Anti-aliasing polygon rasterizer, capable of handling non-uniform grids.
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from typing import Tuple
import numpy
from numpy import logical_and, diff, floor, ceil, ones, zeros, hstack, full_like, newaxis
from scipy import sparse
def raster(vertices: numpy.ndarray,
grid_x: numpy.ndarray,
grid_y: numpy.ndarray
) -> numpy.ndarray:
"""
Draws a polygon onto a 2D grid of pixels, setting pixel values equal to the fraction of the
pixel area covered by the polygon. This implementation is written for accuracy and works with
double precision, in contrast to most other implementations which are written for speed and
usually only allow for 256 (and often fewer) possible pixel values without performing (very
slow) super-sampling.
Polygons are assumed to have clockwise vertex order; reversing the vertex order is equivalent
to multiplying the result by -1.
:param vertices: 2xN ndarray containing x,y coordinates for each vertex of the polygon
:param grid_x: x-coordinates for the edges of each pixel (ie, the leftmost two columns span
x=grid_x[0] to x=grid_x[1] and x=grid_x[1] to x=grid_x[2])
:param grid_y: y-coordinates for the edges of each pixel (see grid_x)
:return: 2D ndarray with pixel values in the range [0, 1] containing the anti-aliased polygon
"""
vertices = numpy.array(vertices)
grid_x = numpy.array(grid_x)
grid_y = numpy.array(grid_y)
min_bounds = floor(vertices.min(axis=1))
max_bounds = ceil(vertices.max(axis=1))
keep_x = logical_and(grid_x >= min_bounds[0],
grid_x <= max_bounds[0])
keep_y = logical_and(grid_y >= min_bounds[1],
grid_y <= max_bounds[1])
if not (keep_x.any() and keep_y.any()): # polygon doesn't overlap grid
return zeros((grid_x.size - 1, grid_y.size - 1))
vertices = create_vertices(vertices, grid_x, grid_y)
parts_grid = get_raster_parts(vertices, grid_x, grid_y).toarray()
result_grid = numpy.real(parts_grid) + numpy.imag(parts_grid).cumsum(axis=0)
return result_grid
def find_intersections(
vertices: numpy.ndarray,
grid_x: numpy.ndarray,
grid_y: numpy.ndarray
) -> Tuple[numpy.ndarray]:
"""
Find intersections between a polygon and grid lines
"""
if vertices.shape[0] != 2:
raise Exception('vertices must be 2xN')
min_bounds = floor(vertices.min(axis=1))
max_bounds = ceil(vertices.max(axis=1))
keep_x = logical_and(grid_x >= min_bounds[0],
grid_x <= max_bounds[0])
keep_y = logical_and(grid_y >= min_bounds[1],
grid_y <= max_bounds[1])
if not (keep_x.any() or keep_y.any()): # polygon doesn't overlap grid
mat_shape = (vertices.shape[1], grid_x.size + grid_y.size)
return zeros(mat_shape), zeros(mat_shape), zeros(mat_shape, dtype=bool)
y_seg_xs = hstack((min_bounds[0], grid_x[keep_x], max_bounds[0])).T
x_seg_ys = hstack((min_bounds[1], grid_y[keep_y], max_bounds[1])).T
'''
Calculate intersections between polygon and grid line segments
'''
xy1b = numpy.roll(vertices, -1, axis=1)
# Lists of initial/final coordinates for polygon segments
xi1 = vertices[0, :, newaxis]
yi1 = vertices[1, :, newaxis]
xf1 = xy1b[0, :, newaxis]
yf1 = xy1b[1, :, newaxis]
# Lists of initial/final coordinates for grid segments
xi2 = hstack((full_like(x_seg_ys, min_bounds[0]), y_seg_xs))
xf2 = hstack((full_like(x_seg_ys, max_bounds[0]), y_seg_xs))
yi2 = hstack((x_seg_ys, full_like(y_seg_xs, min_bounds[0])))
yf2 = hstack((x_seg_ys, full_like(y_seg_xs, max_bounds[1])))
# Perform calculation
dxi = xi1 - xi2
dyi = yi1 - yi2
dx1 = xf1 - xi1
dx2 = xf2 - xi2
dy1 = yf1 - yi1
dy2 = yf2 - yi2
numerator_a = dx2 * dyi - dy2 * dxi
numerator_b = dx1 * dyi - dy1 * dxi
denominator = dy2 * dx1 - dx2 * dy1
# Avoid warnings since we may multiply eg. NaN*False
with numpy.errstate(invalid='ignore', divide='ignore'):
u_a = numerator_a / denominator
u_b = numerator_b / denominator
# Find the adjacency matrix
int_x = xi1 + dx1 * u_a
int_y = yi1 + dy1 * u_a
adjacency = logical_and.reduce((u_a >= 0, u_a <= 1, u_b >= 0, u_b <= 1))
