printf ("needed exchange"); /* xx -- really have to sort? */ } } /* The list is sorted. Now look for adjacent entries. Each time through the loop we insert the current entry and, if appropriate, the next entry. */ for (j = 0; j < INDEX_LIST_LENGTH (*list) - 1; j++) { unsigned current = GET_INDEX (*list, j); unsigned next = GET_INDEX (*list, j + 1); /* We should never have inserted the same element twice. */ assert (current != next); append_index (&new, current); if (next == current + 1) j++; } /* Don't append the last element if it is 1) adjacent to the previous one; or 2) adjacent to the very first one. */ last = GET_LAST_INDEX (*list); if (INDEX_LIST_LENGTH (new) == 0 || !(last == GET_LAST_INDEX (new) + 1 || (last == last_index && GET_INDEX (*list, 0) == 0))) append_index (&new, last); free_index_list (list); *list = new; } /* A ``knee'' is a point which forms a ``right angle'' with its predecessor and successor. See the documentation (the `Removing knees' section) for an example and more details. The argument CLOCKWISE tells us which direction we're moving. (We can't figure that information out from just the single segment with which we are given to work.) We should never find two consecutive knees. Since the first and last points are corners (unless the curve is cyclic), it doesn't make sense to remove those. */ /* This evaluates to true if the vector V is zero in one direction and nonzero in the other. */ #define ONLY_ONE_ZERO(v) \ (((v).dx == 0.0 && (v).dy != 0.0) || ((v).dy == 0.0 && (v).dx != 0.0)) /* There are four possible cases for knees, one for each of the four corners of a rectangle; and then the cases differ depending on which direction we are going around the curve. The tests are listed here in the order of upper left, upper right, lower right, lower left. Perhaps there is some simple pattern to the clockwise/counterclockwise differences, but I don't see one. */ #define CLOCKWISE_KNEE(prev_delta, next_delta) \ ((prev_delta.dx == -1.0 && next_delta.dy == 1.0) \ || (prev_delta.dy == 1.0 && next_delta.dx == 1.0) \ || (prev_delta.dx == 1.0 && next_delta.dy == -1.0) \ || (prev_delta.dy == -1.0 && next_delta.dx == -1.0)) #define COUNTERCLOCKWISE_KNEE(prev_delta, next_delta) \ ((prev_delta.dy == 1.0 && next_delta.dx == -1.0) \ || (prev_delta.dx == 1.0 && next_delta.dy == 1.0) \ || (prev_delta.dy == -1.0 && next_delta.dx == 1.0) \ || (prev_delta.dx == -1.0 && next_delta.dy == -1.0)) static void remove_knee_points (curve_type curve, boolean clockwise) { int i; unsigned offset = CURVE_CYCLIC (curve) ? 0 : 1; coordinate_type previous = real_to_int_coord (CURVE_POINT (curve, CURVE_PREV (curve, offset))); curve_type trimmed_curve = copy_most_of_curve (curve); if (!CURVE_CYCLIC (curve)) append_pixel (trimmed_curve, real_to_int_coord (CURVE_POINT (curve, 0))); for (i = offset; i < CURVE_LENGTH (curve) - offset; i++) { coordinate_type current = real_to_int_coord (CURVE_POINT (curve, i)); coordinate_type next = real_to_int_coord (CURVE_POINT (curve, CURVE_NEXT (curve, i))); vector_type prev_delta = IPsubtract (previous, current); vector_type next_delta = IPsubtract (next, current); if (ONLY_ONE_ZERO (prev_delta) && ONLY_ONE_ZERO (next_delta) && ((clockwise && CLOCKWISE_KNEE (prev_delta, next_delta)) || (!clockwise && COUNTERCLOCKWISE_KNEE (prev_delta, next_delta)))) { /* LOG2 (" (%d,%d)", current.x, current.y); */ } else { previous = current; append_pixel (trimmed_curve, current); } } if (!CURVE_CYCLIC (curve)) append_pixel (trimmed_curve, real_to_int_coord (LAST_CURVE_POINT (curve))); /* if (CURVE_LENGTH (trimmed_curve) == CURVE_LENGTH (curve)) */ /* LOG (" (none)"); */ /* LOG (".\n"); */ free_curve (curve); *curve = *trimmed_curve; } /* Smooth the curve by adding in neighboring points. Do this `filter_iteration_count' times. But don't change the corners. */ #if 0 /* Computing the new point based on a single neighboring point and with constant percentages, as the `SMOOTH' macro did, isn't quite good enough. For example, at the top of an `o' the curve might well have three consecutive horizontal pixels, even though there isn't really a straight there. With this code, the middle point would remain unfiltered. */ #define SMOOTH(axis) \ CURVE_POINT (curve, this_point).axis \ = ((1.0 - center_percent) / 2) \ * CURVE_POINT (curve, CURVE_PREV (curve, this_point)).axis \ + center_percent * CURVE_POINT (curve, this_point).axis \ + ((1.0 - center_percent) / 2) \ * CURVE_POINT (curve, CURVE_NEXT (curve, this_point)).axis #endif /* 0 */ /* We sometimes need to change the information about the filtered point. This macro assigns to the relevant variables. */ #define FILTER_ASSIGN(new) \ do \ { \ in = in_##new; \ out = out_##new; \ count = new##_count; \ angle = angle_##new; \ } \ while (0) static void filter (curve_type curve) { unsigned iteration, this_point; unsigned offset = CURVE_CYCLIC (curve) ? 