You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
275 lines
11 KiB
Python
275 lines
11 KiB
Python
"""
|
|
Module for rasterizing polygons, with float-precision anti-aliasing on
|
|
a non-uniform rectangular grid.
|
|
|
|
See the documentation for raster(...) for details.
|
|
"""
|
|
|
|
from typing import Tuple
|
|
import numpy
|
|
from numpy import logical_and, diff, floor, ceil, ones, zeros, hstack, full_like, newaxis
|
|
from scipy import sparse
|
|
|
|
__author__ = 'Jan Petykiewicz'
|
|
|
|
|
|
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
|
|
|