# Arrange output.
# adjacency[i, j] tells us if polygon segment i intersects with grid line j
# xy[i, j] tells us the x,y coordinates of the intersection in the form x+iy
# normalized_distance[i, j] tells us the fraction of the segment i
# we have to traverse in order to reach the intersection
xy = (int_x + 1j * int_y) * adjacency
normalized_distance = u_a
# print('sparsity', adjacency.astype(int).sum() / adjacency.size)
return normalized_distance, xy, adjacency
def create_vertices(
vertices: numpy.ndarray,
grid_x: numpy.ndarray,
grid_y: numpy.ndarray,
new_vertex_data: Tuple[numpy.ndarray] = None
) -> sparse.coo_matrix:
"""
Create additional vertices where a polygon crosses gridlines
"""
if vertices.shape[0] != 2:
raise Exception('vertices must be 2xN')
if grid_x.size < 1 or grid_y.size < 1:
raise Exception('Grid must contain at least one line in each direction?')
num_poly_vertices = vertices.shape[1]
if new_vertex_data is None:
new_vertex_data = find_intersections(vertices, grid_x, grid_y)
normalized_distance, xy_matrix, adjacency_matrix = new_vertex_data
'''
Insert any polygon-grid intersections as new polygon vertices
'''
# Figure out how to sort each row of the intersection matrices
# based on distance from (xi1, yi1) (the polygon segment's first point)
# This lets us insert them as new vertices in the proper order
sortix = normalized_distance.argsort(axis=1)
sortix_paired = (numpy.arange(num_poly_vertices)[:, newaxis], sortix)
assert(normalized_distance.shape[0] == num_poly_vertices)
# if any new points fall outside the window, shrink them back onto it
xy_shrunken = (numpy.real(xy_matrix).clip(grid_x[0], grid_x[-1]) + 1j *
numpy.imag(xy_matrix).clip(grid_y[0], grid_y[-1]))
# Use sortix to sort adjacency matrix and the intersection (x, y) coordinates,
# and hstack the original points to the left of the new ones
xy_with_original = hstack((vertices[0, :, newaxis] + 1j * vertices[1, :, newaxis],
xy_shrunken[sortix_paired]))
has_intersection = hstack((ones((vertices.shape[1], 1), dtype=bool),
adjacency_matrix[sortix_paired]))