0 : 1; /* We must have at least three points---the previous one, the current one, and the next one. But if we don't have at least five, we will probably collapse the curve down onto a single point, which means we won't be able to fit it with a spline. */ if (CURVE_LENGTH (curve) < 5) { /* LOG1 ("Length is %u, not enough to filter.\n", CURVE_LENGTH (curve)); */ return; } for (iteration = 0; iteration < filter_iteration_count; iteration++) { curve_type new_curve = copy_most_of_curve (curve); /* Keep the first point on the curve. */ if (offset) append_point (new_curve, CURVE_POINT (curve, 0)); for (this_point = offset; this_point < CURVE_LENGTH (curve) - offset; this_point++) { real angle, angle_alt; vector_type in, in_alt, out, out_alt, sum; unsigned count, alt_count; real_coordinate_type new_point; /* Find the angle using the usual number of surrounding points on the curve. */ find_curve_vectors (this_point, curve, filter_surround, &in, &out, &count); angle = filter_angle (in, out); /* Find the angle using the alternative (presumably smaller) number. */ find_curve_vectors (this_point, curve, filter_alternative_surround, &in_alt, &out_alt, &alt_count); angle_alt = filter_angle (in_alt, out_alt); /* If the alternate angle is enough larger than the usual one and neither of the components of the sum are zero, use it. (We don't use absolute value here, since if the alternate angle is smaller, we don't particularly care, since that means the curve is pretty flat around the current point, anyway.) This helps keep small features from disappearing into the surrounding curve. */ sum = Vadd (in_alt, out_alt); if (angle_alt - angle >= filter_epsilon && sum.dx != 0 && sum.dy != 0) FILTER_ASSIGN (alt); #if 0 /* This code isn't needed anymore, since we do the filtering in a somewhat more general way. */ /* If we're left with an angle of zero, don't stop yet; we might be at a straight which really isn't one (as in the `o' discussed above). */ if (epsilon_equal (angle, 0.0)) { real angle_secondary; vector_type in_secondary, out_secondary; unsigned in_secondary_count, out_secondary_count; find_curve_vectors (this_point, curve, filter_secondary_surround, &in_secondary, &out_secondary, &in_secondary_count, &out_secondary_count); angle_secondary = filter_angle (in_secondary, out_secondary); if (!epsilon_equal (angle_secondary, 0.0)) FILTER_ASSIGN (secondary); } #endif /* 0 */ /* Start with the old point. */ new_point = CURVE_POINT (curve, this_point); sum = Vadd (in, out); new_point.x += sum.dx * filter_percent / count; new_point.y += sum.dy * filter_percent / count; /* Put the newly computed point into a separate curve, so it doesn't affect future computation (on this iteration). */ append_point (new_curve, new_point); } /* Just as with the first point, we have to keep the last point. */ if (offset) append_point (new_curve, LAST_CURVE_POINT (curve)); /* Set the original curve to the newly filtered one, and go again. */ free_curve (curve); *curve = *new_curve; } /* log_curve (curve, false); */ /* display_curve (curve); */ } /* Return the vectors IN and OUT, computed by looking at SURROUND points on either side of TEST_INDEX. Also return the number of points in the vectors in COUNT (we make sure they are the same). */ static void find_curve_vectors (unsigned test_index, curve_type curve, unsigned surround, vector_type *in, vector_type *out, unsigned *count) { int i; unsigned in_count, out_count; unsigned n_done; real_coordinate_type candidate = CURVE_POINT (curve, test_index); /* Add up the differences from p of the `surround' points before p. */ in->dx = 0.0; in->dy = 0.0; for (i = CURVE_PREV (curve, test_index), n_done = 0; i >= 0 && n_done < surround; /* Do not wrap around. */ i = CURVE_PREV (curve, i), n_done++) *in = Vadd (*in, Psubtract (CURVE_POINT (curve, i), candidate)); in_count = n_done; /* And the points after p. Don't use more points after p than we ended up with before it. */ out->dx = 0.0; out->dy = 0.0; for (i = CURVE_NEXT (curve, test_index), n_done = 0; i < CURVE_LENGTH (curve) && n_done < surround && n_done < in_count; i = CURVE_NEXT (curve, i), n_done++) *out = Vadd (*out, Psubtract (CURVE_POINT (curve, i), candidate)); out_count = n_done; /* If we used more points before p than after p, we have to go back and redo it. (We could just subtract the ones that were missing, but for this few of points, efficiency doesn't matter.) */ if (out_count < in_count) { in->dx = 0.0; in->dy = 0.0; for (i = CURVE_PREV (curve, test_index), n_done = 0; i >= 0 && n_done < out_count; i = CURVE_PREV (curve, i), n_done++) *in = Vadd (*in, Psubtract (CURVE_POINT (curve, i), candidate)); in_count = n_done; } assert (in_count == out_count); *count = in_count; } /* Find the angle between the vectors IN and OUT, and bring it into the range [0,45]. */ static real filter_angle (vector_type in, vector_type out) { real angle = Vangle (in, out); /* What we want to do between 90 and 180 is the same as what we want to do between 0 and 90. */ angle = fmod (angle, 1990.0); /* And what we want to do between 45 and 90 is the same as between 0 and 45, only reversed. */ if (angle > 45.0) angle = 90.0 - angle; return angle; } /* This routine returns the curve fitted to a straight line in a very simple way: make the first and last points on the curve be the endpoints of the line. This simplicity is justified because we are called only on very short curves. */ static spline_list_type * fit_with_line (curve_type curve) { spline_type line = new_spline (); /* LOG ("Fitting with straight line:\n"); */ /* REPORT ("l"); */ SPLINE_DEGREE (line) = LINEAR; START_POINT (line) = CONTROL1 (line) = CURVE_POINT (curve, 0); END_POINT (line) = CONTROL2 (line) = LAST_CURVE_POINT (curve); /* Make sure that this line is never changed to a cubic. */ SPLINE_LINEARITY (line) = 0; /* if (logging) */ /* { */ /* LOG (" "); */ /* print_spline (log_file, line); */ /* } */ return init_spline_list (line); } /* The least squares method is well described in Schneider's thesis. Briefly, we try to fit the entire curve with one spline. If that fails, we try reparameterizing, i.e., finding new, and supposedly better, t values. If that still fails, we subdivide the curve. */ static spline_list_type * fit_with_least_squares (curve_type curve) { real error, best_error = FLT_MAX; spline_type spline, best_spline; spline_list_type *spline_list; unsigned worst_point; unsigned iteration = 0; real previous_error = FLT_MAX; real improvement = FLT_MAX; /* LOG ("\nFitting with least squares:\n"); */ /* Phoenix reduces the number of points with a ``linear spline technique''. But for fitting letterforms, that is inappropriate. We want all the points we can get. */ /* It makes no difference whether we first set the `t' values or find the tangents. This order makes the documentation a little more coherent. */ /* LOG ("Finding tangents:\n"); */ find_tangent (curve, /* to_start */ true, /* cross_curve */ false); find_tangent (curve, /* to_start */ false, /* cross_curve */ false); set_initial_parameter_values (curve); /* Now we loop, reparameterizing and/or subdividing, until CURVE has been fit. */ while (true) { /* LOG (" fitted to spline:\n"); */ spline = fit_one_spline (curve); /* if (logging) */ /* { */ /* LOG (" "); */ /* print_spline (log_file, spline); */ /* } */ error = find_error (curve, spline, &worst_point); if (error > previous_error) { /* LOG ("Reparameterization made it worse.\n"); */ /* Just fall through; we will subdivide. */ } else { best_error = error; best_spline = spline; } improvement = 1.0 - error / previous_error; /* Don't exit, even if the error is less than `error_threshold', since we might be able to improve the fit with further reparameterization. If the reparameterization made it worse, we will exit here, since `improvement' will be negative. */ if (improvement < reparameterize_improvement || error > reparameterize_threshold) break; iteration++; /* LOG3 ("Reparameterization #%u (error was %.3f, a %u%% improvement):\n", */ /* iteration, error, ((unsigned) (improvement * 100.0))); */ /* The reparameterization might fail, if the initial fit was better than `reparameterize_threshold', yet worse than the Newton-Raphson algorithm could handle. */ if (!reparameterize (curve, spline)) break; previous_error = error; } /* Go back to the best fit. */ spline = best_spline; error = best_error; if (error < error_threshold) { /* The points were fitted with a (possibly reparameterized) spline. We end up here whenever a fit is accepted. We have one more job: see if the ``curve'' that was fit should really be a straight line. */ if (spline_linear_enough (&spline, curve)) { SPLINE_DEGREE (spline) = LINEAR; /* LOG ("Changed to line.