二维点云到网格的双线性加权的最快方法
Fastest method of bilinear weighting of a 2D point cloud onto a grid
给定一个点 (0.8, 0.6) 和一个强度 (3)，可以将单个点双线性“反向插值”到具有整数索引 (0,0) -> (1,1).[=22 的 2x2 网格上=]
点和网格的方向是第一个条目向下增加，第二个条目向右增加。
在这样的网格上，上面坐标的权值变为：
0.08 | 0.12
----------------
0.32 | 0.48
我们可以乘以与坐标相关的强度，得到 2x2 网格双线性加权强度：
0.24 | 0.36
----------------
0.96 | 1.44
可以画成这样：
对于几个点，可以将它们加权到同一个数组中（下面的完整代码）：
points = np.array([(0.8, 0.6), (2.2, 2.6),(5, 1), (3, 4.2), (8.5, 8.2)])
intens = np.array([3, 3, 1, 1, 2])
image, weight = bilinear(points, intens)
对于我的工作，我需要 weights 和 intensity*weights 作为输出数组。我需要在非常大的（数千万个）坐标上执行上述操作，其中坐标的值从 0.0 到 4095.0。我在下面编写了一个 numpy 例程，虽然它对于 100_000 点相当快（1.25 秒），但我更希望它更快，因为我需要在 10_000_000 数据点上多次调用它我有。
我考虑过矢量化 numpy 代码而不是 for 循环，但随后我为每个点生成了一个 4096x4096 几乎为空的数组，然后我将对其求和。那将需要 1000 TB 的内存。
我也在 cupy 中尝试了一个简单的实现，但是由于我使用了 for 循环，它变得太慢了。
在我的代码中，我为每个点生成一个 2x2 加权数组，将该数组乘以强度，然后通过切片将它们添加到主数组。有没有更好的方法？
import numpy as np
def bilinear(points, intensity):
"""Bilinear weighting of points onto a grid.
Extent of grid given by min and max of points in each dimension
points should have shape (N, 2)
intensity should have shape (N,)
"""
floor = np.floor(points)
ceil = floor + 1
floored_indices = np.array(floor, dtype=int)
low0, low1 = floored_indices.min(0)
high0, high1 = floored_indices.max(0)
floored_indices = floored_indices - (low0, low1)
shape = (high0 - low0 + 2, high1-low1 + 2)
weights_arr = np.zeros(shape, dtype=float)
int_arr = np.zeros(shape, dtype=float)
upper_diff = ceil - points
lower_diff = points - floor
w1 = np.prod((upper_diff), axis=1)
w2 = upper_diff[:,0]*lower_diff[:,1]
w3 = lower_diff[:,0]*upper_diff[:,1]
w4 = np.prod((lower_diff), axis=1)
for i, index in enumerate(floored_indices):
s = np.s_[index[0]:index[0]+2, index[1]:index[1]+2]
weights = np.array([[w1[i], w2[i]], [w3[i], w4[i]]])
weights_arr[s] += weights
int_arr[s] += intensity[i]*weights
return int_arr, weights_arr
rng = np.random.default_rng()
N_points = 10_000 # use 10_000 so it is quick
image_shape = (256, 256) # Use 256 so it isn't so big
points = rng.random((N_points, 2)) * image_shape
intensity = rng.random(N_points)
image, weight = bilinear(points, intensity)
为了测试代码，我还提供了以下绘图代码 - 仅使用少量 (~10) 个点，否则散点将覆盖整个图像。
import matplotlib.pyplot as plt
floor = np.floor(points) - 0.5
lower, left = floor.min(0)
upper, right = (floor).max(0) + 2
extent = (left, right, upper, lower)
fig, (ax1, ax2) = plt.subplots(ncols=2, figsize=(6,3))
ax1.scatter(*points[:,::-1].T, c='red')
im1 = ax1.imshow(weight, clim=(image.min(), image.max()), extent=extent)
ax1.set(title='Weight', xlim=(left - 1, right + 1), ylim = (upper + 1, lower - 1))
colorbar(im1)
ax2.scatter(*points[:,::-1].T , c='red')
im2 = ax2.imshow(image, extent=extent)
