JAX 中的广义卷积#
JAX 提供了许多接口来计算数据的卷积，包括
对于基本的卷积操作，jax.numpy 和 jax.scipy 的操作通常就足够了。 如果你想进行更通用的批量多维卷积，那么应该从 jax.lax 函数开始。
基本一维卷积#
基本的一维卷积由 jax.numpy.convolve() 实现，它为 numpy.convolve() 提供了 JAX 接口。 这是一个通过卷积实现一维平滑的简单示例
import matplotlib.pyplot as plt
from jax import random
import jax.numpy as jnp
import numpy as np
key = random.key(1701)
x = jnp.linspace(0, 10, 500)
y = jnp.sin(x) + 0.2 * random.normal(key, shape=(500,))
window = jnp.ones(10) / 10
y_smooth = jnp.convolve(y, window, mode='same')
plt.plot(x, y, 'lightgray')
plt.plot(x, y_smooth, 'black');
mode 参数控制边界条件的处理方式； 这里我们使用 mode='same' 来确保输出与输入大小相同。
有关更多信息，请参阅 jax.numpy.convolve() 文档，或与原始 numpy.convolve() 函数关联的文档。
基本 N 维卷积#
对于 *N* 维卷积，jax.scipy.signal.convolve() 提供了与 jax.numpy.convolve() 相似的接口，并将其推广到 *N* 维。
例如，这里有一个基于高斯滤波器卷积对图像进行去噪的简单方法
from scipy import misc
import jax.scipy as jsp
fig, ax = plt.subplots(1, 3, figsize=(12, 5))
# Load a sample image; compute mean() to convert from RGB to grayscale.
image = jnp.array(misc.face().mean(-1))
ax[0].imshow(image, cmap='binary_r')
ax[0].set_title('original')
# Create a noisy version by adding random Gaussian noise
key = random.key(1701)
noisy_image = image + 50 * random.normal(key, image.shape)
ax[1].imshow(noisy_image, cmap='binary_r')
ax[1].set_title('noisy')
# Smooth the noisy image with a 2D Gaussian smoothing kernel.
x = jnp.linspace(-3, 3, 7)
window = jsp.stats.norm.pdf(x) * jsp.stats.norm.pdf(x[:, None])
smooth_image = jsp.signal.convolve(noisy_image, window, mode='same')
ax[2].imshow(smooth_image, cmap='binary_r')
ax[2].set_title('smoothed');
/tmp/ipykernel_1302/4118182506.py:7: DeprecationWarning: scipy.misc.face has been deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. Dataset methods have moved into the scipy.datasets module. Use scipy.datasets.face instead.
image = jnp.array(misc.face().mean(-1))
与一维情况类似，我们使用 mode='same' 来指定我们希望如何处理边缘。 有关 *N* 维卷积中可用选项的更多信息，请参阅 jax.scipy.signal.convolve() 文档。
通用卷积#
对于在构建深度神经网络时通常更有用的更通用的批量卷积类型，JAX 和 XLA 提供了非常通用的 N 维 **conv_general_dilated** 函数，但如何使用它不是很明显。 我们将给出一些常见用例的示例。
强烈建议阅读关于卷积算子族的综述，卷积算术指南！
让我们定义一个简单的对角边缘核
# 2D kernel - HWIO layout
kernel = jnp.zeros((3, 3, 3, 3), dtype=jnp.float32)
kernel += jnp.array([[1, 1, 0],
[1, 0,-1],
[0,-1,-1]])[:, :, jnp.newaxis, jnp.newaxis]
print("Edge Conv kernel:")
plt.imshow(kernel[:, :, 0, 0]);
Edge Conv kernel:
我们将创建一个简单的合成图像
# NHWC layout
img = jnp.zeros((1, 200, 198, 3), dtype=jnp.float32)
for k in range(3):
x = 30 + 60*k
y = 20 + 60*k
img = img.at[0, x:x+10, y:y+10, k].set(1.0)
print("Original Image:")
plt.imshow(img[0]);
Original Image:
lax.conv 和 lax.conv_with_general_padding#
这些是用于卷积的简单便捷函数
⚠️便捷的 lax.conv 和 lax.conv_with_general_padding 辅助函数假设图像为 NCHW 格式，内核为 OIHW 格式。
from jax import lax
out = lax.conv(jnp.transpose(img,[0,3,1,2]), # lhs = NCHW image tensor
jnp.transpose(kernel,[3,2,0,1]), # rhs = OIHW conv kernel tensor
(1, 1), # window strides
'SAME') # padding mode
print("out shape: ", out.shape)
print("First output channel:")
plt.figure(figsize=(10,10))
plt.imshow(np.array(out)[0,0,:,:]);
out shape: (1, 3, 200, 198)
First output channel:
out = lax.conv_with_general_padding(
jnp.transpose(img,[0,3,1,2]), # lhs = NCHW image tensor
jnp.transpose(kernel,[2,3,0,1]), # rhs = IOHW conv kernel tensor