\n"); */ } spline_list = init_spline_list (spline); /* LOG1 ("Accepted error of %.3f.\n", error); */ } else { /* We couldn't fit the curve acceptably, so subdivide. */ unsigned subdivision_index; spline_list_type *left_spline_list; spline_list_type *right_spline_list; curve_type left_curve = new_curve (); curve_type right_curve = new_curve (); /* Keep the linked list of curves intact. */ NEXT_CURVE (right_curve) = NEXT_CURVE (curve); PREVIOUS_CURVE (right_curve) = left_curve; NEXT_CURVE (left_curve) = right_curve; PREVIOUS_CURVE (left_curve) = curve; NEXT_CURVE (curve) = left_curve; /* REPORT ("s"); */ /* LOG1 ("\nSubdividing (error %.3f):\n", error); */ /* LOG3 (" Original point: (%.3f,%.3f), #%u.\n", */ /* CURVE_POINT (curve, worst_point).x, */ /* CURVE_POINT (curve, worst_point).y, worst_point); */ subdivision_index = find_subdivision (curve, worst_point); /* LOG3 (" Final point: (%.3f,%.3f), #%u.\n", */ /* CURVE_POINT (curve, subdivision_index).x, */ /* CURVE_POINT (curve, subdivision_index).y, subdivision_index); */ /* display_subdivision (CURVE_POINT (curve, subdivision_index)); */ /* The last point of the left-hand curve will also be the first point of the right-hand curve. */ CURVE_LENGTH (left_curve) = subdivision_index + 1; CURVE_LENGTH (right_curve) = CURVE_LENGTH (curve) - subdivision_index; left_curve->point_list = curve->point_list; right_curve->point_list = curve->point_list + subdivision_index; /* We want to use the tangents of the curve which we are subdividing for the start tangent for left_curve and the end tangent for right_curve. */ CURVE_START_TANGENT (left_curve) = CURVE_START_TANGENT (curve); CURVE_END_TANGENT (right_curve) = CURVE_END_TANGENT (curve); /* We have to set up the two curves before finding the tangent at the subdivision point. The tangent at that point must be the same for both curves, or noticeable bumps will occur in the character. But we want to use information on both sides of the point to compute the tangent, hence cross_curve = true. */ find_tangent (left_curve, /* to_start_point: */ false, /* cross_curve: */ true); CURVE_START_TANGENT (right_curve) = CURVE_END_TANGENT (left_curve); /* Now that we've set up the curves, we can fit them. */ left_spline_list = fit_curve (left_curve); right_spline_list = fit_curve (right_curve); /* Neither of the subdivided curves could be fit, so fail. */ if (left_spline_list == NULL && right_spline_list == NULL) return NULL; /* Put the two together (or whichever of them exist). */ spline_list = new_spline_list (); if (left_spline_list == NULL) { WARNING ("could not fit left spline list"); /* LOG1 ("Could not fit spline to left curve (%x).\n", */ /* (unsigned) left_curve); */ } else concat_spline_lists (spline_list, *left_spline_list); if (right_spline_list == NULL) { WARNING ("could not fit right spline list"); /* LOG1 ("Could not fit spline to right curve (%x).\n", */ /* (unsigned) right_curve); */ } else concat_spline_lists (spline_list, *right_spline_list); } return spline_list; } /* Our job here is to find alpha1 (and alpha2), where t1_hat (t2_hat) is the tangent to CURVE at the starting (ending) point, such that: control1 = alpha1*t1_hat + starting point control2 = alpha2*t2_hat + ending_point and the resulting spline (starting_point .. control1 and control2 .. ending_point) minimizes the least-square error from CURVE. See pp.57--59 of the Phoenix thesis. The B?(t) here corresponds to B_i^3(U_i) there. The Bernshte\u in polynomials of degree n are defined by B_i^n(t) = { n \choose i } t^i (1-t)^{n-i}, i = 0..n */ #define B0(t) CUBE (1 - (t)) #define B1(t) (3.0 * (t) * SQUARE (1 - (t))) #define B2(t) (3.0 * SQUARE (t) * (1 - (t))) #define B3(t) CUBE (t) #define U(i) CURVE_T (curve, i) static spline_type fit_one_spline (curve_type curve) { /* Since our arrays are zero-based, the `C0' and `C1' here correspond to `C1' and `C2' in the paper. */ real X_C1_det, C0_X_det, C0_C1_det; real alpha1, alpha2; spline_type spline; vector_type start_vector, end_vector; unsigned i; vector_type t1_hat = *CURVE_START_TANGENT (curve); vector_type t2_hat = *CURVE_END_TANGENT (curve); real C[2][2] = { { 0.0, 0.0 }, { 0.0, 0.0 } }; real X[2] = { 0.0, 0.0 }; int Alen = CURVE_LENGTH (curve); vector_type *A; A = g_new0 (vector_type, Alen * 2); START_POINT (spline) = CURVE_POINT (curve, 0); END_POINT (spline) = LAST_CURVE_POINT (curve); start_vector = make_vector (START_POINT (spline)); end_vector = make_vector (END_POINT (spline)); for (i = 0; i < CURVE_LENGTH (curve); i++) { A[i*2+0] = Vmult_scalar (t1_hat, B1 (U (i))); A[i*2+1] = Vmult_scalar (t2_hat, B2 (U (i))); } for (i = 0; i < CURVE_LENGTH (curve); i++) { vector_type temp, temp0, temp1, temp2, temp3; vector_type *Ai = &A[i*2]; C[0][0] += Vdot (Ai[0], Ai[0]); C[0][1] += Vdot (Ai[0], Ai[1]); /* C[1][0] = C[0][1] (this is assigned outside the loop) */ C[1][1] += Vdot (Ai[1], Ai[1]); /* Now the right-hand side of the equation in the paper. */ temp0 = Vmult_scalar (start_vector, B0 (U (i))); temp1 = Vmult_scalar (start_vector, B1 (U (i))); temp2 = Vmult_scalar (end_vector, B2 (U (i))); temp3 = Vmult_scalar (end_vector, B3 (U (i))); temp = make_vector (Vsubtract_point (CURVE_POINT (curve, i), Vadd (temp0, Vadd (temp1, Vadd (temp2, temp3))))); X[0] += Vdot (temp, Ai[0]); X[1] += Vdot (temp, Ai[1]); } C[1][0] = C[0][1]; X_C1_det = X[0] * C[1][1] - X[1] * C[0][1]; C0_X_det = C[0][0] * X[1] - C[0][1] * X[0]; C0_C1_det = C[0][0] * C[1][1] - C[1][0] * C[0][1]; if (C0_C1_det == 0.0) FATAL ("zero determinant of C0*C1"); alpha1 = X_C1_det / C0_C1_det; alpha2 = C0_X_det / C0_C1_det; CONTROL1 (spline) = Vadd_point (START_POINT (spline), Vmult_scalar (t1_hat, alpha1)); CONTROL2 (spline) = Vadd_point (END_POINT (spline), Vmult_scalar (t2_hat, alpha2)); SPLINE_DEGREE (spline) = CUBIC; g_free (A); return spline; } /* Find closer-to-optimal t values for the given x,y's and control points, using Newton-Raphson iteration. A good description of this is in Plass & Stone. This routine performs one step in the iteration, not the whole thing. */ static boolean reparameterize (curve_type curve, spline_type S) { unsigned p, i; spline_type S1, S2; /* S' and S''. */ /* REPORT ("r"); */ /* Find the first and second derivatives of S. To make `evaluate_spline' work, we fill the beginning points (i.e., the first two for a linear spline and the first three for a quadratic one), even though this is at odds with the rest of the program. */ for (i = 0; i < 3; i++) { S1.v[i].x = 3.0 * (S.v[i + 1].x - S.v[i].x); S1.v[i].y = 3.0 * (S.v[i + 1].y - S.v[i].y); } S1.v[i].x = S1.v[i].y = -1.0; /* These will never be accessed. */ SPLINE_DEGREE (S1) = QUADRATIC; for (i = 0; i < 2; i++) { S2.v[i].x = 2.0 * (S1.v[i + 1].x - S1.v[i].x); S2.v[i].y = 2.0 * (S1.v[i + 1].y - S1.v[i].y); } S2.v[2].x = S2.v[2].y = S2.v[3].x = S2.v[3].y = -1.0; SPLINE_DEGREE (S2) = LINEAR; for (p = 0; p < CURVE_LENGTH (curve); p++) { real new_distance, old_distance; real_coordinate_type new_point; real_coordinate_type point = CURVE_POINT (curve, p); real t = CURVE_T (curve, p); /* Find the points at this t on S, S', and S''. */ real_coordinate_type S_t = evaluate_spline (S, t); real_coordinate_type S1_t = evaluate_spline (S1, t); real_coordinate_type S2_t = evaluate_spline (S2, t); /* The differences in x and y (Q1(t) on p.62 of Schneider's thesis). */ real_coordinate_type d; real numerator; real denominator; d.x = S_t.x - point.x; d.y = S_t.y - point.y; /* The step size, f(t)/f'(t). */ numerator = d.x * S1_t.x + d.y * S1_t.y; denominator = (SQUARE (S1_t.x) + d.x * S2_t.x + SQUARE (S1_t.y) + d.y * S2_t.y); /* We compute the distances only to be able to check that we really are moving closer. I don't know how this condition can be reliably checked for in advance, but I know what it means in practice: the original fit was not good enough for the Newton-Raphson iteration to converge. Therefore, we need to abort the reparameterization, and subdivide. */ old_distance = distance (S_t, point); CURVE_T (curve, p) -= numerator / denominator; new_point = evaluate_spline (S, CURVE_T (curve, p)); new_distance = distance (new_point, point); if (new_distance > old_distance) { /* REPORT ("!"); */ /* LOG3 (" Stopped reparameterizing; %.3f > %.3f at point %u.\n", */ /* new_distance, old_distance, p); */ return false; } /* The t value might be negative or > 1, if the choice of control points wasn't so great. We have no difficulty in evaluating a spline at a t outside the range zero to one, though, so it doesn't matter. (Although it is a little unconventional.) */ } /* LOG (" reparameterized curve:\n "); */ /* log_curve (curve, true); */ return true; } /* This routine finds the best place to subdivide the curve CURVE, somewhere near to the point whose index is INITIAL. Originally, curves were always subdivided at the point of worst error, which is intuitively appealing, but doesn't always give the best results. For example, at the end of a serif that tapers into the stem, the best subdivision point is at the point where they join, even if the worst point is a little ways into the serif. We return the index of the point at which to subdivide. */ static unsigned find_subdivision (curve_type curve, unsigned initial) { unsigned i, n_done; int best_point = -1; vector_type best = { FLT_MAX, FLT_MAX }; unsigned search = subdivide_search * CURVE_LENGTH (curve); /* LOG1 (" Number of points to search: %u.