ax2.set(title='Weight x Intensity', xlim=(left - 1, right + 1), ylim = (upper + 1, lower - 1))
colorbar(im2)
plt.tight_layout()
plt.show()
# If labeling the first point
# ax1.text(*points[0].T, f"({points[0,0]}, {points[0,1]})", va='bottom', ha='center', color='red')
# ax2.text(*points[0].T, f"({points[0,0]}, {points[0,1]}, {intens[0]})", va='bottom', ha='center', color='red')
您想使用 np.add.at。看，https://numpy.org/doc/stable/reference/generated/numpy.ufunc.at.html
def bilinear_2(points, intensity):
# Create empty matrices, starting from 0 to p.max
w = np.zeros((points[:, 0].max().astype(int) + 2, points[:, 1].max().astype(int) + 2))
i = np.zeros_like(w)
# Calc weights
floor = np.floor(points)
ceil = floor + 1
upper_diff = ceil - points
lower_diff = points - floor
w1 = upper_diff[:, 0] * upper_diff[:, 1]
w2 = upper_diff[:, 0] * lower_diff[:, 1]
w3 = lower_diff[:, 0] * upper_diff[:, 1]
w4 = lower_diff[:, 0] * lower_diff[:, 1]
# Get indices
ix, iy = floor[:, 0].astype(int), floor[:, 1].astype(int)
# Use np.add.at. See, https://numpy.org/doc/stable/reference/generated/numpy.ufunc.at.html
np.add.at(w, (ix, iy), w1)
np.add.at(w, (ix, iy+1), w2)
np.add.at(w, (ix+1, iy), w3)
np.add.at(w, (ix+1, iy+1), w4)
np.add.at(i, (ix, iy), w1 * intensity)
np.add.at(i, (ix, iy+1), w2 * intensity)
np.add.at(i, (ix+1, iy), w3 * intensity)
np.add.at(i, (ix+1, iy+1), w4 * intensity)
# Clip (to accomodate image size to be the same as your bilinear function)
iix, iiy = points[:, 0].min().astype(int), points[:, 1].min().astype(int)
i, w = i[iix:, iiy:], w[iix:, iiy:]
return i, w
# At 10_000 samples:
%time image, weight = bilinear(points, intensity)
%time image_2, weight_2 = bilinear_2(points, intensity)
>>>
CPU times: user 178 ms, sys: 3.73 ms, total: 182 ms
Wall time: 185 ms
CPU times: user 9.63 ms, sys: 601 µs, total: 10.2 ms
Wall time: 10 ms
# These tests passes
np.testing.assert_allclose(weight, weight_2)
np.testing.assert_allclose(image, image_2)
# At 100K samples
N_points = 100_000
image_shape = (256, 256)
points = rng.random((N_points, 2)) * image_shape
intensity = rng.random(N_points)
%time image_2, weight_2 = bilinear_2(points, intensity)
CPU times: user 115 ms, sys: 66 ms, total: 181 ms
Wall time: 181 ms
# At 10M samples
N_points = 10_000_000
image_shape = (256, 256)
points = rng.random((N_points, 2)) * image_shape
intensity = rng.random(N_points)
%time image_2, weight_2 = bilinear_2(points, intensity)
CPU times: user 8.23 s, sys: 656 ms, total: 8.88 s
Wall time: 9.31 s
除此之外，这种方法是不可能的。因为整数数组索引不会增量更新。
例如，
a = np.zeros(5)
a[np.array((1,1,2))] += 1
a
>>> array([0., 1., 1., 0., 0.])
但是；
a = np.zeros(5)
np.add.at(a, ([1,1,2]), 1)
a
>>> array([0., 2., 1., 0., 0.])