(1, 1), # window strides
((2,2),(2,2)), # general padding 2x2
(1,1), # lhs/image dilation
(1,1)) # rhs/kernel dilation
print("out shape: ", out.shape)
print("First output channel:")
plt.figure(figsize=(10,10))
plt.imshow(np.array(out)[0,0,:,:]);
out shape: (1, 3, 202, 200)
First output channel:
维度编号定义 conv_general_dilated 的维度布局#
重要的参数是轴布局参数的三元组：(输入布局, 内核布局, 输出布局)
N - 批次维度
H - 空间高度
W - 空间宽度
C - 通道维度
I - 内核*输入*通道维度
O - 内核*输出*通道维度
⚠️ 为了演示维度编号的灵活性，我们为下面的 lax.conv_general_dilated 选择 NHWC 图像和 HWIO 内核约定。
dn = lax.conv_dimension_numbers(img.shape, # only ndim matters, not shape
kernel.shape, # only ndim matters, not shape
('NHWC', 'HWIO', 'NHWC')) # the important bit
print(dn)
ConvDimensionNumbers(lhs_spec=(0, 3, 1, 2), rhs_spec=(3, 2, 0, 1), out_spec=(0, 3, 1, 2))
SAME 填充，无步长，无扩张#
out = lax.conv_general_dilated(img, # lhs = image tensor
kernel, # rhs = conv kernel tensor
(1,1), # window strides
'SAME', # padding mode
(1,1), # lhs/image dilation
(1,1), # rhs/kernel dilation
dn) # dimension_numbers = lhs, rhs, out dimension permutation
print("out shape: ", out.shape)
print("First output channel:")
plt.figure(figsize=(10,10))
plt.imshow(np.array(out)[0,:,:,0]);
out shape: (1, 200, 198, 3)
First output channel:
VALID 填充，无步长，无扩张#
out = lax.conv_general_dilated(img, # lhs = image tensor
kernel, # rhs = conv kernel tensor
(1,1), # window strides
'VALID', # padding mode
(1,1), # lhs/image dilation
(1,1), # rhs/kernel dilation
dn) # dimension_numbers = lhs, rhs, out dimension permutation
print("out shape: ", out.shape, "DIFFERENT from above!")
print("First output channel:")
plt.figure(figsize=(10,10))
plt.imshow(np.array(out)[0,:,:,0]);
out shape: (1, 198, 196, 3) DIFFERENT from above!
First output channel:
SAME 填充，2,2 步长，无扩张#
out = lax.conv_general_dilated(img, # lhs = image tensor
kernel, # rhs = conv kernel tensor
(2,2), # window strides
'SAME', # padding mode
(1,1), # lhs/image dilation
(1,1), # rhs/kernel dilation
dn) # dimension_numbers = lhs, rhs, out dimension permutation
print("out shape: ", out.shape, " <-- half the size of above")
plt.figure(figsize=(10,10))
print("First output channel:")
plt.imshow(np.array(out)[0,:,:,0]);
out shape: (1, 100, 99, 3) <-- half the size of above
First output channel:
VALID 填充，无步长，rhs 内核扩张 ~ 空洞卷积（过度演示）#
out = lax.conv_general_dilated(img, # lhs = image tensor
kernel, # rhs = conv kernel tensor
(1,1), # window strides
'VALID', # padding mode
(1,1), # lhs/image dilation
(12,12), # rhs/kernel dilation
dn) # dimension_numbers = lhs, rhs, out dimension permutation
print("out shape: ", out.shape)
plt.figure(figsize=(10,10))
print("First output channel:")
plt.imshow(np.array(out)[0,:,:,0]);
out shape: (1, 176, 174, 3)
First output channel:
VALID 填充，无步长，lhs=输入扩张 ~ 转置卷积#
out = lax.conv_general_dilated(img, # lhs = image tensor
kernel, # rhs = conv kernel tensor
(1,1), # window strides
((0, 0), (0, 0)), # padding mode
(2,2), # lhs/image dilation
(1,1), # rhs/kernel dilation
dn) # dimension_numbers = lhs, rhs, out dimension permutation
print("out shape: ", out.shape, "<-- larger than original!")
plt.figure(figsize=(10,10))
print("First output channel:")
plt.imshow(np.array(out)[0,:,:,0]);
out shape: (1, 397, 393, 3) <-- larger than original!