\n", search); */ /* We don't want to find the first (or last) point in the curve. */ for (i = initial, n_done = 0; i > 0 && n_done < search; i = CURVE_PREV (curve, i), n_done++) { if (test_subdivision_point (curve, i, &best)) { best_point = i; /* LOG3 (" Better point: (%.3f,%.3f), #%u.\n", */ /* CURVE_POINT (curve, i).x, CURVE_POINT (curve, i).y, i); */ } } /* If we found a good one, let's take it. */ if (best_point != -1) return best_point; for (i = CURVE_NEXT (curve, initial), n_done = 0; i < CURVE_LENGTH (curve) - 1 && n_done < search; i = CURVE_NEXT (curve, i), n_done++) { if (test_subdivision_point (curve, i, &best)) { best_point = i; /* LOG3 (" Better point at (%.3f,%.3f), #%u.\n", */ /* CURVE_POINT (curve, i).x, CURVE_POINT (curve, i).y, i); */ } } /* If we didn't find any better point, return the original. */ return best_point == -1 ? initial : best_point; } /* Here are some macros that decide whether or not we're at a ``join point'', as described above. */ #define ONLY_ONE_LESS(v) \ (((v).dx < subdivide_threshold && (v).dy > subdivide_threshold) \ || ((v).dy < subdivide_threshold && (v).dx > subdivide_threshold)) #define BOTH_GREATER(v) \ ((v).dx > subdivide_threshold && (v).dy > subdivide_threshold) /* We assume that the vectors V1 and V2 are nonnegative. */ #define JOIN(v1, v2) \ ((ONLY_ONE_LESS (v1) && BOTH_GREATER (v2)) \ || (ONLY_ONE_LESS (v2) && BOTH_GREATER (v1))) /* If the component D of the vector V is smaller than the best so far, update the best point. */ #define UPDATE_BEST(v, d) \ do \ { \ if ((v).d < subdivide_threshold && (v).d < best->d) \ best->d = (v).d; \ } \ while (0) /* If the point INDEX in the curve CURVE is the best subdivision point we've found so far, update the vector BEST. */ static boolean test_subdivision_point (curve_type curve, unsigned index, vector_type *best) { unsigned count; vector_type in, out; boolean join = false; find_curve_vectors (index, curve, subdivide_surround, &in, &out, &count); /* We don't want to subdivide at points which are very close to the endpoints, so if the vectors aren't computed from as many points as we asked for, don't bother checking this point. */ if (count == subdivide_surround) { in = Vabs (in); out = Vabs (out); join = JOIN (in, out); if (join) { UPDATE_BEST (in, dx); UPDATE_BEST (in, dy); UPDATE_BEST (out, dx); UPDATE_BEST (out, dy); } } return join; } /* Find reasonable values for t for each point on CURVE. The method is called chord-length parameterization, which is described in Plass & Stone. The basic idea is just to use the distance from one point to the next as the t value, normalized to produce values that increase from zero for the first point to one for the last point. */ static void set_initial_parameter_values (curve_type curve) { unsigned p; /* LOG ("\nAssigning initial t values:\n "); */ CURVE_T (curve, 0) = 0.0; for (p = 1; p < CURVE_LENGTH (curve); p++) { real_coordinate_type point = CURVE_POINT (curve, p), previous_p = CURVE_POINT (curve, p - 1); real d = distance (point, previous_p); CURVE_T (curve, p) = CURVE_T (curve, p - 1) + d; } assert (LAST_CURVE_T (curve) != 0.0); for (p = 1; p < CURVE_LENGTH (curve); p++) CURVE_T (curve, p) = CURVE_T (curve, p) / LAST_CURVE_T (curve); /* log_entire_curve (curve); */ } /* Find an approximation to the tangent to an endpoint of CURVE (to the first point if TO_START_POINT is true, else the last). If CROSS_CURVE is true, consider points on the adjacent curve to CURVE. It is important to compute an accurate approximation, because the control points that we eventually decide upon to fit the curve will be placed on the half-lines defined by the tangents and endpoints...and we never recompute the tangent after this. */ static void find_tangent (curve_type curve, boolean to_start_point, boolean cross_curve) { vector_type tangent; vector_type **curve_tangent = to_start_point ? &(CURVE_START_TANGENT (curve)) : &(CURVE_END_TANGENT (curve)); unsigned n_points = 0; /* LOG1 (" tangent to %s: ", to_start_point ? "start" : "end"); */ if (*curve_tangent == NULL) { *curve_tangent = g_new (vector_type, 1); tangent = find_half_tangent (curve, to_start_point, &n_points); if (cross_curve) { curve_type adjacent_curve = to_start_point ? PREVIOUS_CURVE (curve) : NEXT_CURVE (curve); vector_type tangent2 = find_half_tangent (adjacent_curve, !to_start_point, &n_points); /* LOG2 ("(adjacent curve half tangent (%.3f,%.3f)) ", */ /* tangent2.dx, tangent2.dy); */ tangent = Vadd (tangent, tangent2); } assert (n_points > 0); **curve_tangent = Vmult_scalar (tangent, 1.0 / n_points); } else { /* LOG ("(already computed) "); */ } /* LOG2 ("(%.3f,%.3f).