感谢@armamut 的好回答！启发我看了一下，发现np.bincount，也是用cupy实现的。事实证明，bincount 实现更快，而 cupy 实现真的很快！后者可能会进一步改进，因为我不得不兼顾一些元组才能让它工作。
# Timings
points = np.random.random((10_000_000, 2)) * (256, 256)
intens = np.random.random((10_000_000))
pcupy = cp.asarray(points)
icupy = cp.asarray(intens)
%time bilinear_bincount_cupy(pcupy, icupy)
%time bilinear_bincount_numpy(points, intens)
%time bilinear_2(points, intens)
Wall time: 456 ms
Wall time: 2.57 s
Wall time: 5.37 s
numpy 实现：
def bilinear_bincount_numpy(points, intensities):
"""Bilinear weighting of points onto a grid.
Extent of grid given by min and max of points in each dimension
points should have shape (N, 2)
intensity should have shape (N,)
"""
floor = np.floor(points)
ceil = floor + 1
floored_indices = np.array(floor, dtype=int)
low0, low1 = floored_indices.min(0)
high0, high1 = floored_indices.max(0)
floored_indices = floored_indices - (low0, low1)
shape = (high0 - low0 + 2, high1-low1 + 2)
upper_diff = ceil - points
lower_diff = points - floor
w1 = np.prod((upper_diff), axis=1)
w2 = upper_diff[:,0]*lower_diff[:,1]
w3 = lower_diff[:,0]*upper_diff[:,1]
w4 = np.prod((lower_diff), axis=1)
shifts = np.array([[0, 0], [0, 1], [1, 0], [1, 1]])
indices = floored_indices[:, None] + shifts
indices = (indices * (shape[1], 1)).sum(-1)
weights = np.array([w1, w2, w3, w4]).T
weight_bins = np.bincount(indices.flatten(), weights=weights.flatten())
intens_bins = np.bincount(indices.flatten(), weights=(intensities[:, None]*weights).flatten())
all_weight_bins = np.zeros(np.prod(shape))
all_intens_bins = np.zeros(np.prod(shape))
all_weight_bins[:len(weight_bins)] = weight_bins
all_intens_bins[:len(weight_bins)] = intens_bins
weight_image = all_weight_bins.reshape(shape)
intens_image = all_intens_bins.reshape(shape)
return intens_image, weight_image
以及 cupy 实现：
def bilinear_bincount_cupy(points, intensities):
"""Bilinear weighting of points onto a grid.
Extent of grid given by min and max of points in each dimension
points should be a cupy array of shape (N, 2)
intensity should be a cupy array of shape (N,)
"""
floor = cp.floor(points)
ceil = floor + 1
floored_indices = cp.array(floor, dtype=int)
low0, low1 = floored_indices.min(0)
high0, high1 = floored_indices.max(0)
floored_indices = floored_indices - cp.array([low0, low1])
shape = cp.array([high0 - low0 + 2, high1-low1 + 2])
upper_diff = ceil - points
lower_diff = points - floor
w1 = upper_diff[:, 0] * upper_diff[:, 1]
w2 = upper_diff[:, 0] * lower_diff[:, 1]
w3 = lower_diff[:, 0] * upper_diff[:, 1]
w4 = lower_diff[:, 0] * lower_diff[:, 1]
shifts = cp.array([[0, 0], [0, 1], [1, 0], [1, 1]])
indices = floored_indices[:, None] + shifts
indices = (indices * cp.array([shape[1].item(), 1])).sum(-1)
weights = cp.array([w1, w2, w3, w4]).T
# These bins only fill up to the highest index - not to shape[0]*shape[1]
weight_bins = cp.bincount(indices.flatten(), weights=weights.flatten())