# Now remove all extra entries which don't correspond to new vertices
# (ie, no intersection happened), and then flatten, creating our
# polygon-with-extra-vertices, though some redundant vertices are included,
# which we must later remove manually.
vertices = xy_with_original[has_intersection]
return vertices
def clip_vertices_to_window(
vertices: numpy.ndarray,
min_x: float = -numpy.inf,
max_x: float = numpy.inf,
min_y: float = -numpy.inf,
max_y: float = numpy.inf
) -> numpy.ndarray:
"""
"""
# Remove points outside the window (these will only be original points)
# Since the boundaries of the window are also pixel boundaries, this just
# makes the polygon boundary proceed along the window edge
inside = logical_and.reduce((numpy.real(vertices) <= max_x,
numpy.real(vertices) >= min_x,
numpy.imag(vertices) <= max_y,
numpy.imag(vertices) >= min_y))
vertices = vertices[inside]
# Remove consecutive duplicate vertices
consecutive = numpy.ediff1d(vertices, to_begin=[1 + 1j]).astype(bool)
vertices = vertices[consecutive]
return vertices
def get_raster_parts(
vertices: numpy.ndarray,
grid_x: numpy.ndarray,
grid_y: numpy.ndarray
) -> sparse.coo_matrix:
"""
This function performs the same task as raster(...), but instead of returning a dense array
of pixel values, it returns a sparse array containing the value
(-area + 1j * cover)
for each pixel which contains a line segment, where
cover is the fraction of the pixel's y-length that is traversed by the segment,
multiplied by the sign of (y_final - y_initial)
area is the fraction of the pixel's area covered by the trapezoid formed by
the line segment's endpoints (clipped to the cell edges) and their projections
onto the pixel's left (i.e., lowest-x) edge, again multiplied by
the sign of (y_final - y_initial)
Note that polygons are assumed to be wound clockwise.
The result from raster(...) can be obtained with
raster_result = numpy.real(lines_result) + numpy.imag(lines_result).cumsum(axis=0)
:param vertices: 2xN ndarray containing x,y coordinates for each point in the polygon
:param grid_x: x-coordinates for the edges of each pixel (ie, the leftmost two columns span
x=grid_x[0] to x=grid_x[1] and x=grid_x[1] to x=grid_x[2])
:param grid_y: y-coordinates for the edges of each pixel (see grid_x)
:return: Complex sparse COO matrix containing area and cover information
"""
if grid_x.size < 2 or grid_y.size < 2:
raise Exception('Grid must contain at least one full pixel')
num_xy_px = numpy.array([grid_x.size, grid_y.size]) - 1
vertices = clip_vertices_to_window(vertices,
grid_x[0], grid_x[-1],
grid_y[0], grid_y[-1])
# If the shape fell completely outside our area, just return a blank grid
if vertices.size == 0:
return sparse.coo_matrix(shape=num_xy_px)
'''
Calculate area, cover
'''
# Calculate segment cover, area, and corresponding pixel's subscripts
poly = hstack((vertices, vertices[0]))
endpoint_avg = (poly[:-1] + poly[1:]) * 0.5
# Remove segments along the right,top edges
# (they correspond to outside pixels, but couldn't be removed until now
# because 'vertices' stored points, not segments, and the edge points are needed
# when creating endpoint_avg)
non_edge = numpy.logical_and(numpy.real(endpoint_avg) < grid_x[-1],
numpy.imag(endpoint_avg) < grid_y[-1])
endpoint_avg_final = endpoint_avg[non_edge]
x_sub = numpy.digitize(numpy.real(endpoint_avg_final), grid_x) - 1
y_sub = numpy.digitize(numpy.imag(endpoint_avg_final), grid_y) - 1
cover = diff(numpy.imag(poly), axis=0)[non_edge] / diff(grid_y)[y_sub]
area = (numpy.real(endpoint_avg_final) - grid_x[x_sub]) * cover / diff(grid_x)[x_sub]
# Use coo_matrix(...).toarray() to efficiently convert from (x, y, v) pairs to ndarrays.
# We can use v = (-area + 1j * cover) followed with calls to numpy.real() and numpy.imag() to
# improve performance (Otherwise we'd have to call coo_matrix() twice. It's really inefficient
# because it involves lots of random memory access, unlike real() and imag()).
poly_grid = sparse.coo_matrix((-area + 1j * cover, (x_sub, y_sub)), shape=num_xy_px)
return poly_grid