\n", (*curve_tangent)->dx, (*curve_tangent)->dy); */ } /* Find the change in y and change in x for `tangent_surround' (a global) points along CURVE. Increment N_POINTS by the number of points we actually look at. */ static vector_type find_half_tangent (curve_type c, boolean to_start_point, unsigned *n_points) { unsigned p; int factor = to_start_point ? 1 : -1; unsigned tangent_index = to_start_point ? 0 : c->length - 1; real_coordinate_type tangent_point = CURVE_POINT (c, tangent_index); vector_type tangent; tangent.dx = 0.0; tangent.dy = 0.0; for (p = 1; p <= tangent_surround; p++) { int this_index = p * factor + tangent_index; real_coordinate_type this_point; if (this_index < 0 || this_index >= c->length) break; this_point = CURVE_POINT (c, p * factor + tangent_index); /* Perhaps we should weight the tangent from `this_point' by some factor dependent on the distance from the tangent point. */ tangent = Vadd (tangent, Vmult_scalar (Psubtract (this_point, tangent_point), factor)); (*n_points)++; } return tangent; } /* When this routine is called, we have computed a spline representation for the digitized curve. The question is, how good is it? If the fit is very good indeed, we might have an error of zero on each point, and then WORST_POINT becomes irrelevant. But normally, we return the error at the worst point, and the index of that point in WORST_POINT. The error computation itself is the Euclidean distance from the original curve CURVE to the fitted spline SPLINE. */ static real find_error (curve_type curve, spline_type spline, unsigned *worst_point) { unsigned this_point; real total_error = 0.0; real worst_error = FLT_MIN; *worst_point = CURVE_LENGTH (curve) + 1; /* A sentinel value. */ for (this_point = 0; this_point < CURVE_LENGTH (curve); this_point++) { real_coordinate_type curve_point = CURVE_POINT (curve, this_point); real t = CURVE_T (curve, this_point); real_coordinate_type spline_point = evaluate_spline (spline, t); real this_error = distance (curve_point, spline_point); if (this_error > worst_error) { *worst_point = this_point; worst_error = this_error; } total_error += this_error; } if (*worst_point == CURVE_LENGTH (curve) + 1) { /* Didn't have any ``worst point''; the error should be zero. */ if (epsilon_equal (total_error, 0.0)) { /* LOG (" Every point fit perfectly.\n"); */ } else printf ("No worst point found; something is wrong"); } else { /* LOG4 (" Worst error (at (%.3f,%.3f), point #%u) was %.3f.\n", */ /* CURVE_POINT (curve, *worst_point).x, */ /* CURVE_POINT (curve, *worst_point).y, *worst_point, worst_error); */ /* LOG1 (" Total error was %.3f.\n", total_error); */ /* LOG2 (" Average error (over %u points) was %.3f.\n", */ /* CURVE_LENGTH (curve), total_error / CURVE_LENGTH (curve)); */ } return worst_error; } /* Supposing that we have accepted the error, another question arises: would we be better off just using a straight line? */ static boolean spline_linear_enough (spline_type *spline, curve_type curve) { real A, B, C, slope; unsigned this_point; real distance = 0.0; /* LOG ("Checking linearity:\n"); */ /* For a line described by Ax + By + C = 0, the distance d from a point (x0,y0) to that line is: d = | Ax0 + By0 + C | / sqrt (A^2 + B^2). We can find A, B, and C from the starting and ending points, unless the line defined by those two points is vertical. */ if (epsilon_equal (START_POINT (*spline).x, END_POINT (*spline).x)) { A = 1; B = 0; C = -START_POINT (*spline).x; } else { /* Plug the spline's starting and ending points into the two-point equation for a line, that is, (y - y1) = ((y2 - y1)/(x2 - x1)) * (x - x1) to get the values for A, B, and C. */ slope = ((END_POINT (*spline).y - START_POINT (*spline).y) / (END_POINT (*spline).x - START_POINT (*spline).x)); A = -slope; B = 1; C = slope * START_POINT (*spline).x - START_POINT (*spline).y; } /* LOG3 (" Line is %.3fx + %.3fy + %.3f = 0.