intens_bins = cp.bincount(indices.flatten(), weights=(intensities[:, None]*weights).flatten())
# So we create a zeros array that is big enough
all_weight_bins = cp.zeros(cp.prod(shape).item())
all_intens_bins = cp.zeros_like(all_weight_bins)
# And fill it here
all_weight_bins[:len(weight_bins)] = weight_bins
all_intens_bins[:len(weight_bins)] = intens_bins
weight_image = all_weight_bins.reshape(shape.get())
intens_image = all_intens_bins.reshape(shape.get())
return intens_image, weight_image
给定一个点 (0.8, 0.6) 和一个强度 (3)，可以将单个点双线性“反向插值”到具有整数索引 (0,0) -> (1,1).[=22 的 2x2 网格上=]
点和网格的方向是第一个条目向下增加，第二个条目向右增加。
在这样的网格上，上面坐标的权值变为：
0.08 | 0.12
----------------
0.32 | 0.48
我们可以乘以与坐标相关的强度，得到 2x2 网格双线性加权强度：
0.24 | 0.36
----------------
0.96 | 1.44
可以画成这样：
对于几个点，可以将它们加权到同一个数组中（下面的完整代码）：
points = np.array([(0.8, 0.6), (2.2, 2.6),(5, 1), (3, 4.2), (8.5, 8.2)])
intens = np.array([3, 3, 1, 1, 2])
image, weight = bilinear(points, intens)
对于我的工作，我需要 weights 和 intensity*weights 作为输出数组。我需要在非常大的（数千万个）坐标上执行上述操作，其中坐标的值从 0.0 到 4095.0。我在下面编写了一个 numpy 例程，虽然它对于 100_000 点相当快（1.25 秒），但我更希望它更快，因为我需要在 10_000_000 数据点上多次调用它我有。
我考虑过矢量化 numpy 代码而不是 for 循环，但随后我为每个点生成了一个 4096x4096 几乎为空的数组，然后我将对其求和。那将需要 1000 TB 的内存。
我也在 cupy 中尝试了一个简单的实现，但是由于我使用了 for 循环，它变得太慢了。
在我的代码中，我为每个点生成一个 2x2 加权数组，将该数组乘以强度，然后通过切片将它们添加到主数组。有没有更好的方法？
import numpy as np
def bilinear(points, intensity):
"""Bilinear weighting of points onto a grid.
Extent of grid given by min and max of points in each dimension
points should have shape (N, 2)
intensity should have shape (N,)
"""
floor = np.floor(points)
ceil = floor + 1
floored_indices = np.array(floor, dtype=int)
low0, low1 = floored_indices.min(0)
high0, high1 = floored_indices.max(0)
floored_indices = floored_indices - (low0, low1)
shape = (high0 - low0 + 2, high1-low1 + 2)
weights_arr = np.zeros(shape, dtype=float)
int_arr = np.zeros(shape, dtype=float)
upper_diff = ceil - points
lower_diff = points - floor
w1 = np.prod((upper_diff), axis=1)
w2 = upper_diff[:,0]*lower_diff[:,1]
w3 = lower_diff[:,0]*upper_diff[:,1]
w4 = np.prod((lower_diff), axis=1)
for i, index in enumerate(floored_indices):
s = np.s_[index[0]:index[0]+2, index[1]:index[1]+2]
weights = np.array([[w1[i], w2[i]], [w3[i], w4[i]]])
weights_arr[s] += weights
int_arr[s] += intensity[i]*weights
return int_arr, weights_arr
rng = np.random.default_rng()
N_points = 10_000 # use 10_000 so it is quick
image_shape = (256, 256) # Use 256 so it isn't so big
points = rng.random((N_points, 2)) * image_shape
intensity = rng.random(N_points)
image, weight = bilinear(points, intensity)
为了测试代码，我还提供了以下绘图代码 - 仅使用少量 (~10) 个点，否则散点将覆盖整个图像。
import matplotlib.pyplot as plt
floor = np.floor(points) - 0.5
lower, left = floor.min(0)
upper, right = (floor).max(0) + 2
extent = (left, right, upper, lower)
fig, (ax1, ax2) = plt.subplots(ncols=2, figsize=(6,3))
ax1.scatter(*points[:,::-1].T, c='red')
im1 = ax1.imshow(weight, clim=(image.min(), image.max()), extent=extent)