First output channel:
我们可以使用最后一种方法，例如实现*转置卷积*
# The following is equivalent to tensorflow:
# N,H,W,C = img.shape
# out = tf.nn.conv2d_transpose(img, kernel, (N,2*H,2*W,C), (1,2,2,1))
# transposed conv = 180deg kernel rotation plus LHS dilation
# rotate kernel 180deg:
kernel_rot = jnp.rot90(jnp.rot90(kernel, axes=(0,1)), axes=(0,1))
# need a custom output padding:
padding = ((2, 1), (2, 1))
out = lax.conv_general_dilated(img, # lhs = image tensor
kernel_rot, # rhs = conv kernel tensor
(1,1), # window strides
padding, # padding mode
(2,2), # lhs/image dilation
(1,1), # rhs/kernel dilation
dn) # dimension_numbers = lhs, rhs, out dimension permutation
print("out shape: ", out.shape, "<-- transposed_conv")
plt.figure(figsize=(10,10))
print("First output channel:")
plt.imshow(np.array(out)[0,:,:,0]);
out shape: (1, 400, 396, 3) <-- transposed_conv
First output channel:
一维卷积#
您不仅限于二维卷积，下面是一个简单的一维演示
# 1D kernel - WIO layout
kernel = jnp.array([[[1, 0, -1], [-1, 0, 1]],
[[1, 1, 1], [-1, -1, -1]]],
dtype=jnp.float32).transpose([2,1,0])
# 1D data - NWC layout
data = np.zeros((1, 200, 2), dtype=jnp.float32)
for i in range(2):
for k in range(2):
x = 35*i + 30 + 60*k
data[0, x:x+30, k] = 1.0
print("in shapes:", data.shape, kernel.shape)
plt.figure(figsize=(10,5))
plt.plot(data[0]);
dn = lax.conv_dimension_numbers(data.shape, kernel.shape,
('NWC', 'WIO', 'NWC'))
print(dn)
out = lax.conv_general_dilated(data, # lhs = image tensor
kernel, # rhs = conv kernel tensor
(1,), # window strides
'SAME', # padding mode
(1,), # lhs/image dilation
(1,), # rhs/kernel dilation
dn) # dimension_numbers = lhs, rhs, out dimension permutation
print("out shape: ", out.shape)
plt.figure(figsize=(10,5))
plt.plot(out[0]);
in shapes: (1, 200, 2) (3, 2, 2)
ConvDimensionNumbers(lhs_spec=(0, 2, 1), rhs_spec=(2, 1, 0), out_spec=(0, 2, 1))
out shape: (1, 200, 2)
三维卷积#
import matplotlib as mpl
# Random 3D kernel - HWDIO layout
kernel = jnp.array([
[[0, 0, 0], [0, 1, 0], [0, 0, 0]],
[[0, -1, 0], [-1, 0, -1], [0, -1, 0]],
[[0, 0, 0], [0, 1, 0], [0, 0, 0]]],
dtype=jnp.float32)[:, :, :, jnp.newaxis, jnp.newaxis]
# 3D data - NHWDC layout
data = jnp.zeros((1, 30, 30, 30, 1), dtype=jnp.float32)
x, y, z = np.mgrid[0:1:30j, 0:1:30j, 0:1:30j]
data += (jnp.sin(2*x*jnp.pi)*jnp.cos(2*y*jnp.pi)*jnp.cos(2*z*jnp.pi))[None,:,:,:,None]
print("in shapes:", data.shape, kernel.shape)
dn = lax.conv_dimension_numbers(data.shape, kernel.shape,
('NHWDC', 'HWDIO', 'NHWDC'))
print(dn)
out = lax.conv_general_dilated(data, # lhs = image tensor
kernel, # rhs = conv kernel tensor
(1,1,1), # window strides
'SAME', # padding mode
(1,1,1), # lhs/image dilation
(1,1,1), # rhs/kernel dilation
dn) # dimension_numbers
print("out shape: ", out.shape)
# Make some simple 3d density plots:
def make_alpha(cmap):
my_cmap = cmap(jnp.arange(cmap.N))
my_cmap[:,-1] = jnp.linspace(0, 1, cmap.N)**3
return mpl.colors.ListedColormap(my_cmap)
my_cmap = make_alpha(plt.cm.viridis)
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
ax.scatter(x.ravel(), y.ravel(), z.ravel(), c=data.ravel(), cmap=my_cmap)
ax.axis('off')
ax.set_title('input')
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
ax.scatter(x.ravel(), y.ravel(), z.ravel(), c=out.ravel(), cmap=my_cmap)
ax.axis('off')
ax.set_title('3D conv output');
in shapes: (1, 30, 30, 30, 1) (3, 3, 3, 1, 1)
ConvDimensionNumbers(lhs_spec=(0, 4, 1, 2, 3), rhs_spec=(4, 3, 0, 1, 2), out_spec=(0, 4, 1, 2, 3))
out shape: (1, 30, 30, 30, 1)