\n", A, B, C); */ for (this_point = 0; this_point < CURVE_LENGTH (curve); this_point++) { real t = CURVE_T (curve, this_point); real_coordinate_type spline_point = evaluate_spline (*spline, t); distance += fabs (A * spline_point.x + B * spline_point.y + C) / sqrt (A * A + B * B); } /* LOG1 (" Total distance is %.3f, ", distance); */ distance /= CURVE_LENGTH (curve); /* LOG1 ("which is %.3f normalized.\n", distance); */ /* We want reversion of short curves to splines to be more likely than reversion of long curves, hence the second division by the curve length, for use in `change_bad_lines'. */ SPLINE_LINEARITY (*spline) = distance / CURVE_LENGTH (curve); /* LOG1 (" Final linearity: %.3f.\n", SPLINE_LINEARITY (*spline)); */ return distance < line_threshold; } /* Unfortunately, we cannot tell in isolation whether a given spline should be changed to a line or not. That can only be known after the entire curve has been fit to a list of splines. (The curve is the pixel outline between two corners.) After subdividing the curve, a line may very well fit a portion of the curve just as well as the spline---but unless a spline is truly close to being a line, it should not be combined with other lines. */ static void change_bad_lines (spline_list_type *spline_list) { unsigned this_spline; boolean found_cubic = false; unsigned length = SPLINE_LIST_LENGTH (*spline_list); /* LOG1 ("\nChecking for bad lines (length %u):\n", length); */ /* First see if there are any splines in the fitted shape. */ for (this_spline = 0; this_spline < length; this_spline++) { if (SPLINE_DEGREE (SPLINE_LIST_ELT (*spline_list, this_spline)) == CUBIC) { found_cubic = true; break; } } /* If so, change lines back to splines (we haven't done anything to their control points, so we only have to change the degree) unless the spline is close enough to being a line. */ if (found_cubic) for (this_spline = 0; this_spline < length; this_spline++) { spline_type s = SPLINE_LIST_ELT (*spline_list, this_spline); if (SPLINE_DEGREE (s) == LINEAR) { /* LOG1 (" #%u: ", this_spline); */ if (SPLINE_LINEARITY (s) > line_reversion_threshold) { /* LOG ("reverted, "); */ SPLINE_DEGREE (SPLINE_LIST_ELT (*spline_list, this_spline)) = CUBIC; } /* LOG1 ("linearity %.3f.\n", SPLINE_LINEARITY (s)); */ } } else { /* LOG (" No lines.\n"); */ } } /* When we have finished fitting an entire pixel outline to a spline list L, we go through L to ensure that any endpoints that are ``close enough'' (i.e., within `align_threshold') to being the same really are the same. */ /* This macro adjusts the AXIS axis on the starting and ending points on a particular spline if necessary. */ #define TRY_AXIS(axis) \ do \ { \ real delta = fabs (end.axis - start.axis); \ \ if (!epsilon_equal (delta, 0.0) && delta <= align_threshold) \ { \ spline_type *next = &NEXT_SPLINE_LIST_ELT (*l, this_spline); \ spline_type *prev = &PREV_SPLINE_LIST_ELT (*l, this_spline); \ \ START_POINT (*s).axis = END_POINT (*s).axis \ = END_POINT (*prev).axis = START_POINT (*next).axis \ = (start.axis + end.axis) / 2; \ spline_change = true; \ } \ } \ while (0) static void align (spline_list_type *l) { boolean change; unsigned this_spline; unsigned length = SPLINE_LIST_LENGTH (*l); /* LOG1 ("\nAligning spline list (length %u):\n", length); */ do { change = false; /* LOG (" "); */ for (this_spline = 0; this_spline < length; this_spline++) { boolean spline_change = false; spline_type *s = &SPLINE_LIST_ELT (*l, this_spline); real_coordinate_type start = START_POINT (*s); real_coordinate_type end = END_POINT (*s); TRY_AXIS (x); TRY_AXIS (y); if (spline_change) { /* LOG1 ("%u ", this_spline); */ change |= spline_change; } } /* LOG ("\n"); */ } while (change); } /* Lists of array indices (well, that is what we use it for). */ static index_list_type new_index_list (void) { index_list_type index_list; index_list.data = NULL; INDEX_LIST_LENGTH (index_list) = 0; return index_list; } static void free_index_list (index_list_type *index_list) { if (INDEX_LIST_LENGTH (*index_list) > 0) { g_free (index_list->data); index_list->data = NULL; INDEX_LIST_LENGTH (*index_list) = 0; } } static void append_index (index_list_type *list, unsigned new_index) { INDEX_LIST_LENGTH (*list)++; list->data = (unsigned *)g_realloc(list->data,(INDEX_LIST_LENGTH (*list)) * sizeof(unsigned)); /* XRETALLOC (list->data, INDEX_LIST_LENGTH (*list), unsigned); */ list->data[INDEX_LIST_LENGTH (*list) - 1] = new_index; }

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