ax1.set(title='Weight', xlim=(left - 1, right + 1), ylim = (upper + 1, lower - 1))
colorbar(im1)
ax2.scatter(*points[:,::-1].T , c='red')
im2 = ax2.imshow(image, extent=extent)
ax2.set(title='Weight x Intensity', xlim=(left - 1, right + 1), ylim = (upper + 1, lower - 1))
colorbar(im2)
plt.tight_layout()
plt.show()
# If labeling the first point
# ax1.text(*points[0].T, f"({points[0,0]}, {points[0,1]})", va='bottom', ha='center', color='red')
# ax2.text(*points[0].T, f"({points[0,0]}, {points[0,1]}, {intens[0]})", va='bottom', ha='center', color='red')
您想使用 np.add.at。看，https://numpy.org/doc/stable/reference/generated/numpy.ufunc.at.html
def bilinear_2(points, intensity):
# Create empty matrices, starting from 0 to p.max
w = np.zeros((points[:, 0].max().astype(int) + 2, points[:, 1].max().astype(int) + 2))
i = np.zeros_like(w)
# Calc weights
floor = np.floor(points)
ceil = floor + 1
upper_diff = ceil - points
lower_diff = points - floor
w1 = upper_diff[:, 0] * upper_diff[:, 1]
w2 = upper_diff[:, 0] * lower_diff[:, 1]
w3 = lower_diff[:, 0] * upper_diff[:, 1]
w4 = lower_diff[:, 0] * lower_diff[:, 1]
# Get indices
ix, iy = floor[:, 0].astype(int), floor[:, 1].astype(int)
# Use np.add.at. See, https://numpy.org/doc/stable/reference/generated/numpy.ufunc.at.html
np.add.at(w, (ix, iy), w1)
np.add.at(w, (ix, iy+1), w2)
np.add.at(w, (ix+1, iy), w3)
np.add.at(w, (ix+1, iy+1), w4)
np.add.at(i, (ix, iy), w1 * intensity)
np.add.at(i, (ix, iy+1), w2 * intensity)
np.add.at(i, (ix+1, iy), w3 * intensity)
np.add.at(i, (ix+1, iy+1), w4 * intensity)
# Clip (to accomodate image size to be the same as your bilinear function)
iix, iiy = points[:, 0].min().astype(int), points[:, 1].min().astype(int)
i, w = i[iix:, iiy:], w[iix:, iiy:]
return i, w
# At 10_000 samples:
%time image, weight = bilinear(points, intensity)
%time image_2, weight_2 = bilinear_2(points, intensity)
>>>
CPU times: user 178 ms, sys: 3.73 ms, total: 182 ms
Wall time: 185 ms
CPU times: user 9.63 ms, sys: 601 µs, total: 10.2 ms
Wall time: 10 ms
# These tests passes
np.testing.assert_allclose(weight, weight_2)
np.testing.assert_allclose(image, image_2)
# At 100K samples
N_points = 100_000
image_shape = (256, 256)
points = rng.random((N_points, 2)) * image_shape
intensity = rng.random(N_points)
%time image_2, weight_2 = bilinear_2(points, intensity)
CPU times: user 115 ms, sys: 66 ms, total: 181 ms
Wall time: 181 ms
# At 10M samples
N_points = 10_000_000
image_shape = (256, 256)
points = rng.random((N_points, 2)) * image_shape
intensity = rng.random(N_points)
%time image_2, weight_2 = bilinear_2(points, intensity)
CPU times: user 8.23 s, sys: 656 ms, total: 8.88 s
Wall time: 9.31 s
除此之外，这种方法是不可能的。因为整数数组索引不会增量更新。
例如，
a = np.zeros(5)
a[np.array((1,1,2))] += 1
a
>>> array([0., 1., 1., 0., 0.])
但是；
a = np.zeros(5)
np.add.at(a, ([1,1,2]), 1)
a
>>> array([0., 2., 1., 0., 0.])
感谢@armamut 的好回答！启发我看了一下，发现np.bincount，也是用cupy实现的。事实证明，bincount 实现更快，而 cupy 实现真的很快！后者可能会进一步改进，因为我不得不兼顾一些元组才能让它工作。
# Timings
points = np.random.random((10_000_000, 2)) * (256, 256)
intens = np.random.random((10_000_000))
pcupy = cp.asarray(points)
icupy = cp.asarray(intens)
%time bilinear_bincount_cupy(pcupy, icupy)
%time bilinear_bincount_numpy(points, intens)
%time bilinear_2(points, intens)
Wall time: 456 ms
Wall time: 2.57 s
Wall time: 5.37 s
numpy 实现：
def bilinear_bincount_numpy(points, intensities):
"""Bilinear weighting of points onto a grid.
Extent of grid given by min and max of points in each dimension
points should have shape (N, 2)
intensity should have shape (N,)
"""
floor = np.floor(points)
ceil = floor + 1
floored_indices = np.array(floor, dtype=int)
low0, low1 = floored_indices.min(0)
high0, high1 = floored_indices.max(0)
floored_indices = floored_indices - (low0, low1)
shape = (high0 - low0 + 2, high1-low1 + 2)
upper_diff = ceil - points
lower_diff = points - floor
w1 = np.prod((upper_diff), axis=1)
w2 = upper_diff[:,0]*lower_diff[:,1]
w3 = lower_diff[:,0]*upper_diff[:,1]
w4 = np.prod((lower_diff), axis=1)
shifts = np.array([[0, 0], [0, 1], [1, 0], [1, 1]])
indices = floored_indices[:, None] + shifts
indices = (indices * (shape[1], 1)).sum(-1)
weights = np.array([w1, w2, w3, w4]).T
weight_bins = np.bincount(indices.flatten(), weights=weights.flatten())
intens_bins = np.bincount(indices.flatten(), weights=(intensities[:, None]*weights).flatten())
all_weight_bins = np.zeros(np.prod(shape))
all_intens_bins = np.zeros(np.prod(shape))
all_weight_bins[:len(weight_bins)] = weight_bins
all_intens_bins[:len(weight_bins)] = intens_bins
weight_image = all_weight_bins.reshape(shape)
intens_image = all_intens_bins.reshape(shape)
return intens_image, weight_image
以及 cupy 实现：
def bilinear_bincount_cupy(points, intensities):
"""Bilinear weighting of points onto a grid.
Extent of grid given by min and max of points in each dimension
points should be a cupy array of shape (N, 2)
intensity should be a cupy array of shape (N,)
"""
floor = cp.floor(points)
ceil = floor + 1
floored_indices = cp.array(floor, dtype=int)
low0, low1 = floored_indices.min(0)
high0, high1 = floored_indices.max(0)
floored_indices = floored_indices - cp.array([low0, low1])
shape = cp.array([high0 - low0 + 2, high1-low1 + 2])
upper_diff = ceil - points
lower_diff = points - floor
w1 = upper_diff[:, 0] * upper_diff[:, 1]
w2 = upper_diff[:, 0] * lower_diff[:, 1]
w3 = lower_diff[:, 0] * upper_diff[:, 1]
w4 = lower_diff[:, 0] * lower_diff[:, 1]
shifts = cp.array([[0, 0], [0, 1], [1, 0], [1, 1]])
indices = floored_indices[:, None] + shifts
indices = (indices * cp.array([shape[1].item(), 1])).sum(-1)
weights = cp.array([w1, w2, w3, w4]).T
# These bins only fill up to the highest index - not to shape[0]*shape[1]
weight_bins = cp.bincount(indices.flatten(), weights=weights.flatten())
intens_bins = cp.bincount(indices.flatten(), weights=(intensities[:, None]*weights).flatten())
# So we create a zeros array that is big enough
all_weight_bins = cp.zeros(cp.prod(shape).item())
all_intens_bins = cp.zeros_like(all_weight_bins)
# And fill it here
all_weight_bins[:len(weight_bins)] = weight_bins
all_intens_bins[:len(weight_bins)] = intens_bins
weight_image = all_weight_bins.reshape(shape.get())
intens_image = all_intens_bins.reshape(shape.get())
return intens_image, weight_image